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Die vorliegende Arbeit wurde vom Fachbereich Elektrotechnik - der Universität Kassel als Dissertation zur Erlangung des akademischen Grades eines Doktors-Ingenieurs (Dr. Ing.) angenommen. Erster Gutachter: Prof. Dr.-Ing. G. Kompa Zweiter Gutachter: Prof. Dr.- Ing. H. Früchting Tag der mündlichen Prüfung 11. Dezember 2003 Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.ddb.de abrufbar Zugl.: Kassel, Univ., Diss. 2003 ISBN 3-89958-054-0 © 2004, kassel university press GmbH, Kassel www.upress.uni-kassel.de Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsschutzgesetzes ist ohne Zustimmung des Verlags unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Umschlaggestaltung: 5 Büro für Gestaltung, Kassel Druck und Verarbeitung: Unidruckerei der Universität Kassel Printed in Germany

Gedruckt mit Unterstützung des Deutschen Akademischen

Austauschdienstes

V

Acknowledgements The research work described in this dissertation was a collaborative project between the Department of High Frequency Engineering at the University of Kassel and the Orthopaedic Clinic, Kassel. I first want to thank my supervisor Prof. Dr.-Ing. G. Kompa for his great support, encouragement and insight in the course of this work. He provided a wonderful environment in which I was able to carry out my work successfully, and for that I am very grateful. Special thanks go to Prof. Dr. med. W. Siebert for giving us the opportunity to carry out clinical trials at the Orthopaedic Clinic, and for his acceptance to sit in the examination committee. I am very grateful to Prof. Dr.-Ing. H. Früchting for his time in evaluating this dissertation as the second examiner, and Prof. Dr. H. Hillmer for accepting to be a member of the examination committee. I am also thankful to Prof. Dr. med. F. Kuhn for allowing us to use the high resolution MR imager at the Clinical Centre, Kassel. Many thanks also go to Dr. med. P. Reuter for his time in the acquisition of the MR images. I must express my appreciation to Dr. med. B. Schlangmann for his interest and dedication on the project. Our various discussions at the University as well as in the operating rooms in Orthopaedic Clinic were very vital in the successful completion of this project. I thank him also for his time in looking for patients for the clinical trials and for his assistance in interpreting MR images. I am also very grateful to the patients who took part in the clinical trial. My gratitude goes to Dipl.-Ing. J. Weide for his time and effort during the preparation of the laser radar system. I am thankful to the department’s secretary, Mrs. Hilke Nauditt, for her readiness to assist where possible and for creating a superb environment in her office. I thank my colleague Dipl.-Ing. A. Ghose for his assistance during the clinical trials, and Mr. Edwin Ataro for doing a very good job in proof-reading the dissertation. My thanks also go to all the members of the departments for their team work spirit.

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This work would not have been successful without the financial support of the German Academic Exchange Service (DAAD), and for that I am very grateful. There are many other friends who made my life interesting during my stay in Germany and I want to thank them all. Finally, I would like to thank Liz for the encouragement and love that she gave me at a time when everything looked impossible.

VII

Table of contents

1 Introduction 1

1.1 Registration in Computer-assisted Orthopaedic Surgery………3 1.1.1 Fiducial-based Registration………….…………………3

1.1.2 Anatomy-based Registration…………………………...4 1.2 Motivation…………………………………………………….6 1.3 Organization of the Dissertation ………………………….….8

2 Laser Radar Imaging System 11

2.1 Introduction…………………………..……………………...11 2.2 System Overview …………………………………………...12

2.3 Laser Transmitter………….……………………………..….13 2.4 Photoreceiver .…………….…………………………………17 2.5 Sampling and Data Processing………………………………18 2.6 System Analysis……………………………………………..21 2.6.1 Laser Transmitter Characterization…………………...22 2.6.2 Distance Measurements………………….…………...27

3 Registration Problem Description 31

3.1 Registration Approach………………………………………31 3.2 Registration Datasets………………………………………...32 3.2.1 Preoperative MRI Data……………………………….32 3.2.1.1 MRI Data Segmentation………………...35 3.2.1.2 MRI Surface Extraction…………………40 3.2.2 Intraoperative Laser Radar Dataset…………………...41 3.3 Registration Algorithm………….…………………………...42 3.3.1 The Iterative Closest Point (ICP) Algorithm…………42 3.3.2 3D Registration……………………….……………....45 3.3.3 Modifications to ICP Algorithm ……………………...48 3.4 Registration Evaluation Measures…………………………...51 3.4.1 Registration Accuracy………………………………...51 3.4.2 Robustness of the Registration Technique…………....55

4 Experimental Evaluation of the Registration Technique 57

4.1 Validation of the Registration Algorithm…………………...57 4.1.1 Synthetic Data Experiments…………………………..57 4.1.2 Preoperative Data Experiments……………………….63 4.2 Registration Analysis with Laser Datasets…………………..71

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4.2.1 Accuracy of 3D Laser Imaging…………………….…71 4.2.2 Laser Data to CT Data Registration…………………..76 4.3 Simulation of Laser Data Sensitivity…………………….….89 4.3.1 Clinical Environment Noise…………………....89 4.3.2 Sensitivity due to Clinical Access Limitations...93 5 Clinical Trial 97

5.1 Preoperative data…………………………………………….97 5.1.1 Image Acquisition…………………………………….97

5.1.2 Image Segmentation and Surface Extraction………....98 5.2 Intraoperative Data…………………………………………101 5.3 Registration Analysis………………………………………105 6 Conclusion and Future Work 115

References 117

IX

List of Symbols and Acronyms

Symbols

A 3 x 3 matrix C closest point operator D data shape to be registered to a reference model d a point in data shape D ∆ column vector of a matrix I3 a 3 x 3 identity matrix l distance between a point in D and the model shape M M model shape which is used as the reference dataset m a point in model shape M MC set of closest point operator mc closest point in M that gives the shortest distance between d

and M Q registration procedure qR a rotation unit quaternion

R 3 x 3 rotation matrix

er norm of the rotation error

Resi magnitude of the individual residue Rk rotation matrix at iteration number k T translation vector

et norm of translation error Tk translation vector at iteration number k tr() trace operator µD centroid of D µM centroid of M

X

Acronyms A/D analog to digital converter AlGaAs aluminium gallium arsenide AMCW amplitude modulated continuous wave APD avalanche photodiode CAOS computer assisted orthopaedic surgery CAS computer assisted surgery CT X-ray computed tomography EM Expectation-Maximization FMCW frequency modulated continuous wave FWHM full width at half maximum GaAs gallium arsenide ICP iterative closest point I/O input/output LD laser diode MR magnetic resonance MRE maximum residue error MRI magnetic resonance imaging MSM metal semiconductor metal PCA principal component analysis PET positron emission tomography PIN p-doped/intrinsic/n-doped PRF pulse repetition frequency RAM random access memory RMS root mean squared SH single heterostructure SNR signal to noise ratio SPECT single photon emission computed tomography TE echo time TOF time-of-flight TR repetition time VTK Visualization Toolkit

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Zusammenfassung Medizinische Bilder von herkömmlichen diagnostischen Systemen für chirurgische Anwendungen, wie Computertomographie (CT) und Kernspintomographie (MRI), liefern sehr genaue und informative Einblicke in die interne anatomische Struktur und Funktion. Im allgemeinen enthalten sie jedoch keine Information über das Koordinatensystem des Patienten während der Operation. In einer konventionellen chirurgischen Prozedur müssen Korrespondenzen zwischen den medizinischen Bildern und der tatsächlichen Lage des Patienten im Operationssaal vom Chirurgen erkannt werden. Das klinische Ergebnis einer solchen Operation ist also sehr von der Erfahrung sowie dem Konzentrationszustand des Chirurgen während des chirurgischen Eingriffs abhängig. Diese Einschränkungen sind durch das Aufkommen einer computergestützten Chirurgie (CAS) stark reduziert worden. Ein Rechnersystem kontrolliert genau die Ausführung eines Eingriffs, deren Plan oft aus präoperativen Bildern gewonnen wird. Ein Schlüsselproblem in CAS ist die Registrierung – die Berechnung der Koordinatentransformation zwischen zwei Datensätzen. Die vorliegende Dissertation erkundet die Möglichkeit, eine nicht-invasive Registrierungsmethode mittels eines mit hoher Auflösung arbeitenden gepulsten Laserradarsystems für eine computergestützte Hüftgelenksoperation zu verwenden. Bei dieser Methode wird eine anatomische Struktur vom Laserradar während der Operation abgetastet. Die gewonnenen 3D Laseroberflächenpunkte werden dann mit dem aus den präoperativen MR-Daten extrahierten 3D-Oberflächenmodell mittels eines Oberflächen-basierten Registrierungsprogramms in Übereinstimmung gebracht (Registrierung). Die Registrierungssoftware verwendet den sogenannten Iterative Closest Points (ICP)-Algorithmus. In der vorliegenden Arbeit wurde die Registrierungsanalyse in bezug auf die Hüftgelenkspfanne (Acetabulum) ausgeführt. Die Messzeit des Laserradarsystems für die gegebene Aufgabe (40 x 40 Abtastpunkte) konnte von anfänglich 3.5 Stunden auf 3.8 Minuten bei einer Auflösung von 1 mm reduziert werden. Registrierungsalgorithmen wurden entwickelt und mit Datensätzen idealer Modellkörper sowie auch Anwendungsdatensätzen getestet. Experimentelle Untersuchungen mit verschiedenen Phantomen demonstrierten, daß das Laserradarsystem dazu

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fähig ist, die Oberflächenpunkte für die gegebene Aufgabe mit einer Meßunsicherheit von 1 mm zu messen. Im Operationssaal der orthopädischen Klinik in Kassel wurde die Lage der Hüftgelenkspfanne eines Patienten optisch vermessen. Die Analyse der gewonnen Laseroberflächenpunkte zeigte, daß das entwickelte Laserradar ein funktionsfähiges nicht-invasives System darstellt, das geeignet ist, intraoperative Daten zu erfassen. Es konnte eine hinreichende Anzahl von Laseroberflächenpunkten vom Acetabulum während des klinischen Versuchs gewonnen werden, die sicherstellte, daß die Registrierungsprozedur konvergierte. Es konnte weiterhin gezeigt werden, dass durch einfache Unterdrückung offensichtlich fehlerbehafteter Meßpunkte die Qualität der Registrierung in erheblichen Maße gesteigert werden konnte. Dies macht die Registrierungsmethode unempfindlich gegenüber Störsignalen der Operationsumgebung.

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Abstract Medical images from the commonly used diagnostic tools in surgical applications, such as Computed Tomography (CT) and Magnetic Resonance (MR), provide highly accurate and informative insight into internal anatomical structure and function, but typically contain no information on the world coordinate system of the patient during the operation. In a conventional surgical procedure, any correspondences between the medical images and the actual position of the patient in the operation room have to be drawn mentally by the surgeon. The clinical outcome of such an operation is therefore much dependent on the surgical skill and the state of mind of the surgeon during the surgical intervention. These limitations have been highly minimized by the emergence of Computer Assisted Surgery (CAS), in which a computer system accurately controls the execution of an operation, whose plan is often generated from preoperative images. A key problem in CAS is registration – the process of aligning datasets to the same coordinate frame. This dissertation explores the possibility of employing a non-invasive registration technique using a high resolution pulsed laser radar system in computer assisted hip-joint replacement surgery. The method involves acquiring 3D laser surface points of the anatomical part to be operated on intraoperatively, which are then registered to a 3D surface model from preoperative MR images using a surface based registration program which utilizes an Iterative Closest Point (ICP) algorithm. The registration analysis in this work was carried out with respect to the hip-joint socket (acetabulum). The measurement time of the laser radar system for the given task (40 x 40 scanning points at a resolution of 1 mm) was reduced from 3.5 hours to 3.8 minutes in the course of this work. Registration algorithms were developed and tested with synthetic as well as application datasets. Experiments performed with various phantoms demonstrated that the laser radar system is capable of acquiring the surface points for the given task within an accuracy of 1 mm. Under clinical environment, analysis of the laser surface points collected from a patient’s acetabulum during a hip-joint replacement surgery at Orthopaedic Clinic, Kassel, confirmed that our laser radar is a viable non-contact system of capturing intraoperative data. Enough coverage by the registration points on the acetabulum was obtained during

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the clinical trial, which ensured that the registration procedure converged. It is demonstrated that by simple selection of registration points, the effect of outliers on the parameters of registration transform can be eliminated. This makes the registration technique robust in the presence of spurious signals from the surgical site.

1

Chapter 1 Introduction Over the past decade, many exciting advances in Computer Assisted Surgery (CAS) have emerged. Maturing technologies in robotics, computer graphics, and data visualization have become directly applicable in the operating room. Several trends in surgery are contributing to the growing acceptance of CAS. The main factors include the increasing emphasis on minimally invasive surgical techniques and the widespread availability of 3D image data. Though surgeons are generally dexterous and highly trained to achieve a standardized level of surgical skill, they have limitations, such as a lack of geometric accuracy. For example, they cannot place an instrument at an exact, numerically defined location relative to the patient and then move it through a defined trajectory. Nor are they very good at precisely exerting a predefined force in a particular direction. Furthermore, they may have small hand tremors that limit their ability to operate on very delicate structures. Unfortunately, many of these limitations affect the efficacy of certain surgical procedures, especially in cases where geometric accuracy is required. For example, the three-dimensional position and direction of basic procedures used to modify the bone (including drilling, cutting, and reaming) determine the alignment and fit of the implants. These factors directly influence functional outcomes. One key advantage of CAS is precision and accuracy, or more generally, the ability to use copious, detailed, and quantitative information. The combination of 3D imaging data, computers, and intrasurgical sensors, for example, allows robots to accurately guide instruments to pathological structures deep within the body. Another important difference is that specialized manipulator designs allow robots to work through incisions that are much smaller than would be required for human hands or to work at small scales, where hand tremor poses fundamental limitations. By extending human surgeons’ ability to plan and carry out surgical interventions more accurately and less invasively, CAS systems address a vital need to greatly reduce costs, improve clinical outcomes, and improve the efficiency of health care delivery.

2

In CAS, the relevant portion of the patient’s anatomy is scanned using a medical imaging modality such as X-ray computed tomography (CT) or magnetic resonance imaging (MRI) prior to surgery. The resulting images are cross sections of the anatomy which can be stacked to create a 3D data volume of the corresponding anatomy. These preoperative images are used to generate a plan of the procedure which will be subsequently executed during the operation. A physician uses computer-aided graphical planning tools to construct the preoperative plan. Example surgical tasks in CAS include: biopsy and resection of brain tumors [13], placement of devices such as acetabular and femoral implants in total hip replacement surgery [9], or placement of screws into the vertebral pedicles in spine fusion surgery [7]. Such tasks require that the pre-operative plan specifies the instrument path to be followed (e.g in biopsy) or the location for the placement of the relevant hardware (e.g., prostheses, screws) relative to the anatomy in the medical images. These spatial specifications are only known relative to the coordinate system of the pre-operative medical images. During the operation, precise execution of the tasks specified by the preoperative plan must be achieved. There are many task execution methods in computer-assisted surgery which differ in the amount of direct control which a surgeon has over the underlying surgical tools. The most autonomous mode of computer-assisted surgical execution arises when a robot provides the actuation forces to perform tasks such as drilling, milling or cutting of the patient’s anatomy. A less autonomous alternative is passive navigational guidance in which feedback is provided to the surgeon in the form of computer graphics or medical images. This feedback is derived from the preoperative plan and the locations of conventional surgical tools which are tracked in real-time. Regardless of the mode of surgical task execution, a fundamental problem must be solved before the preoperative plan can be executed. Recall that the pre-operative plan is constructed in a coordinate system attached to the pre-operative medical images. Surgical execution, on the other hand, is performed in a coordinate system relative to a portion of the patient’s anatomy which is assumed to be rigidly fixtured in space. Therefore, before the plan can be executed it is necessary to establish a transformation which maps points in the preoperative plan into corresponding points on the patient. Establishing this transformation requires solution of the registration problem. The aim is to

3

take preoperative images and align them to the physical space of the patient. A particular sub-specialty of CAS is computer-assisted orthopaedic surgery (CAOS). Orthopaedic surgery is a perfect candidate for the union between computers and surgical procedures. This is because the bone is a very rigid tissue, so a rigid-body transformation can be used in registration. A rigid-body transformation has the form

F(x) = Ax + b where x = (x, y, z)T, A is a 3x3 rotation matrix constrained by two equations A

TA = I and det A = 1, and b is a 3x1 translation vector [14]. The constraint

on the rotation A insures that the transformation will be a rigid-body transformation with no shearing or scaling.

1.1 Registration in Computer-assisted Orthopaedic Surgery

The registration techniques applicable to computer-assisted orthopaedic surgery (CAOS) can be divided into two main groups: fiducial-based and anatomy-based techniques. 1.1.1 Fiducial-based Registration

This is the earliest technique of registration in CAOS. The method involves rigidly implanting two to three reference screws or pins (often referred to as fiducial markers) surgically into the patient’s relevant anatomy (e.g., femur), before the acquisition of the preoperative images [1]. Medical images are then taken from the anatomy in question, from which the surgeon constructs a plan of the procedure to be performed during surgery. Along with the surgical plan, the 3D locations of the fiducial markers within the images are extracted. At this juncture, these locations are only known relative to the preoperative images. During the main operation, the markers are physically exposed, and their 3D locations in the patient’s physical space are determined using intra-operative sensing device such as passive robotic arms or optical tracking devices. A computational procedure referred to as fiducial-based registration is then

4

used to determine the required spatial transformation between the two sets of corresponding fiducial marker locations (i.e., from the pre- and intra-operative measurements). The idea is to evaluate the least square rigid body transformation that aligns the two data sets [2]. Once this transformation is known, preoperative plan can be accurately executed during the surgery. The fiducial-based registration technique was the approach primarily designed for Robodoc system for hip replacement surgery by Taylor et al. [3, 4] and applied by others [5]. Though this approach can prove to be very accurate, it has several drawbacks. First, the fixation of the markers can require an invasive non-trivial surgical procedure prior to the acquisition of preoperative images and the primary surgery. Second, the pins may create pain for the patient and may also produce infection. Third, the pins may move with respect to the bone if there is too long a period between imaging and the operation or if the bone is fragile at the pin location, especially if the operation is necessitated by a fracture during an accident. Finally, in most cases the markers must be physically exposed during surgery and are sometimes distant from the primary surgical site. This may result in additional blood loss, may cause additional trauma to the patient and also add to the time required to perform the surgery. In order to alleviate these problems, another class of registration methods evolved: anatomy-based registration, and is described in the following section. 1.1.2 Anatomy-based Registration

In this technique, no external landmarks are used prior to the acquisition of the preoperative images. Rather, features which are intrinsic to the underlying anatomy are used to perform the registration. First, a 3D geometrical surface model of the anatomical structure in question is reconstructed from the preoperative images. In a second step, an intraoperative sensor or device is used to collect the geometric data from the relevant anatomy directly. Finally, a registration algorithm is used to determine the transformation parameters between the preoperative coordinate system and the intraoperative coordinate system.

5

Several anatomy-based registration systems exist whose principles of operation depend on the type of the intraoperative sensor or device used. One technique, referred to as point registration, works by considering anatomical landmarks that can be identified in both the intraoperative space and the preoperative images. Such landmarks, which can correspond to a sharp bump, the extremity of a ridge, the crossing of two ridges, etc, serve the same purpose as the artificial fiducial markers. This approach requires the use of interactive software to identify the points on preoperative images and the use of a 3D localizer to point and digitise the same points during the operation. There are several technologies for the 3D localizers, including passive mechanical arms, ultrasonic devices, electromagnetic sensors, and video systems (optical spatial localizer) [6]. Nolte presents a point registration system for spine surgery in which intraoperative registration is performed by matching point coordinates collected from bony structures of the vertebrae using a 3D optical localizer to a 3D model from preoperative CT data [7]. A conceptually similar technique is presented by Lavallee [8]. Though the registration procedure in this method is fast, it has a big disadvantage in that it requires extensive care and attention from the surgeon during the identification of the anatomical landmarks, both in preoperative images and in the surgical space. If several points are digitized with poor accuracy or mistake, there are not enough points to use efficient statistical methods and unpredictable results may occur. The situation is usually very critical when the anatomical marks are not easy to identify. For example, in spine surgery that occurs after laminectomy, some appropriate reference points that are usually found on the spineous process of normal vertebrae no longer exist and other well defined points are difficult to find. Similarly, when considering anatomical structures like the pelvic bone or the femur, reference points are very difficult to locate accurately. In order to avoid the aforementioned problems, a 3D localizer can be used in a different mode: surface/shape registration. Instead of using a few anatomical landmarks, the whole surface of a reference anatomical structure is taken into account. A computational algorithm then finds the spatial transformation that minimizes the error between the intraoperatively sensed shape and the shape that has been segmented from the preoperative image data. Frequently the surface of interest is digitized by contact using a metallic probe that is optically tracked in three-dimensional space [9, 10].

6

This method has been shown to be accurate in representing surface information; however, it is invasive because points must be accessed directly with the probe. A similar but non-invasive technique is described in literature whereby instead of using a metallic probe, an amplitude mode (A-mode) ultrasound probe is used to collect a number of points on the interface between two tissue types [11, 12]. Other authors describe the use of 2D X-ray images as the intraoperative sensing technique [15, 16, 17]. In this approach, contour-based algorithms are used in which the projection of a segmented surface from the preoperative scan is matched to the outline of the same structure in the X-ray image. The algorithms tend to use physical measurements, such as the mean distance between the 2D X-ray projection and a projection of the 3D segmented surface, as a similarity measure. Such algorithms are efficient to run after feature extraction has occurred. However, automatic, fast and accurate feature extraction from a complex scene is a difficult task and the final registration result is susceptible to errors in segmentation [16, 18]. Though this method is non-invasive, it has also the added cost of radiation exposure to the patient.

1.2 Motivation

The work described in this dissertation was motivated by the need for an accurate and a non-invasive registration technique in computer assisted hip-joint replacement surgery. In total hip-joint replacement surgery, the hip-joint socket is prepared for the fitting of the artificial socket through a milling process. Since the prosthetic fit affects comfort, energy expenditure and utility [81], the preparation of the socket and the positioning of the acetabular implant needs to be precise. Dislocation of a total hip replacement and wear of the artificial socket are mainly caused by the malpositioning of the acetabular component [82] and so there is a great need to improve the accuracy in the placement of the acetabular implants. Only then can the distress to the patient and associated additional treatment costs resulting from the dislocation of the femoral implant be avoided. Accurate fitting of the acetabular implant would only be achieved using computer assisted techniques which would further require accurate registration of the acetabulum during the operation.

7

Laser imaging techniques have become popular as intraoperative data acquisition systems for registration in computer assisted surgery due to the fact that they are absolutely non-invasive and can provide the necessary accuracy. Martelli [83] describes a technique in which registration is established by matching a preoperative surface to a 3D model obtained using a laser range finder during knee-joint replacement surgery. Grimson [67] presents a registration method that aligns a surface model of the patient’s face with points data acquired using a laser scanner during treatment. Audette [84] proposes the application of a laser range finder for a surface based registration of the brain’s cortical surface to preoperative images. A promising registration system using a laser ranger scanner in image-guided liver surgery is described in [85]. Up till now, the only technique that has been applied in registering the position of the acetabulum (hip-joint socket) relative to the preoperative images during the hip-joint replacement surgery uses an optically tracked metallic probe that digitizes the surface by contact [9, 10]. We propose a contactless registration system using high resolution laser radar imaging for the aforementioned task. Fig. 1 illustrates the general principle of the registration procedure. The hip-joint socket of the patient is imaged preoperatively using a high resolution MRI scanner without the attachment of fiducial markers prior to the scanning process. The 3D surface geometry of the hip bone is then extracted from the raw MRI data. At this time, the location of this geometric description is only known relative to the preoperative images. During the surgery, the laser radar system is then used to acquire a 3D surface of the exposed hip-socket. The location of this second geometric description represents the actual intraoperative position of the anatomical structure relative to the laser radar system, whose coordinates are known. These two geometric descriptions are then used as the inputs to a surface-based registration program, which uses an iterative closest point (ICP)-based non-fiducial algorithm ([22], [23]), to obtain the required spatial transformation. This work was carried out in collaboration with the Orthopaedic Clinic, Kassel. The registration accuracy requirement by the clinic was 1 degree in rotation and 1 mm in translation.

8

Fig. 1: Surface-based registration of a preoperative plan to intraoperative execution using laser radar imaging 1.3 Organization of the Dissertation

The remaining part of the dissertation is organized as follows: Chapter 2 gives a description of the used laser radar imaging system and the modifications which were introduced to increase the measuring speed. The characterisation of the parameters of the modified system are discussed. Chapter 3 presents the registration approach which was applied in this work. Details of the registration algorithm as well as the acquisition

9

and processing of the registration datasets are provided. Methods of analysing the performance of the registration technique under varying data conditions are presented. Chapter 4 describes detailed sensitivity and accuracy experiments which were carried out to verify the performance of the registration technique. Experiments performed involved synthetic data as well as measured datasets which are relevant to our application. Chapter 5 provides the results of the clinical trial while conclusion and further work are given in Chapter 6.

11

Chapter 2

Laser Radar Imaging System

2.1 Introduction

In the last several decades, laser radar systems have been used extensively in wide variety of commercial, industrial and military applications which include absolute distance measurements, profile measurements, 3D object scanning, velocimetry, fire control, autonomous missile guidance, air craft navigation and guidance, and remote atmospheric sensing among others [87]. The main feature that makes laser radar systems popular in these applications is their ability to perform high speed non-contact measurements with high resolution. Compared to microwave radar systems, they have several advantages. The relatively large divergence of transmitted beam that is exhibited by microwave radar systems is absent in laser radar systems because apart from the fact that laser beam is less divergent than microwave radiation, the transmitted laser beam can be precisely focussed onto the target using optical lenses. Further, due to the fact that laser radar systems operate at much shorter wavelengths, they are capable of higher accuracy and more precise resolution than microwave radar systems, which make their application in 3D imaging attractive. As in microwave radar systems, the range of an object is often obtained by measuring the time it takes for the laser beam to travel away from the transmitter to the object and back to the receiver. There are essentially three basic methods for measuring this time interval: pulsed time-of-flight (TOF), amplitude modulated continuous wave (AMCW) and frequency modulated continuous wave (FMCW). Pulsed TOF radar systems transmit a laser pulse and directly measure the time taken for the echoed pulse to be received [88]. In AMCW technique, range is determined by measuring the phase difference between an amplitude modulated continuous wave and its received reflection, while the FMCW approach measures the beat frequency between a frequency modulated wave and its reflection [89]. The laser radar system used in this work utilizes the pulsed TOF technique to determine range. One main advantage of the pulsed TOF approach over the other methods is that reliable detection of targets with low reflectance is

12

possible by using laser pulses with high optical peak power, in which the radiation is eye safe, as explained in Section 2.3. Both AMCW and FMCW techniques use continuous wave laser whose typical average power for precise measurements is higher than the power limits recommended in [86] for eye safety considerations. The laser radar system had been realized in the department within the frameworks of a project that was funded by “Bundesministerium fuer Forschung und Technologie (BMFT)” and Thyssen Henschel company, Kassel [76]. The aim was to realize a picosecond laser radar for dynamic online contour measurements with submillimeter accuracies. Optimization of the system for precise 3D imaging was later carried out by Biernat [77]. For the task described in this dissertation, the measurement speed of the laser radar system was found to be inadequate and had to be optimized by increasing the pulse repetition frequency (PRF) of laser transmitter as well as by replacing the sampling oscilloscope with a separate data processing unit, as explained in Section 2.5. Detailed descriptions of the system are given in the following sections.

2.2 System Overview

Fig. 2.1 shows a simplified block diagram of the laser radar system for 3D imaging. An 80 kHz quartz oscillator controls the pumping current generator which modulates the temperature-controlled semiconductor laser. The emitted laser beam from the laser transmitter is focused by two biconvex lenses and directed towards the target via a mirror and galvanometer scanning mirrors. The optical signal is backscattered after impinging onto the target which is assumed to have a diffuse surface. A portion of the diffusely reflected optical signal is received by the scanning mirrors and then collected and focused by the receiver optics onto the avalanche photodiode (APD). A reference signal is also directly coupled out from the transmitter beam and is directly focused onto the same photoreceiver. Using the same photoreceiver is aimed at eliminating temperature drift and various jitters of the laser transmitter and the photodiode. Due to the constant velocity of light, the time interval between the detected reference and measured signal is proportional to the distance of the illuminated surface point on the target. Changes in the optical

13

refractive index caused by atmospheric distortions have no significant influence here. With the knowledge of the mirror positions, a complete 3D image of the target can be obtained.

Fig. 2.1: Block diagram of the laser radar system.

2.3 Laser Transmitter

The laser transmitter consists of a single heterostructure (SH) semiconductor laser diode which is driven by a fast avalanche current pulse generator as illustrated by a simplified circuit diagram in Fig. 2.2 [92]. The production of SH laser diodes has in the meantime been discontinued worldwide. However, picosecond SH laser diodes are being designed and manufactured in the frameworks of an INTAS (International Association) project [96], in which our department is a research partner. The capacitor CL is discharged through the diode if the transistor T is closed. When T is open, the capacitor is recharged due to the DC-Voltage

Current

pulser

Sampling

unitA/D

converterComputer

Temperature

controllerScanner

Op. Fibre / Ref. Pul.

Focussing Optics

Avalanche

Photodiode

Laser diode

14

Vcc. The resistor RM is for internal measurements only and can be left out for normal use. The optical pulses are generated by a current modulation principle based on an extremely fast and strong carrier injection into the active region of a single heterostructure AlGaAs-GaAs laser diode of type LD60 manufactured by Laser Diode Inc. [19]. The technique is based on the use of high dynamic carrier injection into the active zone of a Fabry-Perot laser structure to modulate the optical refractive index of the optical waveguide. The optical refractive index can be changed in such a way that the laser diode emits light pulses with FWHM between 30 to 40 ps and peak power up to around two orders of magnitude larger than by using normal carrier injection.

Fig. 2.2: Simplified circuit diagram of the used avalanche current pulse generator.

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15

Fig. 2.3: Laser diode structure of LD60.

Fig. 2.3 presents the cross-section of the single heterostructure AlGaAs-GaAs laser diode, LD60, which has a rated optical power output of 2.3 W. The single heterostructure laser has a weak vertical optical waveguiding because of the small refractive index step at the GaAs p-n junction. Fig. 2.4 shows a schematic view of the band structure of the diode, the refractive index profile, as well as the optical field distribution.

Fig. 2.4: A) Schematic view of the band structure of LD 60 laser diode under forward bias. B) Refractive index profile. C) Optical field distribution (after [19]).

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Under normal operating conditions, i.e. at moderate pumping currents, the index profile of the asymmetric optical waveguide is only weakly influenced by the concentration of the injected carriers in the center p-layer. The laser diode gives an emission wavelength of about 904 nm at room temperature as expected under such circumstances. However, as is shown by the broken line in Fig. 2.4(B), very strong carrier injection into the laser active area provides a substantial negative contribution to the real part of the refractive index. This has effect that the commonly observed laser mode is suppressed. Regarding the dynamics, this operating condition can only be attained if the starting phase of the carrier injection is sufficiently fast, so that the index profile is changed considerably within the normal time of lasing. The typical rise time of the pumping current used is about 2 ns with a FWHM of 3.4 ns. Then, keeping up carrier injection due to the suppression of the normal mode gives rise to a further increase of the gain in the laser active area. After an additional delay time of roughly 4.5 ns, the laser radiates a very powerful and short single pulse at shorter wavelengths. Fig. 2.5 shows the measured laser impulses from LD60 laser diode at low and high carrier injection. Fig. 2.5: Laser impulses at low and high carrier injection (after [77]).

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Laser pulses with an optical peak power of 128.2 W, a rise time of 32 ps and pulse duration of 44 ps have been obtained with a 2.032 m 76.2 m emitting area, which exceeds the rated optical power by a factor of about 55 (at a supply voltage of 290 V and laser diode temperature of 23 °C). The small emission stripe of the used laser diode leads to a highly focussed laser beam which makes it possible to obtain a measurement spot of 1mm in our measurement system at a distance of up to 1 m. Reliable detection of targets with low reflectance is made possible by the generation of laser pulses with high optical peak power. Eye safety is guaranteed by the low average power of the laser emission due to the short duration of the transmitted laser pulses. For exposure to high repetition rate pulse trains, the average power must not exceed the accessible emission limit (AEL) of class 1 for an equivalent CW exposure [86]. The AEL of class 1 for laser emission with a wavelength of 904 nm is equal to 0.998 mW, as evaluated from [86]. This is the maximum accessible emission level permitted within this class. At our operating pulse repetition frequency (PRF) of 80 kHz, the measured average optical power is 0.3 mW, which is less than the AEL of class 1. The average power of the laser transmitter was determined using an optical power meter (Model 66XLA from Photodyne Inc.).

2.4 Photoreceiver

An important component of the laser radar system is the photoreceiver. It essentially consists of a photodetector, e.g. MSM-, PIN- or avalanche photodiode, followed by a low noise amplifier matched to it. For the given application, the photoreceiver should be fast and highly sensitive. These requirements highly contradict each other because while the sensitivity increases with an increase of the active area of the photodiode, the speed decreases with the resulting increase in capacitance. As a compromise, the laser radar system employs a fast avalanche photodiode from Silicon Sensors (SSO-AD-230), which is relatively slow but highly sensitive as opposed to the MSM- or PIN- photodiodes, which are less sensitive but very fast [95]. The used avalanche photodiode has a high gain (50 to 60) at low bias voltage, a rise time of about 180 ps, and a large active area (0.042 mm2).

18

2.5 Sampling and Data Processing

For the application described in this work, the speed of signal processing in distance measurements was paramount. The speed of measurements in our system depends on two factors: the pulse repetition frequency of the laser transmitter and the speed of the sampling and processing unit. An increase in pulse repetition frequency would result in an increase in the speed provided the sampling and processing unit can measure all the available pulses. In order to measure and process the output signals from the avalanche photodiode, a 50 GHz sampling oscilloscope (HP 54120B from Hewlett Packard) had initially been used in the laser radar imaging system. The maximum pulse repetition frequency of the laser transmitter while using the sampling oscilloscope had been set at an optimum value of 20 kHz. Due to the slow signal processing and data transfer from the oscilloscope to the measuring computer, increase in PRF above 20 kHz had absolutely no effect on the speed of measurements. The use of the oscilloscope ensured submillimeter accuracy in the distance measurements but at the expense of the measuring speed. A distance measurement uncertainty of 0.3 mm at a distance of up to 1.5 m had been obtained using averaging over 16 measurements ([19], [21]) in our laboratory. One point distance measurement required an average of 8 seconds. At this speed, the scanning of a surface of 40 mm х 40 mm (the estimated size of the given task) at a resolution of 1 mm took 3.5 hours. There was therefore need to adapt the system to the given task such that a complete intraoperative image could be acquired in an acceptable time. Various efforts were made to increase the measuring speed while using the sampling oscilloscope. Since only 1 mm uncertainty was demanded for our application, compromises could be made by reducing the number of measurement averages and the number of sampling points. However, the increase of measuring speed through this measures was not very drastic. Normally, it must be ensured that there is no overlapping of the reflected signal and the reference signal during measurements, so as to avoid evaluation errors. An attempt was made to make measurements while the two signals are overlapping in time domain. This resulted in a drastically reduced sampling range, giving rise to a considerable increase of the measuring speed. In this case however, the time interval measurement could obviously not be performed from the double pulse consisting of reference and the reflected signals. Rather, the reference signal would be

19

acquired at the beginning of the 3D-imaging with the reflected signal blocked, after which only the reflected signal without the reference signal would be measured. The scanning time was reduced to 18 minutes. But because of the drift of the measuring system, this attempt was not optimum. A substantial improvement in distance measurement speed was eventually achieved by increasing the pulse repetition frequency of the laser transmitter from 20 kHz to 80 kHz, and then using a separate high bandwidth sampling bridge developed in the department at the output of the avalanche photodiode. The system employs a sequential sampling technique known as extended-time sampling [20]. In this technique, sampling pulses sample the received pulses in a sequential manner with a shift of a fixed time interval t at each sampling point. This operation causes the sampled output to be extended by a factor in the order of 106 in time or to be downconverted in the frequency domain from the GHz to the kHz range. In the designed laser radar system, time-of-flight principle is used to calculate the range. The range is measured by transmitting a pulse and timing the return of this pulse from a target. The elapsed time t between the pulse and the target return is directly proportional to the distance to the target according to:

c

Rt

2 (2.1)

where R is the target range and c is the speed of light in free space. It then follows that, to obtain a measurement accuracy of ±1 mm, the value of t is given by:

ps7.6m/s8103

m3102

t (2.2)

Sampling process is carried out at the pulse repetition frequency (PRF) of 80 kHz, i.e. the sampling period Ts is 12.5 s. Since for every Ts the sampling part on the received signal is shifted by t, the extending time factor k can be obtained from:

20

610866.1ps7.6

µs5.12

t

sTk (2.3)

The sampled signal is amplified by a factor of 34. Fig. 2.6 shows the measured output signal of the avalanche photodiode after extended-time sampling and amplification. A small overshoot can be seen which was due to impedance mismatch at the input of sampling bridge. Ti is the time interval between the time-significant point of the two pulses (50% level). The full width half maximum (FWHM) of the pulses is about 1.12 ms. Due to the low bandwidth of the avalanche photodiode, its signal response to the laser pulses has a rise time of 210 ps and a FWHM of about 600 ps, as measured using an HP 54120B digital sampling scope which has a 50 GHz sampling head. The time conversion factor corresponds to the value calculated in Equation 2.3. Since the downconversion process in the frequency domain of the picosecond received pulses occurs right at the ‘front-end’ of the receiver, signal processing and amplification of the sampled signal are carried out in the low frequency range using low frequency ‘off-the-shelf’ components such as operational amplifiers. This in turn helps to reduce the number of high-frequency components needed at the front end, which reduces the overall cost of the sensor. An 8-bit analog-to-digital (A/D) converter is used to convert the amplified, sampled pulses into digital format. The A/D converter is controlled via an I/O card on a computer. Specially written software is used to read the data from the RAM for range calculations. The same software controls a dual-axis scanner (Model XY30-G3B, General Scanning Inc.). In the scanning process, the software registers the positions of the scanning mirrors which are used together with the range data in the extraction of the 3D-image. With the current system, one point measurement takes 143 ms at 8 averages. This gives a distance measurement uncertainty of less than 1 mm at a distance of 1 m, which satisfies the accuracy requirements of our application (See Sections 2.6.2, 4.2.1 and 4.2.2).

21

Fig. 2.6: Measured output signal of the avalanche photodiode after the extended-time sampling.

2.6 System Analysis

As has already been explained in Section 2.5, before this work, the laser radar system had been set to operate at a maximum pulse repetition frequency (PRF) of 20 kHz while utilizing a high bandwidth sampling oscilloscope for data acquisition. The characterization of the system from the previous works in the department [19, 21, 76, 79], had therefore been carried out using the aforementioned setup. After a four fold increase of the PRF and the replacement of the sampling oscilloscope with the sampling bridge, there was need to re-characterize the laser radar imaging system. Any change as a result of the drastic increase in PRF would most likely be in the laser transmitter and thus the parameters pertaining to the unit were first investigated. The overall performance of the laser radar system after

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the integration of the new data acquisition unit was then analyzed. The following sections give a detailed description of the measurements which were carried out. 2.6.1 Laser Transmitter Characterization

The current pulses obtained from the avalanche pulse generator form the basis of the optical beam generation from the laser diode, as explained in Section 2.3. The parameters of the current pulses plays a major role in determining the quality of the emitted laser pulses. For example, if the maximum amplitude of the current pulse fails to reach a given optimum value, there is a likelihood of getting unwanted spikes in the generated laser pulses [77] and the intended laser pulse power output may not be attained. Fig. 2.7 presents the measured current pulses at 20 kHz and 80 kHz. They were acquired by measuring the voltage across RM (two 0.5 Ω thin film resistors in parallel) using a high bandwidth sampling oscilloscope (HP 54120B). The temperature of the laser diode and the driving voltage of the current pulse generator were set at 23 °C and 290 V respectively. From the graphs, it can be seen that the current pulses at the two frequencies have the same relationship, with slight differences in peak amplitude and delay time. The current pulse at a PRF of 20 kHz had a peak amplitude of 19.64 A, a FWHM of 2.88 ns and a rise time of 2.41 ns, while the current pulse at a PRF of 80 kHz had a peak amplitude of 19.72 A, a FWHM of 2.83 ns and a rise time of 2.39 ns. The delay time between the two pulses was 0.2 ns. The slight deviations between the two pulses can be explained by the fact that the actual temperature of the laser diode was bound to be higher at 80 kHz than at 20 kHz. The same phenomenon had been observed in [77] when the laser diode was operated at different controlled temperatures. The average optical power output of the laser transmitter was then measured at the two PRFs using an optical power meter (66XLA from Photodyne Inc.). The peak optical power was evaluated, and was found to be 127.9 W at a PRF of 20 kHz, and 128.2 W at a PRF of 80 kHz. The slight deviation in the optical power output can be attributed to the small

23

increase in the peak amplitude of the pumping current with the increase in frequency [76]. The laser pulse response at a PRF of 80 kHz was the same as that illustrated in Fig. 2.5. There was no observable difference between the laser pulse responses at the two PRFs. The next step was to investigate whether the laser transmitter was still operating at optimum settings as regards the DC supply voltage Vcc and the laser diode temperature.

In order to carry out this analysis, the peak optical power of the laser pulses was evaluated from the measured average optical power at varying supply voltage. The supply voltage was then set at a fixed value, and peak power was measured at varying temperatures of the laser diode. Fig. 2.8 shows the evolution of the peak power with the supply voltage. It can be observed that there is a large increase in the peak optical power between supply voltage of 270 V and 290 V. This corresponds with the supply voltage

Fig. 2.7: Current pulses at PRFs of 20 and 80 kHz. The supply voltageVcc was 290 V while the temperature of the laser diode was 23 °C.

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Fig. 2.8: Laser peak optical power versus supply voltage. Temperature of the laser diode was 23 °C.

Fig. 2.9: Full wave half maximum (FWHM) of the laser pulses versus supply voltage. Temperature of the laser diode was 23 °C.

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range in which powerful single laser spikes are generated as a result of strong carrier injection in the active region of the laser diode. As shown in Figs. 2.9 and 2.10, the FWHM and the rise time of the emitted laser pulses decrease abruptly in the same supply voltage range, and this is also attributable to the emergence of the short laser spikes.

At a biasing voltage of 290 V, the laser pulse parameters were measured at varying temperatures of the laser diode. The peak optical power was found to increase with an increase in temperature up to 23 °C after which it started dropping gradually (Fig. 2.11). On the hand, the FWHM and the rise time of the generated laser pulses decreases continuously with increasing temperature, as illustrated in Fig. 2.12. The largest change in these parameters was observed to occur between 20 and 21 °C. Volpe [76] observed that this occurs due to an introduction of a delay in the start of laser emission as temperature is increased. Because of the time delay, it is possible to get more carrier injection in the laser active area before laser emission, which finally results in radiation of more powerful and shorter single pulses than at lower temperatures.

Fig. 2.10: Rise time of the laser optical pulses versus supply voltage. Temperature of the laser diode was 23 °C.

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Fig. 2.11: Peak optical power versus the temperature of the laser diode.The supply voltage was 290 V.

Fig. 2.12: Variations of FWHM and rise time of the laser pulses againsttemperature. The supply voltage was 290 V.

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From the results obtained in this section, it can be seen that laser pulses with high optical peak power up to around 180 W (at Vcc=330 V) could be obtained from our laser transmitter. However, to avoid overheating of the laser transmitter components at 80 kHz, the supply voltage was set at 290 V while an optimum operating temperature of 23 °C was selected for our application. The laser temperature of 23 °C gave rise to the highest peak optical power of the generated laser pulses at a supply voltage of 290 V, as observed from Fig. 2.11. The available peak optical power at these settings was 128.2 W and was found to be adequate for our applications as illustrated by measurement results in Section 2.4.2 and in Chapter 4, Sections 4.2.1 and 4.2.2. 2.6.2 Distance Measurements

The accuracy of the modified laser radar system was first analyzed by carrying out single point distance measurements on a diffuse target at distances ranging between 62 cm and 100 cm. The application of the system in scanning the hip-joint socket required a scanning distance ranging between 50 cm and 100 cm. However, the minimum scanning distance was dictated by the need to avoid the overlapping of the reference and reflected signals which was found to occur at distances less than 62 cm. Fig. 2.13 presents the attained measurement uncertainty at a distance of 1 m. Each point distance was obtained using an averaging of 8 measurements. A distance measurement uncertainty of 0.875 mm was attained, which is less than the requirement of 1 mm for our intended application. The dependence of the measurement accuracy on the magnitude of the reflected signal was investigated by evaluating distance measurements made on a target having varying gray levels. By using the target shown in Fig. 2.14, it was possible to obtain varying magnitudes of the signal between 0.17 V and 2.2 V. The voltages were converted to the corresponding responses of the avalanche photodiode by dividing the values by the amplification factor of the data processing unit. Fig. 2.15 shows the evolution of the measurement error as a function of the received signal amplitude. The error is presented as the difference between the measured result and the exact distance of the target, which was in this case 1 m.

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Fig. 2.13: Distance measurement uncertainty versus the number ofmeasured range data at a distance of 1 m. Averaging of 8measurements was used in the distance evaluation.

Fig. 2.14: Target with varying gray levels used to realize varyingresponses of the avalanche photodiode .

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Substantial distance measurement errors of up to 6 mm were observed at response voltages below 25 mV. This can be attributed to the considerable decrease in the rise time of the detected pulse at signal magnitude below 25 mV due to the non-linearity response of the used avalanche photodiode [80], as well as due to decrease of signal to noise ration (SNR). Above 25 mV, the measurement error was found to lie within 1 mm. As expected in a pulsed TOF laser radar, the measurement error did not vary with target distance in the selected range (62 to 100 cm). The analysis of the accuracy of the laser radar system in 3D scanning of an ideal surface, as well as a phantom relevant to our application, is carried out in Sections 4.2.1 and 4.2.2. A dual-axis scanner (Model XY30-G3B, General Scanning Inc.) was used in the given measurements. The scanner’s accuracy of angular position is specified as 0.3 % of full field. Since the scanner’s full field is 40 degrees, this implies that the lateral beam

Fig. 2.15: Distance measurement error as a function of the magnitude ofthe avalanche photodiode response.

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positioning error for a target at a distance of 1 m from the scanning mirrors is about 2 mm. The actual beam positioning accuracy of the scanner was estimated in this work using a graduated scale positioned at 1 m from the scanning mirrors. A computer program that controls the scanner could move the laser beam at specified steps in X and Y (horizontal and vertical ) directions. The specified step sizes were then compared with the obtained beam position measurements on the graduated scale. At steps of 1 mm, the estimated beam positioning error was found to be about ± 0.1 mm in both X and Y directions. The system used in this work utilizes raster scanning for 3D data acquisition. The largest step size thus occurs at the end of each linear sweep whereby the X scanning mirror reverses direction and returns to the starting position. This step size is determined by the maximum width of the target. For our application in the scanning of a patient’s acetabulum during surgery, the laser accessible area was approximated to be about 40 mm × 40 mm. The beam positioning error for a step size of 40 mm was found to be about 0.5 mm. The estimated beam positioning errors of the scanner lie within the required accuracy of 1 mm for our application.

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Chapter 3

Registration Problem Description

3.1 Registration Approach

The aim of the registration in this work is to determine the best possible alignment between 3D datasets of the same subject acquired using two different imaging modalities: one extracted from pre-surgical volumetric data and the other from the intraoperative laser radar imaging. This requires a three-dimensional registration approach. There are several techniques for achieving three-dimensional registration depending on the application. The methods are based on feature extraction of primitives (points [24] or curves [25]), surfaces [26], and volumes [27]. In the primitive-based approach, points or curves that are likely to be stable and descriptive of the datasets are extracted. This is then followed by a search for consistent correspondences between the features extracted from the two datasets. Some differential properties invariant to rigid transformation such as points with maximum Gaussian curvature along isosurfaces of the volumetric data are often used. On the other hand, in the surface-based approach, the data is considered to be a surface and the goal is to determine the transformation by minimizing a criterion relating the distance between the two surfaces. The alignment of the two datasets in this approach is achieved by searching in pose space, namely the components of the transformation, instead of correspondence space i.e. the associations of features in the two datasets. The volume registration approaches also search in pose space, but in this case they minimize a correlation metric that is computed in the whole volume.

In this work, surface-based registration approach was selected for the alignment of the two datasets, rather than features or volumes, for the following reasons:

Our intraoperative dataset from laser radar imaging produces only surface data and not volumes. The data is in the form of 3D points on the scanned surface. The reference dataset from a medical scanner,

32

such as MRI, is volumetric but a surface can be extracted from the data by an isosurface routine after segmentation.

Surfaces offer a reliable and computationally attractive means for registration. Registration of volume data is usually performed in intensity space resulting in a large amount of data which make the registration process computationally expensive. Though it is possible to reduce the complexity of the matching process in primitive-based approach, this is done at the cost of feature extraction and correspondence search. Since feature extraction involves several processing steps, such as multiple derivatives for points of high curvature, it is susceptible to noise and may thus require high signal-to-noise ratios as well as high resolution. Depending on the reliability of the feature extraction process, the correspondence search required for performing the registration may also be computationally expensive.

Surfaces support the measurement of physical distance between two datasets to be matched. Such distances are actually the direct property we wish to minimize during the registration process. While feature-based approaches use similar metrics, volume matching approaches generally measure correlation (possibly non-linear) between voxel values, a measure of how different the two datasets are, but not necessarily how far apart they are.

Details of registration datasets and description of the registration algorithm used in this work are provided in the following sections.

3.2 Registration Datasets

The 3D surface datasets needed for the registration in this work are acquired by two methods: magnetic resonance imaging (MRI) and laser radar imaging. Description of the datasets and the steps used in processing them is given in the next sections. 3.2.1 Preoperative MRI Data

In this work, magnetic resonance imaging (MRI) was used for the acquisition of the preoperative images. MRI was selected primarily due to

33

its superior soft-tissue contrast as compared to computed tomography (CT). Since any remaining cartilage on the acetabum will be captured by the laser radar imaging, it is imperative that the same structure be extracted from the preoperative images. Further, MRI introduces no ionizing radiation to the patient as well as the operators, and has no known side-effects. Medical imaging scanners typically output data in the form of 3D arrays of volume elements appropriately called voxels. Each voxel is a scalar value, often an integer, representative of a physical property that the scanner is sensitive to. For example, CT images are a map of X-ray attenuation coefficients produced by sending X-ray beams through the body at all angles and measuring the output radiation on the flip side. On the other hand, MRI measures the density of atomic nuclei magnetic spins induced by external magnetic fields at specific frequencies. It exploits the principle that any nucleus with an odd number of protons has a dipole moment and therefore behaves as a miniature magnet. For imaging the body, which contains a large amount of water, MRI frequencies are generally tuned to image hydrogen atoms ( 1H). When the body is placed in a strong magnetic field, a fraction of the hydrogen atoms are reoriented to align with the field. Imaging is performed by momentarily generating a different magnetic field and measuring the radio signal emitted by the dipoles as they relax back to their equilibrium positions. The MR image is usually acquired as a set of contiguous slices to yield a volumetric description of magnetization density. Voxel resolution within a slice is usually square, on the order of 1 mm, and slice thickness is about 1-3 mm. Higher resolution can be achieved, but requires longer acquisition times and a trade-off against signal-to-noise ratio (SNR). A wide range of acquisition parameters yields different acquisition times, ranging from a few minutes to about half an hour. More details on the physics and mathematics of 3D imaging in general and MRI in particular can be found in [28, 29, 30]. Though MRI has many attractive features, it suffers from possible geometric distortion which may be caused by several factors [31]:

Magnet calibration errors. A special concern arises with MR images because of geometrical distortions caused by magnetic field inhomogeneity arising from imperfections in the magnet system. The MR scanner consists of a constant-field main magnet and varying-field gradient coils. Scaling errors and geometric distortion occur if

34

the produced magnetic fields vary from the intended field strengths. Techniques for shimming the main magnet and calibrating the complete apparatus have minimized the impact of magnetic field errors. Eddy currents may also be introduced by the gradient coil, but proper shielding minimizes their effect.

Chemical shift. Hydrogen atoms in water and fat respond at slightly

different frequencies which results in spatial shifts between fat regions and soft tissue regions. These artifacts, though, can be eliminated by MR pulse sequences which selectively eliminate the fat signal for imaging the water signal only.

Magnetic susceptibility variations in the anatomy. Magnetic susceptibility refers to the induced magnetic field in materials in the presence of an external field. This induced field acts to vary the overall magnetic field at each point in the volume. Since different materials may have different magnetic susceptibilities they may appear offset relative to each other in one direction, the frequency-encoding direction, of the image. This effect is particularly apparent at boundaries of materials with high susceptibility differences. Air/tissue boundaries are a common example of this phenomenon in body imaging, leading to a possible offset of up to 2 mm for typical image acquisition parameters. These effects are difficult to correct since they are dependent on the imaged structures, but techniques have been developed to minimize their impact by taking multiple scans [32].

The geometric distortions cannot be neglected especially when registering images from other modalities to MRI. Some researchers register images using an affine transformation to simultaneously estimate rigid body and correct for image distortion [33, 34 ]. However, the affine transformation may introduce distortions that are not really in the image, especially if a small number of features is being used for registration [35]. It is thus preferable that image distortions be avoided by careful quality assurance or corrected when possible before registration, and that registrations be performed using a rigid body transformation whenever feasible and appropriate. In other words, the image distortion correction should be decoupled from the process of determining the transformation parameters.

35

This would simplify registration solutions, particularly approaches based on 3D surfaces such as the one we are dealing with in this work. For the registration approach used in this work, there is a need to extract the surface of the anatomy of interest from the volumetric data originating from the MRI scanner. In order to do this, the volume data is typically processed in two steps: data segmentation and surface extraction. These steps are addressed in the following subsections. 3.2.1.1 MRI Data Segmentation

Data segmentation of a 3D digital medical imagery is the process of labeling individual voxels in an input gray-level image by tissue type in the anatomical structure. The labeled image is referred to as the “segmentation” of the input image. Segmentation is usually an off-line process whose goal is to generate well defined regions (segmentations) of key anatomical structures which can later be rendered in three dimensions. An example input gray-scale image is a Magnetic Resonance Imaging (MRI) scan of a human hip-joint that is shown in Fig. 3.1. The figure illustrates a set of two-dimensional axial cross-sectional slices comprising a 3D hip-joint MRI scan. This is the view of an MRI scan that is printed on film and mounted on a “light-box” and is traditionally available to clinicians to aid them in making diagnosis and planning therapy. Though manual segmentation (one in which an expert anatomist manually delineates the boundaries of different structures in medical images) is possible, it is a very tedious process considering the fact that a typical scan usually consists of millions of voxels across approximately a hundred cross-sectional images that need to be classified. Further, it has been shown that manual segmentations vary significantly between experts as well as when an expert segments the same image at different times. Variations of up to 15% and between 15 - 22% have been reported in the segmentation of cortical gray matter of the brain in segmentations by five different experts [36] and in segmentation of brain tumors by multiple experts [37] respectively. These considerable variations make the manual solution problematic in tasks requiring high accuracy such as the patient registration

36

and the neuroscience studies, and underscore the need for accurate automatic segmentation approaches.

Fig. 3.1: A set of two-dimensional axial cross-sectional slices comprising a 3D hip-joint MRI scan.

In automatic segmentation, the process is generally a sensor-specific procedure based on the properties of the observed intensities as well as known anatomical information about the imaged organs. Specifically, a segmentation method for a given anatomical structure of interest aims at

37

obtaining sufficient information about the domain to be able to answer the following four questions:

What is the intensity or gray-level appearance of the structure in the image?

What are the characteristics of the imaging modality that was used to

acquire the image that influence the gray-level appearance of different structures in the image?

What is the shape of the structure of interest?

What is the geometric relationship of the structure to other structures

in the image?

Most techniques approach the medical image segmentation problem as a knowledge-based task that uses some or all of the four types of models, one for each piece of domain knowledge mentioned above, namely [38]:

Intensity models that describe the gray-level appearance of individual structures. These encode information such as “fluid appears bright in T2-weighted MRI”. This is because the MRI sequence is designed to show water or fluid in high contrast to the surrounding tissues.

Imaging models that describe characteristics of the medical imaging modality that is used to create the imagery. These encode information such as the presence of inhomogeneity in MRI images due to limitations in the imaging equipment.

Shape models that describe the shape of structure in a given population. These encode information such as “the ventricles are bent in the middle in healthy adults”.

Geometric models that describe the spatial relationships between

structures in a given population. These encode information such as “cartilage lines the articular surface of acetabulum in healthy adults”.

Frequently, automatic segmentation algorithms use some form of intensity

models to describe the gray-level appearance of anatomical structures.

38

Typically used intensity models are in the form of probability distributions on individual voxels that belong to a particular structure. For example, the intensities of white and gray matter in brain MRI have been represented using the Gaussian distribution [39, 40] and also as mixtures of Gaussian distributions [41]. Richer intensity models capture higher order correlations in gray-levels i.e. describe the appearance of local patches of the structure instead of individual voxels. For example, the mean and variance of local patches are used to characterize the appearance of trabecular bone in MRI [42], and tensors are used to describe the appearance of thin bone in CT images [43].

The use of explicit imaging models to account for the distortions in medical imagery that are attributable to limitations of different modalities such as MRI, CT, PET, SPECT, X-ray, ultrasound, is not very common in automatic segmentation algorithms. Segmentation methods for MRI, CT and X-ray images that incorporate knowledge of the imaging modality report significant improvements in their results compared to methods that do not account for the distortions in the signal that are due to the characteristics of the imaging equipment [44, 45]. Wells [40, 44] introduced a Bayesian method for segmentation of MRI images that is often referred to as Expectation-Maximization (EM)-segmentation. It is an intensity-based segmentation method that corrects for a slowly-varying gain field that corrupts MR images. The technique uses statistical knowledge of tissue properties and gain inhomogeneities to dynamically solve for two components at each voxel: tissue labeling and the gain field. The Expectation-Maximization algorithm is then used to iterate the solution of the two components. The resultant segmentation successfully adapts to intensity variations that are due to spatial inhomogeneities in the sensitivity of the imager, intensity variations associated with different scanners, and those due to the varying appearance of tissues across patients. Other authors have reported methods that correct for spatial distortions due to magnetic susceptibility differences between different tissue types. One way to reduce the distortion is to acquire two images with the phase encoding applied in opposite directions. The distortions will likewise be in opposite directions and a mathematical correction can be calculated and applied [32]. The use of Shape models in medical image segmentation is still an ongoing research. The aim here is to obtain salient characteristics that define the

39

shape of a structure across a population. This technique is quite challenging because the underlying anatomy that is to be captured is complex in shape and varies across subjects. Due to the fact that anatomical structures do not possess regular geometric shapes such as straight lines, rectangles or circles, it is not easy to obtain their mathematical representations. Though parametric models have been realized for specific applications [39, 46], there is still no specific definition of shape that can be applied to anatomical structures in general. If available, such shape models could be used in biasing a segmentation algorithm towards a particular shape, especially in situations where the gray-level appearance is not sufficient to uniquely segment the structure. Most techniques of shape modeling [47-49] apply a combination of deformable models [50] and principal component analysis (PCA) [51-53]. Deformable models are variants of the classical “snake" that was originally introduced in [54]. Their use in segmentation lies in incorporating constraints of local smoothness. On the other hand, principle components capture the directions of largest variation in a given set of points, and are used to characterize the space of possible deformations for a structure in a given population. Generally, in the PCA-driven deformable model style of shape modeling, the models are represented using principle components of normalized spatial locations of feature points that lie on a structure, from a set of training images for a given population. Given a novel image, a deformable model is initialized with the mean shape from the training data. The deformable model is evolved under local constraints of smoothness, and global constraints that require that its shape remains within the space restricted by the principle components. Kapur [38] introduced a class of models referred to as Geometric Models

that are used for the purpose of encoding notions of spatial relationships between structures in an image. The main motivating factor for these models is that while some structures can be segmented from medical images by using methods from low-level image processing, e.g., skin surface can be reproducibly segmented from head MRI using a combination of thresholding, connectivity, and morphological operations, there are other structures, such as the brain tissue in head MRI, that do not have as salient a combination of intensity and topology as the skin, and are harder to segment using low-level methods. A “coarse to fine” strategy for segmentation is provided by geometric models in which the easily identifiable or coarse structures are first segmented automatically and their

40

geometry is then used to bootstrap the segmentation of other fine structures in the image. For the registration approach in this work, we are interested in extracting the 3D surface of the acetabulum from the MRI volumetric data of the hip joint. This surface should include any cartilage present on the bone because, as earlier explained, it will also be captured during the intraoperative laser radar imaging. The segmentation process was carried out using the 3D Slicer software, a medical visualization program created at the MIT AI Lab in collaboration with the Surgical Planning Laboratory at the Brigham and Women’s Hospital [55]. The 3D Slicer is freely available, open-source software whose segmentation techniques include a gain-correcting classifier by Wells [44]. Though semi-automatic segmentation technique is possible using this software, manual interventions were necessary in order to achieve accurate segmentation appropriate to our application. This is mainly because most internal structures of the hip joint such as ligaments, cartilage and bone have some overlapping image intensity values. Fig. 3.2 shows an example slice of an MRI image of a human hip joint in which the acetabulum on the right-hand-side has been segmented. The segmentation process is performed in all the slices within the region of interest. 3.2.1.2 MRI Surface Extraction

The result of the segmentation process is a set of label maps, where pixels take on values corresponding to tissue type. In order to obtain the surface data from the segmented data, the bounding surfaces of the label maps are extracted and represented as a collection of triangles using Marching Cubes algorithm [56]. Decimation is then used to reduce the number of triangles to a quantity that can be more quickly rendered with minimal observable loss in detail [57]. For example, a typical acetabulum surface is reduced from approximately 60,000 triangles to 20,000 triangles. A 3D polygonal surface extracted from the segmented dataset is illustrated in Chapter 5, Section 5.1.2.

41

Fig. 3.2: A single MRI slice of the hip-joint showing the segmentation of the acetabulum on the right-hand-side (Segmented section is shown in red).

3.2.2 Intraoperative Laser Radar Dataset

Our intraoperative data acquired using the laser radar system is direct surface data which can be used as is, generally represented as a set of 3D points, without applying any smoothing algorithm. Using data objects in Visualization Toolkit (VTK) [58], the 3D points could be represented as polygonal data consisting of vertices and triangles. For visualization purposes, “splatting” technique [58] was applied. This method involves using a filter to inject input points into a structured points dataset. As each point is injected, it "splats" or distributes values to neighboring voxels in the structured points dataset. Data is distributed using a Gaussian distribution function which can be modified using scalar values. A 3D surface generated from the laser surface points using the splatting technique is illustrated in Chapter 4, Section 4.2.2 (Fig. 4.18(b)).

42

3.3 Registration Algorithm

The alignment process of two 3D datasets involves two main steps, which include finding the corresponding points between the two datasets followed by the 3D registration of the corresponding point pairs. In this work, a modified version of Iterative Closest Point (ICP) algorithm by Besl and Mckay [22] was used in identifying the corresponding point pairs between the two datasets. A modification of the algorithm was necessary in order to avoid erroneous results in registration as discussed in Section 3.3.3. For the registration purposes, quarternion rotations as employed by Horn [59] were applied. The description of the algorithms as implemented in the registration task in this work is given in the following sections. 3.3.1 The Iterative Closest Point (ICP) Algorithm

The ICP algorithm is a general purpose, representation independent method which can be used to register 3D datasets such as point sets, curves and surfaces. It has been applied successfully in medical imaging fields [60] and in industrial or biometric (face recognition) applications. The technique has many advantages, which include:

It is intuitive and simple. It can handle a reasonable amount of noise. It does not require any local feature extraction.

The algorithm was designed to register two datasets under the assumption that one dataset is a subset of the other dataset. The technique finds corresponding points iteratively on each of the two surfaces and minimizes their distance, also via iteration. The iterations stop when a previously set criterion for the least squared differences is met. The basic ICP algorithm is well known [22], but it is briefly restated here for completeness. Suppose that we have two 3D geometric shapes, data shape D and model shape M, which correspond to a single shape but in different orientations. And we want to match the data shape D to the model shape M. Shape D may be decomposed into a 3D point set if it is not already in a point set form while M is decomposed into any of the allowable forms (segments, points or triangles) depending upon the model representation. A distance

43

metric l between an individual data point d D and the model shape M is defined as

dmMm

Mdl

min),( . (3.1)

The closest point in M that gives the minimum distance is defined as cm such that ),(),( Mdlcdl m where Mcm . If we denote cM as the resulting set of closest points in M corresponding to each point in D, we can define a closest point operator C such that ),( MDCcM . (3.2)

The basic principle of correspondence is illustrated in Fig. 3.3, with the model shape M shown as a curve, the points from data shape D as circles (o), while the closest point set cM is represented by cross ( x ) markings. Notice that the data points, which may represent actual surface measurements, acts as control points. The least squares registration is computed using a registration procedure Q , defined such that ),(),,( DcMQl TR (3.3) returns the rotation matrix R and translation vector T which registers the corresponding point sets cM and D optimally. The procedure also outputs the mean square distance between the two point sets. Q can be realized using any of the registration techniques described in Section 3.3.2.

44

The ICP algorithm can be summarized as follows [22]:

1. Initialize the cumulative transformation parameters R and T to the identity transformation.

2. Reset the iteration counter, k, to zero. 3. Repeat:

a) For each discrete point Di in the data set, compute the closest

point (in terms of Euclidean distance) Mi which lies on the surface of the model.

Fig. 3.3: Illustration of the determination of the corresponding points for the ICP algorithm.

X

X

X

X

X

X

X

XX

Closest points M

Model M shape

Data D shape

C

45

b) Using the correspondences from a), find a rotation, Rk, and a translation, Tk, which minimize l via the corresponding point registration technique Q .

c) Apply the incremental transformation from step b) to all data points, Di. Update the cumulative transformation parameters R

and T based upon the incremental transformations, Rk and Tk. d) Increment k.

4. Until the change in mean squared error lklk 1 falls below a preset threshold 0 .

In a surgical application, the model shape M corresponds to the pre-operative surface data, while the data shape D corresponds to the intraoperative surface data. The experimental work presented in this dissertation uses two surface representations: triangle meshes from the preoperative data as the model and sets of discrete surface points from laser radar imaging as the data.

3.3.2 3D Registration

The 3D registration deals with the alignment of two sets of corresponding point pairs. This requires applying a rigid 3D transformation (3D rotation R and translation T) to the data shape D to bring it into an optimal alignment with model shape M based on a preset threshold . The operation can be expressed algebraically as determining R and T such that TR DM . (3.4) Given two corresponding point sets mi and di, i = 1 …N, we can denote the two point sets as M = m1, …, mN and D = d1, …, dN, with each point mi corresponding to the di with the same index. The goal is then to find the rotation matrix R and translation vector T such that TRdm ii . (3.5)

46

where mi and di are 3 element vectors, R is a 3 3 rotation matrix and T a 3 element translation vector. It is good to note that unless the data is perfect, it is not possible to obtain the rotation and translation in order to satisfy equation 3.5 at each and every point. Rather, there will be an error at each point that is given by )( TRdme iii . (3.6)

Generally, and for the work in this dissertation, registration algorithms attempt at obtaining the registration parameters (R and T) that minimizes a least-squared distance metric given by

N

iii

1)(22 TRdm . (3.7)

Various techniques have been proposed for deriving a closed form solution which minimizes Equation 3.7. The most popular approaches are quaternion-based methods [59] and singular value decomposition (SVD) based methods [74]. In this work, an analytical solution for determining the required rotation matrix using a quaternion representation of rotations as proposed in [59] was implemented. The technique is briefly described below. A quaternion can be thought of as a vector with four components (composed of a scalar and an ordinary vector), or as a complex number with three different imaginary parts, and can be used to represent 2- and 3-D rotations. A rotation unit quaternion may be denoted as

3210 qqqqR q , where q0 0, and 12

3

2

2

2

1

2

0 qqqq .

47

The registration process starts by first translating each point set such that their centroids coincide with the coordinate origin. This is achieved in two steps as follows:

I. Evaluate the centroids

Ni i

NM 1

1mµ and

N

i iN

D 1

1dµ .

II. Construct the point sets M and D such that Mii µmm and

Dii µdd .

The quaternion technique is applied after the above steps as illustrated by the following algorithm (See [59] for details).

1. Determine the covariance matrix N

i iiMD 1dm , where

superscript T implies matrix transposition.

2. Create the matrix MD MDA .

3. Build from matrix the column vector 123123∆ AAA .

4. Construct the symmetric matrix Q as follows:

∆

3)(

)(∆

MD MD MDtr

MDtr

IQ ,

where tr() is the trace operator and I3 is a 33 identity matrix.

5. Determine the unit eigenvector 3210 qqqqR q of Q

corresponding to the largest positive eigenvalue. 6. Calculate the orthonormal rotation matrix R from qR according to

)2031(2

22

21

23

20

)1032(2

)3021(2

)1032(2

23

21

22

20

23

22

21

20

)2031(2)3021(2

qqqq

qqqq

qqqq

qqqq

qqqq

qqqq

qqqq

qqqq

qqqqR .

(3.8)

(3.9)

48

After the calculation of R, the translation vector T is evaluated from DM RµµT .

The rigid 3D rotation matrix R can be decomposed into three elements, (rx, ry, rz), which represent rotations about the X, Y, and Z axes respectively. And since the translation vector T has three components, (tx, ty, tz) along the three axes, it follows that the 3D rigid transformation can be represented by 6 parameters, namely, (tx, ty, tz, rx, ry, rz).

3.3.3. Modifications to ICP Algorithm

One of the key assumption made by ICP algorithm is that one geometric data description is a proper subset of the other (i.e., for each point or region described by the data, there must be a corresponding point or region in the model). This is quite useful for purposes of pose estimation where measured data is to be matched against a known model. However, if this condition is violated such that there are points in each set which have no corresponding point in the other, erroneous pose estimates will be obtained. This is the main limitation of the ICP algorithm. The data which usually violate this assumption are outliers to the registration process. Outliers are caused by various factors, which include: collection of data in regions which are not overlapping, noise in the data collection process, and errors introduced during the extraction of the geometric surface data (e.g., errors in the segmentation of the preoperative data). In order to avoid getting false correspondences as a result of the outliers, it is imperative that the outliers are identified and eliminated during the registration process. Several modifications to the ICP algorithm for solving the outlier problem have been proposed, most of them in the form of constraints applied to the matched point pairs. Some authors have suggested the use of a least median of squares estimator to overcome the ICP method’s intolerance to outlier data [61], however, the computation cost for implementing this technique is high. Others have proposed a robust technique which uses M-estimation to

(3.10)

49

reduce the effect of outliers [62, 63]. In this method, correspondences with small errors are given more weighting than those with large errors. Several researchers propose outlier elimination techniques in which any matched points separated by a distance greater than a certain threshold are rejected, whereby the threshold is based upon desired accuracy or estimated noise in the data. For example, Zhang proposed a semi-automatic method of determining the match distance threshold, based on the variance of the lengths of distances between corresponding points [64]. Pulli suggested a similar technique but in which the threshold is expressed as a multiple of the estimated (or known) scanning accuracy [65]. The threshold-based outlier detection technique has demonstrated to work well in practice [26, 66] and was implemented in this work. The algorithm of this technique can be briefly stated as follows:

I. Execute the usual ICP algorithm until convergence occurs. II. Determine all the corresponding point pair residuals. If any of these

residuals is larger than a user specified threshold, remove a fraction of these outlier data points.

III. Iterate steps I and II until there are no longer any residuals which are greater than the set threshold.

The other main weakness of the ICP correspondence technique is that it requires the two datasets to be roughly aligned to each other before the ICP algorithm is applied [22]. The two datasets should be within 30 degrees of each other [71]. If this requirement is not fulfilled, there is a tendency for the algorithm to converge into local minima as a result of false correspondences. There are two types of local minima that can be encountered during registration: those close to the global minimum, and those distant from it. The latter condition does not occur provided a sufficient good initial alignment is available. However, in instances where the distant local minimum is a problem, techniques proposed by various researchers can be applied to guide the search towards the global minimum. Grimson proposed point-to-surface matching by progressively refining the registration using a series of objective functions [67]. Although it was not stated explicitly, their final objective function is the Huber [68] estimate. Another technique of solving the problem using geometric hashing is proposed in [69]. Others have suggested the use of exhaustive search for corresponding points [70]. In this work, good initial pose estimates are

50

obtained by solving the corresponding point problem using manual specification of anatomical landmarks in the two datasets to determine correspondence. The method involves identifying three anatomical landmark points from the 3D preoperative model of the acetabulum and the corresponding landmark points on the surface model from the intraoperative laser radar imaging. An initial pose estimate is then computed using the corresponding point registration method as described by Horn [59]. The resulting transformation estimate is guaranteed to align the two datasets to within 30 degrees of each other such that local minima that are distant from the global minimum are avoided. In order to solve the problem of getting trapped in the local minima that are near the global minimum, various researchers have proposed repeating the registration process several times, each time randomly perturbing the solution about the current optimum [67, 72]. This technique has demonstrated to work well in practice [26, 73] and a similar approach has been implemented in this work as outlined in the algorithm below. 1) Repeat:

a) Run the ICP algorithm and determine the resulting pose transformation (R, T). If the resulting least-squared error as evaluated from Equation 3.7 is less than those from the previous ICP runs, set the optimum pose transformation (Ro, To) to this pose transformation (R, T).

b) Generate a new pose by perturbing the data shape randomly about

the new optimum pose transformation (Ro, To). 2) Until there is no additional improvement in the resulting value of the

least-squared error.

51

The perturbations are generated using a function that returns uniformly distributed random values within specified maximum translation magnitude tp(max) and rotation angle rp(max) from angle-axis representation. Though the technique works well in practice, it is not always guaranteed to eliminate local minima especially when dealing with noisy data. However, it is easy to identify the presence of local minima at the end of a registration run by checking the magnitudes of the individual residuals given by )( TRdmRes iii . If a large number of residuals are found to be above a certain threshold, there exists a high possibility of convergence to local minima.

3.4 Registration Evaluation Measures

Several measures may be used to evaluate the performance of a registration technique. In this work, two aspects of performance were analyzed, namely, accuracy and robustness of the registration approach. These terminologies are described briefly in the following subsections. 3.4.1 Registration Accuracy

The registration accuracy can be defined as the degree to which physically corresponding points on the model and data surface have been brought into alignment after a registration trial. There are two classes of measures that can be used to quantify registration accuracy. The first class of measures is referred to as ground-truth accuracy measures which relies on the availability of a true registration transformation. This demands the use of a highly accurate calibration. For this to be achieved, a fiducial-based registration is performed in parallel with the surface-based registration in order to obtain a highly-accurate registration transformation which is used as the reference or the ground-truth of the registration process. The ground-truth information is only useful during the validation of a registration

(3.11)

52

technique in the laboratory and is usually not available in the course of the actual registration in the field. Without the availability of the ground-truth information (e.g. in the field), a second class of measures referred to as non-ground-truth accuracy measures can be used to quantify the registration accuracy. In this case, the least-squared distance metric given by Equation 3.7 and the magnitude of the individual residuals from Equation 3.11, become the basis for the various registration accuracy measures that belong to this class. At the termination of a registration trial, the individual residual distances Resi (Equation 3.11) between every data point and its corresponding closest model point can be determined, from which the following four non-ground accuracy measures can be defined:

Normalized least-squared error, N

ii

NLSE 21

Res

Normalized root-mean-squared error, N

ii

NRMS 21

Res

Average residual errors, N

ii

NARE Res

1

and maximum residual errors, )( i

i

maxMRE Res .

The main advantage of the above four measures is that they do not require a true registration transformation and are thus useful in the field in evaluating the results of the registration process. Though these measures are a necessity in the real applications, they have a principal disadvantage in that they do not guarantee optimum registration accuracy. It is good to note that the magnitude of the MRE depends highly on the noise presence in the data or model shape. In this work, there was no possibility of using fiducial markers to obtain the ground-truth information during development and testing of our

(3.12)

(3.13)

(3.14)

(3.15)

53

registration technique. For the purposes of the validation of the algorithm, the ground-truth was estimated by using a simulated data shape. This involves constructing the data shape by sampling discrete points from the model shape (or copying the whole model shape) and then applying a known transformation to it. The ground-truth is then determined by registering the synthetic data and comparing the resulting transformation with the known transformation. In order to analyze the performance of the registration technique with the application datasets, the ground-truth was estimated by running the registration algorithm under the most ideal matching conditions in which the measured data is processed by removing outliers where possible in order to make it closely resemble the model shape. The individual residues were then evaluated at the end of the registration process. If the value of the individual residues happened to lie within 1 mm, the obtained registration transformation would be regarded as the ground-truth. Fig. 3.4 illustrates how the error of registration transformation is determined using ground-truth transformation. The estimate of the transformation obtained from the registration process between the model and the data is represented by eT

m while the ground-truth transformation between the two datasets is given by gT

m . The difference between the estimated and ground-truth transformations provides the error of the registration transformation, eT

g . This transformation is a function of six parameters just like the 3D rigid transformation as explained in Section 3.3.2. The parameters can be represented by zyxzyx ,,,,, , where zyx ,, are rotations about the X, Y, and Z axes, respectively, and zyx ,, are translations along the newly rotated X, Y and Z axes. These six parameters or the norms of the rotation and translation error components can form the basis for evaluating the registration accuracy [22]. The norms are given by:

222 zyxet

222zyxer

(3.16)

(3.17)

54

where et and er represents the norms of the translation and rotation components of the registration transformation error, eT

g , respectively. The validity of Equation 3.17 exists only for small rotation errors in which the relationship )sin( holds. But in case of a large rotation error (greater than 5 degrees), the error can be converted into what is referred to as an angle-axis representation [22, 75] to provide a more accurate measure. In this representation, any 3D rotation can be represented as a single rotation about a given axis. Instead of using the rotation error norm as given by Equation 3.17, the rotation angle from the angle-axis representation is used to evaluate the accuracy for large rotation errors. Fig. 3.4: Determination of the registration transformation error using ground-truth transformation.

model

data(ground truth)

data(estimate)

T

T

T

(Error of registrationtransformation)

m

mg

g

e

e

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3.4.2 Robustness of the Registration Technique

Robustness of a registration technique can be defined as its ability to withstand a range of data variations resulting from outliers described in Section 3.3.3. In this dissertation, robustness was evaluated by analyzing the effect of a controlled data variation parameter on the registration accuracy. In order to carry out this evaluation, the data shape was constructed by copying and perturbing the model shape by a controlled amount. By using such a simulated data shape, the correct ground-truth registration pose is known and therefore the pose error after registration can be measured at varying perturbations that are applied to the data shape prior to the registration process. The following three types of perturbations were used in this work:

Addition of clinical environment noise. This involves introducing noise, whose magnitude is larger than the sensor noise, at random 3D locations of the original data shape. The goal of this simulation is to analyze the effect the spurious structures appearing in the intraoperative imaging would have on the registration accuracy. In our application, such structures would include torn soft tissues and blood, which would be captured by the intraoperative laser radar imaging but would not be present in the preoperative MRI surface model.

Removal of surface patch. A localized neighborhood of surface

points is removed from the data shape in this procedure. The process simulates the sensitivity of the registration technique to incomplete overlaps of the surfaces being registered. This simulation was necessary for our application due to the fact that the intraoperative 3D surface from laser radar imaging covers a fraction of the preoperative MR data.

Addition of sensor noise. The created data shape is corrupted with

random Gaussian noise with an aim of obtaining data points, which more closely resemble real measurements. The magnitude of the added noise corresponds to the magnitude of the sensor noise, in this case the uncertainty of the laser radar imaging system.

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The simulation evaluations were complemented with practical experiments carried out on a phantom but without addition of noise to the data shape. An exact ground-truth registration pose was not available in these cases but the registration sensitivity with respect to the first two controlled data deviations could be performed with respect to an estimated ground-truth, as explained in Section 3.4.1. The same type of evaluations were finally performed on application experiments in which both the model and data shapes are representative clinical data. Results of these evaluations as well as the description of the magnitudes of data variations applied are described in the following chapters.

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Chapter 4 Experimental Evaluation of the Registration Technique The following sections present various experiments which were carried out to verify the performance of the registration technique by analysing the registration evaluation measures as described in Section 3.4. The registration algorithm was first tested using synthetic data, after which real data was applied in evaluating the sensitivity of the registration technique to controlled data variations.

4. 1 Validation of the Registration Algorithm

4.1.1 Synthetic Data Experiments

In these experiments, a simple model of a prism to represent the model shape was generated using a C++ program within VTK [58], while the data shape was created by copying and applying a known transform to the model shape. The prism had a square base of 40 mm 40 mm and a height of 40 mm. The synthetic prism was generated by inserting points only on its two sloping faces at a vertical spacing of 1 mm, giving a total of 3321 points. That means the base and the two triangles at the sides of the prism were left blank. The physical dimensions of the model were designed to correspond with those that would be captured by the laser radar system in the imaging of the prism (see Section 4.2). The data shape was registered to the model shape in four runs using the registration parameters given in Table 4.1. In this experiment, no initial pose of transformation was necessary. Figs. 4.1, 4.2 and 4.3 shows the variation of rotation, translation, and root-mean-squared error of the registration (RMS) with respect to the number of the ICP iteration respectively for a transformation of 15 degrees rotation around the X axis. At the end of each run, the results were compared with the true registration parameters (ground-truth transformations) which are already known. The mean of the translation error norm, et , was found to be 1.2 10-5 mm, while the mean of the rotation error norm, er , was 2.7 10-6 degrees.

58

Registration Parameter Value

Max. no. of iterations 40

Transformation offset of data shape 0 - 15 degrees rotation around X axis

(at steps of 5 degrees)

Minima suppression Disabled

Outlier elimination Disabled

Noise addition Disabled

Table 4.1: Registration parameters for the synthetic prism experiment. The mean RMS error at the termination of the iterations was 4.2 10-10 mm. These error values, which represent our ground-truth transformation errors, are negligible in practice, and demonstrate that under perfect matching conditions (i.e. with good initial alignment and no noise), the algorithm effectively yields an accurate result. The experiment was repeated with the same datasets but this time with varying levels of Gaussian noise added to the point coordinates of the data shape. The registration algorithm used the parameters of Table 4.2 in which minima suppression was enabled and the transformation offset of the data shape was kept fixed while the noise level was varied. The maximum level of the added noise was dictated by the maximum uncertainty of the laser radar imaging system which corresponds to 1.0 mm (see Section 2.6.2). A total of 10 registration experiments were performed with each data configuration using expected noise magnitudes ranging from 0.1 mm to 1.0 mm in 0.1 mm increments. The errors of the registration transformation parameters were evaluated relative to the ground-truth transformation errors for each noise level at the termination of the individual registration process. Fig. 4.4 shows the evolution of translation and rotation error as a function of the noise level while Fig. 4.5 presents the variation the root-mean-squared error (RMS) versus the noise level.

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Fig 4.1: Rotation values versus iteration number for the syntheticprism test. The values are given relative to the pose of the prism after a rotation of 15 degrees around the X axis.

0 5 10 15 20 25 30 35 40

-16

-15

-14

-13

-12

-11

-10

-9

X R

ota

tio

n (

De

gre

es)

Iteration Number

0 5 10 15 20 25 30 35 40

-10

-8

-6

-4

-2

0

2

4

6

8

10

Y R

ota

tio

n (

x1

0-7 D

eg

ree

s)

Iteration Number

0 5 10 15 20 25 30 35 40

-3

-2

-1

0

1

2

3

Z R

ota

tio

n (

x1

0-7 D

eg

ree

s)

Iteration Number

60

0 5 10 15 20 25 30 35 40

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Z T

ran

sla

tio

n (

mm

)

Iteration Number

0 5 10 15 20 25 30 35 40

0.0

0.1

0.2

0.3

0.4

0.5

X T

ran

sla

tio

n (

mm

)

Iteration Number

0 5 10 15 20 25 30 35 40

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Y T

ran

sla

tio

n (

mm

)

Iteration Number

0 5 10 15 20 25 30 35 40

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Z T

ran

sla

tio

n (

mm

)

Iteration Number

Fig 4.2: Translation values versus iteration number for the syntheticprism test. The values are given relative to the pose of the prism after arotation of 15 degrees around the X axis.

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Registration Parameter Value

Maximum number of iterations 40

Transformation offset of data shape 15 degrees rotation around X axis

Minima suppression Enabled

Maximum number of iterations within

minima suppression procedure

5

Maximum angle of the random

rotational perturbation for minima

suppression

7.5 degrees

Maximum value of the random

translational perturbation for minima

suppression

1.2 mm

Noise addition Enabled

Added noise magnitude 0.1-1.0 mm by 0.1 mm

Outlier elimination Disabled

Table 4.2: Registration parameters for the synthetic prism with added noise.

Fig 4.3: Root-mean-squared error versus iteration number for thesynthetic prism test.

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As expected, the rotation and translation errors increase as a function of increasing data noise. The registration accuracy (as measured here by the rotation and translation errors) degrades gracefully and almost predictably over the range of our input noise levels, which confirms that the registration algorithm is accurate. registration algorithm is accurate.

Fig 4.4: Rotation and translation errors versus the noise level on thedata shape for the synthetic prism experiment. The values are givenrelative to those obtained without noise.

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The rotation error norm,

er , and the translation error norm, et , at the

highest noise level of 1.0 mm are 0.08 degrees and 0.41 mm respectively, and lies within the acceptable range. The maximum value of the RMS error in these experiments was 2.4 10-6 mm. Fig. 4.5 indicates that there is no discernible relationship between the RMS error and the magnitude of the data noise. This shows that though the RMS error is necessary in checking the results of registration, it alone can not be used to quantify the registration accuracy. 4.1.2 Preoperative Data Experiments

These experiments, which are similar to those carried out in Section 4.1.1, aim at evaluating the performance of the registration algorithm with real data that are relevant to the application of this work. In this case, the model

Fig 4.5: Root-mean-squared error (RMS) versus the noise level on the data shape for the synthetic prism test.

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shape is an actual CT image. The data shape is acquired by copying the model shape and applying a known transformation to it. The CT image was obtained by scanning a cadaver socket bone from a cow, with soft tissue, using a Siemens CT scanner at Orthopaedic Clinic, Kassel. Axial slices were taken with intra-slice resolution of 1 mm x 1 mm and 1 mm inter-slice distance. It would have been preferable to use a Magnetic resonance (MR) scanner in these experiments but a high resolution MR scanner was not available at the time this evaluation was carried out. However, the CT scanner served the purpose well. Though it is usually difficult to identify accurately soft tissues on bones inside the body from the CT images, this is not a problem when the bone is imaged in air. This is because the attenuation of X-ray in air will be much less than that through the soft tissues and so the outer surface of the soft tissues can be accurately identified.

Fig. 4.6 shows a single slice from a 3D CT image of cadaver socket bone from a cow. In this figure, all the anatomical structures of the acetabulum are discernible, including the soft tissues. In order to generate a 3D surface from the slices, no elaborate segmentation procedure was carried out apart from removing the structures which were

Fig. 4.6: A single slice from a CT image of a cadaver socket bone from a cow, with soft tissues. Imaging resolution: 1.0 1.0 1.0 mm.

Ligament

Cartilage

Cancellous bone

Cortical bone

65

used to support the bone in places where there were physical contacts. A Marching Cubes algorithm [56] was used to create a 3D model which carried only the outer surface information. Fig. 4.7 shows the surface which was extracted from the CT slices of the cow’s cadaver socket bone. The surface had a total of 90351 triangles. After applying decimation techniques [57], these triangles were reduced to 35678 which allowed faster rendering of the dataset without compromising the surface details of the acetabulum.

The decimated dataset became the model shape of the registration experiments from which the data shape was generated. The first set of experiments involved uncorrupted data shape using the registration parameters of Table 4.1 and were carried out using the same procedure as described in Section 4.1.1. Rotations were performed about a coordinate system with the origin at the center of the acetabulum. The resulting registration values are given in Figs. 4.8, 4.9 and 4.10 for the largest transformation of 15 degrees rotation around the X axis. The graphs show

Fig. 4.7: Extracted 3D surface model from CT imaging of a cadaver socket bone from a cow with soft tissues.

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that the errors of the registration transformation parameters converge more monotonically towards zero than in the cube registration experiment (see Section 4.1.1). This can be explained by the fact that there are curvature regions in the CT datasets which give them specific orientations as opposed to the flat faces in the cube datasets. In these experiments, the mean of the translation error norm, et , and the mean of the rotation error norm,

er ,

were found to be 5.2 10-5 mm and 7.3 10-5 degrees respectively, while the mean RMS error at the termination of the iterations was zero. As in the cube registration experiment, the registration transformation errors obtained here are negligible and proves that the registration algorithm is accurate even with complex data under perfect matching conditions (i.e. provided there is good initial alignment and in the absence of noise). The next set of registration experiments was performed to simulate the effects of sensor noise in the process of acquisition of the data shape. This involved adding varying levels of Gaussian noise to the data shape following the procedure described in Section 4.1.1, and then carrying out the registration process for each noise level using the parameters specified in Table 4.2. Based on the ground-truth transformation errors obtained above with the uncorrupted data shape, the errors of the registration transformation parameters were calculated for each noise level at the termination of the individual registration process. The results are shown in Fig. 4.11 which gives the variation of the rotation and translation errors with the noise level added to the data shape. Even here, as observed in Section 4.1.1, the registration accuracy is seen to degrade gracefully and almost predictably over the range of our input noise levels. At the highest noise level of 1 mm, the rotation error norm,

er , and the translation error

norm, et , are 0.046 degrees and 0.49 mm respectively, and lies within the acceptable range. The rotation error norm is observed to be almost a half of the value obtained in the cube registration experiment which can be explained by the presence of distinct landmarks in the CT data shape. These results demonstrate clearly that our registration algorithm effectively returns accurate results with complex datasets under the expected noise levels and is thus appropriate for our application.

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Fig 4.8: Rotation values versus iteration number for the simulation experiment with CT surface. The values are given relative to the poseof the CT data after a rotation of 15 degrees around the X axis.

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Fig 4.9: Translation values versus iteration number for the simulationexperiment with CT surface. The values are given relative to the poseof the CT data after a rotation of 15 degrees around the X axis.

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Fig 4.10: Root-mean-squared error versus iteration number for the simulation experiment with CT data.

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Fig 4.11: Rotation and translation errors versus the noise level on the data shape for the simulation experiment with CT data.

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4.2 Registration Analysis with Laser Datasets

4.2.1 Accuracy of 3D Laser Imaging

This section aims at investigating the accuracy of our laser radar imaging in scanning 3D objects. In order to carry out this analysis, a simple model of a prism, whose geometrical dimensions are exactly known, was used as a calibration body. The prism used had the same physical dimensions as those described in Section 4.1.1 (a square base of 40 mm 40 with a height of 40 mm) and was precisely manufactured using aluminium. It was sprayed with a thin layer (about 0.1 mm) of a grey anti-rust paint to make the surface diffuse to the laser beam. The laser radar imaging system was then used to acquire 3D surface points of the calibration body at a distance of 90 cm with steps of 1 mm in both X and Y axes of an XY coordinate system. Each point on the target was measured with 8 averages and no smoothing algorithm was applied on the acquired surface points. The prism was placed with its base fixed on a support such that the two sloping sides of the prism faced the laser radar system during the scanning process. In that case, only the surface points from the two faces of the prism were relevant. Fig. 4.12 shows the acquired laser surface points. The data was processed by removing some of the surface points which were captured from the supporting structure. A computer model of the prism was also generated using the same technique described in Section 4.1.1.

Fig 4.12: Laser surface points of a prism for calibration experiment. Thesurface data include the background of the supporting structure.

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Registration Parameter Value

Maximum number of iterations 200

Transformation offset of data shape None applied

Minima suppression Enabled

Maximum number of iterations

within minima suppression

procedure

6

Maximum angle of the random

rotational perturbation for minima

suppression

2.5 degrees

Maximum value of the random

translational perturbation for minima

suppression

1.4 mm

Noise addition Disabled

Outlier elimination Disabled

Table 4.3: Registration parameters for the calibration experiment. The evaluation of the accuracy of the laser radar system in 3D scanning was carried out by first registering the measured surface data (as data shape) to the 3D computer model (as model shape) and then computing the distance differences between the points of the two datasets after the termination of the registration process. In order to compute the distance differences, the measured points dataset was converted into a surface representation and then the locations of the points on the surface were subtracted from the corresponding points in the synthetic model after matching the two datasets. The surface representation of the measured points dataset was achieved by using “splatting” technique (See Section 3.2.2 for details). The parameters shown in Table 4.3 were used in the registration process whereby no perturbation was applied to the data shape. The computer model of the prism had been generated in such a way that it would be in an

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approximate orientation with the measured surface data as regards rotational view, and therefore no initial pose alignment was necessary in the registration process.

Fig 4.13: Rotation and translation (relative to the initial pose of thelaser data) versus iteration number during the registration of the lasersurface points of a prism to its computer model.

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Fig. 4.13 shows the variation of the rotation and translation against ICP iteration number. The values of the rotation and translation parameters are plotted relative to the initial pose of the measured data, whereby the rotations are given by a coordinate system with the origin at the center of the target. It can be deduced from the rotation and the translation graphs that the initial pose of the data shape (measured data) was within about 4 degrees and 14 mm of the final pose. These values are reasonable considering the initial orientation of the synthetic data with respect to the measured data as described above.

The variation of the root-mean-squared (RMS) error with the ICP iteration number is shown in Fig. 4.14. It can be seen that the RMS error converges monotonically to zero. To check the presence of outliers in the data shape, the maximum residual error (MRE) was evaluated at every iteration. The evolution of the MRE as a function of the ICP iteration number is plotted in

Fig 4.14: Root-mean-squared error versus iteration number for the registration of the laser surface points of a prism to its computer model.

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Fig. 4.15. It is evident that there are no outliers in the data shape since the MRE value upon convergence is roughly 1.3 mm, a value that is close to the expected data noise magnitude of 1.0 mm. These results indicate that an optimum matching between the two datasets was achieved in this registration procedure. Fig. 4.16 demonstrates the distance differences between the model and the data shapes of the prism after the termination of the registration process. The blue colour represents points with distance differences of less than 1 mm while the red colour represents points where the distance differences were greater than 1 mm. From a total of 3321 points, 3227 points were found to lie within a distance difference of 1 mm, representing about 97 percent of the measured data. The maximum distance difference corresponded with the maximum residual error of 1.3 mm. The mean of the distance differences over the whole measured surface was found to be 0.52 mm while the standard deviation was 0.14 mm. This demonstrates that our laser radar scanning system is capable of reproducing 3D surface geometries of targets within an accuracy of 1 mm and meets the requirements of our application.

Fig 4.15: Maximum residue error versus iteration number in theregistration of the laser surface points of a prism to its computer model.

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4.2.2 Laser Data to CT Data Registration

The experiments in this section were meant to simulate the performance of our registration technique using 3D surface datasets which closely approximate the actual application datasets. In order to carry out this task, a phantom was scanned using a medical scanner to obtain a model which represents the preoperative data (model shape) in the actual registration procedure. Our laser radar imaging system was then used to obtain a 3D surface geometry of the relevant section of the phantom to represent the intraoperative dataset (data shape). The phantom which was used in these experiments, was exactly the same cadaver socket bone from a cow, with soft tissues, whose 3D model extraction from the CT images is described and illustrated in Section 4.1.2. It is this surface geometry which became the model shape for the registration analysis which was carried out throughout this section. The

Fig 4.16: Illustration of the distance differences between the points ofthe measured data and the model of the prism after registration (Blue:distance < 1mm, Red: distance > 1 mm, maximum distancedifference: 1.3 mm)

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laser radar scanning of the phantom was performed immediately after the CT imaging. Care was taken in transporting the phantom to make sure that no deformation occurred to the soft tissues on the bone. This was to ensure that the surface geometry of the phantom during the laser radar scanning remained much the same as that during the CT image acquisition. The acquisition of the 3D surface of the phantom by the laser radar system was carried out at a distance of 90 cm from the scanning mirrors using a resolution of 1mm in both X and Y axes of an XY coordinate system. As in the imaging of the prism in Section 4.2.1, each point measurement on the target was evaluated using 8 averages without applying any smoothing algorithm on the acquired surface points. The scanned area was 110 mm 110 mm and it was selected such that it covered the acetabulum plus the surrounding regions. The signal response of the laser radar system to the laser beam reflected from the surface of the phantom was first investigated before performing the scanning process. Fig. 4.17 shows the maximum and minimum amplitudes of the detected signals from the acetabular surface. The signals were measured directly from the avalanche photodiode using a 50 GHz sampling oscilloscope, HP 54120B from Hewlett Packard (see Chapter 2 for detailed description of the laser radar system). From the graphs of the pulse responses, it can be seen that the maximum and the minimum voltages of the received signal was about 58 mV and 32 mV respectively. These results demonstrate that the power of the reflected laser pulses is adequate to make accurate measurements over the whole bone surface. Experiments performed to analyse the effect of the magnitude of the signal response on the measurement accuracy had shown that, for a reliable measurement to be carried out using our laser radar system, the detected signal need to have an amplitude of not less than 25 mV (See Section 2.6.2). Fig. 4.18 (a) shows the surface points of the phantom which were acquired using our laser radar imaging system. The 3D dataset excludes the background surface points from the supporting structure which were removed using a cutting filter in Visualization Toolkit (VTK) [58]. The filter cuts through the 3D polygonal surface using a defined implicit function to remove the surface points which are not required. The same filter was used to remove sections of the acquired surface such that only the acetabulum’s data was left. In order to obtain

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anatomical features of the phantom from the surface points which are easily apparent to the eye, a surface representation was constructed using “splatting” techniques as described in Section 3.2.2. Fig. 4.18 (b) illustrates the generated surface representation of the acetabulum from the surface data points of the phantom.

Fig. 4.17: The measured maximum and minimum amplitudes of thedetected signals from the surface of cadaver socket bone from a cow,with soft tissues.

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Fig. 4.18: 3D surface model acquired using laser radar imaging: (a)surface points and (b) surface representation generated through“splatting” technique.

(a)

(b)

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By comparing visually the extracted 3D surface model from the CT imaging shown in Fig. 4.7 with the surface generated from the laser radar imaging illustrated in Fig. 4.18(b), one can observe various anatomical features which are common to the two models. These features are important in providing the initial pose estimate for the registration process. As explained in Section 3.3.3, a good initial pose estimate is necessary to bring the two datasets into a rough alignment with each other before applying the ICP algorithm, in order to avoid being trapped in local minima that are far from the global minimum. The initial alignment was estimated by first identifying three anatomical landmark points from the CT model of the acetabulum and the corresponding landmark points on the surface model from the laser radar imaging, as demonstrated in Fig. 4.19. This was performed by clicking with the mouse on an anatomical landmark in the CT model and then clicking a corresponding landmark point on the surface data from the laser radar imaging on the visualisation screen. The process was repeated for all the three corresponding points. With each mouse click, a marker in the form of a sphere would be placed on the selected 3D location. This technique was useful in indicating whether the intended anatomical landmark was selected or not. After the identification of the corresponding landmarks, the transformation between the two sets was then computed using the corresponding point registration method according to Horn [59]. The obtained transformation estimate was used as the starting pose for the surface-based registration process. Table 4.4 shows the registration parameters that were applied in this registration experiment. This time, both local minima suppression and outlier elimination methods were turned on. Fig. 4.20 presents the variation of translation and rotation values with the ICP iteration number for a single run of the ICP algorithm. The translation and rotation values are given with respect to the initial pose estimate determined through the corresponding point registration. The rotations are given about a coordinate system with the origin at the center of the CT model.

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(a)

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Fig. 4.19: Identification of anatomical landmarks for the correspondingpoint registration (a) CT model (b) surface model from laser radarimaging of cadaver socket bone from a cow.

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The graphs illustrate that the starting pose estimate was within around 4 degrees and 10 mm of the final evaluated registration transformation. This shows that a good initial pose estimate was established using the anatomical landmark correspondence technique described above. Fig. 4.21 shows the variation of the root-mean-squared (RMS) error as well as the maximum residual error (MRE), against the ICP iteration number. As in the previous registration experiments, the RMS error converges monotonically towards zero. But the MRE remains at around 4.8 mm after convergence, which indicates the presence of outliers as the value is about five times the expected sensor noise magnitude of 1 mm. In order to analyze the extent of the outliers in the data shape, distance differences between the laser radar acquired surface points and the surface extracted from the CT imaging were evaluated in the same manner described in Section 4.2.1 after one run of the ICP algorithm. Fig. 4.22 presents the surface points from laser radar imaging overlaid on the CT model after the convergence of the registration algorithm and evaluation of the distance differences. The colour on the dots denotes residue errors (distance differences) between the laser points and the CT surface. The blue colour represents points with distance differences of less than 1 mm with the red colour representing points where the distance differences were greater than 1 mm. It can be seen that the number of outliers was quite small. A total of 4283 points from laser radar imaging were used in this experiment, and out of this, only 214 points were found to have a distance difference beyond 1 mm, a mere 5 percent of the overall points. The maximum distance difference was 4.8 mm and corresponded with the calculated maximum residual error. The mean of the distance differences over the area covered by the laser surface points was found to be 0.56 mm while the standard deviation was 0.26 mm.

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Registration Parameter Value

Maximum number of iterations 200

Transformation offset of data shape Corresponding point registration

provided the initial pose estimate

Minima suppression Enabled

Maximum number of iterations

within minima suppression

procedure

6

Maximum angle of the random

rotational perturbation for minima

suppression

3 degrees

Maximum value of the random

translational perturbation for minima

suppression

2.4 mm

Noise addition Disabled

Outlier elimination Enabled

Residue error threshold above which

a data point is regarded as an outlier

1 mm

Fraction of outliers eliminated per

ICP run

10 percent

Table 4.4: Registration parameters for the CT/laser data experiment.

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Fig 4.20: Rotation and translation (relative to the starting pose estimate)versus iteration number for a single run of the ICP algorithm in theregistration experiment between CT model and laser surface points of a cadaver socket bone from a cow.

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Fig. 4.21: Root-mean-squared (RMS) error and maximum residualerror (MRE) versus ICP iteration number for a single run of the ICPalgorithm in the registration experiment between CT model and lasersurface points of a cadaver socket bone from a cow.

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Fig. 4.22: Laser surface points overlaid on the CT surface after a singletrial of the ICP algorithm in the registration experiment between CTmodel and laser surface points of a cadaver socket bone from a cow.The laser data is shown as coloured dots, where the colour presentsdistance from the laser surface point to the underlying CT surface (Blue:distance < 1mm, Red: distance > 1 mm, maximum distance difference:4.8 mm).

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The effect of the outliers on the registration accuracy was investigated by applying the outlier elimination procedure in the registration process, and then evaluating changes in the registration parameters as the outlier points were removed. The use of outlier elimination technique requires multiple runs of the ICP algorithm (See Section 3.3.3). In this experiment, 10 percent of the outlier points were removed from the data points (laser surface points) between the ICP runs until all the outliers were eliminated. Fig 4.23 shows the variation of the maximum residue error (MRE) and the root-mean-squared (RMS) error as a function of the percentage of outlier points eliminated from the laser surface points. From the graphs, it can be seen that the RMS and MRE values for the final ICP run are reasonably smaller after the application of the outlier elimination method. But the changes in the transformation parameters as a result of outlier elimination were found to be in the order of 10-4 mm for the translations and 10-6 degrees for the rotations, which are negligible. This implies that the outliers didn’t have any substantial effect on the registration accuracy and can be explained by the fact that the percentage of the outliers in the data shape was small. The possible source of the outliers could be CT segmentation errors, tissue deformation or CT imaging errors. The MRE after the complete elimination of the outliers was found to be 0.92 mm, which lies within the expected sensor noise of around 1 mm. From the results obtained in this experiment, it is evident that an optimum matching between the CT model and the laser surface points was obtained after the registration process. The registration transformation obtained in this registration procedure was used as a ground-truth for the simulation of the registration sensitivity to data perturbations as described in Section 4.3.

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4.3 Simulation of Laser Data Sensitivity

The experiments performed in Section 4.2.2 used almost an ideal data shape due to the fact that the laser surface data was acquired on a surface which was a replica of the geometry extracted from the CT imaging. The laser points therefore approximated very well the CT surface. In practice, however, the situation is somehow different. On one hand, the intraoperative laser points would be acquired on an acetabulum with some torn tissues and blood. These structures would not appear on the preoperative model but will be captured by the laser radar system. Though efforts can be made to dry the blood and remove some torn tissues, there are bound to be some remnants. On the other hand, due to clinical access limitations, it is not possible to acquire all the points on the regions of acetabulum as shown in Fig. 4.18. It would be especially difficult to collect points on the periphery of the acetabulum due to the presence of ligaments. This section aims at analyzing the registration performance with controlled data variations to the data shape (laser points) that are meant to simulate the two adverse conditions described above. The controlled variations were applied to the laser surface points obtained from Section 4.2.2, in which outliers had been eliminated. 4.3.1. Clinical Environment Noise Sensitivity

These experiments were meant to evaluate the ability of the registration technique to withstand the clinical environment noise (soft tissues and blood) added to the laser surface points. A pessimistic noise magnitude of 5 mm was chosen in this analysis. The area covered by the noise was a function of the number of points in the laser surface dataset, starting from 10 to 100 percent in increments of 10 percent. For every simulation experiment at a particular fraction of the laser surface points, Gaussian noise was added at random in the original laser surface dataset. The corrupted laser points for each experiment were registered to the CT surface model which appears in Sections 4.1.2 and 4.2.2. Table 4.5 presents the registration parameters that were used in this experiment.

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Registration Parameter Value

Maximum number of iterations 200

Transformation offset of data shape Corresponding point registration

provided the initial pose estimate

Minima suppression Enabled

Maximum number of iterations

within minima suppression

procedure

6

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rotational perturbation for minima

suppression

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Maximum value of the random

translational perturbation for minima

suppression

5.2 mm

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Added noise magnitude 5 mm

Region on data shape covered by

noise

10 to 100 %, Increment per trial:

10%

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Table 4.5: Registration parameters for the clinical environment noise experiment.

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The results of the registration transformation parameters at every percentage of the laser surface points covered by the clinical environment noise were compared with the transformation from Section 4.2.2, which was regarded as the ground-truth in these experiments. Fig. 4.24 illustrates the evolution of the rotation and translation errors with the percentage of the region of the laser surface points covered by the noise level. From the graphs, it is evident that the registration transformation errors increases substantially when the clinical environment noise covered more than 30 percent of the laser surface points, reaching about 0.14 degrees in rotation and about 2 mm in translation. At 30 percent, the rotation error norm,

er , and the translation error norm, et , were found to be 0.03

degrees and 0.29 mm respectively. The required accuracy for our application is 1 degree in rotation and 1 mm in translation, and so the transformation errors at 30 percent of the region covered by noise lies within an acceptable range. It can then be concluded that our registration technique is reliable when the clinical environment noise covers up to 30 percent of the acetabulum’s region. This is feasible in practice.

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Fig 4.24: Rotation and translation errors as a function of thepercentage of the region on the acetabulum covered by clinicalenvironment noise in the registration experiment between CT modeland laser surface points of a cadaver socket bone from a cow.

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4.3.2 Sensitivity due to Clinical Access Limitations

The sensitivity of our registration technique to incomplete overlaps of the datasets being registered, due mainly to the clinical access limitations, is investigated in these experiments. In order to carry out this analysis, points were removed from laser surface dataset in fractions of the original laser surface points, starting from 10 to 70 percent in increments of 10 percent. The process of points removal was started from the rim of the acetabulum because that is one region on the socket that would be impossible to capture using laser radar imaging due to the presence of ligaments. It then proceeded towards the center of the acetabulum. At every step of the points removal, points would be removed uniformly from all around the periphery of the acetabulum represented by the laser surface points. For each experiment, the clipped laser dataset was registered to the CT surface model. The registration parameters that were used in this experiment are given in Table 4.6. Based on the estimated ground- truth transformation from Section 4.2.2, errors of the registration transformation parameters were evaluated at every step of the points removal from the laser dataset. The errors in the rotation and translation parameters are presented in Fig. 4.25. It can be seen that the errors were negligible up to the removal of 50 percent of the laser surface points. Above this value, the errors increased substantially, reaching about 2 degrees in rotation and about 8 mm in translation. This can be explained by the fact that below 50 percent, the absence of anatomical landmarks causes the laser points to easily slide or rotate around the CT surface model during the registration procedure. Our registration technique is therefore accurate when the region captured by the laser radar imaging is at least 50 percent of the area of the acetabulum. In practice, the laser radar system would be able to cover around 80 percent of the acetabulum’s surface.

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Registration Parameter Value

Maximum number of iterations 200

Transformation offset of data shape Corresponding point registration

provided the initial pose estimate

Minima suppression Enabled

Maximum number of iterations

within minima suppression

procedure

6

Maximum angle of the random

rotational perturbation for minima

suppression

3 degrees

Maximum value of the random

translational perturbation for minima

suppression

1.2 mm

Noise addition Disabled

Outlier elimination Disabled

Removal of laser surface points 10 to 70 %, Increment per trial:

10%

Table 4.6: Registration parameters for the clinical access limitations experiment.

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Fig 4.25: Rotation and translation errors as a function of the percentageof the surface removed from the laser dataset.

10 20 30 40 50 60 70

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Chapter 5

Clinical Trial The performance of the laser radar system was tested in clinical environment where it was used to scan an acetabulum of a female osteoarthritic patient who was undergoing a total hip replacement surgery at the Orthopaedic Clinic, Kassel. The goal was to analyze the efficacy of our registration technique under the real operating conditions. In this trial, the preoperative data of the affected hip joint was acquired using an MR scanner one day before the operation. The laser radar system was then used to obtain 3D points of the acetabulum during the surgical procedure once part of the pelvis was exposed. The data acquisition procedures and parameters, data extraction as well as the analysis of the registration results are presented in the following sections.

5.1 Preoperative Data

5.1.1 Image acquisition

A 1.5 Tesla high resolution MR imager (MRT-Symphony Quantum, Siemens, Erlangen, Germany) at Clinical Centre, Kassel (Klinikum Kassel) was applied in the acquisition of the MR images of the patient using a fat-suppressed T1-weighted 3D gradient echo pulse sequence. As explained in Section 3.2.1, our registration technique requires that any cartilage present on the acetabulum be included in the segmentation of the MRI data because its surface profile will also be captured by the intraoperative laser radar imaging. This requirement necessitates the use of a pulse sequence which shows articular cartilage with high intensity in contrast to most of the other surrounding tissues such as fat pads, ligaments and synovial fluid. The fat-suppressed T1-weighted 3D gradient echo sequence has proved to be reliable in highlighting the cartilage with respect to the adjacent tissues [90], and was therefore selected for the acquisition of the preoperative images used in this work. Axial slices of the affected hip joint were acquired with an in-plane resolution of 1 mm × 1 mm and an inter-slice thickness of 1mm. The size

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of MR images was 256 × 256 × 87 voxels. Other acquisition parameter settings were as follows: 50 degrees flip angle, 34 ms TR, 4.8 ms TE, and 250 mm field of view. 5.1.2 Image Segmentation and Surface Extraction

An example slice from the 3D MR image of the patient’s hip joint is shown in Fig. 5.1 (a). The cartilage is seen to be in high contrast to the adjacent tissues. Fig. 5.1 (b) shows the same slice after segmenting the acetabulum, including the cartilage, with manually traced boundaries (shown in red). The 3D Slicer software [55] was used in carrying out the segmentation process. As explained in Section 3.2.1.1, manual interventions during segmentation were necessary due to the fact that most internal structures of the hip joint such as ligaments, cartilage and bone, have some overlapping image intensity values. Further, in an osteoarthritic hip whereby some cartilage is completely worn out, the bone of the head of femur appears embedded within the acetabulum bone. MRI slices in such regions show hardly any contrast in the image intensity values between the two anatomical structures. These factors ruled out full automation in segmenting the acetabulum. Rather, much care was taken in manual tracing to define the required boundaries. The segmentation process was later visually verified by a medical imaging expert from the Orthopaedic Clinic, Kassel, to ensure accuracy [97].

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Fig. 5.1: An example slice from the patient’s 3D MRI data: (a) before segmentation, and (b) after segmenting the acetabulum with manually traced boundaries shown in red.

(a)

(b)

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After segmentation of the MRI data, the acetabulum’s surface was extracted using Marching Cubes algorithm [56] followed by Decimation [57] (See Section 3.2.1.2). The decimation technique reduced the number of triangles on the acetabulum’s surface by about a third, from 53,526 to 18,535, in an effort to increase the data rendering and processing speeds. Fig. 5.2 shows the 3D surface model of the patient’s acetabulum extracted from the MRI data. In this surface, the peripheral sections of the hip bone had been removed using a cutting filter in Visualization Toolkit(VTK) [58] that is described in Section 4.2.2.

Fig. 5.2: Extracted 3D surface from the patient’s MRI data.

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5.2 Intraoperative data

During the actual surgical operation, the surgeon generally separates the muscles, ligaments and tendons to access the hip joint. The hip joint is then dislocated by separating the head of femur from the acetabulum. This procedure exposes the acetabulum. For our measurements, tissue which would produce spurious results during laser scanning was carefully removed from the outer edges of the acetabulum as well as from the acetabular fossa. Acetabular fossa is a circular non-articular depression in the floor of the acetabulum superior to the acetabular notch [93] (See illustration of the hip bone in Fig. 5.3). The depression is perforated by numerous apertures, and lodges a mass of fat. Nerves and blood vessels supplying the joint with nutrients are found in this depression. For example, the ligament of the head of femur (ligamentum teres) carries an artery that supplies the femoral head directly from a more proximal route by coursing through the acetabular fossa. Since these tissues get torn during the process of separating the femoral head from the acetabulum, there was a need to remove them, where possible, before collecting surface data using the laser radar system. Fig. 5.3 : Illustration of a normal human acetabulum (after [93]) .

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The acetabular exposure was optimized by using metallic retractors as shown in Fig. 5.4, such that the entire acetabular margin could be observed. The laser radar system was then used to acquire 3D points of the exposed acetabulum from a distance of about 85 cm from the scanning mirrors at a resolution of 1 mm in both X and Y axes of an XY coordinate system. The scanned area was 40 mm × 40 mm and it was selected such that it covered the entire laser beam accessible region of the acetabulum. Fig. 5.5 shows the setup of the patient and the laser radar system in the operating room during measurements, while Fig. 5.6 presents the acquired laser surface points. Each point measurement on the target was evaluated using 8 averages.

Fig. 5.4: Opening of the patient’s hip joint after dislocation. Themetallic retractors were applied to optimize the acetabular exposure.

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Fig. 5.5: Setup of the patient and the laser radar system in theoperating room during the scanning of the acetabulum.

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During the laser scanning of the acetabulum, the control program evaluated at every pixel the peak voltages corresponding to the response of the avalanche photodiode to the received signal. The minimum and maximum voltages were found to be 35 mV and 67 mV respectively. These voltages are above 25 mV which means accurate distance measurements could be resolved within the scanned area of the acetabulum, as explained in Section 2.6.2. The evaluated voltage values are found to be higher than those obtained while scanning the cow’s bone, whereby the minimum and maximum voltages were 32 mV and 58 mV respectively (See Section 4.2.2). This can be explained by the fact that the imaging range in this case was shorter (85 cm) than the one that was used in scanning the bovine bone (90 cm). Further, the patient’s acetabulum was bound to have more moisture content as compared to that from a slaughtered cow. At close range, the intensity of the received laser beam in a pulsed laser radar system has been observed to increase in most cases with the increase of the moisture content of the reflecting surface [91]. A small section in the region of the acetabular fossa (measuring about 2 mm × 2 mm) produced no detectable signal. This could be explained by the presence of a clot, which could have attained a dark brown colour. The dark brown colour has been found to have a poor spectral reflectance [94] and explains why no

Fig. 5.6: 3D laser surface points from the patient’s acetabulum.

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surface points could be obtained in that section of the acetabular fossa. This phenomena was not observed in any other section of the acetabulum.

5.3 Registration Analysis

A registration trial was carried out using the laser surface points as the data shape and the extracted surface from the MRI data as the model shape. A rough initial alignment was obtained by first identifying three anatomical landmark points from the MRI model of the acetabulum and the corresponding landmark points on the laser surface points, using the technique described in Section 4.2.2. The transformation between the two sets was then computed using the corresponding point registration method according to Horn [59]. The resulting transformation estimate was used as the starting pose for the ICP algorithm which was used in the surface-based registration process to accurately realize the best laser to MRI transformation. Table 5.1 shows the registration parameters that were applied in this registration experiment. Both local minima suppression and outlier elimination methods were turned on. Fig. 5.7 shows the variation of translation and rotation values with the ICP iteration number for a single run of the ICP algorithm. The translation and rotation parameters are given with respect to the initial pose estimate determined through the corresponding point registration. The rotations are represented about a coordinate system with the origin at the center of the acetabulum in the MRI model. The graphs show that the starting pose estimate was within around 7 degrees and 30 mm of the final evaluated registration transformation, indicating that a good rough initial alignment was established using the anatomical landmark correspondence technique. As explained in Section 3.3.3, the ICP algorithm requires that the two datasets be within 30 degrees of each other so as to avoid local minima that are distant from the global minimum.

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Registration Parameter Value

Maximum number of iterations 200

Transformation offset of data shape Corresponding point registration

provided the initial pose estimate

Minima suppression Enabled

Maximum number of iterations

within minima suppression

procedure

6

Maximum angle of the random

rotational perturbation for minima

suppression

3.5 degrees

Maximum value of the random

translational perturbation for minima

suppression

4.7 mm

Noise addition Disabled

Outlier elimination Enabled

Residue error threshold above which

a data point is regarded as an outlier

1 mm

Fraction of outliers eliminated per

ICP run

10 percent

Table 5.1: Registration parameters for the patient’s MRI/laser data experiment.

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mm

)

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X Translation

Y Translation

Z Translation

Fig 5.7: Rotation and translation (relative to the starting pose estimate) versus iteration number for a single run of the ICP algorithm during the registration of the patient’s MRI model and laser surface points.

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Fig. 5.8 presents the evolution of the root-mean-squared (RMS) error as well as the maximum residual error (MRE), as a function of the ICP iteration number. The RMS error is seen to converge monotonically towards zero. However, as in the experiment in Section 4.2.2, the MRE does not settle at the expected sensor noise magnitude of 1 mm, but remains at a value of around 9.3 mm after convergence. This indicates the presence of outliers. The extent of the outliers in the data shape was analyzed by evaluating the distance differences between the laser surface points and the surface extracted from the MRI data after one run of the ICP algorithm. Fig. 5.9 shows the laser surface points overlaid on the 3D MRI model after the convergence of the registration algorithm and evaluation of the distance differences. The points are colour coded based on the distance to the MRI model. The blue colour represents distances up to 1 mm, green colour represents distances above 1 mm up to 3 mm, while red colour represents distances above 3 mm. The maximum distance difference was 9.3 mm and corresponded with the calculated maximum residual error. The mean of the distance differences was found to be 1.338 mm while the standard deviation was 1.039 mm. The laser surface used in this experiment had 1474 points and covered about 71 percent of the acetabulum’s surface. Out of these surface points, 633 points were found to have a distance difference beyond 1 mm and were regarded as outliers in the subsequent ICP runs. 186 points had distance differences above 3 mm. The outlier elimination procedure was applied in the registration process to investigate the overall effect of the outliers on the registration accuracy. Changes in the registration parameters were evaluated as the outlier points were removed at a rate of 10 percent between the ICP runs until all the outliers were eliminated (See Section 3.3.3 for details on the outlier elimination technique). Fig. 5.10 presents the variation of changes in translation and rotation values as a function of the percentage of outlier points eliminated from the laser surface points. The changes in translation and rotation parameters are given with respect to the values obtained after a single run of the ICP algorithm without application of outlier elimination procedure.

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0 50 100 150 200

0

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35

40

Err

ors

(m

m)

Iteration Number

RMS Error

MRE

Fig. 5.8: Root-mean-squared (RMS) error and maximum residual error(MRE) versus ICP iteration number for a single run of the ICP algorithmduring the registration of the patient’s MRI model and laser surfacepoints.

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Fig. 5.9: Patient’s laser surface points overlaid on the MRI surface afterthe first registration trial. The laser data is shown as coloured dots, where the colour presents distance from the laser surface points to theunderlying MRI model (Blue: distance < 1 mm, Green: distance between1 and 3 mm, Red: distance > 3 mm, maximum distance difference: 9.3mm).

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Fig. 5.10: Deviation of rotation and translation values againstpercentage of outlier points eliminated from the data shape. The values are given relative to the parameters obtained after a single runof the ICP algorithm.

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From the graphs, it can be seen that the deviations in registration parameters were much prominent between 0 and 40 percent of the removal of the outliers. This corresponds to the region where points with the largest errors (above 3 mm) were eliminated. The graphs also demonstrate that after 40 percent of the outliers were removed (leaving about 26 percent of the total data points as outlier), there was little variation in the registration parameters. The results here approximate the simulation results obtained in Section 4.3.1, whereby addition of 20 percent of the expected clinical noise had no substantial effect on the registration parameters. After the elimination of all the outliers, the deviations of the rotation error norm,

er , and the translation error norm, et , were found to be 0.30

degrees and 0.33 mm respectively. Considering that the required accuracy of registration lies within 1 degree rotation and 1 mm translation, the errors introduced by the outliers cannot be ignored. However, most of the outliers which give rise to the largest residues can be removed from the laser surface points before the registration process. Fig. 5.9 demonstrates that the majority of the large residues lies on the regions where spurious soft tissues would be found during the intraoperative laser imaging, i.e., on the rim of the acetabulum and at the acetabular fossa. Because these points are localized, especially at the acetabular fossa, they can easily be eliminated using a cutting filter similar to the one in VTK [58]. 57.3 percent of the outliers, including points with the largest residues, were found to lie on the acetabular fossa. It then follows that, based on the graphs shown in Fig. 5.10 as well as simulation results obtained in Section 4.3.1, complete removal of the laser data points acquired from the region of the acetabular fossa will eliminate the large effect of the outliers on the registration parameters. From the results which were applied in plotting graphs in Fig. 5.10, the remaining outliers would cause deviations of 0.0054 degrees and 0.0127 mm in the rotation error norm,

er , and the translation error norm,

et , respectively. The laser surface points covered about 71 percent of the acetabular surface. From the experiments in Section 4.3.2, which simulates the effect of clinical access limitations, we expect negligible registration errors due to lack of the full coverage by the laser surface points. The main factor that would have a bearing on the registration accuracy would be the intraoperative sensor noise. From the simulation results obtained in Section 4.1.2 (Fig. 4.11), it can be surmised that the maximum expected

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registration errors as a result of sensor noise would be 0.046 degrees and 0.49 mm in rotation and translation respectively. However, due to the fact that a ground-truth transformation was not available in the course of this work, it was not possible to determine quantitatively the overall accuracy of the registration process. Steps were taken to minimize MR imaging and segmentation errors. The resolution of MR images affects the accuracy of the segmentation process, and hence the registration. Therefore, high resolution MR images, with a voxel size of 1 mm × 1 mm × 1 mm, were used in this work. The patient was informed to remain still in the course of image acquisition in order to minimize imaging artifacts. And since the image acquisition time was only 8 minutes, patient movements could be avoided.

.

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Chapter 6

Conclusion and Future Work

This dissertation has presented work performed in the development and analysis of a non-invasive registration method for computer assisted hip-joint replacement surgery. The technique involves acquiring 3D surface points of the anatomical part to be operated on intraoperatively using an eye-safe high resolution pulsed laser radar system. The intraoperative data points are then registered to a 3D surface model from preoperative MR images using a surface based registration program which utilizes an ICP algorithm. The registration analysis in this work was carried out with respect to the hip-joint socket (acetabulum). The measurement time of the laser radar system for the given task (40 x 40 scanning points at a resolution of 1 mm) was reduced from 3.5 hours to 3.8 minutes in the course of this work. Registration algorithms were developed and tested with synthetic as well as application datasets. Analysis of surface data acquired using the laser radar system demonstrated that the system is capable of capturing simple as well as complex surfaces of phantoms within an accuracy of 1 mm. Under clinical environment, the laser surface points collected from a patient’s acetabulum during surgery showed that the laser radar system is a viable non-contact method of capturing intraoperative data. Enough coverage by the registration points on the acetabulum (71 percent) was obtained during the clinical trial, which ensured that the registration procedure converged. Though there were substantial outliers on the patient’s laser surface points (about 42 percent) as a result of soft tissue, it was demonstrated that, the majority of the outliers could be easily eliminated from the registration process by removing data points acquired on the region of acetabular fossa and outside the rim of the acetabulum. This measure was found to reduce the errors on the registration parameters as a result of the outliers to negligible values. Simulation results of our registration technique showed that the maximum expected registration errors as a result of sensor noise would be 0.046 degrees and 0.49 mm in rotation and translation respectively. These values lie within the required accuracy of 1 degree in rotation and 1 mm in

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translation. However, due to the fact that a ground-truth transformation was not available in the course of this work, it was not possible to determine quantitatively the overall accuracy of the registration process. Estimation of the total registration error using a ground-truth transformation should be a future goal. Ground-truth experiments usually employ phantoms in validating medical systems. Though phantoms provide useful means of testing the accuracy of the system, they often fail to account for the complexity of the actual environment in the operating room. Validation of the system in this way is thus a useful measure, but does not imply that the same degree of accuracy will be met during the surgical operation. We therefore recommend determination of registration error employing a phantom as well as in vivo using a system that is well tested in operation rooms. The optical localization system (OPTOTRAK, Northern Digital, Ontario, Canada) has been recognized as an accurate registration system ([9], [85]) and can be used to gauge the total registration error of our system. For this purpose, the accuracy of the laser radar system in 3D data acquisition needs to be optimized by employing the department’s newly acquired dual-axis scanner (Model XY30-M3ST, GSI Lumnonics). The scanner has a much better accuracy of angular position ( 0.005 % of full field) as compared to that of the older scanner (Model XY30-G3B, General Scanning Inc. ), which is specified as 0.3 % of full field. Further increase of the measurement speed of the laser radar system would be desirable. The first option would be to raise the PRF of the laser transmitter from 80 kHz to 240 kHz, without compromising the eye-safety of the laser radiation. Secondly, implementation of an impedance matching network between the photodiode and the sampling bridge would decrease the signal noise, which would allow fewer measurement averages to be used during data acquisition.

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