Rigid Registration for Image-Guided Interventions

Rigid Registration for Image-Guided Interventions

Ziv Yaniv Computer Aided Interventions and Medical Robotics,

Georgetown University

Last Modified September 2008(Image-guided interventions Tutorial MICCAI’08)

Why?

• Rigid registration is a basic human activity:

Why? (in our context)

– Combine multiple information sources to improve interventional outcome (e.g. PET/CT).

– Alignment of virtual/image space and the physical patient space. – An initial step for non-rigid registration.

Tumor location is barely visible on CT, but clearly visible in fused PET/CT.

Sources of Data

• Images:– Diagnostic (primarily 3D): anatomical (CT,MR…), functional

(PET, SPECT…)– Interventional (primarily 2D): projective (X-ray, video…),

tomographic (US, CT-Fluorocsopy)

• Digitized points:– 3D digitized points: contact-based digitization, laser range

scanning, stereo imaging.

3D/3D Registration

Definition

Given two data sets defined over the domains

with, possibly, additional side information (e.g. pairing between corresponding features) find the rigid transformation:

where

Specific instances:

are the identity map, we have point set to point set registration.

map locations to intensity values, we have image to image registration.

are the identity map, and point correspondences are known.

Paired-Point Methods

• Analytic least squares solutions exist (e.g. Arun et al. 1987, Horn 1987), and are optimal if it is assumed that the measurement process is corrupted by additive isotropic i.i.d. zero mean Gaussian noise.

• Correspondence is assumed to be correct (breakdown point of 1).

• Require that points be detected in both coordinate systems – anatomical landmarks or fiducials.

• Fiducials: • accurate localization by design• number of points is independent of the anatomical structures.

IGI - Assumptions can Kill

• What happens if the data acquisition process doesn’t match our assumptions? • An acquisition which is biased by a rigid transformation can have serious implications as the algorithm cannot detect this situation and will compensate for the bias:

Head and Hat

• Align two surface representations of the brain segmented from complementary modalities (Pelizzari et al. 1989).

• Head – higher resolution data, represented as stacked 2D closed contours.• Hat – lower resolution data, represented as a set of points.

• Minimize (Powell’s direction set method) the sum of distances between the head surface and the rays emanating from the head centroid and going through the hat points.

• Point matching strategy suited for spherical objects.

Head and Hat

head

hat

Iterative Closest/Corresponding Point (ICP)

• Input: point set P, surface S t>0 - improvement threshold n - maximal number of iterations

1. Initialization:1. Set cumulative transformation, and apply to points.2. Find corresponding points and compute similarity (e.g. root mean

square distance).2. Iterate:

1. Compute incremental transformation using the current correspondences (e.g. analytic least squares solution), update cumulative transformation and apply to points.

2. Find corresponding points and compute similarity. If improvement in similarity is less than t, or number of iterations has

reached n terminate.

A natural extension of the analytic paired-point algorithms (Besl and McKay 1992, Zhang 1994, Chen and Medioni 1992).

ICP – It almost works

1. Convergence is local, requiring an initial transformation near the optimum.

2. Point pairing is a computationally expensive operation.

3. The use of an analytic least squares method for computing the incremental transformations implies that ICP assumes additive isotropic i.i.d. zero mean Gaussian noise, and that there are no outliers.



Improve probability of convergence by: – gross manual alignment of the data– initial gross localization of anatomical landmarks (Ma and Ellis 2003)– use multiple starting points in parameter space (Besl and McKay 1992) – simulated annealing (Penney et al. 2001)






Accelerate run time by:– kD spatial data structure (Besl and McKay 1992)– Cache closest points (Simon 1996)– Approximate nearest neighbor searching (Greenspan and Yurik 2003)– Parallel nearest neighbor searching (Langis et al. 2001).– Hierarchical data sampling (Jost and Hugli 2003)






Replace analytic least squares methods with:– M-estimators (Ma and Ellis 2003, Kaneko et al. 2003)– Least median of squares (Trucco et al. 1999, Masuda and Yokoya 1995)– Weighted least squares (Turk and Levoy 1994, Maurer et al. 1998)– Least trimmed squares (Chetverikov et al. 2005)– Use a fraction of the data (Phillips et al. 2007)

Standard Optimization

• Replace the ICP match-optimize approach with standard optimization (Fitzgibbon 2003).

• Implicitly represent point set/surface using a precomputed uniform distance map.

• Minimize using Levenberg-Marquardt with M-estimators instead of sum of squares.

Intensity based Registration

• Cast registration as an optimization task, requiring three components:

1. Similarity measure (the objective function).

2. Optimization algorithm.

3. Interpolation scheme (intensity values are only given at a discrete set of locations).

Popular Similarity Measures

Relationship between intensity values

Similarity Measure

Identity Sum of Squared Differences (SSD)

Identity Sum of Absolute

Differences (SAD)

Affine Normalized Cross Correlation (NCC)

General Functional

Correlation Ratio (CR)

Stochastic Mutual Information

(MI)

Stochastic Normalized Mutual Information (NMI)

Which Similarity Measure

• Computational complexity.• Robustness.• Capture range.• Accuracy


• Computational complexity.– Theoretical analysis.– Optimize implementation as the similarity measure will be

evaluated many times (improve the constant factor).

• Robustness.– Theoretical analysis (e.g. SAD more robust than SSD)

• Capture range.• Accuracy


• Computational complexity.• Robustness.• Capture range.

– Assess registration performance as a function of the similarity measure – evaluates the combination of the similarity measure and the optimization algorithm.

• Accuracy– Explore the behavior of the similarity measure as a function of

the transformation parameters – axis aligned orthogonal slices through the parameter space

The blind men and the elephant:


• Computational complexity.• Robustness.• Capture range.• Accuracy

– Exhaustively sample the parameter space along diameters of a 6D hypersphere centered on the gold standard parameters (Škerl et al. 2006).

– Analyze the function behavior based on these observations (e.g. distinctiveness of optimum).

Optimization

• General purpose optimization methods.

• No single algorithm is optimal for all similarity measures (no-free-lunch theorem).

• Improve convergence range using heuristic search algorithms (e.g. simulated annealing) – double edged sword. Sometimes the desired pose is a local optimum and not a global one.

Interpolation

• Linear interpolation – a compromise between accuracy and computational complexity.

• May have adverse effects on the similarity measure (Pluim et al. 2000).

• B-spline interpolation - best tradeoff between accuracy and computational complexity (Lehmann et al. 1999, Meijering et al. 2001).

2D/3D Registration

Definition

Given two data sets defined over the domains

and camera matrices, , where is a rigid transformation and a perspective projection matrix, find the rigid transformation:

where

• Closely related to the pose estimation task studied in computer vision. • A single image of a known scene is acquired and we seek the camera’s pose relative to the scene.• Use point and line features which arise in man made environments.

Camera Model

• X-ray fluoroscopy and endoscopic video are modeled using the standard perspective camera model.

• Images are distorted for X-ray fluoroscopy utilizing image intensifiers, and for standard endoscopes.

Iterative Algorithms - Initialization

1. Manual initialization via visual inspectionUser interactively explores the parameters space

2. Clinical setupIntersect bounding pyramids from multiple

viewing angles.

3. Approximate paired-point registration• Localize anatomical landmarks

1. The same anatomical landmarks (>3) in multiple projection images.

2. Digitize with a tracked pointer if already exposed.

• Localize skin adhesive fiducials.

2D/3D Head and Hat

• A variation of the head and hat is used for patient alignment in radiation therapy, X-ray/CT, (Murphy 1997).

• A digitally reconstructed radiograph (DRR), simulated X-ray is generated and the optimization is performed in 2D.

• The X-ray serves as the head and the DRR as the hat.

• Edges (skull) are detected along rays emanating from the center of the X-ray image (head centroid) and going towards the image boundary – both in DRR and X-ray.

• The distance between edge points on the rays is minimized, along with the mean grey level value along the ray.

2D/3D ICP

• Minimize the distance between back-projected rays and a 3D model, formulation is in the context of computer vision (Wunsch and Hirzinger 1996).

• In the medical context this approach requires extracting a 3D surface model and detecting 2D edge points that were generated by the anatomical structure.

camera

back-projected rays

2D/3D Standard Optimization

• Minimize the sum of squared distances between a surface model and back-projected rays, CT/X-ray, (Lavallée and Szeliski 1995).

• Surface is implicitly represented using an octree based distance map (rediscovered in computer graphics, Frisken et al. 2000, Adaptively Sampled

Distance Fields).

• Minimization using Levenberg-Marquardt.

Intensity based Registration

• Most often X-ray/CT, in some cases X-ray/MR via mapping MR values to CT, and video/CT.

• Simulate the 2D image generation process.

• Maximize the similarity between the simulated and actual 2D images – choice depends on the realism of the simulation.

• Primary challenge - fast generation of simulated images, either X-ray or surface/volume rendering in the case of video.

Generation of Simulated Images

• Use established volume rendering techniques:– Ray casting.– Splatting.– Shear warp.

• Image based rendering - Use a set of images to render a new image from a novel camera pose.– In our context, simulate a set of images from specific camera poses and

use them to generate all other simulated images (LaRose 2001, Knaan and Joskowicz 2003, Russakoff 2005).

• GPU based rendering (Khamene 2006, LaRose 2001).

• Standard rendering of polygonal models (Mori et al 2002, Turgeon et al. 2005).

fluoroscopy ray casting with mask

Other Approaches

• Gradient based registration – directly relate the volumetric gradients to the X-ray image gradients (Livyatan et. al 2003, Tomaževič et al. 2003) :

• Perform cone-beam reconstruction from a few X-ray images (2-15) and then we are back to 3D/3D registration (Tomaževič et al. 2006) [Quality of reconstruction is poor, required the introduction of a novel similarity measure].

Evaluation

Ideal registration algorithm:– Fast:

The result is obtained in real time (<0.1sec).– Accurate:

The distance between corresponding points after applying the estimated transformation is less than 0.1mm.

– Robust: More than half of the data elements must be outliers in

order to throw the registration out of reasonable bounds (breakdown point of N/2).

– Automatic:No user interaction required.

– Reliable:Given the expected input the registration always

succeeds.

Evaluation

• Empirically evaluate using standard data sets with known “gold standard” registration (preferably in vivo):

– Retrospective Image Registration Evaluation (in-vivo head CT/MR, PET/MR) [Fitzpatrick]:

http://www.insight-journal.org/rire/

– 2D/3D Standardized Evaluation Methodology (in-vitro spine X-ray/MR/CT/CBCT) [Van De Kraats]:

http://www.isi.uu.nl/Research/Databases/GS/

– Gold standard data for evaluation of 2D-3D registration (in-vitro spine X-ray/MR/CT) [Tomaževič]:

http://lit.fe.uni-lj.si/tools.php?lang=eng

Accuracy

• Most important criterion. Directly addresses the goal of registration, alignment of corresponding points.

• Spatially variant.

• When it is not possible to digitize the target to report actual target registration error, report the expected target registration error (e.g. first order approximation [Fitzpatrick et al. 1998], second order approximation [Moghari and Abolmaesumi 2006]).

Take Home Message

• Be explicit about your assumptions.• Evaluate using all criteria from the definition of the ideal

algorithm.• Usability is context dependent, registration is not a goal

but a means to an end:

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Rigid Registration for Image-Guided Interventions