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  • Luc Bidaut et al., preprint to be published in CERN Yellow Reports

    3D IMAGE RECONSTRUCTION IN MEDICINE AND BEYOND

    L. Bidaut+, C. Morel+ LFMI, Radiology and Surgery Departments, University Hospitals, Geneva, Switzerland Institute for High Energy Physics, University of Lausanne, Switzerland

    Abstract In the medical field, 3D volumes are reconstructed mainly by tomographic techniques. Transmission or emission projection data sets are acquired and processed to reconstruct slices across the volume of interest, e.g., the patients body. Although initially based on 2D acquisitions, current reconstruction techniques for emission modalities (e.g. PET) use more sensitive 3D acquisition data sets processed through modified or entirely original algorithms. Beyond the simple 3D imaging that single tomographies permit, multimodality approaches and equipment actually help reconstruction algorithms to perform better. Mixing these technical developments with complex clinical imaging protocols provides the foundation for a refined and open-ended multidimensional and multisensor approach to either diagnosis or therapy planning and follow-up.

    1. INTRODUCTION

    Since the advent of X-ray-based Computed Tomography (CT) [1,2], other medical imaging modalities have been developed which at least initially used a similar acquisition and reconstruction principle.

    For CT, data are initially acquired by measuring the attenuation of an X-ray beam through the body at various locations around the body during the synchronized rotation of the X-ray tube and detecting equipment. Reconstruction is then a simple filtered back-projection of the acquired sinograms to best recreate a map of the attenuation coefficients (linked to the tissues' densities) inside the field of view (FOV).

    For emission tomography (e.g., Single Photon Emission CT (SPECT) or Positron Emission Tomography (PET)), the acquisition is based on the detection of the radioactive decay within the FOV. Similarly to CT, this detection is rearranged in sinograms which are later filtered and back-projected to estimate a map of the radioactive activity inside the FOV.

    Due to the scope of this report, we shall concentrate on emission tomographies, and even more so on PET. For such modalities, several factors affect the quality of the results at various stages.

    For example in PET, acquisitions were initially performed in 2D, with mechanical septas to focus the rays and prevent too many scattered events from being detected. Because such a design was also getting rid of many useful signals, the septas were eventually discarded which increased the S/N ratio by a factor of about 4 to 6 with the same detector design. Of course, the acquisition data sets were not 2D anymore and this new paradigm forced reconstruction techniques to be modified or even totally re-invented to take care of the new dimensionality as well as of the increased scatter and noise components.

    Another major factor affecting data quality for emission tomographies is the attenuation of the rays through the various objects they intersect before being detected. These obstacles not only include the table or various mechanical holders, but also the body itself which can significantly attenuate the signal, even at the higher energies stemming from positron decay. Classically, attenuation which is generally closely linked to the material's density has been corrected prior to reconstruction by

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    estimating the attenuation map through either direct measurement (e.g., with an external source of constant activity), or simple (e.g., ROI based) simulation of the various objects in the FOV. Multimodality techniques and more recently even machines now allow the attenuation to be estimated with greater accuracy and resolution directly from the actual morphology of the patient. The latest developments for iterative reconstruction techniques actually incorporate morphology in the non linear fit both as a parameter for correcting attenuation or other density-linked effects, and also as an added constraint for the calculations.

    This report will attempt to succinctly present the rationale and evolution of tomography, and its outcome through techniques such as multimodality and advanced clinical protocols. Magnetic Resonance Imaging (MRI) has been voluntarily left aside because its acquisition and imaging principles are actually much closer to frequency spectrum analysis than transmission or emission tomographies are.

    2. TOMOGRAPHY IN MEDICINE 2.1 Principles

    Medical imaging aims to get in vivo pictures of the interior of the body. However, direct imaging of a whole slice through a body using a regular camera is not possible since visible light does not penetrate deeply in human tissues. Fortunatly, except in the visible part of the spectrum, living matter is mostly transparent to electromagnetic radiations. Thus, medical images can be obtained by using sources of light at lower or higher energies than visible light, either transmitted through the body or directly emitted from the body. In both cases, when using either X-rays for transmission tomography or gamma rays for emission tomography, only those rays which escape the body can be detected. Consequently, the picture that a scanner using X-rays or gamma rays can see of a slice through a body is not actually represented in the direct space, but rather by its projections from all around the body.

    2.2 2D concepts

    The problem of image reconstruction is to obtain a representation of an object in the direct space from its representation in a projection space.

    Figure 1: Radon and Fourier transforms

    To make it simple, let us consider the case of parallel projections as they can be constructed from the detection of annihilation pairs in PET. In two dimensions (2D), the whole set of parallel projections which can be built for every projection angle around the object is a 2D representation of the object in a projection space called sinogram, where, by convention, each line corresponds to a parallel projection of the slice at a different angle (Figs. 1 and 2).

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    The analytic solution of the 2D image reconstruction has been known for nearly 80 years and was established by Radon [3], who gave his name to the mathematical transformation which gives the representation of an object in the sinogram space from its representation in the direct space.

    As indicated in Figure 1, the Radon Transform is equivalent in 2D to the X-Ray Transform which accounts for the description of line integrals through the object represented in direct space. An object can also be described by its spatial frequencies. For this, a Fourier Transform is applied to its representation in direct space. There is a close relationship between the representation of an object in the sinogram space and its representation in a spatial frequency space which permits by inversion of the Fourier Transform to reconstruct a representation of an object in direct space f x, y( ) from its projections p s,( ). This relationship is expressed by the central slice theorem which connects the 1D Fourier Transform of a parallel projection P s,( ) to the 2D Fourier Transform of the object image F x ,y( ) along an axis perpendicular to the projection direction:

    P s,( )= F s cos,s sin( )

    (1)

    x

    y

    P(,) = F(cos,sin)

    x

    y

    s

    t

    s

    p(s,)

    Figure 2: Central Slice Theorem: spatial projection (left) vs. frequency space (right)

    Consequently, as shown in Figure 2, measuring the projections all around the object is equivalent to measuring the 2D Fourier Transform of the object using a polar coordinate system. Thus, the representation of the object in direct space can be obtained by inverting this frequency space representation, being aware that the inverse Fourier Transform has to be applied in a Cartesian coordinate system. Therefore, a Jacobean must be introduced to hold for the change of variables in the frequency space from polar coordinates to Cartesian coordinates. This ends up in multiplying the frequency space representation of the object obtained from the 1D Fourier Transform of the measured projections by the absolute value of the frequency s . In other words, a ramp filter is applied to the measured projections, and the representation of the object in the direct space is obtained after backprojection of these filtered projections onto the lines of projections.

    f x, y( ) = d ds s ei 2s s

    0

    , s = x cos + y sin (2)

    This filtered backprojection (FBP) algorithm is a unique analytic solution to the problem of the inversion of the 2D Radon Transform and allows for reconstructing the image of a 2D object from its

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    projections. This solution is purely analytical in the sense that projections are assumed to be continuous functions measured with an infinite accuracy.

    The ramp filter s used in FBP is not at all related to sampling considerations. In order to reduce the amplification of high frequencies the ramp filter causes (statistical noise lies in the high frequencies), another low pass filter has to be added to it and windowed in the frequency space for taking into account the fact that projections are not continuously measured but sampled with a finite sampling step given by the scanner [4] (Fig. 3).

    Figure 3: Filters (or none) for 2D tomography

    2.3 Extension to 3D acquisition

    Eventually, 2D PET machines were extended to 3D acquisition by removing the septas (physical collimators) which

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