Forward Projection for Use with Iterative Reconstruction

Forward Projection for Use with Iterative Reconstruction

Raja Guedouar (Higher School of Health Sciences and Technics of Monastir, Tunisia) and Boubaker Zarrad (Higher School of Health Sciences and Technics of Monastir, Tunisia)
Copyright: © 2012 |Pages: 29
DOI: 10.4018/978-1-61350-326-3.ch003
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Modelling the forward projection or reprojection, that is defined as the operation that transforms a 3D volume into series of 2D set of line integrals, is of interest in several medical imaging applications as iterative tomographic reconstruction (X-ray, Computed Tomography [CT], Positron Emission Tomography [PET], Single Photon Emission Computed Tomography [SPECT]), dose-calculation in radiotherapy and 3D-display volume-rendering. As forward projection is becoming widely used, iterative reconstruction algorithms and their characteristics may affect the reconstruction quality; its accuracy and performance needs more attention. The aim of this chapter is to show the importance of the modelling of the forward projection in the accuracy of medical tomographic data (CT, SPECT and PET) reconstructed with iterative algorithms. Therefore, we first present a brief overview on the iterative algorithms used in tomographic reconstruction in medical imaging. Second, we focus on the projection operators. Concepts and implementation of the most popular projection operators are discussed in detail. Performance of the computer implementations is shown using the well-known Shepp_Logan phantom. In order to avoid possibly confounding perspective effects implied by fan or cone-beam, this study is performed in parallel acquisition geometry.
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Iterative Reconstruction Techniques

During the last decades, various algorithms have been proposed for both 2D and 3D tomographic reconstruction such as the analytical and the iterative methods. The analytical algorithms, the most used, have advantage to be fast, but they are not able to model the characteristics of the data acquisition process. Iterative tomographic reconstruction which is the process of recovering 3D image data from a set of integrals of that data over 2D subspaces, provide an attractive solution for tomographic imaging modalities over analytic techniques and they have been successfully used in medical imaging (Ziegler, 2008; Suetens, 2002), including computed tomography (CT), single photon emission computed tomography (SPECT), positron emission tomography (PET), tomosynthesis and projection mode 2D magnetic resonance imaging (MRI). The iterative methods aim to minimize or maximize a cost function between reconstructed slices T and measured projection P and have the advantage to incorporate imaging geometry and physics effects into the forward projection operator R that results in quantitatively improved reconstruction images. All iterative methods begin with initial guess for solution and successively improve it until solution is as accurate as desired. In theory, infinite number of iterations might be required to converge to exact solution. In practice, iteration terminates when some measure of error is as small as desired. Figure 1 illustrates steps of implementation of an iterative reconstruction algorithm where both forward projection matrix and back projection matrix (the reverse model of forward projection) are needed to achieve one iteration.

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