This chapter describes the 2D and 3D image reconstruction algorithms used in PET and the most important evolutions in the last ten years: the introduction of 3D acquisition and reconstruction and the increasing role of iterative algorithms. As will be seen, iterative algorithms improve image quality by allowing more accurate modeling of the data acquisition. This model includes the detection, the photon transport in the tissues, and the statistical distribution of the acquired data, i.e. the noise properties. The popularity of iterative methods dates back to the seminal paper of Shepp and Vardi on the maximum-likelihood (ML) estimation of the tracer distribution. Practical implementation of this algorithm has long been hindered by the size of the collected data, which has increased more rapidly than the speed of computers. Thanks to the introduction of fast iterative algorithms in the nineties, such as the popular Ordered Subset Expectation Maximization (OSEM) algorithm, iterative reconstruction has become practical. Reconstruction time with iterative methods nevertheless remains an issue for very large 3D data sets, especially when multiple data sets are acquired in whole-body or dynamic studies. Speed, however, is not the only reason why filtered-back-projection (FBP) remains important: analytic algorithms are linear and thereby allow an easier control of the spatial resolution and noise correlations in the reconstruction, a control which is mandatory for quantitative data analysis.

The chapter is organized as follows. First, the organization of the data acquired in 2D mode is described,

and the reconstruction problem is defined. The third section reviews the classical analytic reconstruction of 2D tomographic data and describes the FBP method, which remains a workhorse of tomography. Iterative reconstruction is presented in the following section, where the accent is set on the key concepts and on their practical implications. Owing to the wide variety of iterative methods, only the popular ML-EM and OSEM methods are described in detail, though this does not entail any claim that these algorithms are optimal. The last sections concern the reconstruction of data acquired in 3D mode. Three-dimensional FBP is described, as well as fast rebinning algorithms, which reduce the redundant 3D data set to synthetic 2D data that can be processed by analytic or iterative 2D algorithms. Hybrid algorithms combining rebinning with a 2D iterative algorithm are introduced, and the chapter concludes with a discussion of the practical aspects of fully 3D iterative reconstruction.

Presented here as a separate chapter, image reconstruction cannot be understood independently of the other steps of the data-processing chain, including data acquisition, data corrections (described in chapters 2, 3, 5), as well as the quantitative or qualitative analysis of the reconstructed images. The variety of algorithms for PET reconstruction arises from the fact that there is no such thing as an optimal reconstruction algorithm. Different algorithms may be preferred depending on factors such as the signal-to-noise ratio (number of collected coincident events in the emission and transmission scans), the static or dynamic character of the tracer distribution, the practical constraints on the processing time, and, most importantly, the specific clinical task for which the image is reconstructed. It is

* Figures 4.1-4.11 are reproduced from Valk PE, Bailey DL, Townsend DW, Maisey MN. Positron Emission Tomography: Basic Science and Clinical Practice. Springer-Verlag London Ltd 2003, 91-114.

important to keep this observation in mind when discussing reconstruction as an isolated topic.

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