Emission computed tomography (ECT) is a medical imaging modality that combines conventional nuclear imaging techniques with methods for image reconstruction from projections.1 Depending on the radionuclide used, ECT can be divided into two major categories, positron emission tomography (PET) and single- photon emission computed tomography (SPECT). In PET, radiopharmaceuticals labeled with radionuclides that emit positrons are used. The annihilation of a positron and an electron results in two 511 keV photons traveling in directions that are 180° apart. Coincidence detection of the pair of 511 keV photons forms a line of response (LoR). The multiple detectors and coincidence electronic circuitry collect LoRs in many orientations. The collection of LoRs is then resorted and rebinned to form the projection dataset used in image reconstruction to compute the tomo-graphic images.2,3 In SPECT, radiopharmaceuticals labeled with radio-nuclides that emit gamma-ray photons are used. They include many of the agents that are routinely used in nuclear medicine clinics. Depending on the design geometry, a collimator that is placed in front of a position sensitive detector, typically a scintillation camera, accepts photons traveling in certain directions. The most commonly used parallel-hole collimator accepts photons incident from directions perpendicular to the detector face. By rotating the collimator-detector around the patient, projection data from different views are collected. The projection data from the different views are then used in image reconstruction.4-6

Emission computed tomography techniques were conceived early in the development of nuclear medicine imaging. A notable development was the MARK IV brain scanner by Kuhl and Edwards in the early 1960's.7 The system consisted of four linear arrays of eight collimated detectors arranged

* Drs. B.M.W. Tsui and E.C. Frey, Division of Medical Imaging Physics, The Russell H. Morgan Department of Radiology and Radiological Science, Johns Hopkins University, Baltimore, Maryland 21287, USA

in a square configuration. By rotating the arrays of detector around the patient's head, projection data were collected. At that time, the investigators were unable to solve the problem of image reconstruction from projections and the simple backprojection method was used to reconstruct the image data. The results were unimpressive and the full development of ECT was delayed until the development of computed tomography (CT) methods.

The advancement of ECT was boosted by the landmark development of both x-ray CT and image reconstruction techniques resulting in the award of the Nobel Prize in medicine in 1979.8,9 At the same time, continuing improvements in both PET and SPECT systems and data acquisition methods were made by academic groups and commercial companies. These activities culminated in the introduction of the first commercial SPECT system by GE Medical Systems in 1981 and PET system designed at EG&G ORTEC in collaboration with Phelps and Hoffman in 1978. Since gaining FDA approval in 1999 for cancer staging and further fueled by the recent development of PET/CT, there has been a surge in interest in the clinical applications of PET and PET/CT. Today, SPECT and SPECT/CT systems are the major equipment in any clinical nuclear medicine clinic. The number of installed PET and PET/CT systems continues to increase at a healthy pace.

In this chapter, we review the fundamentals of image reconstruction from projections. Specifically, we describe the analytical image reconstruction methods which were the impetus for the development of ECT and have continued to play a major role in clinical ECT applications. The development of analytical image reconstruction methods can be traced back to Bracewell's work in radioastronomy in the 1950s. Finally, we discuss the limitations of analytical image reconstruction methods which have spurred the development of a new class of statistical image reconstruction methods and compensation techniques which are gaining much attention today (see chapter 4).

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