The problem of forming cross-sectional or tomographic images of the attenuation characteristics of objects arises in a variety of contexts, including medical x-ray computed tomography (CT) and nondestructive evaluation of objects in industrial inspection. In the context of emission imaging, such as positron emission tomography (PET) [1, 2], single photon emission computed tomography (SPECT) , and related methods used in the assay of containers of radioactive waste , it is useful to be able to form "attenuation maps," tomographic images of attenuation coefficients, from which one can compute attenuation correction factors for use in emission image reconstruction. One can measure the attenuating characteristics of an object by transmitting a collection of photons through the object along various paths or "rays" and observing the fraction that pass unabsorbed. From measurements collected over a large set of rays, one can reconstruct tomographic images of the object. Such image reconstruction is the subject of this chapter.
In all the above applications, the number of photons one can measure in a transmission scan is limited. In medical x-ray CT, source strength, patient motion, and absorbed dose considerations limit the total x-ray exposure. Implanted objects such as pacemakers also significantly reduce transmissivity and cause severe artifacts . In industrial applications, source strength limitations, combined with the very large attenuation coefficients of metallic objects, often result in a small fraction of photons passing to the detector unabsorbed. In PET and SPECT imaging, the transmission scan only determines a "nuisance" parameter of secondary interest relative to the object's emission properties, so one would like to minimize the transmission scan duration. All the above considerations lead to "low-count" transmission scans. This chapter discusses algorithms for reconstructing attenuation images from low-count transmission scans. In this context, we define low-count to mean that the mean number of photons per ray is small enough that traditional filtered-backproject on (FBP) images, or even methods based on the Gaussian approximation to the distribution of the Poisson measurements (or logarithm thereof), are inadequate. We focus the presentation in the context of PET and SPECT transmission scans, but the methods are generally applicable to all low-count transmission studies. See  for an excellent survey of statistical approaches for the emission reconstruction problem.
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