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In many imaging systems, recorded observations have the physical meaning of a quantity of detected photons. The photons are counted at different spatial locations and in this way form an image of the object. This sort of scenario is typical for imaging in digital photography and for many problems in medicine, including positron emission tomography (PET) and single-photon emission tomography (SPET) [131], [170]; in gamma astronomy; and in microscopy [95], [207]. For instance, confocal microscopy is used to obtain volume images of small fluorescent objects with high-spatial resolutions [176]. The microscope performs a 3D scan of the object, and at each point of the scan a photon multiplier tube measures the emission of fluorescence from the object, essentially acting as a photon counter. Due to the geometry of these microscopes, a “blurring” is introduced into the measurement process. The Poisson distribution is a conventional probabilistic model used for a random number of photons appearing during an exposure time. In this chapter we concentrate on Poissonian models and simulation experiments demonstrating the performance of the developed algorithms. In Sec. 12.1 we consider the direct observations where the filtering of Poissonian data is the main goal. The remaining sections deal with the complex problems associated with indirect observations when the filtering is complicated by the required deconvolution.
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