Expectation-maximization restoration of band-limited, truncated point-process intensities with application in microscopy

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Abstract
The problem posed in this paper is that of restoring a Poisson-point-process intensity that has been degraded by a band-limiting filter followed by a truncation of the signal. The approach is to derive a maximum-likelihood estimate from the count data of the degraded point process. The expectation-maximization algorithm is used to realize this estimate, while the derivation of this algorithm is an extension to previous developments by Shepp and Vardi [ IEEE Trans. Med. Imaging MI-2, 113 ( 1982)], Snyder et al. [ IEEE Trans. Nucl. Sci. NS-28, 3575 ( 1981)], and others used for positron-emission tomography. We also extend our own work reported earlier by considering the truncated signal, which is analogous to practical situations in both two- and three-dimensional microscopy in which the image of the specimen has been truncated. Computer simulations with one-dimensional and two-dimensional signals demonstrate such a reconstruction with reasonable success. The plausibility of doing such a reconstruction is explained in that for the noiseless case the transformation characterizing the degradation is invertible.

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