Applications of the EM Algorithm to Linear Inverse Problems with Positivity Constraints
Linear inverse problems with pos itivity constraints (LININPOS problems, for short) are common in science and technology. Examples include deconvolution problems, signal recovery from linearly-distorting input-output systems, and more. In the absence of statistical noise, such problems are purely deterministic problems. It is therefore surprising that the EM algorithm (Baum et al., 1970, Dempster et al., 1977, and others) which was specifically developed as a tool for deriving maximum likelihood estimates (MLE’s) for complicated statistical models involving incomplete data, is also a useful tool for attacking LININPOS problems. This paper reviews the EM algorithm as a methodology for solving LININPOS problems and demonstrates it on two very different applications. The first is in the area of image analysis (specifically, “motion deblurring”) and the second is in the area of statistical inference from networks (specifically, ‘network tomography’).
KeywordsPoint Spread Function Probability Vector Moment Equation Positivity Constraint Picture Frame
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