Abstract. It happens that a solution of a problem lacks the property of being positive or of being spatially smooth. Based on proximal projections, we explore techniques in how to incorporate spatial regularization, including the nonnegativity, into that solution. We introduce the general algorithm and more subsequent ones that impose the nonnegativity into a solution and enforce the spatial regularization while using both norms, the 2-norm and the 1-norm, and also via segmentation. The new constrained nonnegative solution possesses theoretical properties that are stated and proved. We validate numerically these algorithms to solve an inverse medical imaging problem.

Received: May 20, 2012

AMS Subject Classification: 49N45, 49N60, 62G05, 65K10, 90C06, 90C40, 93C41

Key Words and Phrases: estimation, stochastic filtering, Kalman filter, optimal filtering, convex optimization, proximal, projection, state constraint, recursive, optimal, nonnegative, regularization, reconstruction, iterative algorithm, median, segmentation, Holder, Tikhonov, medical image, dynamic SPECT, time varying, time dependent, cross entropy, nonnegative reconstruction, Markov, expectation maximization, maximum likelihood, temporal regularization

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