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Restoration of edges under Poisson noise using convex constraints with application to confocal microscopy

A new algorithm is presented which uses maximum likelihood (ML) estimation and convex constraints to restore edge information in a robust and accurate way for microscope images. The convex constraints are spatially variant bounds on the image intensities and the directions of gradients, and they are extracted from an image restored with strong smoothing constraints. The high resolution estimate is obtained by maximizing the ML objective under the convex constraints and relaxed smoothing, using conjugate gradient and constrained optimization techniques. Relaxed smoothing allows edges in the final estimate, while the bound constraints preserve the robustness of the estimate. The convex constraints can be considered as a generalization of the positivity constraint which has been widely used in image restoration. The method does not impose any constraints on the profile of the edges, and it compares favorably with first order Bayesian methods which are frequently used for edge enhanced restoration in medical imaging. The performance of the method is demonstrated by restoring an image obtained from a confocal microscope.