Tradeoffs in regularized maximum-likelihood image restoration

All algorithms for three-dimensional deconvolution of fluorescence microscopical images have as a common goal the estimation of a specimen function (SF) that is consistent with the recorded image and the process for image formation and recording. To check for consistency, the image of the estimated SF predicted by the imaging operator is compared to the recorded image, and the similarity between them is used as a figure of merit (FOM) in the algorithm to improve the specimen function estimate. Commonly used FOMs include squared differences, maximum entropy, and maximum likelihood (ML). The imaging operator is usually characterized by the point-spread function (PSF), the image of a point source of light, or its Fourier transform, the optical transfer function (OTF). Because the OTF is non-zero only over a small region of the spatial-frequency domain, the inversion of the image formation operator is non-unique and the estimated SF is potentially artifactual. Adding a term to the FOM that penalizes some unwanted behavior of the estimated SF effectively ameliorates potential artifacts, but at the same time biases the estimation process. For example, an intensity penalty avoids overly large pixel values but biases the SF to small pixel values. A roughness penalty avoids rapid pixel to pixel variations but biases the SF to be smooth. In this article we assess the effects of the roughness and intensity penalties on maximum likelihood image estimation.