Reconstruction of biomedical images and sparse stochastic modelling

We propose a novel statistical formulation of the image-reconstruction problem from noisy linear measurements. We derive an extended family of MAP estimators based on the theory of continuous-domain sparse stochastic processes. We highlight the crucial roles of the whitening operator and of the Lévy exponent of the innovations which controls the sparsity of the model. While our family of estimators includes the traditional methods of Tikhonov and total-variation (TV) regularization as particular cases, it opens the door to a much broader class of potential functions (associated with infinitely divisible priors) that are inherently sparse and typically nonconvex. We also provide an algorithmic scheme—naturally suggested by our framework—that can handle arbitrary potential functions. Further, we consider the reconstruction of simulated MRI data and illustrate that the designed estimators can bring significant improvement in reconstruction performance. Index Terms — sparse stochastic processes, statistical estimation, sparsity-promoting regularization, nonconvex optimization. 1