title: Sparse stochastic processes and discretization of linear inverse problems.
abstract: We present a novel statistically-based discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving ill-conditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuous-domain signals as solutions of linear stochastic differential equations. Accordingly, we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions. Our estimators not only cover the well-studied methods of Tikhonov and l1-type regularizations as particular cases, but also open the door to a broader class of sparsity-promoting regularization schemes that are typically nonconvex. We provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution, magnetic resonance imaging, and X-ray tomographic reconstruction problems. Finally, we compare the performance of estimators associated with models of increasing sparsity.
authors.affiliation: Biomedical Imaging Group, École Polytechnique fédérale de Lausanne, Lausanne CH–1015, Switzerland. emrah.bostan@gmail.com
authors.forename: Emrah
authors.initials: E
authors.lastname: Bostan
authors.forename: Ulugbek S
authors.initials: US
authors.lastname: Kamilov
authors.forename: Masih
authors.initials: M
authors.lastname: Nilchian
authors.forename: Michael
authors.initials: M
authors.lastname: Unser
mesh_terms: D000465:Algorithms; D001499:Bayes Theorem; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D016014:Linear Models; D008279:Magnetic Resonance Imaging; D008954:Models, Biological; D009474:Neurons / Q000502:physiology; D019047:Phantoms, Imaging; D012815:Signal Processing, Computer-Assisted; D013234:Stem Cells / Q000502:physiology; D013269:Stochastic Processes; D014057:Tomography, X-Ray Computed