Block delayed Majorize-Minimize subspace algorithm for large scale image restoration ^*

Abstract

In this work, we propose an asynchronous Majorization-Minimization (MM) algorithm for solving large scale differentiable non-convex optimization problems. The proposed algorithm runs efficient MM memory gradient updates on blocks of coordinates, in a parallel and possibly asynchronous manner. We establish the convergence of the resulting sequence of iterates under mild assumptions. The performance of the algorithm is illustrated on the restoration of 3D images degraded by depth-variant 3D blur, arising in multiphoton microscopy. Significant computational time reduction, scalability and robustness are observed on synthetic data, when compared to state-of-the-art methods. Experiments on the restoration of real acquisitions of a muscle structure illustrate the qualitative performance of our approach and its practical applicability.

A novel variational approach for multiphoton microscopy image restoration: from PSF estimation to 3D deconvolution

“…and we have used the fact that D + ϵ 1 I 3 0. Finally, given the definition of Φ in (12), it is clear the lower bound in (36) goes to +∞ when D F → +∞. Therefore, F admits a minimizer.…”

“…To illustrate the performance of our heteroscedastic constrained formulation for the MPM image restoration tast, we first conducted an experiment using simulated data. For this experiment, we chose an image of a fly brain from [12] as the object of interest x. This image, with dimensions M = 128 × 128 × 40, was artificially degraded through a convolution operator H mimicking the effect of a normalized 3D Gaussian kernel with inverse covariance matrix parameterized by the angles θ = 5π/6 rad, φ = 0 rad, and eigenvalues s = (50,50,20).…”

A Kurdyka-Lojasiewicz property for stochastic optimization algorithms in a non-convex setting

“…When (1.1) is solved using a deterministic scheme, the main advantage of KL condition lies in its ability to promote interesting asymptotic behavior when F is non-necessarily convex. In this context, KL has been used to prove convergence of proximal point algorithms [1,2], of simple splitting algorithms such as the forward-backward algorithm and its variants [2,16,7,24,12,44,8,9], as well as other algorithms based on the majorization-minimization principle [15,13,11]. A natural question is to investigate the transfer of the proof techniques from the deterministic setting, to the stochastic setting, for asymptotic analysis including a.s. convergence of stochastic processes.…”