An introduction to continuous optimization for imaging

Abstract

A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.

Accelerated Alternating Descent Methods for Dykstra-Like Problems

“…Since alternating minimizations are a variant of Forward-Backward splitting methods, it is clear that one can expect good convergence rates by adapting standard methods [3,19,12]. This is what we establish in Theorem 1, extending a result of [11] in the non strongly convex case.…”

“…2.2.2] will provide efficient convergence rates. A derivation from an equality similar to (14) is provided in [12,Appendix B]. We provide here an adaption of that proof to our particular situation (only the parameters are slightly differing, so that we will sketch most of the arguments).…”

“…This is done in a similar way as in [19,Chap. 2] and [12,Appendix B]. A starting point is the update equation for t k+1 , which is chosen as the nonnegative root of:…”

“…In this paper, we extend these results in two directions: first, we consider strongly convex objectives and show how one can obtain nearly optimal linear convergence rates (in the sense of the lower bounds of [18,19]) in the framework of alternating minimizations or descent. The case of minimizations is particular, as it can also be reduced into a forward-backward splitting method applied to auxiliary functions and we could just refer to [19,12] where a rate analysis is performed, however it is roughly equivalent to perform an analysis adapted to the alternating minimizations algorithm.…”

Faster PET reconstruction with a stochastic primal-dual hybrid gradient method

“…An algorithm that can cope well with the aforementioned challenges is the primal-dual hybrid gradient algorithm (PDHG) [3][4][5][6] which can decompose the problem into a sequence of easy operations such as matrix-vector products and proximal operators that have closed-form solutions. However, as the PDHG updates are parallel, every iteration of PDHG is computationally demanding for the large variable sizes (easily exceeding 100 million) encountered with modern PET scanners.…”