On-the-fly Approximation of Multivariate Total Variation Minimization

In the context of change-point detection, addressed by Total Variation minimization strategies, an efficient on-the-fly algorithm has been designed leading to exact solutions for univariate data. In this contribution, an extension of such an on-the-fly strategy to multivariate data is investigated. The proposed algorithm relies on the local validation of the Karush-Kuhn-Tucker conditions on the dual problem. Showing that the non-local nature of the multivariate setting precludes to obtain an exact on-the-fly solution, we devise an on-the-fly algorithm delivering an approximate solution, whose quality is controlled by a practitioner-tunable parameter, acting as a trade-off between quality and computational cost. Performance assessment shows that high quality solutions are obtained on-the-fly while benefiting of computational costs several orders of magnitude lower than standard iterative procedures. The proposed algorithm thus provides practitioners with an efficient multivariate change-point detection on-the-fly procedure

Total variation on a tree

We consider the problem of minimizing the continuous valued total variation subject to different unary terms on trees and propose fast direct algorithms based on dynamic programming to solve these problems. We treat both the convex and the non-convex case and derive worst case complexities that are equal or better than existing methods. We show applications to total variation based 2D image processing and computer vision problems based on a Lagrangian decomposition approach. The resulting algorithms are very efficient, offer a high degree of parallelism and come along with memory requirements which are only in the order of the number of image pixels.Comment: accepted to SIAM Journal on Imaging Sciences (SIIMS

The taut string approach to statistical inverse problems: theory and applications

A novel solution approach to a class of nonlinear statistical inverse problems with finitely many observations collected over a compact interval on the real line blurred by Gaussian white noise of arbitrary intensity is presented. Exploiting the nonparametric taut string estimator, we prove the state recovery strategy is convergent to a solution of the unnoisy problem at the rate of $n^{−1/2}$ as the number of observations n grows to infinity. Illustrations of the method\u27s application to real-world examples from hydrology, civil & electrical engineering are given andan empirical study on the robustness of our approach is presented

A note on the dual treatment of higher order regularization functionals

In this paper, we apply the dual approach developed by A. Chambolle for the Rudin-Osher-Fatemi model to regularization functionals with higher order derivatives. We emphasize the linear algebra point of view by consequently using matrix-vector notation. Numerical examples demonstrate the differences between various second order regularization approaches