Multiscale estimation and Interface detection – Multisc-In

An in-depth study of the impact of discretization in the discrete Mumford-Shah model has been initiated. We consider both the discrete Mumford-Shah model and the Ambrosio-Tortorelli model. Both can be formulated as a minimization involving the sum of a data fidelity term, a term requiring smoothing everywhere in the image except at the location of the edges, and a third term that penalizes the length of the outlines. The relation between discrete calculus and the choice of linear operators in the coupling term as well as in the penalization term within the framework of Ambrosio-Tortorelli has been clarified.

We have proposed proximal alternating minimization schemes with convergence guarantees (PALM and SL-PAM) as well as closed forms of the involved proximal operators. An in-depth comparison between the discrete Mumford-Shah and discrete Ambrosio-Tortorelli models solved with PALM and SL-PAM is underway to assess the performance of each scheme in terms of interface detection and perimeter computation.

Numerical comparisons between PALM and SL-PAM show that it is preferable, when possible, to replace a gradient descent step with a proximal step. In order to analyze this behavior, we carried out a preliminary theoretical analysis within the framework of the convex and differentiable functions between various approaches based on gradient or prox. This study will now be continued for the study of the PALM and SL-PAM schemes.

We have addressed the crucial question of the estimation of hyperparameters by SURE approach (Stein Unbiaised Risk Estimator) for the detection of discontinuities in signals, then in the context of segmentation of textured images, and finally for the estimation of hyperparameters of the discrete Mumford-Shah model.

- SLPAM and PALM for discrete Mumford-Shah and Ambrosio-Tortorelli
- Theoretical and numerical comparisons between gradient-based and prox-based proximal approaches.
- Estimation of hyperparameters in the Mumford-Shah model
- Design of an unrolled algorithm (in the form of a neural network) for image restoration.

- PALM and SL-PAM theoretical guarantees and comparisons.
- Multi-scale proximal algorithms.
- Unfolded algorithms (Neural network) for interface detection.
- Interfaces detection in textured images. Application to multiphase flow data.
- “Graph” formulation of the Mumford-Shah model.

M. Jiu and N. Pustelnik, A deep primal-dual proximal network for image restoration, accepted to IEEE JSTSP Special Issue on Deep Learning for Image/Video Restoration and Compression, 2021, arXiv:2007.00959.

B. Pascal, N. Pustelnik, P. Abry, J.-C. Géminard , V. Vidal Parameter-free and fast nonlinear piecewise fitering. Application to experimental physics, Annals of Telecommunications, 75, 655-671, Oct. 2020, arXiv:2006.03297.

B. Pascal, N. Pustelnik, and P. Abry, How Joint Fractal Features Estimation and Texture Segmentation can be cast into a Strongly Convex Optimization Problem ?, accepté à ACHA, 2021, arXiv:1910.05246.

B. Pascal, S. Vaiter, N. Pustelnik, and P. Abry, Automated data-driven selection of the hyperparameters for Total-Variation based texture segmentation, submitted, 2020, arXiv:2004.09434.

L. M. Briceño-Arias and N. Pustelnik, Proximal or gradient steps for cocoercive operators, submitted, 202, arXiv:2101.06152.

Interface detection can be viewed as the image processing counterpart of change-point detection, a well-studied subject in signal processing. Several solutions exist to provide both accurate detection of the change-points and handling a large dataset by means of on-the-fly algorithms or dynamic programming algorithms, and with effective strategies to automatically adjust the hyperparameters. Most of these strategies are related to the l2-Potts model or its convex relaxation known as total variation (TV) denoising. Several extensions to image analysis have been provided, but the double aim of having accurate interface detection and being adapted to large scale data is still a challenging open question, especially due to the difficult definition of a « contour » in the discrete setting. This question is of major interest for a large panel of applications going from geophysics research to societal studies. The common point to these applications is the willingness to have an interface detection at a fine scale, possibly with subpixel accuracy, in order to extract interpretable parameters (e.g. physical or societal), from high resolution data.

This project is devoted to innovative image processing tools relying both on optimization and multiresolution techniques in order to provide a new paradigm for the interface detection on large scale data. This project essentially relies on:
* a deep theoretical study of the discrete Mumford-Shah model to perform accurate interface detection, and thus measure precise interface length;
* the design of multiscale proximal algorithms to make possible the implementation of the discrete Mumford-Shah model of large scale data;
* the construction of a flexible joint estimation/interface detection procedure allowing us to accurately detect interface into textured images.

This project aims to combine advanced mathematical tools which are multiresolution analysis and continuous optimization, already combined for solving inverse problems by means of nonsmooth regularization terms in the beginning of the 21st century and whose combination is revisited in this project in order to design multiscale proximal algorithms. The theoretical contributions of this project relies essentially on optimization and multiresolution formalism but will also benefit from recent advances on manifold theory and optimal transportation.

Performance resulting from the theoretical developments explored in this project will be evaluated both on synthetic data and on real data. The real data, we propose to analyse, are coming from research in geophysics (multiphasic flow study) and human science (vote transfer matrices estimation) studied in the Laboratoire de Physique de l'ENS de Lyon (LPENSL)

The two main scientific impacts of the project Multisc-In are (i) the design of a new class of algorithms named multiscale proximal algorithms, whose major interest is to run proximal algorithms on huge dataset, but also give the opportunity to practitioners to analyse data at a coarse scale and then focus on some specific areas at a finer scale in real time starting from the coarser analysis and (ii) a better understanding of the interface detection with subpixel accuracy with a quantitative rule relating the level of expected accuracy and the resolution of the (textured or not-textured) data. Such a rule is of almost importance for Physicists, when they design an experiments, but also in medical research area (tissue analysis) or climate studies (sensors location and number) to name just a few.