Stefania Petra

A class of quasi-variational inequalities for adaptive image denoising and decomposition

We introduce a class of adaptive non-smooth convex variational problems for image denoising in terms of a common data fitting term and a support functional as regularizer. Adaptivity is modeled by a set-valued mapping with closed, compact and convex values, that defines and steers the regularizer depending on the variational solution. This extension gives rise to a class of quasi-variational inequalities. We provide sufficient conditions for the existence of fixed points as solutions, and an algorithm based on solving a sequence of variational problems. Denoising experiments with spatial and spatio-temporal image data and an adaptive total variation regularizer illustrate our approach.

On a semismooth least squares formulation of complementarity problems with gap reduction

We present a nonsmooth least squares reformulation of the complementarity problem and investigate its convergence properties. The global and local fast convergence results (under mild assumptions) are similar to some existing equation-based methods. In fact, our least squares formulation is obtained by modifying one of these equation-based methods (using the Fischer–Burmeister function) in such a way that we overcome a major drawback of this equation-based method. The resulting nonsmooth Levenberg–Marquardt-type method turns out to be significantly more robust than the corresponding equation-based method. This is illustrated by our numerical results using the MCPLIB test problem collection. E-mail: petra@mathematik.uni-wuerzburg.de

Projected filter trust region methods for a semismooth least squares formulation of mixed complementarity problems

A reformulation of the mixed complementarity problem as a box constrained overdetermined system of semismooth equations or, equivalently, a box constrained nonlinear least squares problem with zero residual is presented. On the basis of this reformulation, a trust region method for the solution of mixed complementarity problems is considered. This trust region method contains elements from different areas: a projected Levenberg–Marquardt step in order to guarantee local fast convergence under suitable assumptions, affine scaling matrices which are used to improve the global convergence properties, and a multidimensional filter technique to accept a full step more frequently. Global convergence results as well as local superlinear/quadratic convergence is shown under appropriate assumptions. Moreover, numerical results for the MCPLIB indicate that the overall method is quite robust.

TomoPIV Meets Compressed Sensing

We study the discrete tomography problem in Experimental Fluid Dynamics—Tomographic Particle Image Velocimetry (TomoPIV)—from the viewpoint of Compressed Sensing (CS). The problem results in an ill‐posed image reconstruction problem due to undersampling. Ill‐posedness is also intimately connected to the particle density. Higher densities ease subsequent flow estimation but also aggravate ill‐posedness of the reconstruction problem. A theoretical investigation of this trade‐off is studied in the present work.

3D Tomography from Few Projections in Experimental Fluid Dynamics

We study the tomographic problem of reconstructing particle volume functions in experimental fluid dynamics from the general viewpoint of compressed sensing, which is a central theme of current research in applied mathematics. The probability of exact reconstructions from few projections is studied empirically and shown to resemble provable results for idealized mathematical measurement setups. Application of our reconstruction algorithm to noisy projections outperforms the state-of-the-art both with respect to accuracy and runtime.

Variational Adaptive Correlation Method for Flow Estimation

A variational approach is presented to the estimation of turbulent fluid flow from particle image sequences in experimental fluid mechanics. The approach comprises two coupled optimizations for adapting size and shape of a Gaussian correlation window at each location and for estimating the flow, respectively. The method copes with a wide range of particle densities and image noise levels without any data-specific parameter tuning. Based on a careful implementation of a multiscale nonlinear optimization technique, we demonstrate robustness of the solution over typical experimental scenarios and highest estimation accuracy for an international benchmark data set (PIV Challenge).

Image Labeling by Assignment

We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.

Variational image denoising with adaptive constraint sets

We propose a generalization of the total variation (TV) minimization method proposed by Rudin, Osher and Fatemi. This generalization allows for adaptive regularization, which depends on the minimizer itself. Existence theory is provided in the framework of quasi-variational inequalities. We demonstrate the usability of our approach by considering applications for image and movie denoising.

On Sparsity Maximization in Tomographic Particle Image Reconstruction

This work focuses on tomographic image reconstruction in experimental fluid mechanics (TomoPIV), a recently established 3D particle image velocimetry technique. Corresponding 2D image sequences (projections) and the 3D reconstruction via tomographical methods provides the basis for estimating turbulent flows and related flow patterns through image processing. TomoPIV employs undersampling to make the high-speed imaging process feasible, resulting in an ill-posed image reconstruction problem. We address the corresponding basic problems involved and point out promising optimization criteria for reconstruction based on sparsity maximization, that perform favorably in comparison to classical algebraic methods currently in use for TomoPIV.

Phase Transitions and Cosparse Tomographic Recovery of Compound Solid Bodies from Few Projections

We study unique recovery of cosparse signals from limited-view tomographic measurements of two-and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality l with respect to the discrete gradient operator. We relate l both to the number of measurements and to a nullspace condition with respect to the measurement matrix, so as to achieve unique recovery by linear programming. These results are supported by comprehensive numerical experiments that show a high correlation of performance in practice and theoretical predictions. Despite poor properties of the measurement matrix from the viewpoint of compressed sensing, the class of uniquely recoverable signals basically seems large enough to cover practical applications, like contactless quality inspection of compound solid bodies composed of few materials.