Martin Burger

An Iterative Regularization Method for Total Variation-Based Image Restoration

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.

Convergence rates of convex variational regularization

The aim of this paper is to provide quantitative estimates for the minimizers of non-quadratic regularization problems in terms of the regularization parameter, respectively the noise level. As usual for ill-posed inverse problems, these estimates can be obtained only under additional smoothness assumptions on the data, the so-called source conditions, which we identify with the existence of Lagrange multipliers for a limit problem. Under such a source condition, we shall prove a quantitative estimate for the Bregman distance induced by the regularization functional, which turns out to be the natural distance measure to use in this case. We put a special emphasis on the case of total variation regularization, which is probably the most important and prominent example in this class. We discuss the source condition for this case in detail and verify that it still allows discontinuities in the solution, while imposing some regularity on its level sets.

A survey on level set methods for inverse problems and optimal design

The aim of this paper is to provide a survey on the recent development in level set methods in inverse problems and optimal design. We give introductions on the general features of such problems involving geometries and on the general framework of the level set method. In subsequent parts we discuss shape sensitivity analysis and its relation to level set methods, various approaches on constructing optimization algorithms based on the level set approach, and special tools needed for the application of level set based optimization methods to ill-posed problems. Furthermore, we provide a review on numerical methods important in this context, and give an overview of applications treated with level set methods. Finally, we provide a discussion of the most challenging and interesting open problems in this field, that might be of interest for scientists who plan to start future research in this field.

A level set method for inverse problems

This paper is devoted to the solution of shape reconstruction problems by a level set method. The basic motivation for the setup of this level set algorithm is the well-studied method of asymptotic regularization, which has been developed for ill-posed problems in Hilbert spaces. Using analogies to this method, the convergence analysis of the proposed level set method is established and it is shown that the evolving level set converges to a solution in the symmetric difference metric as the artificial time evolves to infinity. Furthermore, the regularizing properties of the level set method are shown, if the discrepancy principle is used as a stopping rule. The numerical implementation of the level set method is discussed and applied to some examples in order to compare the numerical results with theoretical statements. The numerical results demonstrate the power of the level set method, in particular for examples where the number of connected components the solution consists of is not known a priori.

A framework for the construction of level set methods for shape optimization and reconstruction

The aim of this paper is to develop a functional-analytic framework for the construction of level set methods, when applied to shape optimization and shape reconstruction problems. As a main tool we use a notion of gradient flows for geometric configurations such as used in the modelling of geometric motions in materials science. The analogies to this field lead to a scale of level set evolutions, characterized by the norm used for the choice of the velocity. This scale of methods also includes the standard approach used in previous work on this subject as a special case. Moreover, we apply this framework to some (inverse) model problems for elliptic boundary value problems. In numerical experiments we demonstrate that an appropriate choice of norms (dependent on the problem) yields stable and fast methods.

Phase-Field Relaxation of Topology Optimization with Local Stress Constraints

We introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase-field method. In the basic formulation we have a PDE-constrained optimization problem, where the finite element and design analysis are solved simultaneously. The starting point of the relaxation is a reformulation of the material problem involving linear and 0-1 constraints only. The 0-1 constraints are then relaxed and approximated by a Cahn-Hilliard-type penalty in the objective functional, which yields convergence of minimizers to 0-1 designs as the penalty parameter decreases to zero. A major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter. The relaxation scheme yields a large-scale optimization problem with a high number of linear inequality constraints. We discretize the problem by finite elements and solve the arising finite-dimensional programming problems by a primal-dual interior point method. Numerical experiments for problems with local stress constraints based on different criteria indicate the success and robustness of the new approach.

Error estimation for Bregman iterations and inverse scale space methods in image restoration

SummaryIn this paper, we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. Burger in Inverse Prob 20: 1411–1421, 2004; Resmerita in Inverse Prob 21: 1303–1314, 2005; Inverse Prob 22: 801–814, 2006), but so far it was not applied to image restoration in a systematic way. Due to the flexibility of the Bregman distances, this approach is particularly attractive for imaging tasks, where often singular energies (non-differentiable, not strictly convex) are used to achieve certain tasks such as preservation of edges. Besides the discussion of the variational image restoration schemes, our main goal in this paper is to extend the error estimation approach to iterative regularization schemes (and time-continuous flows) that have emerged recently as multiscale restoration techniques and could improve some shortcomings of the variational schemes. We derive error estimates between the iterates and the exact image both in the case of clean and noisy data, the latter also giving indications on the choice of termination criteria. The error estimates are applied to various image restoration approaches such as denoising and decomposition by total variation and wavelet methods. We shall see that interesting results for various restoration approaches can be deduced from our general results by just exploring the structure of subgradients.

Accurate EM-TV algorithm in PET with low SNR

PET measurements of tracers with a lower dose rate or short radioactive half life suffer from extremely low SNRs. In these cases standard reconstruction methods (OSEM, EM, filtered backprojection) deliver unsatisfactory and noisy results. Here, we propose to introduce nonlinear variational methods into the reconstruction process to make an efficient use of a-priori information and to attain improved imaging results. We illustrate our technique by evaluating cardiac H215O measurements. The general approach can also be used for other specific goals allowing to incorporate a-priori information about the solution with Poisson distributed data.

Nonlinear inverse scale space methods for image restoration

In this paper we generalize the iterated refinement method, introduced by the authors in [8],to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. n nThe inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known. n nThe inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow accurate, efficient and straightforward implementation.

Primal and Dual Bregman Methods with Application to Optical Nanoscopy

Measurements in nanoscopic imaging suffer from blurring effects modeled with different point spread functions (PSF). Some apparatus even have PSFs that are locally dependent on phase shifts. Additionally, raw data are affected by Poisson noise resulting from laser sampling and “photon counts” in fluorescence microscopy. In these applications standard reconstruction methods (EM, filtered backprojection) deliver unsatisfactory and noisy results. Starting from a statistical modeling in terms of a MAP likelihood estimation we combine the iterative EM algorithm with total variation (TV) regularization techniques to make an efficient use of a-priori information. Typically, TV-based methods deliver reconstructed cartoon images suffering from contrast reduction. We propose extensions to EM-TV, based on Bregman iterations and primal and dual inverse scale space methods, in order to obtain improved imaging results by simultaneous contrast enhancement. Besides further generalizations of the primal and dual scale space methods in terms of general, convex variational regularization methods, we provide error estimates and convergence rates for exact and noisy data. We illustrate the performance of our techniques on synthetic and experimental biological data.