Simon Setzer

Fast alternating direction optimization methods

Alternating direction methods are a common tool for general mathematical programming and optimization. These methods have become particularly important in the field of variational image processing, which frequently requires the minimization of nondifferentiable objectives. This paper considers accelerated (i.e., fast) variants of two common alternating direction methods: the alternating direction method of multipliers (ADMM) and the alternating minimization algorithm (AMA). The proposed acceleration is of the form first proposed by Nesterov for gradient descent methods. In the case that the objective function is strongly convex, global convergence bounds are provided for both classical and accelerated variants of the methods. Numerical examples are presented to demonstrate the superior performance of the fast methods for a wide variety of problems.

Deblurring Poissonian images by split Bregman techniques

The restoration of blurred images corrupted by Poisson noise is an important task in various applications such as astronomical imaging, electronic microscopy, single particle emission computed tomography (SPECT) and positron emission tomography (PET). In this paper, we focus on solving this task by minimizing an energy functional consisting of the I-divergence as similarity term and the TV regularization term. Our minimizing algorithm uses alternating split Bregman techniques (alternating direction method of multipliers) which can be reinterpreted as Douglas-Rachford splitting applied to the dual problem. In contrast to recently developed iterative algorithms, our algorithm contains no inner iterations and produces nonnegative images. The high efficiency of our algorithm in comparison to other recently developed algorithms to minimize the same functional is demonstrated by artificial and real-world numerical examples.

Operator Splittings, Bregman Methods and Frame Shrinkage in Image Processing

We examine the underlying structure of popular algorithms for variational methods used in image processing. We focus here on operator splittings and Bregman methods based on a unified approach via fixed point iterations and averaged operators. In particular, the recently proposed alternating split Bregman method can be interpreted from different points of view—as a Bregman, as an augmented Lagrangian and as a Douglas-Rachford splitting algorithm which is a classical operator splitting method. We also study similarities between this method and the forward-backward splitting method when applied to two frequently used models for image denoising which employ a Besov-norm and a total variation regularization term, respectively. In the first setting, we show that for a discretization based on Parseval frames the gradient descent reprojection and the alternating split Bregman algorithm are equivalent and turn out to be a frame shrinkage method. For the total variation regularizer, we also present a numerical comparison with multistep methods.

Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage

We examine relations between popular variational methods in image processing and classical operator splitting methods in convex analysis. We focus on a gradient descent reprojection algorithm for image denoising and the recently proposed Split Bregman and alternating Split Bregman methods. By identifying the latter with the so-called Douglas-Rachford splitting algorithm we can guarantee its convergence. We show that for a special setting based on Parseval frames the gradient descent reprojection and the alternating Split Bregman algorithm are equivalent and turn out to be a frame shrinkage method.

An Optimal Control Approach to Find Sparse Data for Laplace Interpolation

Finding optimal data for inpainting is a key problem in the context of partial differential equation-based image compression. We present a new model for optimising the data used for the reconstruction by the underlying homogeneous diffusion process. Our approach is based on an optimal control framework with a strictly convex cost functional containing an L 1 term to enforce sparsity of the data and non-convex constraints. We propose a numerical approach that solves a series of convex optimisation problems with linear constraints. Our numerical examples show that it outperforms existing methods with respect to quality and computation time.

Inpainting by Flexible Haar-Wavelet Shrinkage

We present novel wavelet-based inpainting algorithms. Applying ideas from anisotropic regularization and diffusion, our models can better handle degraded pixels at edges. We interpret our algorithms within the framework of forward-backward splitting methods in convex analysis and prove that the conditions for ensuring their convergence are fulfilled. Numerical examples illustrate the good performance of our algorithms.

Combined ℓ2 data and gradient fitting in conjunction with ℓ1 regularization

We are interested in minimizing functionals with ℓ2 data and gradient fitting term and ℓ1 regularization term with higher order derivatives in a discrete setting. We examine the structure of the solution in 1D by reformulating the original problem into a contact problem which can be solved by dual optimization techniques. The solution turns out to be a ’smooth’ discrete polynomial spline whose knots coincide with the contact points while its counterpart in the contact problem is a discrete version of a spline with higher defect and contact points as knots. In 2D we modify Chambolle’s algorithm to solve the minimization problem with the ℓ1 norm of interacting second order partial derivatives as regularization term. We show that the algorithm can be implemented efficiently by applying the fast cosine transform. We demonstrate by numerical denoising examples that the ℓ2 gradient fitting term can be used to avoid both edge blurring and staircasing effects.

Restoration of images with rotated shapes

Methods for image restoration which respect edges and other important features are of fundamental importance in digital image processing. In this paper, we present a novel technique for the restoration of images containing rotated (linearly transformed) rectangular shapes which avoids the round-off effects at vertices produced by known edge-preserving denoising techniques. Following an idea of Berkels et al. our approach is also based on two steps: the determination of the angles related to the rotated shapes and a subsequent restoration step which incorporates the knowledge of the angles. However, in contrast to Berkels et al., we find the smoothed rotation angles of the shapes by minimizing a simple quadratic functional without constraints which involves only first order derivatives so that we finally have to solve only a linear system of equations. Moreover, we propose to perform the restoration step either by quadratic programming or by solving an anisotropic diffusion equation. We focus on a discrete approach which approximates derivatives by finite differences. Particular attention is paid to the choice of the difference filters. We prove some relations concerning the preservation of rectangular shapes for our discrete setting. Finally, we present numerical examples for the denoising of artificial images with rotated rectangles and parallelograms and for the denoising of a real-world image.

On vector and matrix median computation

The aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them. We prove relations between vector and matrix means and medians, particularly regarding the classical structure tensor. Moreover, we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2i?2 matrices, which frequently arise in image processing. Our findings are explained and illustrated by numerical examples. To solve the corresponding minimization problems, we propose several algorithms. Existing approaches include Weiszfelds algorithm for the computation of ? 2 vector medians and semi-definite programming, in particular, second order cone programming, which has been used for matrix median computation. In this paper, we adapt Weiszfelds algorithm for our setting and show that also two splitting methods, namely the alternating direction method of multipliers and the parallel proximal algorithm, can be applied for generalized vector and matrix median computations. Besides, we compare the performance of these algorithms numerically and apply them within local median filters.

Restoration of matrix fields by second-order cone programming

SummaryWherever anisotropic behavior in physical measurements or models is encountered matrices provide adequate means to describe this anisotropy. Prominent examples are the diffusion tensor magnetic resonance imaging in medical imaging or the stress tensor in civil engineering. As most measured data these matrix-valued data are also polluted by noise and require restoration. The restoration of scalar images corrupted by noise via minimization of an energy functional is a well-established technique that offers many advantages. A convenient way to achieve this minimization is second-order cone programming (SOCP). The goal of this article is to transfer this method to the matrix-valued setting. It is shown how SOCP can be applied to minimize various energy functionals defined for matrix fields. These functionals couple the different matrix channels taking into account the relations between them. Furthermore, new functionals for the regularization of matrix data are proposed and the corresponding Euler–Lagrange equations are derived by means of matrix differential calculus. Numerical experiments substantiate the usefulness of the proposed methods for the restoration of matrix fields.