First order algorithms in variational image processing

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on tasks like denoising, deblurring, inpainting, segmentation, super-resolution, disparity, and optical flow estimation. The overall structure of such approaches is of the form ${\cal D}(Ku) + \alpha {\cal R} (u) \rightarrow \min_u$ ; where the functional ${\cal D}$ is a data fidelity term also depending on some input data $f$ and measuring the deviation of $Ku$ from such and ${\cal R}$ is a regularization functional. Moreover $K$ is a (often linear) forward operator modeling the dependence of data on an underlying image, and $\alpha$ is a positive regularization parameter. While ${\cal D}$ is often smooth and (strictly) convex, the current practice almost exclusively uses nonsmooth regularization functionals. The majority of successful techniques is using nonsmooth and convex functionals like the total variation and generalizations thereof or $\ell_1$-norms of coefficients arising from scalar products with some frame system. The efficient solution of such variational problems in imaging demands for appropriate algorithms. Taking into account the specific structure as a sum of two very different terms to be minimized, splitting algorithms are a quite canonical choice. Consequently this field has revived the interest in techniques like operator splittings or augmented Lagrangians. Here we shall provide an overview of methods currently developed and recent results as well as some computational studies providing a comparison of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure

Splitting Methods in Image Processing

It is often necessary to restore digital images which are affected by noise (denoising), blur (deblurring), or missing data (inpainting). We focus here on variational methods, i.e., the restored image is the minimizer of an energy functional. The first part of this thesis deals with the algorithmic framework of how to compute such a minimizer. It turns out that operator splitting methods are very useful in image processing to derive fast algorithms. The idea is that, in general, the functional we want to minimize has an additive structure and we treat its summands separately in each iteration of the algorithm which yields subproblems that are easier to solve. In our applications, these are typically projections onto simple sets, fast shrinkage operations, and linear systems of equations with a nice structure. The two operator splitting methods we focus on here are the forward-backward splitting algorithm and the Douglas-Rachford splitting algorithm. We show based on older results that the recently proposed alternating split Bregman algorithm is equivalent to the Douglas-Rachford splitting method applied to the dual problem, or, equivalently, to the alternating direction method of multipliers. Moreover, it is illustrated how this algorithm allows us to decouple functionals which are sums of more than two terms. In the second part, we apply the above techniques to existing and new image restoration models. For the Rudin-Osher-Fatemi model, which is well suited to remove Gaussian noise, the following topics are considered: we avoid the staircasing effect by using an additional gradient fitting term or by combining first- and second-order derivatives via an infimal-convolution functional. For a special setting based on Parseval frames, a strong connection between the forward-backward splitting algorithm, the alternating split Bregman method and iterated frame shrinkage is shown. Furthermore, the good performance of the alternating split Bregman algorithm compared to the popular multistep methods is illustrated. A special emphasis lies here on the choice of the step-length parameter. Turning to a corresponding model for removing Poisson noise, we show the advantages of the alternating split Bregman algorithm in the decoupling of more complicated functionals. For the inpainting problem, we improve an existing wavelet-based method by incorporating anisotropic regularization techniques to better restore boundaries in an image. The resulting algorithm is characterized as a forward-backward splitting method. Finally, we consider the denoising of a more general form of images, namely, tensor-valued images where a matrix is assigned to each pixel. This type of data arises in many important applications such as diffusion-tensor MRI

Coupling Image Restoration and Segmentation: A Generalized Linear Model/Bregman Perspective

We introduce a new class of data-fitting energies that couple image segmentation with image restoration. These functionals model the image intensity using the statistical framework of generalized linear models. By duality, we establish an information-theoretic interpretation using Bregman divergences. We demonstrate how this formulation couples in a principled way image restoration tasks such as denoising, deblurring (deconvolution), and inpainting with segmentation. We present an alternating minimization algorithm to solve the resulting composite photometric/geometric inverse problem.We use Fisher scoring to solve the photometric problem and to provide asymptotic uncertainty estimates. We derive the shape gradient of our data-fitting energy and investigate convex relaxation for the geometric problem. We introduce a new alternating split- Bregman strategy to solve the resulting convex problem and present experiments and comparisons on both synthetic and real-world images

Coupling Image Restoration and Segmentation: A Generalized Linear Model/Bregman Perspective

We introduce a new class of data-fitting energies that couple image segmentation with image restoration. These functionals model the image intensity using the statistical framework of generalized linear models. By duality, we establish an information-theoretic interpretation using Bregman divergences. We demonstrate how this formulation couples in a principled way image restoration tasks such as denoising, deblurring (deconvolution), and inpainting with segmentation. We present an alternating minimization algorithm to solve the resulting composite photometric/geometric inverse problem. We use Fisher scoring to solve the photometric problem and to provide asymptotic uncertainty estimates. We derive the shape gradient of our data-fitting energy and investigate convex relaxation for the geometric problem. We introduce a new alternating split-Bregman strategy to solve the resulting convex problem and present experiments and comparisons on both synthetic and real-world image

A Variational Model for Joint Motion Estimation and Image Reconstruction

The aim of this paper is to derive and analyze a variational model for the joint estimation of motion and reconstruction of image sequences, which is based on a time-continuous Eulerian motion model. The model can be set up in terms of the continuity equation or the brightness constancy equation. The analysis in this paper focuses on the latter for robust motion estimation on sequences of twodimensional images. We rigorously prove the existence of a minimizer in a suitable function space setting. Moreover, we discuss the numerical solution of the model based on primal-dual algorithms and investigate several examples. Finally, the benefits of our model compared to existing techniques, such as sequential image reconstruction and motion estimation, are shown.The work of the first author was also supported by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, M¨unster, German

4D imaging in tomography and optical nanoscopy

Diese Dissertation gehört zu den Gebieten mathematische Bildverarbeitung und inverse Probleme. Ein inverses Problem ist die Aufgabe, Modellparameter anhand von gemessenen Daten zu berechnen. Solche Probleme treten in zahlreichen Anwendungen in Wissenschaft und Technik auf, z.B. in medizinischer Bildgebung, Biophysik oder Astronomie. Wir betrachten Rekonstruktionsprobleme mit Poisson Rauschen in der Tomographie und optischen Nanoskopie. Bei letzterer gilt es Bilder ausgehend von verzerrten und verrauschten Messungen zu rekonstruieren, wohingegen in der Positronen-Emissions-Tomographie die Aufgabe in der Visualisierung physiologischer Prozesse eines Patienten besteht. Standardmethoden zur 3D Bildrekonstruktion berücksichtigen keine zeitabhängigen Informationen oder Dynamik, z.B. Herzschlag oder Atmung in der Tomographie oder Zellmigration in der Mikroskopie. Diese Dissertation behandelt Modelle, Analyse und effiziente Algorithmen für 3D und 4D zeitabhängige inverse Probleme. This thesis contributes to the field of mathematical image processing and inverse problems. An inverse problem is a task, where the values of some model parameters must be computed from observed data. Such problems arise in a wide variety of applications in sciences and engineering, such as medical imaging, biophysics or astronomy. We mainly consider reconstruction problems with Poisson noise in tomography and optical nanoscopy. In the latter case, the task is to reconstruct images from blurred and noisy measurements, whereas in positron emission tomography the task is to visualize physiological processes of a patient. In 3D static image reconstruction standard methods do not incorporate time-dependent information or dynamics, e.g. heart beat or breathing in tomography or cell motion in microscopy. This thesis is a treatise on models, analysis and efficient algorithms to solve 3D and 4D time-dependent inverse problems

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness