Joint methods in imaging based on diffuse image representations

This thesis deals with the application and the analysis of different variants of the Mumford-Shah model in the context of image processing. In this kind of models, a given function is approximated in a piecewise smooth or piecewise constant manner. Especially the numerical treatment of the discontinuities requires additional models that are also outlined in this work. The main part of this thesis is concerned with four different topics. Simultaneous edge detection and registration of two images: The image edges are detected with the Ambrosio-Tortorelli model, an approximation of the Mumford-Shah model that approximates the discontinuity set with a phase field, and the registration is based on these edges. The registration obtained by this model is fully symmetric in the sense that the same matching is obtained if the roles of the two input images are swapped. Detection of grain boundaries from atomic scale images of metals or metal alloys: This is an image processing problem from materials science where atomic scale images are obtained either experimentally for instance by transmission electron microscopy or by numerical simulation tools. Grains are homogenous material regions whose atomic lattice orientation differs from their surroundings. Based on a Mumford-Shah type functional, the grain boundaries are modeled as the discontinuity set of the lattice orientation. In addition to the grain boundaries, the model incorporates the extraction of a global elastic deformation of the atomic lattice. Numerically, the discontinuity set is modeled by a level set function following the approach by Chan and Vese. Joint motion estimation and restoration of motion-blurred video: A variational model for joint object detection, motion estimation and deblurring of consecutive video frames is proposed. For this purpose, a new motion blur model is developed that accurately describes the blur also close to the boundary of a moving object. Here, the video is assumed to consist of an object moving in front of a static background. The segmentation into object and background is handled by a Mumford-Shah type aspect of the proposed model. Convexification of the binary Mumford-Shah segmentation model: After considering the application of Mumford-Shah type models to tackle specific image processing problems in the previous topics, the Mumford-Shah model itself is studied more closely. Inspired by the work of Nikolova, Esedoglu and Chan, a method is developed that allows global minimization of the binary Mumford-Shah segmentation model by solving a convex, unconstrained optimization problem. In an outlook, segmentation of flowfields into piecewise affine regions using this convexification method is briefly discussed

A CONVEX AND SELECTIVE VARIATIONAL MODEL FOR IMAGE SEGMENTATION

Selective image segmentation is the task of extracting one object of interest from an image, based on minimal user input. Recent level set based variational models have shown to be effective and reliable, although they can be sensitive to initialization due to the minimization problems being nonconvex. This sometimes means that successful segmentation relies too heavily on user input or a solution found is only a local minimizer, i.e. not the correct solution. The same principle applies to variational models that extract all objects in an image (global segmentation); however, in recent years, some have been successfully reformulated as convex optimization problems, allowing global minimizers to be found. There are, however, problems associated with extending the convex formulation to the current selective models, which provides the motivation for the proposal of a new selective model. In this paper we propose a new selective segmentation model, combining ideas from global segmentation, that can be reformulated in a convex way such that a global minimizer can be found independently of initialization. Numerical results are given that demonstrate its reliability in terms of removing the sensitivity to initialization present in previous models, and its robustness to user input

High Order Schemes for Gradient Flows

First, two new classes of energy stable, high order accurate Runge-Kutta schemes for gradient flows in a very general setting are presented: a class of fully implicit methods that are unconditionally energy stable and a class of semi-implicit methods that are conditionally energy stable. The new schemes are developed as high order analogs of the minimizing movements approach for generating a time discrete approximation to a gradient flow by solving a sequence of optimization problems. In particular, each step entails minimizing the associated energy of the gradient flow plus a movement limiter term that is, in the classical context of steepest descent with respect to an inner product, simply quadratic. A variety of existing stable numerical methods can be recognized as (typically just first order accurate in time) minimizing movement schemes for their associated evolution equations, already requiring the optimization of the energy plus a quadratic term at every time step. Therefore, our methods give a painless way to extend the existing schemes to high order accurate in time schemes while maintaining their stability. Additionally, we extend the schemes to gradient flows with solution dependent inner product. Here, the stability and consistency conditions of the methods are given and proved, specific examples of the schemes are given for second and third order accuracy, and convergence tests are performed to demonstrate the accuracy of the methods. Next, two algorithms for simulating mean curvature motion are considered. First is the threshold dynamics algorithm of Merriman, Bence, and Osher. The algorithm is only first order accurate in the two-phase setting and its accuracy degrades further to half order in the multi-phase setting, a shortcoming it has in common with other related, more recent algorithms. As a first, rigorous step in addressing this shortcoming, two different second order accurate versions of two-phase threshold dynamics are presented. Unlike in previous efforts in this direction, both algorithms come with careful consistency calculations. The first algorithm is consistent with its limit (motion by mean curvature) up to second order in any space dimension. The second achieves second order accuracy only in dimension two but comes with a rigorous stability guarantee (unconditional energy stability) in any dimension -- a first for high order schemes of its type. Finally, a level set method for multiphase curvature motion known as Voronoi implicit interface method is considered. Here, careful numerical convergence studies, using parameterized curves to reach very high resolutions in two dimensions are given. These tests demonstrate that in the unequal, additive surface tension case, the Voronoi implicit interface method does not converge to the desired limit. Then a variant that maintains the spirit of the original algorithm is presented. It appears to fix the non-convergence and as a bonus, the new variant extends the Voronoi implicit interface method to unequal mobilities.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162894/1/azaitzef_1.pd

Mini-Workshop: Geometries, Shapes and Topologies in PDE-based Applications

The aim of the workshop was to study geometrical objects and their sensitivities in applications based on partial differential equations or differential variational inequalities. Focus topics comprised analytical investigations, numerical developments, issues in applications as well as new and future directions. Particular emphasis was put on: (i) combined shape and topological sensitivity; (ii) extended topological expansions and their numerical realization; (iii) level set based shape and topology optimization