On Median Filters for Motion by Mean Curvature

The median filter scheme is an elegant, monotone discretization of the level set formulation of motion by mean curvature. It turns out to evolve every level set of the initial condition precisely by another class of methods known as threshold dynamics. Median filters are, in other words, the natural level set versions of threshold dynamics algorithms. Exploiting this connection, we revisit median filters in light of recent progress on the threshold dynamics method. In particular, we give a variational formulation of, and exhibit a Lyapunov function for, median filters, resulting in energy based unconditional stability properties. The connection also yields analogues of median filters in the multiphase setting of mean curvature flow of networks. These new multiphase level set methods do not require frequent redistancing, and can accommodate a wide range of surface tensions.Comment: 41 pages, 8 figure

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

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation