A level set method with Sobolev Gradient and Haralick Edge Detection

Variational level set methods, which have been proposed with various energy functionals, mostly use the ordinary L type gradient in gradient descent algorithm to minimize the energy functional. The gradient flow is influenced by both the energy to be minimized and the norms, which are induced from inner products, used to measure the cost of perturbation of the curve. However, there are many undesired properties related to the gradient flows due to the 2 L type inner products. For example, there is not any regularity term in the definition of this inner product that causes non-smooth flows and inaccurate results. Therefore, in this work, Sobolev gradient has been used that is more efficient than the 2 L type gradient for image segmentation and has powerful properties such as regular gradient flows, independency to parameterization of curves, less sensitive to local features and noise in the image and also faster convergence rate than the standard gradient. In addition, Haralick edge detector has been used instead of the edge indicator function in this study. Because, the traditional edge indicator function, which is the absolute of the gradient of the convolved image with the aussian function, is sensitive to noise in level set methods. Experimental results on real images , which are abdominal magnetic resonance images, have been obtained for spleen and kidney segmentation. Quantitative analyses have been performed by using different measurements to evaluate the performance of the proposed approach, which can ignore topological noises and detect boundaries successfully

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

Trends in Mathematical Imaging and Surface Processing

Motivated both by industrial applications and the challenge of new problems, one observes an increasing interest in the field of image and surface processing over the last years. It has become clear that even though the applications areas differ significantly the methodological overlap is enormous. Even if contributions to the field come from almost any discipline in mathematics, a major role is played by partial differential equations and in particular by geometric and variational modeling and by their numerical counterparts. The aim of the workshop was to gather a group of leading experts coming from mathematics, engineering and computer graphics to cover the main developments

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

Foundations, Inference, and Deconvolution in Image Restoration

Image restoration is a critical preprocessing step in computer vision, producing images with reduced noise, blur, and pixel defects. This enables precise higher-level reasoning as to the scene content in later stages of the vision pipeline (e.g., object segmentation, detection, recognition, and tracking). Restoration techniques have found extensive usage in a broad range of applications from industry, medicine, astronomy, biology, and photography. The recovery of high-grade results requires models of the image degradation process, giving rise to a class of often heavily underconstrained, inverse problems. A further challenge specific to the problem of blur removal is noise amplification, which may cause strong distortion by ringing artifacts. This dissertation presents new insights and problem solving procedures for three areas of image restoration, namely (1) model foundations, (2) Bayesian inference for high-order Markov random fields (MRFs), and (3) blind image deblurring (deconvolution). As basic research on model foundations, we contribute to reconciling the perceived differences between probabilistic MRFs on the one hand, and deterministic variational models on the other. To do so, we restrict the variational functional to locally supported finite elements (FE) and integrate over the domain. This yields a sum of terms depending locally on FE basis coefficients, and by identifying the latter with pixels, the terms resolve to MRF potential functions. In contrast with previous literature, we place special emphasis on robust regularizers used commonly in contemporary computer vision. Moreover, we draw samples from the derived models to further demonstrate the probabilistic connection. Another focal issue is a class of high-order Field of Experts MRFs which are learned generatively from natural image data and yield best quantitative results under Bayesian estimation. This involves minimizing an integral expression, which has no closed form solution in general. However, the MRF class under study has Gaussian mixture potentials, permitting expansion by indicator variables as a technical measure. As approximate inference method, we study Gibbs sampling in the context of non-blind deblurring and obtain excellent results, yet at the cost of high computing effort. In reaction to this, we turn to the mean field algorithm, and show that it scales quadratically in the clique size for a standard restoration setting with linear degradation model. An empirical study of mean field over several restoration scenarios confirms advantageous properties with regard to both image quality and computational runtime. This dissertation further examines the problem of blind deconvolution, beginning with localized blur from fast moving objects in the scene, or from camera defocus. Forgoing dedicated hardware or user labels, we rely only on the image as input and introduce a latent variable model to explain the non-uniform blur. The inference procedure estimates freely varying kernels and we demonstrate its generality by extensive experiments. We further present a discriminative method for blind removal of camera shake. In particular, we interleave discriminative non-blind deconvolution steps with kernel estimation and leverage the error cancellation effects of the Regression Tree Field model to attain a deblurring process with tightly linked sequential stages