Level Set Methods for MRE Image Processing and Analysis

Ph.DDOCTOR OF PHILOSOPH

Nonlinear Evolutionary PDE-Based Refinement of Optical Flow

The goal of this paper is to propose two nonlinear variational models for obtaining a refined motion estimation from an image sequence. Both the proposed models can be considered as a part of a generalized framework for an accurate estimation of physics-based flow fields such as rotational and fluid flow. The first model is novel in the sense that it is divided into two phases: the first phase obtains a crude estimate of the optical flow and then the second phase refines this estimate using additional constraints. The correctness of this model is proved using an Evolutionary PDE approach. The second model achieves the same refinement as the first model, but in a standard manner, using a single functional. A special feature of our models is that they permit us to provide efficient numerical implementations through the first-order primaldual Chambolle-Pock scheme. Both the models are compared in the context of accurate estimation of angle by performing an anisotropic regularization of the divergence and curl of the flow respectively. We observe that, although both the models obtain the same level of accuracy, the two-phase model is more efficient. In fact, we empirically demonstrate that the single-phase and the two-phase models have convergence rates of order $O(1/N^2)$ and $O(1/N)$ respectively

Recent Advances in Signal Processing

The signal processing task is a very critical issue in the majority of new technological inventions and challenges in a variety of applications in both science and engineering fields. Classical signal processing techniques have largely worked with mathematical models that are linear, local, stationary, and Gaussian. They have always favored closed-form tractability over real-world accuracy. These constraints were imposed by the lack of powerful computing tools. During the last few decades, signal processing theories, developments, and applications have matured rapidly and now include tools from many areas of mathematics, computer science, physics, and engineering. This book is targeted primarily toward both students and researchers who want to be exposed to a wide variety of signal processing techniques and algorithms. It includes 27 chapters that can be categorized into five different areas depending on the application at hand. These five categories are ordered to address image processing, speech processing, communication systems, time-series analysis, and educational packages respectively. The book has the advantage of providing a collection of applications that are completely independent and self-contained; thus, the interested reader can choose any chapter and skip to another without losing continuity

Variational Multi-Task Models for Image Analysis: Applications to Magnetic Resonance Imaging

This thesis deals with the study and development of several variational multi-task models for solving inverse problems in imaging, with a particular focus on Magnetic Resonance Imaging (MRI). In most image processing problems, one usually deals with the reconstruction task, i.e., the task of reconstructing an image from indirect measurements, and then performs various operations, one after the other (i.e. sequentially), to improve the quality of the reconstruction and to extract useful information. However, recent developments in a variational context, have shown that performing those tasks jointly (i.e. in a multi-task framework) offers great benefits, and this is the perspective that we follow in this thesis. We go beyond traditional sequential approaches and set a new basis for variational multi-task methods for MRI analysis. We demonstrate that by sharing representation between tasks and carefully interconnecting them, one can create synergies across challenging problems and reduce error propagation. More precisely, firstly we propose a multi-task variational model to tackle the problems of image reconstruction and image segmentation using non-convex Bregman iteration. We describe theoretical and numerical details of the problem and its optimisation scheme. Moreover, we show that our multi-task model achieves better results in several examples and MRI applications than existing approaches in the same context. Secondly, we show that our approach can be extended to a multi-task reconstruction and segmentation model for the nonlinear inverse problem of velocity-encoded MRI. In this context, the aim is to estimate not only the magnitude from MRI data, but also the phase and its flow information, whilst simultaneously identify regions of interest through the segmentation task. Finally, we go beyond two-task frameworks and introduce for the first time a variational multi-task model to handle three imaging tasks. To this end, we design a variational multi-task framework addressing reconstruction, super-resolution and registration for improving the quality of MRI reconstruction. We demonstrate that our model is theoretically well-motivated and it outperforms sequential models whilst requiring less computational cost. Furthermore, we show through experimental results the potential of this approach for clinical applications

Modern Regularization Methods for Inverse Problems

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research. In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.Leverhulme Trust Early Career Fellowship ‘Learning from mistakes: a supervised feedback-loop for imaging applications’ Isaac Newton Trust Cantab Capital Institute for the Mathematics of Information ERC Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse German Ministry for Science and Education (BMBF) project MED4D EPSRC grant EP/K032208/

Entropy in Image Analysis II

Image analysis is a fundamental task for any application where extracting information from images is required. The analysis requires highly sophisticated numerical and analytical methods, particularly for those applications in medicine, security, and other fields where the results of the processing consist of data of vital importance. This fact is evident from all the articles composing the Special Issue "Entropy in Image Analysis II", in which the authors used widely tested methods to verify their results. In the process of reading the present volume, the reader will appreciate the richness of their methods and applications, in particular for medical imaging and image security, and a remarkable cross-fertilization among the proposed research areas