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Title: Variational methods for image segmentation
Author: Spencer, Jack A.
ISNI:       0000 0004 6059 6280
Awarding Body: University of Liverpool
Current Institution: University of Liverpool
Date of Award: 2016
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The work in this thesis is concerned with variational methods for two-phase segmentation problems. We are interested in both the obtaining of numerical solutions to the partial differential equations arising from the minimisation of a given functional, and forming variational models that tackle some practical problem in segmentation (e.g. incorporating prior knowledge, dealing with intensity inhomogeneity). With that in mind we will discuss each aspect of the work as follows. A seminal two-phase variational segmentation problem in the literature is that of Active Contours Without Edges, introduced by Chan and Vese in 2001, based on the piecewise-constant formulation of Mumford and Shah. The idea is to partition an image into two regions of homogeneous intensity. However, despite the extensive success of this work its reliance on the level set method means that it is nonconvex. Later work on the convex reformulation of ACWE by Chan, Esedoglu, and Nikolova has led to a burgeoning of related methods, known as the convex relaxation approach. In Chapter 4, we introduce a method to find global minimisers of a general two-phase segmentation problem, which forms the basis for work in the rest of the thesis. We introduce an improved additive operator splitting (AOS) method based on the work of Weickert et al. and Tai et al. AOS has been frequently used for segmentation problems, but not in the convex relaxation setting. The adjustment made accounts for how to impose the relaxed binary constraint, fundamental to this approach. Our method is analogous to work such as Bresson et al. and we quantitatively compare our method against this by using a number of appropriate metrics. Having dealt with globally convex segmentation (GCS) for the general case in Chapter 4, we then bear in mind two important considerations. Firstly, we discuss the matter of selective segmentation and how it relates to GCS. Many recent models have incorporated user input for two-phase formulations using piecewise-constant fitting terms. In Chapter 5 we discuss the conditions for models of this type to be reformulated in a similar way. We then propose a new model compatible with convex relaxation methods, and present results for challenging examples. Secondly, we consider the incorporation of priors for GCS in Chapter 8. Here, the intention is to select objects in an image of a similar shape to a given prior. We consider the most appropriate way to represent shape priors in a variational formulation, and the potential applications of our approach. We also investigate the problem of segmentation where the observed data is challenging. We consider two cases in this thesis; in one there is significant intensity inhomogeneity, and in the other the image has been corrupted by unknown blur. The first has been widely studied and is closely related to the piecewise-smooth formulation of Mumford and Shah. In Chapter 6 we discuss a Variant Mumford- Shah Model by D.Chen et al. that uses the bias field framework. Our work focuses on improving results for methods of this type. The second has been less widely studied, but is more commonly considered when there is knowledge of the blur type. We discuss the advantages of simultaneously reconstructing and segmenting the image, rather than treating each problem separately and compare our method against comparable models. The aim of this thesis is to develop new variational methods for two-phase image segmentation, with potential applications in mind. We also consider new schemes to compute numerical solutions for generalised segmentation problems. With both approaches we focus on convex relaxation methods, and consider the challenges of formulating segmentation problems in this manner. Where possible we compare our ideas against current approaches to determine quantifiable improvements, particularly with respect to accuracy and reliability.
Supervisor: Chen, K. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral