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Title: Diffeomorphic variational models and their fast algorithms for image registration problems
Author: Zhang, D.
Awarding Body: University of Liverpool
Current Institution: University of Liverpool
Date of Award: 2019
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Image registration aims to find an optimal geometric transformation to overlay two or more images of the same scene which are taken at different times, from different perspectives or various sensors. As one of the most fundamental tasks in image processing, image registration has a wide range of applications in computer vision, biological imaging, remote sensing, and medical imaging. In general, image registration involves two images, where one called template is kept transformed, and the other one called reference is kept unchanged. To measure the similarity between the deformed template and the reference, a fidelity term should be chosen. However, only minimizing/maximizing the fidelity term is ill-posed in the sense of Hadamard since it is not sufficient to ensure the uniqueness and continuity of the solution. To overcome this difficulty, regularization is indispensable. Hence, in this thesis, we mainly consider the variational framework. Nowadays, there exist many kinds of different regularizers. Unfortunately, when we solve the variational model for image registration, although we can obtain a visually satisfied deformed template, the corresponding transformation often has foldings and is not physically correct. Hence, over the last decade, more and more researchers have focused on diffeomorphic image registration, whose aim is to find an accurate diffeomorphic mapping, namely the transformation is continuously differentiable, and it has a continuously differentiable inverse. Here, we consider choosing the first-discretize-thenoptimize method to solve the variational model for image registration, namely, first directly discretize the variational model and obtain a finite dimensional optimization problem then choose suitable optimization methods to solve the resulting optimization problem. However, in image registration, the number of variables is usually huge, and the dimension of the resulting optimization problem is huge as well. For example, when the size of the given images is 128 x 64 x 128, the number of unknowns is over 3 million. Hence, designing an efficient and converging solver is also an important issue. This thesis can be mainly classified into two parts: one is how to design the diffeomorphic registration model leading to a diffeomorphic transformation, and the other one is how to develop highly efficient and effective solvers. In the first part, we first propose a new variational model with a special regularizer, based on the quasi-conformal theory, which can guarantee that the registration map is diffeomorphic. Also, since the Beltrami coefficient uses complex analysis and is only defined in 2D space, we further present two new formulations in 3D space motivated by the Beltrami concept and then set our new registration models. We propose a converging Gauss-Newton iterative method to solve the resulting nonlinear optimization problems and prove its convergence. Numerical experiments can demonstrate that the new 2D and 3D models can not only get diffeomorphic registrations even when the deformations are large, but also possess the accuracy in comparing with the state-of-the-art models. In the second part, we consider using the subspace method to accelerate the algorithm. Here, we propose two simple techniques to improve the performance of the standard multilevel Gauss-Newton algorithm. The first technique consists of the possible use of a second step within each iteration of the Gauss-Newton method. This step is computed by minimizing a quadratic approximation of the objective function over a two-dimensional subspace. This subspace is spanned by the steepest descent direction and the L-BFGS direction concerning the current point given by the Gauss-Newton step. The second technique is a modification of the standard coarse-to-fine multilevel strategy. At each level, instead of using the interpolated solution of the previous level directly as the initial point, we try to find a better initial point by minimizing a quadratic approximation of the objective function over the subspace spanned by the interpolated solutions of all the previous levels. Numerical experiments illustrate that the subspace method can significantly reduce the running time compared with the standard multilevel GaussNewton method. Overall, this thesis is concerned with the diffeomorphic variational models and their fast algorithms for image registration problems.
Supervisor: Chen, Ke Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral