Algorithms and applications of semidefinite space tensor conic convex program

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Algorithms and applications of semidefinite space tensor conic convex program

 

Author: Xu, Yi
Title: Algorithms and applications of semidefinite space tensor conic convex program
Degree: Ph.D.
Year: 2013
Subject: Imaging systems -- Mathematics.
Hong Kong Polytechnic University -- Dissertations
Department: Dept. of Applied Mathematics
Pages: xx, 83 p. : ill. ; 30 cm.
Language: English
InnoPac Record: http://library.polyu.edu.hk/record=b2653068
URI: http://theses.lib.polyu.edu.hk/handle/200/7267
Abstract: This thesis focuses on studying the algorithms and applications of positive semi-definite space tensors. A positive semi-definite space tensors are a special type semi-definite tensors with dimension 3. Positive semi-definite space tensors have some applications in real life, such as the medical imaging. However, there isn't an algorithm with good performance to solve an optimization problem with the positive semi-definite space tensor constraint, and the structure of positive semi-definite space tensors is not well explored. In this thesis, firstly, we try to analysis the properties of positive semi-definite space tensors; Then, we construct practicable algorithms to solve an optimization problem with the positive semi-definite space tensor constraint; Finally we use positive semi-definite space tensors to solve some medical problems. The main contributions of this thesis are shown as follows. Firstly, we study the methods to verify the semi-definiteness of space tensors and the properties of H-eigenvalue of tensors. As a basic property of space tensors, the positive semi-definiteness show significant importance in theory. However, there is not a good method to verify the positive semi-definiteness of space tensors. Based upon the nonnegative polynomial theory, we present two methods to verify whether a space tensor positive semi-definite or not. Furthermore, we study the smallest H-eigenvalue of tensors by the relationship between the smallest H-eigenvalue of tensors and their positive semi-definiteness.
Secondly, we consider the positive semi-definite space tensor cone constrained convex program, its structure and algorithms. We study defining functions, defining sequences and polyhedral outer approximations for this positive semi-definite space tensor cone, give an error bound for the polyhedral outer approximation approach, and thus establish convergence of three polyhedral outer approximation algorithms for solving this problem. We then study some other approaches for solving this structured convex program. These include the conic linear programming approach, the nonsmooth convex program approach and the bi-level program approach. Some numerical examples are presented. Thirdly, we apply positive semi-definite tensors into medical brain imagining. Because of the well-known limitations of diffusion tensor imaging (DTI) in regions of low anisotropy and multiple fiber crossing, high angular resolution diffusion imaging (HARDI) and Q-Ball Imaging (QBI) are used to estimate the probability density function (PDF) of the average spin displacement of water molecules. In particular, QBI is used to obtain the diffusion orientation distribution function (ODF) of these multiple fiber crossing. The ODF, as a probability distribution function, should be nonnegative. We propose a novel technique to guarantee nonnegative ODF by minimizing a positive semi-definite space tensor convex optimization problem. Based upon convex analysis and optimization theory, we derive its optimality conditions. And then we propose a gradient descent algorithm for solving this problem. We also present formulas for determining the principal directions (maxima) of the ODF. Numerical examples on synthetic data as well as MRI data are displayed to demonstrate our approach.

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