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 semidefinite space tensors. A positive semidefinite space tensors are a special type semidefinite tensors with dimension 3. Positive semidefinite 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 semidefinite space tensor constraint, and the structure of positive semidefinite space tensors is not well explored. In this thesis, firstly, we try to analysis the properties of positive semidefinite space tensors; Then, we construct practicable algorithms to solve an optimization problem with the positive semidefinite space tensor constraint; Finally we use positive semidefinite 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 semidefiniteness of space tensors and the properties of Heigenvalue of tensors. As a basic property of space tensors, the positive semidefiniteness show significant importance in theory. However, there is not a good method to verify the positive semidefiniteness of space tensors. Based upon the nonnegative polynomial theory, we present two methods to verify whether a space tensor positive semidefinite or not. Furthermore, we study the smallest Heigenvalue of tensors by the relationship between the smallest Heigenvalue of tensors and their positive semidefiniteness. Secondly, we consider the positive semidefinite space tensor cone constrained convex program, its structure and algorithms. We study defining functions, defining sequences and polyhedral outer approximations for this positive semidefinite 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 bilevel program approach. Some numerical examples are presented. Thirdly, we apply positive semidefinite tensors into medical brain imagining. Because of the wellknown limitations of diffusion tensor imaging (DTI) in regions of low anisotropy and multiple fiber crossing, high angular resolution diffusion imaging (HARDI) and QBall 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 semidefinite 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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