Dissertation/Thesis Abstract

Wavelet frames on the sphere, high angular resolution diffusion imaging and l1-regularized optimization on Stiefel manifolds
by Weiqiang, Chen, M.Sc., National University of Singapore (Singapore), 2015, 161; 10005982
Abstract (Summary)

In the past two decades, wavelet frames are preferred over wavelet bases in image and signal processing applications as they yield redundant andflexible representations of squareintegrable functions. As such, Chapter 1 provides preliminaries on MRA-based wavelet frames for L2(R), wavelet frame-based image restoration models. To facilitate the discussion in subsequent chapters, introductions to the spherical harmonic functions and Sobolev spaces on the sphere are also provided.

Building upon the wavelet bases constructions for Hilbert spaces in [46], Chapter 2 constructs tight wavelet frames for the space of (symmetric) square-integrable real-valued functions denied on the unit sphere, by considering special linear and weighted combinations of (modified) spherical harmonic.

In Chapter 3, we describe how these wavelet frames can be applied to denoise signals in High Angular Resolution Diffusion Imaging (HARDI) [83], a relatively recent non-invasive brain imaging technique. Tight framelet filters can also be used to impose spatial regularization of HARDI signals to improve denoising performances. The proposed wavelet frame-based approach generally denoises highly corrupted HARDI signals more cost-effectively than the spherical harmonics-based and spherical ridgelets-based approaches.

In Chapter 4, the HARDI denoising performances are further improved through adaptive spatial regularization, which can be modelled by optimization on Stiefel manifolds, i.e., orthogonality constrained problems. The resulting optimization problems are solved by the proximal alternating minimized augmented Lagrangian (PAMAL) method, which is a hybridization of the augmented Lagrangian method and the proximal alternating minimization method. Convergence analysis is also provided for the PAMAL method.

In Chapter 5, the PAMAL method is applied to a class of ℓ1-regularized optimization problems with orthogonality constraints, which includes the compressed modes problem [69]. Convergence analysis of the PAMAL method is also provided in this case. Numerical results illustrate that the PAMAL method is noticeably faster than the splitting of orthogonality constraints (SOC) method [53] in producing compressed modes with comparable quality.

Indexing (document details)
School: National University of Singapore (Singapore)
Department: Mathematics
School Location: Republic of Singapore
Source: DAI-B 77/06(E), Dissertation Abstracts International
Subjects: Mathematics
Publication Number: 10005982
ISBN: 9781339438542
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