Dissertation
Sampling and recovery on parametric manifolds
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2021
DOI: 10.17077/etd.006077
Abstract
Recent studies show the great potential of using deep-learned image regularization priors in computational imaging in a variety of application areas including remote sensing, microscopy, and medical imaging. The obtained image quality is often superior to approaches that rely on union of subspaces (UoS) priors, which are either hand-crafted (e.g. wavelet sparsity) or shallow-learned (e.g. dictionary learning). However, the theoretical understanding of deep-learning based reconstruction algorithms is lagging behind UoS schemes. The main objective of this thesis is to introduce a union of surfaces framework, which models high-dimensional data as points on a union of low-dimensional surfaces or manifolds. We will develop novel sampling theoretic results and algorithmic tools for the learning of signals and functions on surfaces. The computational structure of these methods bear remarkable similarity to deep learning architectures, while being more amenable to theoretical analysis.
We first investigate the learning of a Union of Surfaces model from noisy and incomplete data. We develop novel algorithms with sampling guarantees to learn a union of surfaces representation from (a) few fully training samples, (b) and few partially observed samples. We introduce bounds on the approximation error in representing an arbitrary surface, and introduce algorithms that can learn the model from noisy and missing data.
Next, we consider the learning of multidimensional functions on Union of Surfaces. We introduce a novel framework for the representation of multidimensional functions, when the domain is restricted to union of surfaces. Unlike conventional reproducing kernel Hilbert space setting, the representation only involves a sparse combination of training samples, whose number depends on the surface complexity. Motivated by deep architectures, we will investigate the factorization of complex functions into simpler ones, thus facilitating their compact representation and efficient learning.
Finally, we develop reconstruction algorithms, which combine the learned generative models as priors with the imaging physics. The framework is then used to enable free breathing and ungated multicontrast cardiac MRI reconstructions from highly undersampled measurements.
Details
- Title: Subtitle
- Sampling and recovery on parametric manifolds
- Creators
- Qing Zou
- Contributors
- Mathews Jacob (Advisor)Xueyu Zhu (Committee Member)Weiyu Xu (Committee Member)Sanvesh Srivastava (Committee Member)Prashant Nagpal (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Applied Mathematical and Computational Sciences
- Date degree season
- Spring 2021
- DOI
- 10.17077/etd.006077
- Publisher
- University of Iowa
- Number of pages
- xxiv, 160 pages
- Copyright
- Copyright 2021 Qing Zou
- Comment
- This thesis has been optimized for improved web viewing. If you require the original version, contact the University Archives at the University of Iowa: https://www.lib.uiowa.edu/sc/contact/
- Language
- English
- Description illustrations
- illustrations (some color)
- Description bibliographic
- Includes bibliographical references (pages 147-160).
- Public Abstract (ETD)
The last decade has witnessed extensive research on computational imaging, where faster and cheaper acquisition methods are combined with computational algorithms to dramatically improve spatial and temporal resolution of images, while reducing the cost of scan and hardware. The standard practice is to pose the recovery as an optimization problem, where the cost function is sum of a data consistency term and an image prior. Hand-crafted (e.g. sparsity in the wavelet domain) and shallow-learning (e.g. dictionary learning, low-rank methods) priors have been extensively studied. Most of these methods rely on a union of subspaces signal representation, which assumes that the signal can be represented by the linear combination of vectors from union of subspaces. These approaches are now well-understood with theoretical guarantees, fast algorithms, and commercial products in several areas, including magnetic resonance imaging (MRI). However, recent empirical studies have shown the significantly superior performance of non-linear deep architectures for recovery and inference in a wide range of application areas. The improved performance of these algorithms over union of subspaces schemes may be attributed to their ability of the associated prior terms to exploit the complex non-linear redundancies in the data. Unfortunately, current deep architectures do not enjoy a sound theoretical understanding compared to union of subspaces methods.
In this thesis, we develop a continuous-domain framework to exploit the nonlinear dependencies in high-dimensional image data, with the focus of using it in computational imaging applications. We model the data as points on a union of surfaces. The focus of this thesis is to bridge the gap between well-understood union of subspaces (compressed-sensing) frameworks and deep learning methods that offer great empirical performance. The proposed framework is then extended to yield a more efficient and theoretically-founded framework for MRI data with non-linear structure.
- Academic Unit
- Interdisciplinary Graduate Program in Applied Mathematical & Computational Sciences
- Record Identifier
- 9984097368402771
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