Parsimonious models for inverse problems

Pfister, Luke

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https://hdl.handle.net/2142/106309 Copy

Description

Title

Parsimonious models for inverse problems

Author(s)

Pfister, Luke

Issue Date

2019-08-20

Director of Research (if dissertation) or Advisor (if thesis)

Bresler, Yoram

Doctoral Committee Chair(s)

Bresler, Yoram

Committee Member(s)

• Bhargava , Rohit
• Dokmanic, Ivan
• Carney, P. Scott

Department of Study

Electrical & Computer Eng

Discipline

Electrical & Computer Engr

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Inverse problems
• adaptive filter banks
• sparsifying transforms
• inverse scattering
• tomography

Abstract

This dissertation can be coarsely divided into two parts: Chapters 1 and 2 study the problem of the multidimensional filter bank design and data-driven adaptation, while Chapters 3 to 5 focus on variations of optical tomography. Chapter 1 describes a fast way to estimate the extremal values of a trigonometric polynomial given samples from the polynomial. This work came about from a simple question: Can we determine whether the Discrete-Time Fourier Transform of a multidimensional discrete index signal reaches zero, given only its Discrete Fourier Transform? The answer is yes— provided that the signal has small support and its samples do not vary too much. This property unlocks new possibilities for the numerical design of multidimensional, multirate, perfect reconstruction filter banks; we conclude by designing a curvelet-like filter bank. Chapter 2 focuses on data-adaptive sparse representations; that is, a sparse representation learned directly from the data itself. These representations are usually described as modeling and acting on small image patches. We show that many of the existing sparse representations can instead be thought of as filter banks, thus linking the local properties of a patch-based model to the global properties of a convolutional model. We then use the results on trigonometric polynomials developed in Chapter 1 as the foundation for a new algorithm to learn perfect reconstruction filter banks that sparsify data. Our learned model outperforms local, patch- based transform learning approaches in image denoising tasks while benefiting from additional flexibility in the design process. Chapter 3 marks the transition to the second family of topics in this dissertation. In this chapter, we review a particular optical tomographic imaging: Interferometric Synthetic Aperture Microscopy (ISAM). ISAM allows for rapid, non-invasive imaging of quasi-transparent objects in three spatial dimensions from measurements of back-scattered light. In this modality, volumetric images are formed by solving the inverse scattering problem using perturbative methods. The resulting image reconstruction algorithms have efficient numerical implementations. The usual ISAM image reconstruction algorithms are well-suited for data collected from a single focal plane, with Tikhonov regularization, and/or if Gaussian noise is present. In these situations a non-iterative image reconstruction algorithm is applicable. However, when an iterative solution is required, the perturbative ISAM model leads to artifacts in the reconstructed image. In Chapter 4, we present a new approximation to the ISAM forward model. This model facilitates the combination of fast numerical algorithms and iterative image reconstruction. We construct the singular value decomposition of our new approximate ISAM operator and investigate the resolution of the imaging system. In Chapter 5, we combine ISAM with imaging spectroscopy to determine spatial morphology and chemical composition in three spatial dimensions. We assume the target has a low-rank structure; physically, this implies the target is composed of a few distinct chemical species. We call this the N-species approximation. We use this low-rank structure to reduce the amount of data needed to solve the inverse scattering problem.

Graduation Semester

2019-12

Type of Resource

text

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http://hdl.handle.net/2142/106309

Copyright and License Information

Copyright 2019 Luke Pfister

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