Pixels, Priors, and Precision: Bayesian Approaches to Image Reconstruction Öffentlichkeit

Abstract

With a growing reliance on data analytics to draw insights and conclusions from real-world phenomena, ensuring accuracy no longer suffices --- computational methods are essential for handling large-scale datasets and characterizing uncertainty, enabling reconstructions that are not only precise but also trustworthy. This thesis seeks to examine the existing regularization methods, Bayesian frameworks, and computational tools used to stabilize the ill-posedness of inverse problems, in the context of seismic imaging. Specifically, methods for spectral filtering in a Tikhonov formulation (e.g. General Cross Validation minimization), prior specification, and Markov Chain Monte Carlo methods for exploring complex non-Gaussian distributions are applied to synthetic seismic and tomographic data, with the goal of gauging fidelity, reliability, and computational feasibility. Numerical experiments demonstrate that these approaches --- especially when combined with iterative solvers --- succeed in mitigating noise and recovering high-fidelity reconstructions, while remaining computationally feasible. Furthermore, the comparison of hierarchical sampling schemes and fixed parameter methods reveals how hyperparameter inference can refine the solution space, yielding significantly better estimates at the cost of computational complexity.

Table of Contents

1 Introduction 1

1.1 Forward and Inverse Problems.......................... 1 1.2 Applications of Inverse Problems ........................ 4 1.3 Computational Challenges to Inverse Problems. . . . . . . . . . . . . . . . . 6

2 Computational Inverse Problems 10

2.1 Regularization Methods ............................. 10 2.1.1 Regularization Parameter Selection Methods . . . . . . . . . . . . . 14 2.2 Bayesian Framework Formalization ....................... 15 2.2.1 Different Priors .............................. 17 2.2.2 Conjugate Gradient............................ 22

3 Computational Methods for UQ and Sampling 24

3.1 Randomized GCV Curve Estimation ...................... 25 3.2 MAP Estimation ................................. 28 3.3 Edge Preserving Prior .............................. 29 3.4 Fixed Parameter Sampling............................ 31 3.5 Hierarchical Gibbs ................................ 32

4 Numerical Experiments 36

4.1 Experiment #1: Regularization Parameter Selection . . . . . . . . . . . . . . 37 4.1.1 GCV: A Deeper Dive........................... 40 4.2 Experiment #2: Different Priors......................... 43 4.3 Experiment #3: MCMC Methods........................ 48 4.3.1 Computational Complexity........................ 53

5 Conclusions and Future Work 55

5.1 Future Work.................................... 56 5.1.1 Real-World Data and Varying Sensor Placement . . . . . . . . . . . . 57 5.1.2 Preconditioning and Other Iterative Methods . . . . . . . . . . . . . . 57 5.1.3 Boundary Conditions and Priors..................... 58

Bibliography ....59 

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Stichwort	Image Reconstruction Bayesian Methods
Committee Chair / Thesis Advisor	Julianne Chung, Emory University
Committee Members	Levon Nurbekyan, Emory University Elizabeth Newman, Emory University Kevin McAlister, Emory University

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