Efficient Solvers for Nonlinear Problems in Imaging Open Access

Abstract

Nonlinear inverse problems arise in numerous imaging applications, and solving them is often difficult due to ill-posedness and high computational cost. In this work, we introduce tailored solvers for several nonlinear inverse problems in imaging within a Gauss-Newton optimization framework.

We develop a linearize and project (LAP) method for a class of nonlinear problems with two (or more) sets of coupled variables. At each iteration of the Gauss-Newton optimization, LAP linearizes the residual around the current iterate, eliminates one block of variables via a projection, and solves the resulting reduced dimensional problem for the Gauss-Newton step. The method is best suited for problems where the subproblem associated with one set of variables is comparatively well-posed or easy to solve. LAP supports iterative, direct, and hybrid regularization and supports element-wise bound constraints on all the blocks of variables. This offers various options for incorporating prior knowledge of a desired solution. We demonstrate the advantages of these characteristics with several numerical experiments. We test LAP for two and three dimensional problems in super-resolution and MRI motion correction, two separable nonlinear least-squares problems that are linear in one block of variables and nonlinear in the other. We also use LAP for image registration subject to local rigidity constraints, a problem that is nonlinear in all sets of variables. These two classes of problems demonstrate the utility and flexibility of the LAP method. 

We also implement an efficient Gauss-Newton optimization scheme for the problem of phase recovery in bispectral imaging, a univariate nonlinear inverse problem. Using a fixed approximate Hessian, matrix-reordering, and stored matrix factors, we accelerate the Gauss--Newton step solve, resulting in a second-order optimization method which outperforms first-order methods in terms of cost per iteration and solution quality. 

Table of Contents

1 Introduction

2 Background

2.1 Nonlinear Inverse Problems

2.2 Gauss-Newton Method

2.2.1 Gauss-Newton Step

2.2.2 Armijo Line Search

2.2.3 Stopping Criteria

2.2.4 Gauss-Newton Algorithm

2.2.5 Convergence

2.3 Projected Gauss-Newton

2.4 Regularization

2.4.1 Regularization for Nonlinear Problems

2.4.2 Regularization for the Gauss-Newton Step

2.5 Imaging

3 Linearize and Project Method for Coupled Problems

3.1 Coupled Nonlinear Inverse Problems

3.2 LAP Method

3.2.1 LAP with Projected Gauss-Newton

3.3 Alternatives Methods for Coupled Problems

3.3.1 Block Coordinate Descent for Coupled Problems

3.3.2 Variable Projection for Separable Least-squares

4 Motion Correction Problems

4.1 Motion Correction Imaging Problem

4.2 Super-Resolution

4.2.1 Two Dimensional Super-Resolution

4.2.2 Three Dimensional Super-Resolution

4.3 Motion Correction for Magnetic Resonance Imaging (MRI)

4.4 Discussion

5 Locally Rigid Image Registration

5.1 Locally Rigid Registration Problem

5.1.1 Eulerian and Lagrangian Frameworks

5.1.2 Hyperelastic Regularizer

5.1.3 Numerical Implementation

5.2 Numerical Results

5.2.1 2D Knee Registration

5.2.2 3D Block Registration

5.2.3 Discussion

6 Phase Recovery from the Bispectrum

6.1 Bispectral Imaging Problem

6.1.1 Calculating the Object Power Spectrum

6.1.2 Accumulating the Object Bispectrum

6.1.3 Phase Recovery Problem

6.2 Numerical Results

7 Conclusions and Future Work

Appendices

A.1 Linear Interpolation in Higher Dimensions

A.1.1 Bilinear Interpolation

A.1.2 Trilinear Interpolation

A.1.3 Bilinear and Trilinear Interpolation for Multiple Points

A.2 Rigid Transformations

A.2.1 Rigid Transformations in 2D

A.2.2 Rigid Transformations in 3D

Bibliography

 

About this Dissertation

Rights statement

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Applied Mathematics
Keyword	variable projection motion correction coupled problems VarPro super-resolution projection MRI projected Gauss-Newton LAP block coordinate descent linearize and project nonlinear optimization Gauss-Newton magnetic resonance imaging hybrid regularization registration imaging inverse problems
Committee Chair / Thesis Advisor	Ruthotto, Lars, Emory University
Committee Members	Kong, Jun, Emory University Nagy, James, Emory University

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