Mathematical Models and Numerical Methods for Wavefront Reconstruction Public
Chu, Qing (2013)
Published
Abstract
Obtaining high resolution images of space objects from ground based telescopes is challenging, and often requires computational post processing methods to remove blur caused by atmospheric turbulence. In order for an image deblurring (deconvolution) algorithm to be effective, it is important to have a good approximation of the blurring operator. In space imaging, the blurring operator is defined in terms of the wavefront of light, and how it is distorted as it propagates through the atmosphere.
In this thesis we consider new mathematical models and algorithms to reconstruct the wavefront, which requires solving a large scale ill-posed inverse problem. We show that by exploiting and fusing information from multiple measurements, we are able to obtain better reconstructed wavefronts than existing methods. In addition, to fulfill the large scale requirement for astronomical uses, we present results of a parallel implementation utilizing the Trilinos project, a mathematical software library for solving problems from many academic and research fields.
Moreover, we study an symmetric successive over-relaxation (SSOR) preconditioner for this image reconstruction problem. Numerical results for different image reconstruction systems under variety of seeing conditions indicate good behavior of the SSOR preconditioner with respect to iteration numbers and computational time.
Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Linear Least Squares Formulation . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Ill-Posedness and Regularization of Inverse Problems . . . . . . . . . . . . .3
1.1.3 Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Outline of the Work . . . . . . . . . . . . . . . . . . . . . . . 12
2 Image Restoration in Astronomy . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 The Effect of Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 The Blurring Kernel Model . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Diffraction Limited Imaging Model . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Wavefront Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Frozen Flow Hypothesis and the Linear Formulation . . . . . . . . . . . . . . . . . . . . .22
2.3.1 Frozen Flow Hypothesis . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Linear Model of the Wavefront Motion . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 Wavefront Motion in the Multi-Layered Assumption . . . . . . . . . . . . . . . . . . . .29
2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Parameter Settings . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Experiment 1: Good Seeing Conditions . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.3 Experiment 2: Poor Seeing Conditions . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.4 Experiment 3: Extremely Poor Seeing Conditions . . . . . . . . . . . . . . . . . . . . .41
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 45
3 Parallel Implementation . . . . . . . . . . . . . . . . . . . . . . . 48
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Overview of Trilinos . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Epetra . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.2 EpetraExt . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.3 Belos . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.4 Teuchos . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Detailed Implementation and Results . . . . . . . . . . . . . . . . . . . . . . . 53
4 Preconditioning . . . . . . . . . . . . . . . . . . . . . . . 58
4.1 Preconditioning LS systems . . . . . . . . . . . . . . . . . . . . . . . 58
4.2 General Techniques for Choosing Preconditioners . . . . . . . . . . . . . . . . . . . . . .60
4.2.1 Classical Iterative Scheme as Preconditioners . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.2 Incomplete Factorizations . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.3 Approximate inverse . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 SSOR Preconditioner . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Other Preconditioning Techniques . . . . . . . . . . . . . . . . . . . . . . . 74
5 Wavefront Screen Simulation . . . . . . . . . . . . . . . . . . . . . . . 77
5.1 Optics Models . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.1 Sinlge Layer Wavefront Screen Generation . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.2 Multi-Layered Wavefront Screen Generation . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Remarks and Future Directions . . . . . . . . . . . . . . . . . . . . . . . 86
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . 92
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