Inverse Problems in Hyperspectral Imaging Open Access

Abstract

In hyperpsectral imaging, multiple images of the same scene are obtained over a contiguous range of wavelengths in the electromagnetic spectrum. Hyperspectral images represent observations of a scene at many different wavelengths and most importantly associate to each pixel in the imaged scene a full spectral vector or spectral signature. However, due to the presence of spectral mixtures (at different scales) in the scene and/or low spatial resolution of the hyperspectral sensor, the acquired spectral vectors of each pixel are actually a mixture of the spectra of the various materials present in the spatial coverage area of the corresponding pixel, and they also contain additional degradations caused by atmospheric blurring.We present a numerical approach for deblurring and sparse unmixing of space objects taken by ground based telescopes. A major challenge for deblurring hyperspectral images is that of estimating the overall blurring operator, taking into account the fact that the blurring operator point spread function (PSF) can be wavelength dependent and depend on the imaging system as well as the effects of atmospheric turbulence. We formulate the PSF estimation as a nonlinear least squares problem, which is solved using a variable projection Gauss-Newton method. Our analysis shows that the Jacobian can be potentially very ill-conditioned. To deal with this ill-conditioning, we use a combination of subset selection and regularization. We then incorporate the PSF estimation scheme with a preconditioned alternating direction method of multipliers to solve the deblurring and sparse unmixing problem. Experimental results illustrate the effectiveness of the resulting numerical schemes.

Table of Contents

List of Figures xi

List of Tables xiv

1 Introduction 2

1. 1.1 Inverse Problems ............................. 5

   1. 1.1.1 Regularization........................... 7

      1. 1.1.1.1 SVD Analysis...................... 8

      2. 1.1.1.2 Regularization by Filtering .............. 9

      3. 1.1.1.3 Spectral Value Decomposition............. 10

      4. 1.1.1.4 Variational Regularization............... 11

      5. 1.1.1.5 Iterative Regularization ................ 12

      6. 1.1.1.6 Regularization Parameter ............... 13

   2. 1.1.2 Separable Nonlinear Inverse Problems. . . . . . . . . . . . . . 14

      1.1.2.1 Solution Methods.................... 14

   3. 1.1.3 Preconditioning.......................... 20

      1.1.3.1 Considerations for Ill-posed Inverse Problems . . . . 21

2. 1.2 Outline of Work.............................. 23

3. 1.3 Contributions ............................... 24

2 Hyperspectral Imaging 26

2.1 Electromagnetic Spectrum ........................ 27

2.2 Structure of HSI Data .......................... 29

2.3 Measuring HSI Data ........................... 31

2.4 Spectral Reflectance ........................... 33

2.5 Mixed Spectra............................... 36

2.6 Hyperspectral Mixing Models ...................... 38

3 Estimation of Hyperspectral PSF Parameters 41

1. 3.1 Mathematical Framework ........................ 42

   3.1.1 PSF Star Image Formation Model................ 42

   3.1.2 Circular Moffat .......................... 43

   3.1.3 Elliptical Moffat.......................... 45

3.2 Optimization Problem .......................... 47

3.2.1 Variable Projection........................ 50

3.2.2 Jacobian Matrix.......................... 52

3.2.3 Subset Selection.......................... 57

4 Deblurring and Sparse Unmixing of Hyperspectral Images using Multiple PSFs 59

1. 4.1 Numerical Scheme for the Single PSF Case . . . . . . . . . . . . . . . 61

2. 4.2 Numerical Scheme for the Multiple PSF Case . . . . . . . . . . . . . 64

3. 4.3 Conjugate Gradient Preconditioner ................... 67

4.3.1 Approximation Quality of the Preconditioner . . . . . . . . . . 68

5 Numerical Results 70

5.1 Hyperspectral PSF Parameter Estimation Results . . . . . . . . . . . 70

1. 5.1.1 CircularMoffatResults...................... 72

2. 5.1.2 EllipticalMoffatModel...................... 75

3. 5.1.3 Hyperspectral Unmixing and Deblurring Results . . . . . . . . 79

6 Conclusions and Future Work 90

Bibliography 93

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics Computer Science
Keyword	deblurring psf parameter estimation alternating direction methods linear spectral unmixing hyperspectral imaging subset selection
Committee Chair / Thesis Advisor	Nagy, James, Emory University
Committee Members	Veneziani, Alessandro, Emory University Plemmons, Robert, Wake Forest University Ettinger, Bree, Emory University

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