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Inverse Problems in Image Processing: Blind Image Restoration University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2005 Inverse Problems in Image Processing: Blind Image Restoration Ravi Viswanathan University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Engineering Commons Recommended Citation Viswanathan, Ravi, "Inverse Problems in Image Processing: Blind Image Restoration. " Master's Thesis, University of Tennessee, 2005. https://trace.tennessee.edu/utk_gradthes/2546 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Ravi Viswanathan entitled "Inverse Problems in Image Processing: Blind Image Restoration." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Engineering Science. Mohamed R. Mahfouz, Major Professor We have read this thesis and recommend its acceptance: Anthony E. English, Aly Fathy Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a thesis written by Ravi Viswanathan entitled "Inverse Problems in Image Processing: Blind Image Restoration." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Master of Science, with a major in Engineering Science. Mohamed R. Mahfouz__ Major Professor We have read this thesis and recommend its acceptance: Anthony E. English_______________ Aly Fathy ______________________ Accepted for the Council: Anne Mayhew___________ Vice Chancellor and Dean of Graduate Studies (Original signatures are on file with official student records.) INVERSE PROBLEMS IN IMAGE PROCESSING: BLIND IMAGE RESTORATION A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville Ravi Viswanathan August 2005 Acknowledgements I am indebted to my energetic advisor Prof. Mohamed R. Mahfouz for constantly encouraging me in my research focus and making my UT experience truly enlightening. I am also grateful to my other thesis committee members Prof. Anthony E. English and Prof. Aly Fathy for stimulating discussions and valuable comments. Thanks also go to UT faculty and students whose creative energy and ideas helped discover the creative self in me. Many thanks to my group of friends who have helped keep my spirits high. Thanks mom, dad and Jai. You guys were my prime source of strength. ii Abstract Blind Image Restoration pertains to the estimation of degradation in an image, without any prior knowledge of the degradation system, and using this estimation to help restore the original image. Original Image, in this case, refers to that version of the image before it experienced degradation. In this thesis, after estimating the degradation system in the form of Gaussian blur and noise, we employ Deconvolution to help restore the original image. In this thesis, we use a Redundant Wavelet based technique to estimate blur in the image using high-frequency information in the image itself. Lipschitz exponent – a measure of local regularity of signals, is computed using the evolution of wavelet coefficients of singularities across scales. It has been shown before that this exponent is related to the blur in the image and we use it in this case to estimate the standard deviation of the Gaussian blur. The properties of wavelets enable us to compute the noise variance in the image. In this thesis, we employ two cases of deconvolution – A strictly Fourier domain Regularized Iterative Wiener filtering approach and A Fourier-Wavelet Cascaded approach with Regularized Iterative Wiener filtering - to compute an estimate of the image to be restored using the blur and noise variance information that was earlier computed. The estimated value of standard deviation of the blur helped obtain robust estimates with deconvolution. It can be observed from the results that Fourier domain Regularized Iterative Wiener filtering provides a more stable output estimate than the Iterative Filtering with Additive Correction methods, especially when the number of iterations employed is more. The Fourier-Wavelet Cascaded deconvolution seems to be iii image dependent with regards to performance although it outperforms the strictly Fourier domain deconvolution approach in some cases, as can be gauged from the visual quality and Mean Squared Error. iv Contents 1 Introduction 1 1.1 Blind Image Restoration………………………………………………………..1 1.2 Image Deconvolution: Biomedical Applications……………………………….3 2 PSF Estimation 4 2.1 Blur – A Mathematical Representation……………………………….………..4 2.2 Principle for Blur Estimation…………………………………………………...5 2.3 The Wavelet Transform………………………………………………………...7 2.3.1 Why Wavelets…………………………………………………………….9 2.3.2 The Discrete Wavelet Transform………………………………………...10 2.3.4 Economy of Wavelet Representations…………………………………...12 2.4 Mallat’s Relationships – The Lipschitz Exponent…...…………………………12 2.4.1 Definition I……………………………………………………………….13 2.4.2 Definition II……………………………………………………………...14 2.4.3 Vanishing Moments……………………………………………………...15 2.5 Singularity Detection…………………………………………………………...15 2.6 Wavelet Transform Modulus Maxima………………………………………….16 2.7 Modulus Maxima and Lipschitz Exponent – The Relationship………..……….16 2.8 Computation of Lipschitz Exponent…………………………………...……….17 2.9 Blur and Lipschitz Exponent – The Relationship…………………………...….20 2.10 Computation of Constants…………………………………………….……….22 2.11 Implementation Summary……………………………………………………..26 v 3 Fourier Domain Deconvolution 29 3.1 Problem Statement…………………………………………………………......29 3.2 Computation of Noise Variance…………………………………………….....30 3.3 Linear Time Invariant Wiener Filtering………………………………………..30 3.4 Regularized Deconvolution………………………………………….………...32 3.4.1 Choice of Shrinkage Term……………………………………………….33 3.5 Fourier Domain Filtering…………………………………………….………...33 3.5.1 Inverse Filtering ………………………………………………………....34 3.5.2 Wiener Filtering ………………………………………………………....34 3.5.3 Regularized Fourier Domain Wiener Filtering……………………….….35 3.5.4 Iterative Wiener Filtering…………………………………………….….37 3.5.5 Iterative Wiener Filtering with Additive Correction…………………….40 4 Fourier Wavelet Cascaded Deconvolution 44 4.1 Deconvolution – A Cascaded Approach…...………………………………....44 4.2 Experiment Outline…………………………………………………………...45 4.3 The Experiment…………………………………………………….………....45 4.3.1 Noise Computation…………………………………………………......46 4.3.2 Initial Experimental Set-up………………………………………….....46 4.3.3 Fourier Domain Shrinkage……………………………………………..48 4.3.4 Need for Wavelet Domain Shrinkage…………………………………..48 4.3.5 Wavelet Domain Set-up………………………………………………...49 4.3.6 Wavelet Domain Deconvolution………………………………………..49 4.3.6.1 Hard Thresholding……………………………………………...50 4.3.6.2 Soft Thresholding………………………………………….…...51 4.3.6.3 Donoho Approach………………………………………….…...51 4.3.6.4 Multi-resolution Wiener Filtering………………………….…...52 4.3.7 Wiener Wavelet Filtering……………………………………………......53 vi 4.3.8 Final Filtering Mechanism………………………………………….….54 4.4 Technical Summary…………………………………………………………..57 5 Results and Conclusion 60 5.1 PSF Estimation…………………………………………………………….60 5.2 Deconvolution Results…………………………………………………….65 5.2.1 Fourier Domain Wiener Filtering…………………………………..65 5.2.2 Fourier Wavelet Cascaded Deconvolution………………………....68 5.3 Deconvolution on the Test Image………………………………………....70 5.4 Conclusion……………………………………………………….………..74 Bibliography 79 Vita 83 vii List of Tables 5.1 Estimated values of standard deviation for training images………………………63 5.2 Standard Deviation values of applied blur and estimated blur for test image…….65 viii List of Figures 2.1 Degradation of an image by LTI system followed by additive noise……………..4 2.2 Evolution of wavelet coefficients across scales…………………………………...6 2.3 Experimental flow………………………………………………………………....7 2.4 Signal analysis in different domains……………………………………………...8 2.5 Discrete Wavelet Transform on an image………………………………………..12 2.6 Fitting at dyadic powers………………………………………………………….18 2.7 Exponential fitting curve…………………………………………………………22 2.8 Double Exponential fitting curve………………………………………………...24 2.9 Image Restoration results for training image with different blur estimates……...25 2.10 Image Restoration results for test image with different blur estimates………....27 3.1 Image degradation process….…………………………………………………....33 3.2 Trivial Wiener filtering results…………………………………………………...36 3.3 Iterative Wiener filtering estimates………………………………………………43 4.1 Fourier-Wavelet Cascaded Deconvolution model………………………………..44 4.2: Corrupting a zero intensity image with Gaussian noise…………………………50 4.3 Fourier and Fourier-Wavelet Cascaded filtering estimates………………………58 5.1 Applied blur vs. estimated blur for training images using exp1 fitting …....….....61 5.2 Applied blur vs. estimated blur for training images using exp2 fitting …..…..….62 5.3 Actual Blur vs. Estimated Blur for the test image………………………………..64 5.4 Fourier Domain Iterative Wiener Filtering for training image …………………..66 5.5 RMSE between original image and
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