
Image Denoising of Gaussian and Poisson Noise Based on Wavelet Thresholding A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY (Ph.D.) in the School of Electronic and Computing Systems University of Cincinnati Cincinnati OH 45221 USA 2013 by Jin Quan Bachelor of Science in Engineering, Tongji University Shanghai, China, 2007 Committee chair: Dr. William G. Wee Abstract Noise on images is generally undesirable and disturbing. It always plays a negative role on higher level processing tasks such as image registration and segmentation. Thus,image denoising becomes a fundamental step necessarily required for better im- age understanding and interpretation. During the last couple of years, wavelet has been extensively employed in the application of suppressing noise and proven to be a successful tool which outperforms many conventional denoising filters due to its pre- ferred properties. Therefore, in this dissertation, the wavelet transform is applied to develop our denoising strategy. Basically, two generic scenarios occur during the acquisition of images. First, when the detected intensities on the image are sufficiently high, the noise can be suitably modeled as following an additive independent Gaussian distribution. Second, when only a few photons are detected, this observed image is usually modeled as a Poisson process and the intensities to be estimated are assumed to be the underlying Poisson parameters. In this dissertation, these two scenarios are discussed respectively in Part IandPartII. In part I, we consider to reduce the typical additive white Gaussian noise (AWGN). Our driving principle is to decrease the upper bound of the error restricted by the soft-thresholding strategy between the investigated image and noise-free image. Thus we develop a new context modeling method to group coefficients with similar statis- ii tics and construct a smoothed version of the noisy image prior to the actual denoising operation. Then, we propose an optimized soft-thresholding denoising function with parameters derived from a modification of a closed form solution which has a more flexible shape and is adaptively pointwise. Furthermore, we extend it to its overcom- plete representation by employing the “cycle spinning” method so that the property of shift invariance is achieved which leads to a boost of the denoising performance. By combining these strategies, the denoising results in our experiments confirm that the approach is very competitive to some state-of-the-art denoising methods in terms of quantitative measurements and computational simplicity. In Part II, a new denoising method for Poisson noise corrupted images is proposed which is based on the variance stabilizing transformation (VST) with a new inverse. The VST is used to approximately convert the Poisson noisy image into Gaussian distributed, so that the denoising methods aiming at Gaussian noise can be applied subsequently. The motivation for the improved inverse comes from a main drawback existing in the conventional VSTs such as the Anscombe transformation: its efficiency degrades significantly when the pixel intensities of the observed images are very low due to the biased errors generated by its inverse transformation. In order to correct the biased errors, we introduce a polynomial regression model based on weighted least squares as an alternate to its inverse. Moreover, we incorporate our developed wavelet thresholding strategy for Gaussian noise presented in Part I into the proposed method. We also extend it to the overcomplete representation to suppress the Pseudo-Gibbs phenomena and therefore gains additional denoising effects. Experimental analysis indicates that this method is very competitive. iii iv Acknowledgments I would like to express my deepest gratitude and most sincere appreciation to my advisor Dr. William G. Wee for all his continuous guidance, constant encouragement and precious support during my stay at the University of Cincinnati. In every sense, he was always a fruitful source of inspiration from which I tremendously benefited. I am also very grateful to the members of the dissertation committee: Dr. Chia Y. Han, Dr. Xuefu Zhou, Dr. Raj Bhatnagar and Dr. Ali Minai for kindly sharing their scientific knowledge, devoting priceless time and providing constructive advice and comments on my research. I learned a lot from enthusiastic discussions with them. In addition, many thanks go to the former and current members of the Multimedia and Augmented Reality lab at the University of Cincinnati for their help, kindness and valuable suggestions. Most of all, I owe my warmest thanks to my father Dr. Shuhai Quan and my mother Mrs. Yulan Ruan. Nothing could have been possible without their everlasting love, understanding and support. v Table of Contents Abstract ii Acknowledgments v List of Figures ix List of Tables xiv 1 Introduction 1 1.1ImageDenoising.............................. 1 1.2ProblemStatement............................. 2 1.3 Research Scope ............................... 4 1.4Contributions................................ 5 1.5DissertationOrganization......................... 6 Part I Image Denoising for Additive Gaussian Noise 9 2 Background 10 2.1 Additive Gaussian Noise Model ...................... 10 2.2ImageQualityEvaluation......................... 11 2.2.1 Objective Image Quality Evaluation ................ 11 vi 2.2.2 Subjective Image Quality Evaluation ............... 13 2.3 Summary .................................. 14 3 Literature Review 15 3.1SpatialDomainApproaches........................ 15 3.1.1 LinearFilters............................ 16 3.1.2 Nonlinear Filters .......................... 17 3.2TransformedDomainApproaches..................... 20 3.2.1 FourierTransformDenoising.................... 20 3.2.2 WaveletTransformDenoising................... 21 3.2.3 Data-AdaptiveTransformDenoising............... 50 3.3OtherAlternativeApproaches....................... 51 3.4 Summary .................................. 53 4 The Proposed Gaussian Denoising Method: CWMT 55 4.1Motivation.................................. 55 4.2Overview................................... 58 4.3 Denoising Operation 1–Improved Context Modeling ........... 59 4.4ExperimentalResultsofDenoisingOperation1 ............. 63 4.5 Denoising Operation 2–The Optimization of the Soft Thresholding Func- tion..................................... 66 4.6ExperimentalResultsofDenoisingOperation2 ............. 69 4.7 Experimental Results of Combining Two Denoising Operations ..... 71 4.8TheStepsoftheDenoisingProposedMethod:CMWT......... 75 4.9 Summary .................................. 79 5 Expansion to the Overcomplete Representation 81 5.1 Overview of Applying Overcomplete Expansion ............. 81 vii 5.2 Overcomplete Expansion Procedure .................... 83 5.3 Experimental Results for Overcomplete Expansion ............ 83 5.4 Summary .................................. 91 Part II Image Denoising for Poisson Noise 92 6 Background on Poisson Denoising 93 6.1 Modeling of Low Intensity Images ..................... 93 6.2PoissonNoiseModel............................ 94 6.3RelatedWork................................ 95 6.3.1 Variance Stabilization ....................... 95 6.3.2 HypothesisTesting......................... 98 6.3.3 WaveletFiltering.......................... 99 6.3.4 BayesianBasedApproach..................... 100 6.4AnscombeTransformationandItsInversions............... 101 6.5 Summary .................................. 104 7 The Proposed Poisson Denoising Method: CMWT-IAT 105 7.1InvestigationoftheBiasedErrors..................... 106 7.2NewInverseTransformationforAnscombeTransformation....... 108 7.3CombinationoftheDenoisingMethodCMWTforAWGN....... 112 7.4 Summary .................................. 113 8 Performance Evaluation 114 8.1ComparisonswithTwoConventionalInversions............. 115 8.2ComparisonswithSURE-LETUsingtheProposedInversion...... 115 8.3ComparisonswithState-of-the-ArtDenoisingMethods......... 117 8.4 Summary .................................. 121 viii 9 Conclusion and Perspectives 125 9.1Conclusion.................................. 125 9.2Perspectives................................. 126 Bibliography 130 ix List of Figures 1.1 (a) A noise-free Image Pepper,(b)Anoisyversionofit......... 2 3.1 Original Image Lena anditsFourierdecomposition........... 21 3.2Onescaleofwaveletdecomposition.................... 22 3.3Somefamouswavelets........................... 23 3.4 Original Image Lena and its first level decomposition by using db4 wavelet 24 3.5 Hard-thresholding function ......................... 29 3.6 Soft-thresholding function ......................... 30 3.7 Semisoft-thresholding function ....................... 31 4.1Awaveletdenoisingflowchart....................... 57 4.2 The parent-child relationship of a three level wavelet decomposition. 60 4.3 Subband designations ............................ 61 4.4 Six standard testing images used on our experiments (a) Lena,(b)Boat, (c) Goldhill,(d)Barbara,(e)Couple,(f)Man. .............. 64 4.5 Subbands of the 2D orthogonal wavelet transform ............ 65 4.6 Sensitivity of the denoising function with respect to variations of T on Lena ..................................... 69 4.7 Sensitivity of the denoising function with respect to variations of T on Boat ..................................... 70 x 4.8 Visual comparison between original soft-thresholding and the optimized soft-thresholding functions
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