Image restoration with MMSE nonlocal means filtering

다국어 초록 (Multilingual Abstract)

The advancement of digital imaging technology enables a variety of imaging devices to capture images of high resolutions. As the resolution of an image increases, the amount of photons that enter the unit area of the sensor decreases and should be amplified with a larger gain. This amplification process increases noise components as well, degrading the image quality. However, the performances of image processing applications, such as image compression, stereo matching, and object segmentation, are affected by the fidelity of an input image. Therefore, denoising, which is the procedure to recover the original image that is degraded by noise components, has been an important research topic in image processing communities. Furthermore, although there are various noise sources with different characteristics, noise components in captured images can be well approximated by the Gaussian noise model or the Poisson noise model according to light conditions. Therefore, it is essential to take these noise characteristics into account in the denoising procedure.

In this dissertation, we propose two nonlocal minimum mean square error (MMSE) image denoising algorithms for the Gaussian noise model and the Poisson noise model, respectively.

For the Gaussian noise model, based on the Bayesian estimation theory, we first derive that the conventional nonlocal means filter is an MMSE estimator in the special case of noise-free nonlocal neighbors. Then, we develop the nonlocal MMSE denoising filter that can minimize the mean square error (MSE) of a denoised block in more general cases of noisy nonlocal neighbors. Note that, as we search a wider region for nonlocal neighbors, we can increase the probability to find more similar blocks and improve the denoising performance. Therefore, the proposed algorithm selects nonlocal neighbors from the entire input image and also optionally from an external database, while discarding dissimilar neighbors. A wider search range, however, demands higher computational complexity. To reduce the computational burden, we propose a probabilistic tree search method, which can identify similar nonlocal neighbors efficiently by probabilistically traversing a binary tree that contains candidate neighbors. Simulation results demonstrate that the proposed algorithm provides significantly better denoising performance than the conventional nonlocal means filter. Specifically, even when the external database is not used, the proposed algorithm provides 0.22$~1.31 dB higher PSNR's on the test images from the VisTex database, the USC-SIPI image database, and the Berkeley segmentation dataset. When the external database is used, the PSNR is further improved by the maximum of 0.31 dB and 0.07 dB on average.

We also propose another nonlocal means algorithm for the Poisson noise model by extending the nonlocal MMSE denoising filter for the Gaussian noise model. First, we derive two distance measures, which indicate how dissimilar two blocks are from each other in the presence of Poisson noise. Then, similarly to the case for Gaussian noise, we develop the MMSE nonlocal means denoising filter that can minimize the MSE of a denoised block based on the Bayesian estimation theory. However, we notice that it is intractable to obtain the optimal MMSE nonlocal means filter in practice due to its high computational complexity. Therefore, based on the observation on the statistical properties of general images, we provide a practical approximation of the MMSE nonlocal means filter for fast implementation. Simulation results show that the proposed algorithm provides significantly better denoising performance than the conventional nonlocal means filter and its recent extensions for Poisson noise removal. More specifically, the proposed algorithm provides 0.21~1.09 dB higher PSNR's on the test images from the USC-SIPI image database.

목차 (Table of Contents)

• Abstract i
• Contents iv
• List of Figures vii
• List of Tables ix
• Abbreviations x
• 1 Introduction 1
• 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 1
• 1.2 Organization of the Dissertation . . . . . . . . . . . . . . 6
• 2 Related Work and Motivations 8
• 2.1 Nonlocal Means Filtering . . . . . . . . . . . . . . . . . . 8
• 2.2 MMSE Denoising . . . . . . . . . . . . . . . . . . . . . . . 10
• 2.3 Fast Algorithms for Nonlocal Means Filtering . . . . . . . . 11
• 2.4 Denoising Using Multiple Images . . . . . . . . . . . . . . 12
• 2.5 Denoising Algorithms for Poisson Noise Removal . . . . . . . 14
• 3 MMSE Nonlocal Means Denoising Filter 16
• 3.1 Theoretical Derivation of the Filter . . . . . . . . . . . . 16
• 3.1.1 Block Generation Model . . . . . . . . . . . . . . . . 17
• 3.1.2 MMSE Filtering . . . . . . . . . . . . . . . . . . . . 20
• 3.1.3 Case I: Noise-Free Nonlocal Neighbors . . . . . . . . 24
• 3.1.4 Case II: Noisy Nonlocal Neighbors . . . . . . . . . . 24
• 3.2 Filter Implementation . . . . . . . . . . . . . . . . . . . 27
• 3.2.1 Mean-Removed Processing . . . . . . . . . . . . . . . 28
• 3.2.2 Binary Tree Construction . . . . . . . . . . . . . . . 28
• 3.2.3 Probabilistic Tree Search . . . . . . . . . . . . . . 29
• 3.2.4 Overlapping Block Filtering . . . . . . . . . . . . . 33
• 3.3 Experimental Results . . . . . . . . . . . . . . . . . . . . 34
• 3.3.1 Synthetic Noise . . . . . . . . . . . . . . . . . . . 37
• 3.3.2 Captured Noise . . . . . . . . . . . . . . . . . . . . 48
• 3.3.3 Computational Complexity . . . . . . . . . . . . . . . 49
• 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 52
• 4 MMSE Nonlocal Means Filter for Poisson Noise Removal 55
• 4.1 Preliminary Work . . . . . . . . . . . . . . . . . . . . . . 56
• 4.1.1 Nonlocal Means Denoising . . . . . . . . . . . . . . . 56
• 4.1.2 Distance Measures Under Poisson Noise Model . . . . . 57
• 4.2 Proposed MMSE Denoising Filter for Poisson Noise . . . . . . 58
• 4.2.1 Block Generation Model . . . . . . . . . . . . . . . . 58
• 4.2.2 MMSE Filtering . . . . . . . . . . . . . . . . . . . . 60
• 4.2.3 Approximation for Practical Implementation . . . . . . 61
• 4.3 Experimental Results . . . . . . . . . . . . . . . . . . . . 64
• 4.3.1 Denoising Performance . . . . . . . . . . . . . . . . 66
• 4.3.2 Computational Complexity . . . . . . . . . . . . . . . 75
• 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 75
• 5 Conclusions 77
• Appendix A Mathematical Derivations 80
• A.1 Derivation of ({\bf V}^T{\bf V}+d\sigma_n^2{\bf H}_M)^{-1} in (3.18) . 80
• A.2 Derivation of Formulae in (3.20) and (3.21) . . . . . . . . 82
• Bibliography 83
• Acknowledgments 93