在影像清晰的條件底下，我們能從其中獲得大量資訊。但時常由於硬體的設定或人為地因
素，使得拍下照片變得模糊。由於無法清楚得知模糊的原因，因此在重建影像時需要同時估計模糊核。在只有一張模糊照片的情況下，同時重建影像及模糊核是個不適定性問題。雖然已經有許多演算法可以重建出相對清晰的影像，但這些方法裡時常忽略或是不清楚初始模糊核的估計。此篇論文探討利用 MAPk 的方法以廣義的狄拉克核為初始核來估計一個相對較接近真實的模糊核。我們會詳細推導其中的理論，並且應用於真實世界的影像上。

Blind deconvolution is the recovery of a sharp version of a blurred image when the blur kernel is unknown. This is an ill-posed problem since both true kernel with latent image and delta kernel with blur image can minimize the energy function. Therefore a good initial kernel that helps avoiding blur image with delta kernel for deblur process plays an essential role. Many research have already developed reasonable algorithms, but among them the initial blur kernel required for the algorithms are often unclear. This paper look into the MAPk algorithm based on Bayesian Inference, while the initial kernel of the process is always a delta kernel. The goal of this paper is to analyze and evaluate the blind image deconvolution method-MAPk both theoretically and experimentally. We derive the method mathematically and implement it on a real world images.

Contents
摘要 i
Abstract ii
List of Figures iv
1 Background 1
1.1 Image Model 1
1.2 Probability Assumptions 3
2 Fitting the Image Prior 6
2.1 The Expectation-Maximization Algorithm 6
2.2 Why it works 9
3 MAPx,k and MAPk 11
3.1 Maximum A Posteriori estimation 11
3.2 MAPx,k 12
3.3 MAPk 14
3.3.1 Difference between MAPk and MAP via loss function perspective 15
3.3.2 EM based algorithm of MAPk 17
4 Non blind deconvolution 23
4.1 Fast TV-l1 Deconvolution 24
5 Numerical Results 27
6 Conclusions 44
References 45

[1] C. M. Bishop. Pattern recognition. Machine learning, 128(9), 2006.
[2] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1):1–22, 1977.
[3] R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman. Removing camera shake from a single photograph. pages 787–794, 2006.
[4] D. J. Field. What is the goal of sensory coding? Neural computation, 6(4):559–601, 1994.
[5] R. C. Gonzalez and R. E. Woods. Digital image processing 4th edition, global edition. 2018.
[6] O. Kupyn, V. Budzan, M. Mykhailych, D. Mishkin, and J. Matas. Deblurgan: Blind motion deblurring using conditional adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 8183–8192, 2018.
[7] A. Levin, Y. Weiss, F. Durand, and W. T. Freeman. Understanding and evaluating blind deconvolution algorithms. pages 1964–1971, 2009.
[8] A. Levin, Y. Weiss, F. Durand, and W. T. Freeman. Effcient marginal likelihood optimization in blind deconvolution. pages 2657–2664, 2011.
[9] Y. Li, M. Tofghi, V. Monga, and Y. C. Eldar. An algorithm unrolling approach to deep image deblurring. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 7675–7679. IEEE, 2019.
[10] Y. Nan, Y. Quan, and H. Ji. Variational-em-based deep learning for noise-blind image deblurring. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3626–3635, 2020.
[11] Y. Wang, J. Yang, W. Yin, and Y. Zhang. A new alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences, 1(3):248–272, 2008.
[12] L. Xu and J. Jia. Two-phase kernel estimation for robust motion deblurring. pages 157–170, 2010.
[13] L. Xu, S. Zheng, and J. Jia. Unnatural l0 sparse representation for natural image deblurring. pages 1107–1114, 2013.
[14] J. Yang, Y. Zhang, and W. Yin. An effcient tvl1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM Journal on Scientifc Computing, 31(4):2842–2865, 2009.
[15] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pages 2223–2232, 2017