本論文研究了在高光譜影像中的逆問題。近年來，深度學習被證實是一種有發展前途的圖像恢復方法。然而，有效的深度學習通常需要大量的訓練數據。此外，高光譜影像(HSI)可能會受到各種雜訊的汙染，如高斯白雜訊、稀疏雜訊(sparse noises)、條紋(stripes)和死線(dead lines)。最近，根據研究指出，使用在深度圖像先驗(DIP)的核心元素-卷積神經網路可以不需要預先訓練而獲取圖像統計特性進行圖像恢復。儘管如此，深度圖像先驗(DIP)與高光譜影像(HSI)中最先進的方法(例如:低秩(low-rank)模型)之間在性能上存在一些差異。我們透過強健統計(robust statistics)中的最不利分布(least favorable distribution)所推導出的Huber loss function應用在深度圖像先驗，提出一種新的非監督式去除雜訊的演算法，稱為HLF-DIP，不需要預先訓練且不涉及正則化器。此外，我們採用可微分的正則化器以降低資料的維度，再將我們提出的去除雜訊演算法應用到到高光譜修復(HI)問題上。基於所提出的框架，我們又提出一種新的HI演算法，DLRHyln和具有對抗異常值的強健 DLRHyln (R-DLRHyln)，兩者的區別僅在於後者使用對混合雜訊具健健性的HLF。實驗結果証實我們提出的演算法在不同場景中顯著地勝過其他最先進的演算法。

This thesis considers the inverse problem in hyperspectral imaging. Recently, it has been shown that deep learning is a promising approach to image restoration. However, deep learning to be effective usually needs a massive amount of training data. Moreover, in a practical scenario, hyperspectral images (HSIs) may get contaminated by different kinds of degradation such as Gaussian and sparse noises, stripes, and dead lines. Lately, it has been reported that the convolutional neural network (CNN), the core element used by deep image prior (DIP), is able to capture image statistical characteristics without the need of pre-training, i.e., restore the clean image blindly. Nonetheless, there exists some performance gap between DIP and state-of-the-art methods in HSIs (e.g., low-rank models). By applying the Huber loss function (HLF), which is derived through a least favorable distribution in robust statistics, to DIP, we propose a novel unsupervised denoising algorithm, referred as to the HLF-DIP, free from pre-training and without involving any other explicit regularization. Moreover, we generalized and modified our proposed denoising algorithm to the HSI inpainting (HI) problem by adapting differentiable regularization for the data rank. Based on the proposed framework, we come up with a novel HI algorithm (denoted as DLRHyIn), and a robust DLRHyIn (denoted as R-DLRHyIn) which is robust against outliers, where the latter differs from the former only in HLF (which has been justified robust to mixed noise) used instead. Extensive experimental results are provided to demonstrate that the proposed algorithms significantly outperform other state-of-the-art algorithms in different scenarios.

Abstract (Chinese)--------------------------------------------------------------------- I
Abstract--------------------------------------------------------------------------------II
Acknowledgements----------------------------------------------------------------------- III
Contents--------------------------------------------------------------------------------IV
List of Figures-------------------------------------------------------------------------VII
List of Tables-------------------------------------------------------------------------- X
List of Notations------------------------------------------------------------------------XI
1 Introduction---------------------------------------------------------------------------1
1.1 Motivation-------------------------------------------------------------------------- 1
1.2 Thesis Contributions and Organization------------------------------------------------5
2 Related Backgrounds--------------------------------------------------------------------8
2.1 Inverse Problems-------------------------------------------------------------------- 8
2.2 Deep Image Prior--------------------------------------------------------------------10
2.3 Robust Statistics-------------------------------------------------------------------10
2.3.1 M-estimators----------------------------------------------------------------------11
2.3.2 Least Favorable Distribution------------------------------------------------------12

3 Proposed HLF-DIP Algorithm for Hyperspectral Denoising------------------------------- 14
3.1 The Proposed HLF-DIP Denoising Method --------------------------------------------- 14
3.1.1 Image Degradation Model --------------------------------------------------------- 14
3.1.2 The Statistical Approach -------------------------------------------------------- 15
3.1.3 Convolutional Neural Network Architecture ----------------------------------------17
3.1.4 Summary ------------------------------------------------------------------------- 18
3.2 Experimental Results -------------------------------------------------------------- 19
3.2.1 Experimental Settings------------------------------------------------------------ 19
3.2.2 Quantitative Results ------------------------------------------------------------ 22
3.2.3 Real Noisy Data Experiments ------------------------------------------------------34
3.2.4 Parameter Analysis--------------------------------------------------------------- 38
3.2.4.1 Choice of Parameter δ for HLF ------------------------------------------------ 38
3.2.4.2 Choice of the Maximum Number of Iterations tmax ------------------------------- 39

4 Proposed DLRHyIn and R-DLRHyIn for Hyperspectral Inpainting ------------------------- 43
4.1 Image Degradation Model ----------------------------------------------------------- 43
4.2 Low-Rank Prior -------------------------------------------------------------------- 44
4.3 Proposed Methods ------------------------------------------------------------------ 45
4.3.1 Deep Low-Rank Hyperspectral Inpainting ------------------------------------------ 45
4.3.2 Robust Deep Low-Rank Hyperspectral Inpainting ----------------------------------- 46
4.4 Experimental Results -------------------------------------------------------------- 47
4.4.1 Simulated Dataset ----------------------------------------------------------------47
4.4.2 Performance Evaluation ---------------------------------------------------------- 47
4.4.3 Hyperspectral Inpainting -------------------------------------------------------- 49
4.4.4 Robust Hyperspectral Inpainting ------------------------------------------------- 51
4.4.5 Real Data Experiment ------------------------------------------------------------ 53

5 Conclusion -------------------------------------------------------------------------- 55

A Proof of Proposition 1--------------------------------------------------------------- 57

Bibliography--------------------------------------------------------------------------- 60

[1] J. M. Bioucas-Dias, A. Plaza, G. Camps-Valls, P. Scheunders, N. Nasrabadi,
and J. Chanussot, “Hyperspectral remote sensing data analysis and future
challenges,” IEEE Geosci. Remote Sens. Mag., vol. 1, no. 2, pp. 6–36, 2013.

[2] G. Lu and B. Fei, “Medical hyperspectral imaging: A review,” J. Biomed.
Opt., vol. 19, no. 1, pp. 1–24, 2014.

[3] D.-W. Sun, Hyperspectral Imaging for Food Quality Analysis and Control.
Elsevier, 2010.

[4] C.-I. Chang, Hyperspectral Imaging: Techniques for Spectral Detection and
Classification. Springer Science & Business Media, 2003, vol. 1.

[5] B. Rasti, B. Koirala, P. Scheunders, and P. Ghamisi, “How hyperspectral
image unmixing and denoising can boost each other,” Remote Sens., vol. 12,
no. 11, p. 1728, 2020.

[6] T. Akgun, Y. Altunbasak, and R. M. Mersereau, “Super-resolution recon-
struction of hyperspectral images,” IEEE Trans. Image Process., vol. 14,
no. 11, pp. 1860–1875, 2005.

[7] W.-K. Ma, J. M. Bioucas-Dias, T.-H. Chan, et al., “A signal processing
perspective on hyperspectral unmixing: Insights from remote sensing,” IEEE
Signal Process. Mag., vol. 31, no. 1, pp. 67–81, 2013.

[8] B. Rasti, P. Scheunders, P. Ghamisi, G. Licciardi, and J. Chanussot, “Noise
reduction in hyperspectral imagery: Overview and application,” Remote Sens.,
vol. 10, no. 3, p. 482, 2018.

[9] A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image de-
noising,” in Proc. IEEE CVPR, vol. 2, 2005, pp. 60–65.

[10] L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise
removal algorithms,” Physica D: Nonlinear Phenomena, vol. 60, no. 1-4,
pp. 259–268, 1992.

[11] H. Othman and S.-E. Qian, “Noise reduction of hyperspectral imagery using
hybrid spatial-spectral derivative-domain wavelet shrinkage,” IEEE Trans.
Geosci. Remote Sens., vol. 44, no. 2, pp. 397–408, 2006.

[12] M. Maggioni, V. Katkovnik, K. Egiazarian, and A. Foi, “Nonlocal transform-
domain filter for volumetric data denoising and reconstruction,” IEEE Trans.
Image Process., vol. 22, no. 1, pp. 119–133, 2012.

[13] H. K. Aggarwal and A. Majumdar, “Hyperspectral image denoising using
spatio-spectral total variation,” IEEE Geosci. Remote Sens. Lett., vol. 13,
no. 3, pp. 442–446, 2016.

[14] J. Xue, Y. Zhao, W. Liao, and S. G. Kong, “Joint spatial and spectral low-
rank regularization for hyperspectral image denoising,” IEEE Trans. Geosci.
Remote Sens., vol. 56, no. 4, pp. 1940–1958, 2017.

[15] J. Xue, Y. Zhao, W. Liao, and J. C.-W. Chan, “Nonlocal low-rank regular-
ized tensor decomposition for hyperspectral image denoising,” IEEE Trans.
Geosci. Remote Sens., vol. 57, no. 7, pp. 5174–5189, 2019.

[16] W. Dong, G. Shi, and X. Li, “Nonlocal image restoration with bilateral
variance estimation: A low-rank approach,” IEEE Trans. Image Process.,
vol. 22, no. 2, pp. 700–711, 2012.

[17] W. He, N. Yokoya, and X. Yuan, “Fast hyperspectral image recovery of dual-
camera compressive hyperspectral imaging via non-iterative subspace-based
fusion,” IEEE Trans. Image Process., vol. 30, pp. 7170–7183, 2021.

[18] L. Zhuang and J. M. Bioucas-Dias, “Fast hyperspectral image denoising
and inpainting based on low-rank and sparse representations,” IEEE J. Sel.
Topics Appl. Earth Observ. Remote Sens., vol. 11, no. 3, pp. 730–742, 2018.

[19] H. Zhang, W. He, L. Zhang, H. Shen, and Q. Yuan, “Hyperspectral image
restoration using low-rank matrix recovery,” IEEE Trans. Geosci. Remote
Sens., vol. 52, no. 8, pp. 4729–4743, 2013.

[20] M. Golbabaee and P. Vandergheynst, “Hyperspectral image compressed sens-
ing via low-rank and joint-sparse matrix recovery,” in Proc. IEEE ICASSP,
2012, pp. 2741–2744.

[21] Y. Wang, J. Peng, Q. Zhao, Y. Leung, X.-L. Zhao, and D. Meng, “Hyper-
spectral image restoration via total variation regularized low-rank tensor
decomposition,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens.,
vol. 11, no. 4, pp. 1227–1243, 2017.

[22] M. Wang, Q. Wang, J. Chanussot, and D. Li, “Hyperspectral image mixed
noise removal based on multidirectional low-rank modeling and spatial–
spectral total variation,” IEEE Trans. Geosci. Remote Sens., vol. 59, no. 1,
pp. 488–507, 2020.

[23] Y.-B. Zheng, T.-Z. Huang, X.-L. Zhao, Y. Chen, and W. He, “Double-
factor-regularized low-rank tensor factorization for mixed noise removal in
hyperspectral image,” IEEE Trans. Geosci. Remote Sens., vol. 58, no. 12,
pp. 8450–8464, 2020.

[24] D. Cerra, R. M ̈uller, and P. Reinartz, “Unmixing-based denoising for de-
striping and inpainting of hyperspectral images,” in Proc. IEEE Int. Geosci.
Remote Sens. Symp., 2014, pp. 4620–4623.

[25] W. Xie and Y. Li, “Hyperspectral imagery denoising by deep learning with
trainable nonlinearity function,” IEEE Geosci. Remote Sens. Lett., vol. 14,
no. 11, pp. 1963–1967, 2017.

[26] Q. Yuan, Q. Zhang, J. Li, H. Shen, and L. Zhang, “Hyperspectral image
denoising employing a spatial–spectral deep residual convolutional neural
network,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 2, pp. 1205–1218,
2018.

[27] W. Dong, H. Wang, F. Wu, G. Shi, and X. Li, “Deep spatial–spectral repre-
sentation learning for hyperspectral image denoising,” IEEE Trans. Comput.
Imag., vol. 5, no. 4, pp. 635–648, 2019.

[28] Q. Shi, X. Tang, T. Yang, R. Liu, and L. Zhang, “Hyperspectral image
denoising using a 3-d attention denoising network,” IEEE Trans. Geosci.
Remote Sens., vol. 59, no. 12, pp. 10 348–10 363, 2021.

[29] K. Wei, Y. Fu, and H. Huang, “3-D quasi-recurrent neural network for hyper-
spectral image denoising,” IEEE Trans. Neural Netw. Learn. Syst., vol. 32,
no. 1, pp. 363–375, 2020.

[30] F. Xiong, J. Zhou, Q. Zhao, J. Lu, and Y. Qian, “MAC-Net: Model aided
nonlocal neural network for hyperspectral image denoising,” IEEE Trans.
Geosci. Remote Sens., vol. 60, pp. 1–14, 2021.

[31] D. Ulyanov, A. Vedaldi, and V. Lempitsky, “Deep image prior,” in Proc.
IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2018, pp. 9446–9454.

[32] O. Sidorov and J. Yngve Hardeberg, “Deep hyperspectral prior: Single-image
denoising, inpainting, super-resolution,” in Proc. IEEE/CVF Int. Conf. Com-
put. Vis. Workshop, 2019, pp. 3844–3851.

[33] H. V. Nguyen, M. O. Ulfarsson, and J. R. Sveinsson, “Hyperspectral image
denoising using SURE-based unsupervised convolutional neural networks,”
IEEE Trans. Geosci. Remote Sens., vol. 59, no. 4, pp. 3369–3382, 2020.

[34] J. Liu, Y. Sun, X. Xu, and U. S. Kamilov, “Image restoration using to-
tal variation regularized deep image prior,” in Proc. IEEE ICASSP, 2019,
pp. 7715–7719.

[35] Z. Sun, F. Latorre, T. Sanchez, and V. Cevher, “A plug-and-play deep image
prior,” in Proc. IEEE ICASSP, 2021, pp. 8103–8107.

[36] G. Mataev, P. Milanfar, and M. Elad, “DeepRED: Deep image prior powered
by RED,” in Proc. IEEE Int. Conf. Comput. Vis. Workshops, 2019, pp. 1–
10.

[37] P. J. Huber, Robust Statistics. John Wiley & Sons, 2004, vol. 523.

[38] P. J. Huber, “Robust estimation of a location parameter,” in Breakthroughs
in Statistics, Springer, 1992, pp. 492–518.

[39] H. Ye, H. Li, B. Yang, F. Cao, and Y. Tang, “A novel rank approximation
method for mixture noise removal of hyperspectral images,” IEEE Trans.
Geosci. Remote Sens., vol. 57, no. 7, pp. 4457–4469, 2019.

[40] K. F. Niresi and C.-Y. Chi, “Unsupervised hyperspectral denoising based
on deep image prior and least favorable distribution,” IEEE J. Sel. Topics
Appl. Earth Observ. Remote Sens., vol. 15, pp. 5967–5983, 2022. doi: 10.
1109/JSTARS.2022.3187722.

[41] A. M. Zoubir, V. Koivunen, E. Ollila, and M. Muma, Robust Statistics for
Signal Processing. Cambridge University Press, 2018.

[42] R. A. Maronna, R. D. Martin, V. J. Yohai, and M. Salibi ́an-Barrera, Robust
Statistics: Theory and Methods (with R). John Wiley & Sons, 2019.

[43] P. J. Rousseeuw and A. M. Leroy, Robust Regression and Outlier Detection.
John wiley & sons, 2005.

[44] C.-Y. Chi, W.-C. Li, and C.-H. Lin, Convex Optimization for Signal Process-
ing and Communications: From Fundamentals to Applications. Boca Raton,
FL, USA: CRC Press, 2017.

[45] T. Luki ́c, J. Lindblad, and N. Sladoje, “Regularized image denoising based on
spectral gradient optimization,” Inverse Problems, vol. 27, no. 8, p. 085 010,
2011.

[46] P. Charbonnier, L. Blanc-F ́eraud, G. Aubert, and M. Barlaud, “Determinis-
tic edge-preserving regularization in computed imaging,” IEEE Trans. Image
Process., vol. 6, no. 2, pp. 298–311, 1997.

[47] D. Geman and G. Reynolds, “Constrained restoration and the recovery of
discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 14, no. 3,
pp. 367–383, 1992.

[48] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in
Proc. Int. Conf. Learn. Represent. (ICLR), 2015, pp. 1–15.

[49] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network
training by reducing internal covariate shift,” arXiv preprint arXiv:1502.03167,
2015.

[50] S. Santurkar, D. Tsipras, A. Ilyas, and A. Madry, “How does batch nor-
malization help optimization?” In Proc. 32nd International Conference on
Neural Information Processing Systems, 2018, pp. 2488–2498.

[51] A. L. Maas, A. Y. Hannun, A. Y. Ng, et al., “Rectifier nonlinearities improve
neural network acoustic models,” in Proc. ICML, 2013, pp. 1–6.

[52] Y.-C. Chen, C. Gao, E. Robb, and J.-B. Huang, “NAS-DIP: Learning deep
image prior with neural architecture search,” arXiv preprint arXiv:2008.11713,
2020.

[53] B. Rasti, M. O. Ulfarsson, and P. Ghamisi, “Automatic hyperspectral im-
age restoration using sparse and low-rank modeling,” IEEE Geosci. Remote
Sens. Lett., vol. 14, no. 12, pp. 2335–2339, 2017.

[54] L. Zhuang, X. Fu, M. K. Ng, and J. M. Bioucas-Dias, “Hyperspectral im-
age denoising based on global and nonlocal low-rank factorizations,” IEEE
Trans. Geosci. Remote Sens., vol. 59, no. 12, pp. 10 438–10 454, 2021.

[55] L. Zhuang and M. K. Ng, “FastHyMix: Fast and parameter-free hyperspec-
tral image mixed noise removal,” IEEE Trans. Neural Netw. Learn. Syst.,
pp. 1–15, 2021. doi: 10.1109/TNNLS.2021.3112577.

[56] A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-
Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans.
Image Process., vol. 17, no. 10, pp. 1737–1754, 2008.

[57] K. Zhang, W. Zuo, and L. Zhang, “FFDNet: Toward a fast and flexible solu-
tion for CNN-based image denoising,” IEEE Trans. Image Process., vol. 27,
no. 9, pp. 4608–4622, 2018.

[58] R. Oten and R. J. de Figueiredo, “Adaptive alpha-trimmed mean filters
under deviations from assumed noise model,” IEEE Trans. Image Process.,
vol. 13, no. 5, pp. 627–639, 2004.

[59] T. Plotz and S. Roth, “Benchmarking denoising algorithms with real pho-
tographs,” in Proc. IEEE Conf. Comput. Vis. Pattern Recognit. (CVPR),
2017, pp. 1586–1595.

[60] A. L. Maas, A. Y. Hannun, A. Y. Ng, et al., “Pytorch: An imperative style,
high-performance deep learning library,” in Proc. Adv. Neural Inf. Process.
Syst., 2019, pp. 8026–8037.

[61] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality as-
sessment: From error visibility to structural similarity,” IEEE Trans. Image
Process., vol. 13, no. 4, pp. 600–612, 2004.

[62] F. A. Kruse, A. Lefkoff, J. Boardman, et al., “The spectral image processing
system (SIPS)—interactive visualization and analysis of imaging spectrom-
eter data,” Remote Sens. Environ., vol. 44, pp. 145–163, 1993.

[63] L. Zhang, L. Zhang, X. Mou, and D. Zhang, “FSIM: A feature similarity
index for image quality assessment,” IEEE Trans. Image Process., vol. 20,
no. 8, pp. 2378–2386, 2011.

[64] E. J. Cand`es, X. Li, Y. Ma, and J. Wright, “Robust principal component
analysis?” Journal of the ACM (JACM), vol. 58, no. 3, pp. 1–37, 2011.

[65] M. Burger, S. Osher, J. Xu, and G. Gilboa, “Nonlinear inverse scale space
methods for image restoration,” in Proc. International Workshop on Vari-
ational, Geometric, and Level Set Methods in Computer Vision, Springer,
2005, pp. 25–36.

[66] Z. Cheng, M. Gadelha, S. Maji, and D. Sheldon, “A Bayesian perspective
on the deep image prior,” in Proc. IEEE/CVF Conf. Comput. Vis. Pattern
Recognit., 2019, pp. 5443–5451.

[67] H. Wang, T. Li, Z. Zhuang, T. Chen, H. Liang, and J. Sun, “Early stopping
for deep image prior,” arXiv preprint arXiv:2112.06074, 2021.

[68] Y. Jo, S. Y. Chun, and J. Choi, “Rethinking deep image prior for denoising,”
in Proc. IEEE Int. Conf. Comput. Vis., 2021, pp. 5087–5096.

[69] J. D’Errico, Inpainting Nan Elements in 3-D. Natick. MA, USA: MathWorks,
MATLAB Central File Exchange, 2008. [Online]. Available:
https://www.mathworks.com/matlabcentral/fileexchange/21214-inpainting-
nan-elements-in-3-d.

[70] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by
sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image
Process., vol. 16, no. 8, pp. 2080–2095, 2007.

[71] C. Lu, X. Peng, and Y. Wei, “Low-rank tensor completion with a new tensor
nuclear norm induced by invertible linear transforms,” in Proc. IEEE/CVF
Conf. Comput. Vis. Pattern Recognit., 2019, pp. 5996–6004.

[72] C.-H. Lin, Y.-C. Lin, and P.-W. Tang, “ADMM-ADAM: A new inverse imag-
ing framework blending the advantages of convex optimization and deep
learning,” IEEE Trans. Geosci. Remote Sens., vol. 60, pp. 1–16, 2022. doi:
10.1109/TGRS.2021.3111007.