# American Institute of Mathematical Sciences

December  2018, 12(6): 1389-1410. doi: 10.3934/ipi.2018058

## Local block operators and TV regularization based image inpainting

 Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: jliu@bnu.edu.cn

Received  January 2018 Revised  July 2018 Published  October 2018

In this paper, we propose a novel image blocks based inpainting model using group sparsity and TV regularization. The block matching method is employed to collect similar image blocks which can be formed as sparse image groups. By reducing the redundant information in these groups, we can well restore textures missing in the inpainting areas. We built a variational framework based on a local SVD operator for block matching and group sparsity. In addition, TV regularization is naturally integrated in the model to reduce artificial effects which are caused by image blocks stacking in the block matching method. Besides, enforcing the sparsity of the representation, the SVD operators in our method are iteratively updated and play the role of dictionary learning. Thus it can greatly improve the quality of the restoration. Moreover, we mathematically show the existence of a minimizer for the proposed inpainting model. Convergence results of the proposed algorithm are also given in the paper. Numerical experiments demonstrate that the proposed model outperforms many benchmark methods such as BM3D based image inpainting.

Citation: Wei Wan, Haiyang Huang, Jun Liu. Local block operators and TV regularization based image inpainting. Inverse Problems & Imaging, 2018, 12 (6) : 1389-1410. doi: 10.3934/ipi.2018058
##### References:

show all references

##### References:
Filling in the missing pixels by different inpainting method
Comparison of details between different inpainting methods
Scratch and text removal by different inpainting methods
Comparison of details between different inpainting methods
The relative error curves as functions of the iteration number on our experiments for the proposed-$\ell_1$ method
The relative error curves as functions of the iteration number on our experiments for the proposed-$\ell_0$ method
PSNR values of the different methods on filling randomly missing pixels
 Image CTM Cubic BPFA IDI-BM3D Proposed-$\ell_1$ Proposed-$\ell_0$ Monarch 23.01 24.18 24.49 26.63 25.15 27.25 Lena 27.21 27.40 28.32 29.63 28.50 29.98 Barbara 25.65 26.24 27.08 28.62 27.59 29.69
 Image CTM Cubic BPFA IDI-BM3D Proposed-$\ell_1$ Proposed-$\ell_0$ Monarch 23.01 24.18 24.49 26.63 25.15 27.25 Lena 27.21 27.40 28.32 29.63 28.50 29.98 Barbara 25.65 26.24 27.08 28.62 27.59 29.69
PSNR values of different inpainting methods on text and scratch removal
 Image Cubic TV BPFA IDI-BM3D Proposed-$\ell_1$ Proposed-$\ell_0$ Barbara 33.25 34.58 37.28 40.16 38.26 40.98 Hill 33.30 33.44 33.84 35.38 34.54 35.61 Baboon 35.87 35.86 35.39 37.77 36.80 38.03
 Image Cubic TV BPFA IDI-BM3D Proposed-$\ell_1$ Proposed-$\ell_0$ Barbara 33.25 34.58 37.28 40.16 38.26 40.98 Hill 33.30 33.44 33.84 35.38 34.54 35.61 Baboon 35.87 35.86 35.39 37.77 36.80 38.03
 [1] Yuan Wang, Zhi-Feng Pang, Yuping Duan, Ke Chen. Image retinex based on the nonconvex TV-type regularization. Inverse Problems & Imaging, 2021, 15 (6) : 1381-1407. doi: 10.3934/ipi.2020050 [2] Emmanuel Frénod. Homogenization-based numerical methods. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i [3] Emmanuel Frénod. An attempt at classifying homogenization-based numerical methods. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : i-vi. doi: 10.3934/dcdss.2015.8.1i [4] Dale McDonald. Sensitivity based trajectory following control damping methods. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 127-143. doi: 10.3934/naco.2013.3.127 [5] Yun Chen, Jiasheng Huang, Si Li, Yao Lu, Yuesheng Xu. A content-adaptive unstructured grid based integral equation method with the TV regularization for SPECT reconstruction. Inverse Problems & Imaging, 2020, 14 (1) : 27-52. doi: 10.3934/ipi.2019062 [6] Michael Hintermüller, Monserrat Rincon-Camacho. An adaptive finite element method in $L^2$-TV-based image denoising. Inverse Problems & Imaging, 2014, 8 (3) : 685-711. doi: 10.3934/ipi.2014.8.685 [7] Alina Toma, Bruno Sixou, Françoise Peyrin. Iterative choice of the optimal regularization parameter in TV image restoration. Inverse Problems & Imaging, 2015, 9 (4) : 1171-1191. doi: 10.3934/ipi.2015.9.1171 [8] Sergio Grillo, Leandro Salomone, Marcela Zuccalli. On the relationship between the energy shaping and the Lyapunov constraint based methods. Journal of Geometric Mechanics, 2017, 9 (4) : 459-486. doi: 10.3934/jgm.2017018 [9] Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013 [10] Yiting Chen, Jia Li, Qingyun Yu. Large region inpainting by re-weighted regularized methods. Inverse Problems & Imaging, 2021, 15 (5) : 827-842. doi: 10.3934/ipi.2021015 [11] Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019 [12] Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems & Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149 [13] Lili Ju, Wei Leng, Zhu Wang, Shuai Yuan. Numerical investigation of ensemble methods with block iterative solvers for evolution problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4905-4923. doi: 10.3934/dcdsb.2020132 [14] Sanjay Khattri. Another note on some quadrature based three-step iterative methods for non-linear equations. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 549-555. doi: 10.3934/naco.2013.3.549 [15] Abdel-Rahman Hedar, Ahmed Fouad Ali, Taysir Hassan Abdel-Hamid. Genetic algorithm and Tabu search based methods for molecular 3D-structure prediction. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 191-209. doi: 10.3934/naco.2011.1.191 [16] Shummin Nakayama, Yasushi Narushima, Hiroshi Yabe. Memoryless quasi-Newton methods based on spectral-scaling Broyden family for unconstrained optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1773-1793. doi: 10.3934/jimo.2018122 [17] Yigui Ou, Haichan Lin. A class of accelerated conjugate-gradient-like methods based on a modified secant equation. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1503-1518. doi: 10.3934/jimo.2019013 [18] Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems & Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028 [19] Zhong Wan, Chaoming Hu, Zhanlu Yang. A spectral PRP conjugate gradient methods for nonconvex optimization problem based on modified line search. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1157-1169. doi: 10.3934/dcdsb.2011.16.1157 [20] Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007

2020 Impact Factor: 1.639