# 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:

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##### 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
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