Fast image inpainting using exponential-threshold POCS plus conjugate gradient

Abstract

Image inpainting can remove unwanted objects and reconstruct the missing or damaged portions of an image. The projection onto convex sets (POCS) is a classical method used in image inpainting. However, the traditional POCS converges slowly due to the linear error threshold. We propose an exponential-threshold scheme, which greatly improves the convergence of the POCS. Although the exponential-threshold POCS can recover the image in about 20 iterations, it cannot reconstruct the image details very well even with hundreds of iterations. Thus, we append the non-local restoration to the exponential-threshold POCS to further refine the image details, and then we solve this objective function using the conjugate gradient. Numerical experiments show that for each iteration, the exponential-threshold POCS and the conjugate gradient have very similar computational efficiencies. For an image with various topologies of the missing areas, our scheme can recover missing pixels simultaneously and obtain a satisfied inpainting result in only 20 iterations of the exponential-threshold POCS and 20 iterations of the conjugate gradient. The proposed method can excellently restore damaged photographs and remove superimposed text. This method has less computational cost than the conjugate gradient and has a higher resolution than the POCS.

Keywords:

• Projection onto convex sets
• Inpainting
• Exponential threshold
• Conjugate gradient
• Non-local restoration

Acknowledgements

We thank Dr Xinguo Wang for providing the rock and scene images taken from Tibet. This research was supported by the National Natural Science Foundation of China (Grant No. 41074092), the Free Exploration project of Minzu University of China (Grant Nos. 1112KYQN38 and 1112KYZY48) and the Humanities and Social Science Planning Fund Supported by Chinese Ministry of Education (Grant No. 12YJA630123).