Coefficient identification in PDEs applied to image inpainting

Abstract

In this paper, we introduce the concept of parameter identification problems, which are inverse problems, as a methodology to inpainting. More specifically, as a first study in this new direction, we generalize the method of harmonic inpainting by studying an elliptic equation in divergence form where we assume that the diffusion coefficient is unknown. As a first step, this unknown coefficient is determined from the information obtained by the known data in the image. Next, we fill in the region with missing data by solving an elliptic equation in divergence form using this obtained diffusion coefficient. An error analysis shows that this approach is promising and our numerical experiments produces better results than the harmonic inpainting.

Microstructure Images Restoration of Metallic Materials Based upon KSVD and Smoothing Penalty Sparse Representation Approach

“…Image restoration is an effective technique for recovering incomplete or damaged images approximate to the ideal images. Up to now, various image restoration strategies have been proposed, which can be classified into several categories: (i) Time domain analysis methods, such as adaptive filter denoising (AFD) [ 6 , 7 ]; (ii) Frequency domain analysis methods, such as wavelet–wavelet packet denoising (W-WPD) technique [ 8 , 9 ]; (iii) Data-driven approaches, such as partial differential equation (PDE) [ 10 , 11 , 12 , 13 ], and wavelet hidden Markov random field (WHMRF) [ 14 ]; (iv) Sparse representation (SR) techniques, such as redundant dictionary and non-convex penalty regularization [ 15 , 16 , 17 , 18 ], etc. Although the damaged images can be restored more precisely by the above methods, the drawbacks are also obvious.…”

A Novel Image Inpainting Technique Based on Isotropic Diffusion

“…where (𝜇 𝐼 , 𝜎 𝐼 ) and (𝜇 𝐼 ′ , 𝜎 𝐼 ′ ) refers to mean and standard deviation of patches in the images 𝐼 and 𝐼 ′ , respectively; 𝜎 𝐼𝐼 ′ : the covariance of 𝐼 and 𝐼 ′ ; 𝑐 1 = (𝑘 1 ℎ) 2 , 𝑐 2 = (𝑘 2 ℎ) 2 such that h = 2 𝑏𝑖𝑡𝑠 𝑝𝑒𝑟 𝑝𝑖𝑥𝑒𝑙 − 1 , 𝑘 1 = 0.01 and 𝑘 2 = 0.03. However, the value of SSIM is range from 1 to 0 and when the value is 1 means the two images are identical while 0 values refer to not identical.…”

“…Inpainting of damaged paintings was first used in art, where it was done by skilled painters. In contrast, inpainting in mathematics relates to the process of filling in damaged areas in an image by propagating information from surrounding areas [2]. In order to make the recognition process as simple and automatic as possible, improving the quality of damaged and noisy images is a vital challenge in image processing.…”

On Geometry of Locally Conformal C_12-Manifolds

“…In the field of art, the concept of picture inpainting was developed. In the field of mathematics, the term "inpainting" refers to the process of filling in damaged regions by propagating information from their surrounding areas in the image [2]. Inpainting of ruined paintings was done by expert artists.…”