We propose a multi-well potential based modified Cahn-Hilliard model for gray-scale image inpainting. An image specific multi-well potential function is constructed from the histogram information of the given image. Existence, uniqueness and the continuous dependency of the solution on the initial data, of the newly proposed modified Cahn-Hilliard equation has been established using Faedo-Galerkin approach. Also the complete discretization error for the numerical scheme based on convexity splitting in time and Fourier spectral method in space has been derived. Numerical simulations have been carried out on some standard gray-scale test images and the metrics PSNR, SNR, SSIM corresponding to the obtained results are compared with those in the literature.

A fourth-order non-linear partial differential equation (PDE) model together with multi-well potential has been proposed for greyscale image segmentation. The multi-well potential is constructed from the histogram of the given image to make the segmentation process fully automatic and unsupervised. Further, the model is refined for effective segmentation of noisy greyscale image. The fourth-order anisotropic term with the multi-well potential is shown to properly segment noisy images. Fourier spectral method in space with semi-implicit convexity splitting in time is used to derive an unconditionally stable scheme. Numerical studies on some standard test images and comparison of results with those in literature clearly depict the superiority of the anisotropic variant of the non-linear PDE model.