Spatially Adapted First and Second Order Regularization for Image Reconstruction: From an Image Surface Perspective

It is well-known that images are comprised of multiple objects at different scales. Thus, we propose a spatially adapted first and second order regularization for image reconstruction to better localize image features. More specifically, we minimize the $L^1$ norm of the Weingarten map of the image surface $(x,f(x))$ for a given image $f:{\rm{\Omega}}\rightarrow \mathbb R$, which is further reformulated into a combined first and second order regularizer with adapted parameters. We analytically prove our model can keep the greyscale intensity contrasts of images and preserve edges. In what follows, we present the numerical solution to the proposed model by employing the alternating direction method of multipliers (ADMM) and analyze the convergence under certain conditions. Various numerical experiments on image denoising, deblurring and inpainting are implemented to demonstrate the effectiveness and efficiency of the proposed regularization scheme. By comparing with several state-of-the-art methods on synthetic and real image reconstruction problems, it is shown that the proposal can enhance image regions containing fine details and smoothness in homogeneous regions, while being simple and efficiently numerically solvable.