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

In this paper, we propose a new variational model for image reconstruction by minimizing the $L^1$ norm of the \emph{Weingarten map} of image surface $(x,y,f(x,y))$ for a given image $f:{\mathrm{\Omega}}\rightarrow \mathbb R$. We analytically prove that the Weingarten map minimization model can not only keep the greyscale intensity contrasts of images, but also preserve edges and corners of objects. The alternating direction method of multiplier (ADMM) based algorithm is developed, where one subproblem needs to be solved by gradient descent. In what follows, we derive a hybrid nonlinear first and second order regularization from the Weingarten map, and present an efficient ADMM-based algorithm by regarding the nonlinear weights as known. By comparing with several state-of-the-art methods on synthetic and real image reconstruction problems, it confirms that the proposed models can well preserve image contrasts and features, especially the spatially adapted first and second order regularization economizing much computational cost.