  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.
