In this paper, we propose a novel low dimensional manifold model (LDMM) and
apply it to some image processing problems. LDMM is based on the fact that the patch manifolds
of many natural images have low dimensional structure. Based on this fact, the dimension of the
patch manifold is used as a regularization to recover the image. The key step in LDMM is to solve
a Laplace-Beltrami equation over a point cloud which is solved by the point integral method. The
point integral method enforces the sample point constraints correctly and gives better results than the
standard graph Laplacian. Numerical simulations in image denoising, inpainting and super-resolution
problems show that LDMM is a powerful method in image processing.