  Inverse imaging problems are inherently under-determined, and hence it is
important to employ appropriate image priors for regularization. One recent
popular prior---the graph Laplacian regularizer---assumes that the target pixel
patch is smooth with respect to an appropriately chosen graph. However, the
mechanisms and implications of imposing the graph Laplacian regularizer on the
original inverse problem are not well understood. To address this problem, in
this paper we interpret neighborhood graphs of pixel patches as discrete
counterparts of Riemannian manifolds and perform analysis in the continuous
domain, providing insights into several fundamental aspects of graph Laplacian
regularization for image denoising. Specifically, we first show the convergence
of the graph Laplacian regularizer to a continuous-domain functional,
integrating a norm measured in a locally adaptive metric space. Focusing on
image denoising, we derive an optimal metric space assuming non-local
self-similarity of pixel patches, leading to an optimal graph Laplacian
regularizer for denoising in the discrete domain. We then interpret graph
Laplacian regularization as an anisotropic diffusion scheme to explain its
behavior during iterations, e.g., its tendency to promote piecewise smooth
signals under certain settings. To verify our analysis, an iterative image
denoising algorithm is developed. Experimental results show that our algorithm
performs competitively with state-of-the-art denoising methods such as BM3D for
natural images, and outperforms them significantly for piecewise smooth images.
