Optimal graph laplacian regularization for natural image denoising

Image denoising is an under-determined problem, and hence it is important to define appropriate image priors for regularization. One recent popular prior is the graph Laplacian regularizer, where a given pixel patch is assumed to be smooth in the graph-signal domain. The strength and direction of the resulting graph-based filter are computed from the graph's edge weights. In this paper, we derive the optimal edge weights for local graph-based filtering using gradient estimates from non-local pixel patches that are self-similar. To analyze the effects of the gradient estimates on the graph Laplacian regularizer, we first show theoretically that, given graph-signal h
^D
is a set of discrete samples on continuous function h(x; y) in a closed region Ω, graph Laplacian regularizer (h
^D
)
^T
Lh
^D
converges to a continuous functional S
Ω
integrating gradient norm of h in metric space G—i.e., (▽h)
^T
G
⁻¹
(▽h)—over Ω. We then derive the optimal metric space G★: one that leads to a graph Laplacian regularizer that is discriminant when the gradient estimates are accurate, and robust when the gradient estimates are noisy. Finally, having derived G★ we compute the corresponding edge weights to define the Laplacian L used for filtering. Experimental results show that our image denoising algorithm using the per-patch optimal metric space G★ outperforms non-local means (NLM) by up to 1.5 dB in PSNR.