Nonlocal Means and Optimal Weights for Noise Removal

Abstract

In this paper, a new denoising algorithm to deal with the additive white Gaussian noise model is described. In the line of work of the Non-Local means approach, we propose an adaptive estimator based on the weighted average of observations taken in a neighborhood with weights depending on the similarity of local patches. The idea is to compute adaptive weights that best minimize an upper bound of the pointwise L 2 risk. In the framework of adaptive estimation, we show that the "oracle" weights are optimal if we consider triangular kernels instead of the commonly-used Gaussian kernel. Furthermore, we propose a way to automatically choose the spatially varying smoothing parameter for adaptive denoising. Under conventional minimal regularity conditions, the obtained estimator converges at the usual optimal rate. The implementation of the proposed algorithm is also straightforward and the simulations show that our algorithm improves significantly the classical NL-means and is competitive when compared to the more sophisticated NL-means filters both in terms of PSNR values and visual quality.

Removing random-valued impulse noise with reliable weight

“…In this section we briefly review the Optimal Weights Filter in order to adapt it for removing the impulse noise. Based on similarities among local patches, the Optimal Weights Filter [25] was initially introduced to deal with the Gaussian noise model,…”

“…The Optimal Weights Filter is constructed by minimizing a tight bound of the quadratic risk. It is shown in [25] that the optimal weights are given by the formula (6) via the triangular kernel (7). This minimization procedure gives also an exact formula for the bandwidth a as stated below.…”

“…However, compared with [21] (which is an improvement on [28]), there are two major differences in the present work: (1) The use of the SROAD statistic rather than the ROAD statistic makes the choice of the parameter H much more stable (it is not sensible to the choice of the size R of the windows used for the calculation of the SROAD statistic); accordingly the choice of the parameter H is less sensitive to the value of p (since the choice of R depends the value of p). (2) In the weighted means here we use the optimal weights obtained in [25] instead of the usual Gaussian weights. Simulation results show that the performance of the filters are very similar for small values of p (≤ 40%), but the new filter performs significantly better for large value of p (p > 40%), especially for p ≥ 60%.…”

“…Recall that the TriF introduced in [18] is also a one-step approach, but not patch-based; the patch-based filter proposed in [28] is an improvement of the TriF. Compared with the filters proposed in [18] and [28], in addition to the two aspects mentioned above, RRWF has one further advantage: the weights w H (x) and w H1 (x) (see (17) and (25)) used here are onedimensional indexed (x ∈ I) rather than two-dimensional indexed ((x 0 , x) ∈ I 2 ), in the sense that here we use the same weights w H (x) and w H1 (x) for each research window centered at x 0 ∈ I, while in [18] and [28] we need to calculate the new weights w H (x 0 , x) depending on both x and x 0 , when the center x 0 of the research window varies. This progress reduces very significantly the calculation time.…”

“…The outline of this paper is as follows. In Section 2, we give a brief review of the Optimal Weights Filter [25] and the TriF with ROAD statistic [18]. In Section 3, we introduce Reliable Weights, construct our new filter and establish its convergence.…”