image, the whole color patch containing the red, green, and blue pixels is compared. Thus, a single weight is obtained for any pair of pixels and used for the denoising of the three channels at this pixel.

The filtering parameter h has been fixed to 0.8s when a noise of standard deviation s is added. Due to the fast decay of the exponential kernel, large Euclidean distances lead to nearly zero weights acting as an automatic threshold (Figure 4).

Visual Comparison: Visual criteria remains essential to decide if the quality of the image has been improved by the denoising method. We display some denoising experiences comparing the NL-means algorithm with local smoothing and frequency domain filters. All experiments have been simulated by adding a Gaussian white noise of standard deviation s to the true image. The objective is to compare the visual quality of the restored images, the nonpresence of artifacts, and the correct reconstruction of edges, texture, and details.

Due to the nature of the algorithm, the most favorable case for NL-means is the textured or periodic case. In this situation, for every pixel i, we can find a large set of samples with a very similar configuration. See Figure 4f for an example of the weight distribution of the NL-means algorithm for a periodic image. Figure 5 illustrates the performance of the NL-means for a natural texture and Figure 6 for an artificial periodic pattern.

Natural images also have enough redundancy to be
restored by NL-means, see Figure 7. Flat zones present
a huge number of similar configurations lying inside
the same object, see Figure 4a. Straight or curved edges
have a complete line of pixels with similar configurations,
see Figure 4b, c. In addition, natural images allow us to find
many similar configurations in far away pixels, as Figure 4f
shows. Figure 7 shows an experiment on a natural image.
This experience must be compared with Figure 3, where we
display the method noise of the original image. The blurred
or degraded structures of the restored images coincide
with the noticeable structures of its method noise. Figure 8
shows that the frequency domain filters are well adapted to
the recovery of oscillatory patterns. Although some artifacts
are noticeable in both solutions, the stripes are well recon-
structed. The DCT transform seems to be more adapted
to this type of texture, and stripes are little better recon-
structed. NL-means also performs well on this type of tex-
ture, due to its high degree of redundancy.
noise-to-noise Criterion: The noise-to-noise principle
requires that a denoising algorithm transforms a white
noise into white noise. This paradoxical requirement seems

figure 5. nL-means denoising experiment with a Brodatz texture image. Left: noisy image with standard deviation 30. Right: nL-means restored image. the fourier transform of the noisy and restored images show how main features are preserved even at high frequencies.

figure 4. on the right-hand side of each pair, we display the weight distribution used to estimate the central pixel of the left image by the nL-means algorithm. (a) In flat zones, the weights are distributed as a convolution filter (as a Gaussian convolution). (b) on straight edges, the weights are distributed in the direction of the edge (like for af). (c) on curved edges, the weights favor pixels belonging to the same contour or level line, which is a strong improvement with respect to af. (d) In a flat neighborhood, the weights are distributed in a gray-level neighborhood (like for a neighborhood filter). In the cases of (e) and (f), the weights are distributed across the more similar configurations, even though they are far away from the observed pixel. this shows a behavior similar to a nonlocal neighborhood filter or to an ideal Wiener filter.

figure 6. Denoising experience on a periodic image. from left to right and from top to bottom: noisy image (standard deviation 35), total variation minimization, neighborhood filter, translation invariant wavelet thresholding, DCt sliding window Wiener filtering, and nL-means.

(a) (b) (c)