Theorem 4. Suppose u ∈ C2(W), and let r, h, a > 0 such that r, h →0 and h = O(r). Let us consider the continuous function �g defined by for t ≠0, where

Let �f be the continuous function defined by
Then, for x ∈ W,

According to Theorem 4 the Yaroslavsky neighborhood filter acts as an evolution PDE with two terms. The first term is proportional to the second derivative of u in the direction x, which is tangent to the level line passing through x. The second term is proportional to the second derivative of u in the direction h, which is orthogonal to the level line passing through x.

The weighting coefficient of the tangent diffusion, uxx, is given by �g ( |Du|). The function �g is positive and decreasing. Thus, there is always diffusion in that direction. The weight of the normal diffusion, uhh, is given by �f ( |Du|). As the function �f takes positive and negative values (see Figure 1), the filter behaves as a filtering/enhanc-ing algorithm in the normal direction depending on |Du|. The intensity of the filtering in the tangent diffusion and the enhancing in the normal diffusion tend to zero when the gradient tends to infinity. Thus, points with a very large gradient are not altered.

The neighborhood filter asymptotically behaves as the Perona–Malik equation, 35 also creating shocks inside smooth regions (see Buades et al. 7 for more details on this comparison).

3. nL-means aLGoRIthm

Given a discrete noisy image v = {v(i ) | i ∈ I}, the estimated value NL[v](i), for a pixel i, is computed as weighted average of all the pixels in the image:

figure 1. magnitude of the tangent diffusion (continuous line) and normal diffusion (dashed line – –) of theorem 4.

0.2

0.15

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0.05

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8

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where the family of weights {w(i, j)}j depend on the similarity between the pixels i and j and satisfy the usual conditions 0 ≤ w(i, j) ≤ 1 and ∑j w(i, j ) = 1.

The similarity between two pixels i and j depends on the similarity of the intensity gray level vectors v(Ni) and v(Nj), where Nk denotes a square neighborhood of fixed size centered at a pixel k. This similarity is measured as a decreasing function of the weighted Euclidean distance, ½½v (Ni) − v(Nj)½½ 2 2, a , where a > 0 is the standard deviation of the Gaussian kernel. The expectation of the Euclidean distance of the noisy neighborhoods is This equality shows the robustness of the algorithm since in expectation the Euclidean distance preserves the order of similarity between pixels.

figure 2. q1 and q2 have a large weight in nL-means because their similarity windows are similar to that of p, while the weight w(p, q3) is much smaller.