The application of a denoising algorithm should not alter the non-noisy images. So the method noise should be very small when some kind of regularity for the image is assumed. If a denoising method performs well, the method noise must look like a noise even with non-noisy images and should contain as little structure as possible. Since even good quality images have some noise, it makes sense to evaluate any denoising method in that way, without the traditional “add noise and then remove it” trick. We shall list formulas permitting to compute and analyze the method noise for several classical local smoothing filters: the Gaussian filtering, 27 the anisotropic filtering, 2, 35 the total variation minimization, 39 and the neighborhood filtering. 46 The formal analysis of the method noise for the frequency domain filters falls out of the scope of this paper. These method noises can also be computed but their interpretation depends on the particular choice of the wavelet or Fourier basis.

the Gaussian filtering: The image isotropic linear filtering
boils down to the convolution of the image by a linear sym-
metric kernel. The paradigm of such kernels is of course
the Gaussian kernel . In that case, Gh has
standard deviation h and it is easily seen that
Theorem 1 (Gabor 1960). The image method noise of the
Gaussian convolution satisfies u − Gh ∗ u = −h2 ∆ u + o(h2).

The Gaussian method noise is zero in harmonic parts of the image and very large near edges or texture, where the Laplacian cannot be small. As a consequence, the Gaussian convolution is optimal in flat parts of the image but edges and texture are blurred.

the anisotropic filtering: The anisotropic filter (AF)
attempts to avoid the blurring effect of the Gaussian by con-
volving the image u at x only in the direction orthogonal to
Du(x). The idea of this filter goes back to Perona and Malik35
and is interpreted in Alvarez et al. 2 It is defined by
for x such that Du(x) ≠ 0 and where (x, y)⊥ = (−y, x) and Gh
is the one-dimensional Gauss function with variance h2.
If one assumes that the original image u is twice continu-
ously differentiable (C2) at x, it is easily shown by a second-
order Taylor expansion that
Theorem 2. The image method noise of AFh satisfies (for
Du(x) ≠ 0)

By curv(u)(x), we denote the curvature, that is, the signed inverse of the radius of curvature of the level line passing by x. This method noise is zero wherever u behaves locally like a straight line and large in curved edges or texture (where the curvature and gradient operators take high values). As a consequence, the straight edges are well restored

while flat and textured regions are degraded. total Variation Minimization: The total variation minimization was introduced by Rudin et al. 39 Given a noisy image v(x), these authors proposed to recover the original image u(x) as the solution of the minimization problem:

TVFλ(v) = arg min u TV(u) + λ∫|v(x) − u(x)|2dx, where TV (u) denotes the total variation of u and λ is a given Lagrange multiplier. The minimum of the above minimization problem exists and is unique. The parameter λ is related to the noise statistics and controls the degree of filtering of the obtained solution.

Theorem 3. The method noise of the total variation minimization is

As in the anisotropic case, straight edges are maintained because of their small curvature. However, details and texture can be over smoothed if λ is too small.

neighborhood filtering: The previous filters are based on a notion of spatial neighborhood or proximity. Neighborhood filters instead take into account gray-level values to define neighboring pixels. In the simplest and more extreme case, the denoised value at pixel i is an average of values at pixels that have a gray-level value close to u(i). The gray-level neighborhood is therefore

B(i, h) = { j ∈ I | u(i) − h < u( j) < u(i) + h}.

This is a fully nonlocal algorithm, since pixels belonging to the whole image are used for the estimation at pixel i. This algorithm can be written in a more continuous form: where W⊂R2 is an open and bounded set, and C(x) =