figure 7. Denoising experience on a natural image. from left to right
and from top to bottom: noisy image (standard deviation s = 20),
Gaussian convolution (h = 1. 8), anisotropic filter (h = 2. 4), total variation
(l = 0.04), the Yaroslavsky neighborhood filter (r = 7, h = 28), and the
nL-means algorithm. Parameters have been set for each algorithm so that
the removed quadratic energy is equal to the energy of the added noise.
figure 8. Denoising experience on a natural 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 filter, and nL-means.
to be the best way to characterize artifact-free algorithms.
The transformation of a white noise into any correlated
signal creates structure and artifacts. Only white noise is
perceptually devoid of visual structure, a principle first
stated by Attneave. 4 Figure 9 shows how denoising methods
transform a white noise.
The convolution with a Gauss kernel is equivalent to
the product in the Fourier domain with a Gauss kernel of
inverse standard deviation. Therefore, convolving the noise
with a kernel reinforces the low frequencies and cancels the
high ones. Thus, the filtered noise actually shows big grains
due to its prominent low frequencies.
Noise filtered by a wavelet thresholding is no more a
white noise. The few coefficients with a magnitude larger
than the threshold are spread all over the image. The
pixels that do not belong to the support of one of these
coefficients are set to zero. The visual result is a constant
image with superposed wavelets as displayed in Figure 9.
It is easy to prove that the denoised noise is spatially highly
Given a noise realization, the filtered value by the neighborhood filter at a pixel i only depends on its value n(i) and
the parameters h and r. The neighborhood filter averages
noise values at a distance from n(i) less or equal than h.
Thus, when the size r of the neighborhood increases,
by the law of large numbers the filtered value tends to the
expectation of the Gauss distribution restricted to the
interval (n(i)− h, n(i)+ h). This filtered value is therefore
a deterministic function of n(i) and h. Independent random variables are mapped by a deterministic function on
independent variables. Thus the noise-to-noise
requirement is asymptotically satisfied by the neighborhood filter. NL-means satisfies the noise-to-noise principle in a
similar manner. However, a mathematical statement and
proof of this property are more intricate and we shall skip
numerical Comparison: Table 1 displays the mean square
error for the denoising experiments given in the paper.
This numerical measurement is the most objective one,
since it does not rely on any visual interpretation. However,
this error is not computable in a real problem and a small
mean square error does not assure a high visual quality. So
all above-discussed criteria seem necessary to compare the
performance of denoising algorithms.
6. extensIons sInCe 2005
Our conclusions on the better denoising performance of
nonlocal methods with respect to former state-of-the-art
algorithms such as the total variation or the wavelet thresholding have been widely accepted. The “method noise”
methodology to compare the denoising performance has
been adopted ever since.
The NL-means algorithm is easily extended to the
denoising of image sequences and video. The denoising
algorithm involves indiscriminately pixels not belonging
only to same frame but also to all frames in the image.