figure 10. Weight distribution of nL-means applied to a movie. In
(a), (b), and (c) the first row shows a five frames image sequence. In
the second row, the weight distribution used to estimate the central
pixel (in white) of the middle frame is shown. the weights are equally
distributed over the successive frames, including the current one.
they actually involve all the candidates for the motion estimation
instead of picking just one per frame. the aperture problem can be
taken advantage of for a better denoising performance by involving
more pixels in the average.
local PDE’s has been discussed in Buades et al., 7, 8 leading to an adaptation of NL-means that avoids the stair casing effect. Yet, the main interest has shifted to defining
nonlocal PDEs. The extension of the NL-means method to
define nonlocal image-adapted differential operators and
nonlocal variational methods starts with Kindermann
et al., 25 who propose to perform denoising and deblurring
by nonlocal functionals. Several articles on deblurring
have followed this variational line22, 23, 32 (for image segmentation) and Lou et al. 28 for deconvolution and tomo-graphic reconstruction.
A particular notion of nonlocal PDE has emerged,
whose coefficients are actually image dependent. For
instance, in Elmoataz et al., 21 the image colorization is
viewed as the minimization of a discrete partial differential functional on the weighted block graph. Thus, it
can be seen either as a nonlocal heat equation on the
image or as a local heat equation on the space of image
The exploration of image redundancy and its application to image restoration has led to new attempts at sparse
image representations by block dictionaries. 24 Algorithms
and application have been developed by Chatterjee and
Milanfar, 11 Mairal et al., 30 and Protter and Elad. 36
This work was partially financed by MCYIT grant number
1. adams, a., Gelfand, n., Dolson, J.,
levoy, M. Gaussian kd-trees for fast
high-dimensional filtering. ACM Trans.
Graphics 28, 3 (2009), 1–12.
2. alvarez, l., lions, P., Morel, J. Image
selective smoothing and edge
detection by nonlinear diffusion. II.
SIAM J. Numer. Anal. 29, 3 (1992),
3. arias, P., Caselles, V., Sapiro, G. a
variational framework for non-local
image inpainting. In Proceedings
of the 7th International
Conference on Energy
Minimization Methods in Computer
Vision and Pattern Recognition
(EMMCVPR´09) (Bonn, Germany,
august 24–27, 2009), Springer,
4. attneave, F. Some informational
aspects of visual perception.
Psychol. Rev. 61, 3 (1954),
5. Boulanger, J., Sibarita, J., Kervrann,
C., Bouthemy, P. non-parametric
regression for patch-based
fluorescence microscopy image
sequence denoising. In 5th IEEE
International Symposium on
Biomedical Imaging: From Nano to
Macro, 2008 (Paris, France,
May 14–17, 2008), ISBI 2008,
6. Buades, a., Coll, B., Morel, J. a review
of image denoising algorithms, with a
new one. Multiscale Model. Simul. 4,
2 (2005), 490–530.
7. Buades, a., Coll, B., Morel,
J. neighborhood filters and
PDe’s. Numer. Math. 105, 1 (2006),
8. Buades, a., Coll, B., Morel, J.
The staircasing effect in
neighborhood filters and its solution.
IEEE Trans. I.P. 15, 6 (2006),
9. Buades, a., Coll, B., Morel, J.
nonlocal image and movie denoising.
Int. J. Comput. Vision 76, 2 (2008),
10. Buades, a., Coll, B., Morel, J.,
Sbert, C. Self-similarity driven color
demosaicking. IEEE Trans. I.P. 18,
6 (2009), 1192–1202.
11. Chatterjee, P., Milanfar, P. Image
denoising using locally learned
dictionaries. Comput. Imag.
VII, SPIE 7246 (2009),
12. Coifman, r., Donoho, D.
In Wavelets and Statistics.
Lecture Notes in Statistics
(Springer Verlag, new york, 1995),
13. Coupé, P., yger, P., Barillot, C.
Fast non local means denoising
for 3D Mr images. In Medical
Image Computing and Computer-
2006. Lecture Notes in Computer
Science (Springer, new york, 2006),