Abstract
Neighborhood filters are nonlocal image and movie filters which reduce the noise by averaging similar pixels. The first object of the paper is to present a unified theory of these filters and reliable criteria to compare them to other filter classes. A CCD noise model will be presented justifying the involvement of neighborhood filters. A classification of neighborhood filters will be proposed, including classical image and movie denoising methods and discussing further a recently introduced neighborhood filter, NL-means. In order to compare denoising methods three principles will be discussed. The first principle, “method noise”, specifies that only noise must be removed from an image. A second principle will be introduced, “noise to noise”, according to which a denoising method must transform a white noise into a white noise. Contrarily to “method noise”, this principle, which characterizes artifact-free methods, eliminates any subjectivity and can be checked by mathematical arguments and Fourier analysis. “Noise to noise” will be proven to rule out most denoising methods, with the exception of neighborhood filters. This is why a third and new comparison principle, the “statistical optimality”, is needed and will be introduced to compare the performance of all neighborhood filters.
The three principles will be applied to compare ten different image and movie denoising methods. It will be first shown that only wavelet thresholding methods and NL-means give an acceptable method noise. Second, that neighborhood filters are the only ones to satisfy the “noise to noise” principle. Third, that among them NL-means is closest to statistical optimality. A particular attention will be paid to the application of the statistical optimality criterion for movie denoising methods. It will be pointed out that current movie denoising methods are motion compensated neighborhood filters. This amounts to say that they are neighborhood filters and that the ideal neighborhood of a pixel is its trajectory. Unfortunately the aperture problem makes it impossible to estimate ground true trajectories. It will be demonstrated that computing trajectories and restricting the neighborhood to them is harmful for denoising purposes and that space-time NL-means preserves more movie details.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alvarez, L., Guichard, F., Lions, P. L., & Morel, J. M. (1993). Axioms and fundamental equations of image processing. Archive for Rational Mechanics and Analysis, 123, 199–257.
Attneave, F. (1954). Some informational aspects of visual perception. Psychological Review, 61, 183–193.
Awate, S. P. & Whitaker, R. T. (2005). Higher-order image statistics for unsupervised, information-theoretic, adaptive, image filtering. In Proceedings of the 2005 IEEE computer society conference on computer vision and pattern recognition (CVPR’05) (Vol. 2, pp. 44–51).
Azzabou, N., Paragios, N., & Guichard, F. (2006). Random walks, constrained multiple hypothesis testing and image enhancement. In ECCV (Vol. 1, pp. 379–390).
Boulanger, J., Kervrann, C., & Bouthemy, P. (2006). Adaptive space-time patch-based method for image sequence restoration. In Proceedings of the ECCV’06 workshop on statistical methods in multi-image and video processing (SMVP’06), Graz, Austria, May 2006.
Brailean, J. C., Kleihorst, R. P., Efsratiadis, S., Katsaggelos, A. K., & Lagendijk, R. L. (1995). Noise reduction filters for dynamic image sequences: a review. Proceedings of the IEEE, 83, 1272–1292.
Buades, A., Coll, B., & Morel, J. M. (2005a). A review of image denoising methods, with a new one. Multiscale Modeling and Simulation, 4(2), 490–530.
Buades, A., Coll, B., & Morel, J. M. (2005b). A non-local algorithm for image denoising, In IEEE international conference on computer vision and pattern recognition.
Buades, A., Coll, B., & Morel, J. M. (2005c). Denoising image sequences does not require motion estimation. Preprint, CMLA, N 2005-18, May 2005. http://www.cmla.ens-cachan.fr/Cmla/.
Colleen Gino, M. (2004). “Noise, noise, noise”. http://www.astrophys-assist.com/educate/noise/noise.htm.
Cremers, D., & Grady, L. (2006). Statistical priors for efficient combinatorial optimization via graph cuts. In European conference on computer vision.
Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. (2006, submitted). Image denoising by sparse 3D transform-domain collaborative filtering. IEEE Transactions on Image Processing.
Donoho, D., & Johnstone, I. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika, 81, 425–455.
Efros, A., & Leung, T. (1999). Texture synthesis by nonparametric sampling. In Proceedings of the international conference on computer vision (ICCV 99) (Vol. 2, pp. 1033–1038).
Gilboa, G., & Osher, S. (2006). Nonlocal linear image regularization and supervised segmentation. UCLA CAM Report 06-47.
Gilboa, G., Darbon, J., Osher, S., & Chan, T. F. (2006). Nonlocal convex functionals for image regularization. UCLA CAM Report 06-57.
Gonzalez, R. C., & Woods, R. E. (2002). Digital image processing (2nd ed.). New York: Prentice Hall.
Horn, B., & Schunck, B. (1981). Determining optical flow. Artificial Intelligence, 17, 185–203.
Howell, S. B. (2000). Handbook of CCD astronomy. Cambridge: Cambridge University Press.
Huang, T. (1981). Image sequence analysis. Berlin: Springer.
Keeling, S. L., & Stollberger, R. (2002). Nonlinear anisotropic diffusion filtering for multiscale edge enhancement. Inverse Problems, 18, 175–190.
Kervrann, C., & Boulanger, J. (2006). Unsupervised patch-based image regularization and representation. In Proceedings of the European conference on computer vision (ECCV’06), Graz, Austria, May 2006.
Kindermann, S., Osher, S., & Jones, P. W. (2005). Deblurring and denoising of images by nonlocal functionals. Multiscale Modeling and Simulation, 4(4), 1091–1115.
Kokaram, A. C. (1993). Motion picture restoration. PhD thesis, Cambridge University.
Lee, J. S. (1980). Digital image enhancement and noise filtering by use of local statistics. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2, 165–168.
Lee, J. S. (1983). Digital image smoothing and the sigma filter. Computer Vision, Graphics and Image Processing, 24, 255–269.
Liu, C., Freeman, W. T., Szeliski, R., & Kang, S. B. (2006). Noise estimation from a single image. In CVPR.
Mahmoudi, M., & Sapiro, G. (2005). Fast image and video denoising via non-local means of similar neighborhoods. IEEE Signal Processing Letters, 12(12), 839–842.
Martinez, D. M. (1986). Model-based motion estimation and its application to restoration and interpolation of motion pictures. PhD thesis, Massachusetts Institute of Technology.
Merriman, B., Bence, J., & Osher, S. (1992). Diffusion generated motion by mean curvature. In Proceedings of the geometry center workshop.
Meyer, Y. (2002). Oscillating patterns in image processing and nonlinear evolution equations. In AMS university lecture series (Vol. 22).
Nagel, H. H. (1983). Constraints for the estimation of displacement vector fields from image sequences. In Proceedings of the eighth international joint conference on artificial intelligence (IJCAI ’83) (pp. 945–951).
Osher, S., Burger, M., Goldfarb, D., Xu, J., & Yin, W. (2005). An iterative regularization method for total variation based image restoration. Multiscale Modelling and Simulation, 4, 460–489.
Ozkan, M. K., Sezan, M. I., & Tekalp, A. M. (1993). Adaptive motion compensated filtering of noisy image sequences. IEEE Transactions on Circuits and Systems for Video Technology, 3, 277–290.
Papenberg, N., Bruhn, A., Brox, T., Didas, S., & Weickert, J. (2006). Highly accurate optic flow computation with theoretically justified warping. International Journal of Computer Vision, 67(2), 141–158.
Rudin, L., Osher, S., & Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D, 60, 259–268.
Samy, R. (1985). An adaptive image sequence filtering scheme based on motion detection. SPIE, 596, 135–144.
Sezan, M. I., Ozkan, M. K., & Fogel, S. V. (1991). Temporally adaptive filtering of noisy sequences using a robust motion estimation algorithm. In Proceedings of the international conference on acoustics, speech, signal processing (Vol. 91, pp. 2429–2432).
Smith, S. M., & Brady, J. M. (1997). Susan—new approach to low level image processing. International Journal of Computer Vision, 23(1), 45–78.
Tadmor, E., Nezzar, S., & Vese, L. (2004). A multiscale image representation using hierarchical (BV,L 2) decompositions. Multiscale Modeling and Simulation, 2, 554–579.
Tomasi, C., & Manduchi, R. (1998). Bilateral filtering for gray and color images. In Sixth international conference on computer vision (pp. 839–846).
Tukey, J. (1977). Exploratory data analysis. Reading: Addison-Wesley.
Weickert, J. (1998). On discontinuity-preserving optic flow. In Proceedings of the computer vision and mobile robotics workshop (pp. 115–122).
Weickert, J., & Schnörr, C. (2001). Variational optic flow computation with a spatio-temporal smoothness constraint. Journal of Mathematical Imaging and Vision, 14, 245–255.
Yaroslavsky, L. P. (1985). Digital picture processing—an introduction. Berlin: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Buades, A., Coll, B. & Morel, JM. Nonlocal Image and Movie Denoising. Int J Comput Vis 76, 123–139 (2008). https://doi.org/10.1007/s11263-007-0052-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-007-0052-1