A class of nonparametric smoothing kernel methods for image processing and filtering that possess edge-preserving properties is examined. The proposed approach is a nonlinearly modified version of the classical nonparametric regression estimates utilizing the concept of vertical weighting. The method unifies a number of known nonlinear image filtering and denoising algorithms such as bilateral and steering kernel filters. It is shown that vertically weighted filters can be realized by a structure of three interconnected radial basis function (RBF) networks. We also assess the performance of the algorithm by studying industrial images.
@article{bwmeta1.element.bwnjournal-article-amcv18i1p49bwm,
author = {Ewaryst Rafaj\l owicz and Miros\l aw Pawlak and Angsar Steland},
title = {Nonlinear image processing and filtering: A unified approach based on vertically weighted regression},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {18},
year = {2008},
pages = {49-61},
zbl = {1243.94008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv18i1p49bwm}
}
Ewaryst Rafajłowicz; Mirosław Pawlak; Angsar Steland. Nonlinear image processing and filtering: A unified approach based on vertically weighted regression. International Journal of Applied Mathematics and Computer Science, Tome 18 (2008) pp. 49-61. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv18i1p49bwm/
[000] Barash D. (2002). A fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation, IEEE Transactions on Pattern Analysis and Machine Intelligence 24(6): 844-847.
[001] Barner K., Sarham A. and Hadie R. (1999). Partition-based weighted sum filters for image restoration, IEEE Transactions on Image Procedings 8(5): 740-745.
[002] Barner K.E. and Arce G.R. (2004). Nonlinear Signal and Image Processing: Theory, Methods, and Applications, CRC Press, Boca Raton, FL.
[003] Buades A., Coll B. and Morel J. (2005). A review of image denoising algorithms, with a new one, SIAM Journal on Multiscale Modeling and Simulation 4(2): 490-530. | Zbl 1108.94004
[004] Chiu C., Glad K., Godtliebsen F. and Marron J. (1998). Edgepreserving smoother for image processing, Journal of the American Statistical Association 93(442): 526-541.
[005] Efromovich S. (1999). Nonparametric Curve Estimation: Methods, Theory and Applications, Springer-Verlag, New York. | Zbl 0935.62039
[006] Elad M. (2002): On the origin of the bilateral filter and ways to improve it, IEEE Transactions on Image Processing 11(10): 1141-1150.
[007] Hall P. and Koch S. (1992). On the feasibility of cross-validation in image analysis, SIAM Journal on Applied Mathematics 52(1): 292-313. | Zbl 0743.60041
[008] Jain A. (1989): Fundamentals of Digital Image Processing, Prentice Hall, New York. | Zbl 0744.68134
[009] Krzyżak A., Rafajłowicz E. and Skubalska-Rafajłowicz E. (2001). Clipped median and space-filling curves in image filtering, Nonlinear Analysis 47(1): 303--314. | Zbl 1042.94503
[010] Lee J. (1983). Digital image smoothing and the sigma filter, Computer Vision, Graphics and Image Processing 24(2): 255-269.
[011] Mitra S. and Sicuranza G. (2001). Nonlinear Image Processing, Academic Press, San Diego.
[012] P. Saint-Marc J.S., and Medioni G. C. (1991). Adaptive smoothing: A general tool for early vision, IEEE Transations on Pattern Analysis and Machine Intelligence 13(6): 514-529.
[013] Pawlak M. and Liao S. X. (2002). On the recovery of a function on a circular domain, IEEE Transactions on Information Theory 48(10): 2736-2753. | Zbl 1062.94002
[014] Pawlak M. and Rafajłowicz E. (1999). Vertically weighted regression - A tool for nonlinear data anlysis and constructing control charts, Statistical Archives 84: 367-388. | Zbl 1117.62499
[015] Pawlak M. and Rafajłowicz E. (2001). Jump preserving signal reconstruction using vertical weighting, Nonlinear Analysis 47(1): 327-338. | Zbl 1043.94521
[016] Pawlak M., Rafajłowicz E. and Steland A. (2004). On detecting jumps in time series: Nonparametric setting, Nonparametric Statistics 16(3-4): 329-347. | Zbl 1065.62200
[017] Polzehl J. and Spokoiny V. (2000). Adaptive weights smoothing with applications to image restoration, Journal of the Royal Statistical Society B 62(2): 335-354.
[018] Smith S. and Brady J. M. (1997). SUSAN - A new approach to low level image processing, International Journal of Computer Vision 23(1): 45-78.
[019] Steland A. (2003). Jump-preserving monitoring of dependent processes using pilot estimators, Statistics and Decision 21(4): 343-366. | Zbl 1041.62076
[020] Steland A. (2005). On the distribution of the clipping median under a mixture model, Statistics and Probability Letters 71(1): 1-13. | Zbl 1058.62018
[021] Takeda H., Farsiu S., and Milanfar P. (2007). Kernel regression for image processing and reconstruction, IEEE Transactions on Image Processing 16(2): 349-366.
[022] Tomasi C. and Manduchi R. (1998). Bilateral filtering for gray and color images, IEEE International Conference on Computer Vision, pp. 839-845.
[023] van der Vaart A. (1998). Asymptotic Statistics, Cambridge University Press, Cambridge, 1998. | Zbl 0910.62001
[024] Wasserman L. (2006). All of Nonparametric Statistics, Springer-Verlag, New York. | Zbl 1099.62029
[025] Yaroslavsky L. (1985). Digital Picture Processing, Springer-Verlag, New York.