Nonlinear Image Processing and Filtering: A Unified Approach Based on Vertically Weighted Regression

Rafajłowicz, Ewaryst; Pawlak, Mirosław; Steland, Angsar

doi:10.2478/v10006-008-0005-z

Abstract

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.

References

Barash D. (2002). A fundamental relationship between bilateral filtering, adaptive smoothing, and the nonlinear diffusion equation, IEEE Transactions on Pattern Analysis and Machine Intelligence24(6): 844-847.10.1109/TPAMI.2002.1008390
Search in Google Scholar
Barner K., Sarham A. and Hadie R. (1999). Partition-based weighted sum filters for image restoration, IEEE Transactions on Image Procedings8(5): 740-745.10.1109/83.760341
Search in Google Scholar
Barner K.E. and Arce G.R. (2004). Nonlinear Signal and Image Processing: Theory, Methods, and Applications, CRC Press, Boca Raton, FL.10.1201/9780203010419
Search in Google Scholar
Buades A., Coll B. and Morel J. (2005). A review of image denoising algorithms, with a new one, SIAM Journal on Multiscale Modeling and Simulation4(2): 490-530.10.1137/040616024
Search in Google Scholar
Chiu C., Glad K., Godtliebsen F. and Marron J. (1998). Edgepreserving smoother for image processing, Journal of the American Statistical Association93(442): 526-541.10.1080/01621459.1998.10473702
Search in Google Scholar
Efromovich S. (1999). Nonparametric Curve Estimation: Methods, Theory and Applications, Springer-Verlag, New York.
Search in Google Scholar
Elad M. (2002): On the origin of the bilateral filter and ways to improve it, IEEE Transactions on Image Processing11(10): 1141-1150.10.1109/TIP.2002.801126
Search in Google Scholar
Hall P. and Koch S. (1992). On the feasibility of cross-validation in image analysis, SIAM Journal on Applied Mathematics52(1): 292-313.10.1137/0152015
Search in Google Scholar
Jain A. (1989): Fundamentals of Digital Image Processing, Prentice Hall, New York.
Search in Google Scholar
Krzyżak A., Rafajłowicz E. and Skubalska-Rafajłowicz E. (2001). Clipped median and space-filling curves in image filtering, Nonlinear Analysis47(1): 303-314.10.1016/S0362-546X(01)00178-X
Search in Google Scholar
Lee J. (1983). Digital image smoothing and the sigma filter, Computer Vision, Graphics and Image Processing24(2): 255-269.10.1016/0734-189X(83)90047-6
Search in Google Scholar
Mitra S. and Sicuranza G. (2001). Nonlinear Image Processing, Academic Press, San Diego.
Search in Google Scholar
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 Intelligence13(6): 514-529.10.1109/34.87339
Search in Google Scholar
Pawlak M. and Liao S. X. (2002). On the recovery of a function on a circular domain, IEEE Transactions on Information Theory48(10): 2736-2753.10.1109/TIT.2002.802627
Search in Google Scholar
Pawlak M. and Rafajłowicz E. (1999). Vertically weighted regression - A tool for nonlinear data anlysis and constructing control charts, Statistical Archives84: 367-388.10.1007/s101820000039
Search in Google Scholar
Pawlak M. and Rafajłowicz E. (2001). Jump preserving signal reconstruction using vertical weighting, Nonlinear Analysis47(1): 327-338.10.1016/S0362-546X(01)00180-8
Search in Google Scholar
Pawlak M., Rafajłowicz E. and Steland A. (2004). On detecting jumps in time series: Nonparametric setting, Nonparametric Statistics16(3-4): 329-347.10.1080/10485250410001656435
Search in Google Scholar
Polzehl J. and Spokoiny V. (2000). Adaptive weights smoothing with applications to image restoration, Journal of the Royal Statistical Society B62(2): 335-354.10.1111/1467-9868.00235
Search in Google Scholar
Smith S. and Brady J. M. (1997). SUSAN - A new approach to low level image processing, International Journal of Computer Vision23(1): 45-78.10.1023/A:1007963824710
Search in Google Scholar
Steland A. (2003). Jump-preserving monitoring of dependent processes using pilot estimators, Statistics and Decision21(4): 343-366.10.1524/stnd.21.4.343.25348
Search in Google Scholar
Steland A. (2005). On the distribution of the clipping median under a mixture model, Statistics and Probability Letters71(1): 1-13.10.1016/j.spl.2004.10.016
Search in Google Scholar
Takeda H., Farsiu S., and Milanfar P. (2007). Kernel regression for image processing and reconstruction, IEEE Transactions on Image Processing16(2): 349-366.10.1109/TIP.2006.88833017269630
Search in Google Scholar
Tomasi C. and Manduchi R. (1998). Bilateral filtering for gray and color images, IEEE International Conference on Computer Vision, pp. 839-845.
Search in Google Scholar
van der Vaart A. (1998). Asymptotic Statistics, Cambridge University Press, Cambridge, 1998.10.1017/CBO9780511802256
Search in Google Scholar
Wasserman L. (2006). All of Nonparametric Statistics, Springer-Verlag, New York.
Search in Google Scholar
Yaroslavsky L. (1985). Digital Picture Processing, Springer-Verlag, New York.10.1007/978-3-642-81929-2
Search in Google Scholar

Nonlinear Image Processing and Filtering: A Unified Approach Based on Vertically Weighted Regression

Abstract

Paradigm

My account