Nonlinear Digital Filters pp 267-311 | Cite as

# Adaptive Nonlinear Filters

## Abstract

The nonlinear filters described in the previous chapters are usually optimized for a specific type of noise and sometimes for a specific type of signal. However, this is not usually the case in many applications of nonlinear filtering, especially in image processing. Images can be modeled as two-dimensional stochastic processes, whose statistics vary in the various image regions. Images are nonstationary processes. Furthermore the noise statistics, e.g., the noise standard deviation and even the noise probability density function, vary from application to application, as was described in chapter 3. Sometimes, the noise characteristics vary in the same application from one image to the next. Such cases are the channel noise in image transmission and the atmospheric noise (e.g., the cloud noise) in satellite images. In these environments non-adaptive filters cannot perform well because their characteristics depend on the noise and signal characteristics, which are unknown. Therefore, adaptive filters are the natural choice in such cases. Their performance depends on the accuracy of the estimation of certain signal and noise statistics, namely the signal mean and standard deviation and the noise standard deviation. The estimation is usually local, i.e., relatively small windows are used to obtain the signal and noise characteristics. An important property of these estimators is their robustness to impulse noise, which is present in many image processing applications. Another reason for using adaptive filters is the fact that edge information is very important for the human eye and must be preserved. Certain filters, e.g., the moving average, perform well in homogeneous image regions but fail close to edges. The opposite is true for other filters, e.g., for the median. A combined filter which performs differently in the image edges than in the image plateaus can be used in such a case. These filters are also called *decision directed filters* because they employ an edge detector to decide if an edge is present or not. Decision directed filtering can also be used in the cases of mixed additive white noise and impulsive noise. Impulses can be detected and removed before the additive noise filtering is performed. Another approach related to decision directed filtering is the *two-component model* filtering. An image is assumed to consist of two components, the low-pass and the high-pass component. The first one is mainly related to homogeneous image regions, whereas the second one is related to edge information. These two components can be processed in different ways. The output of the two corresponding filters can be recombined to give the final filtered image. The two-component image processing model has been used both for noise removal and image enhancement applications.

## Keywords

Less Mean Square Adaptive Filter Impulsive Noise Volterra Series Nonlinear Filter## Preview

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## References

- [1]H.A. David,
*Robust statistics*, Wiley, 1981.Google Scholar - [2]W.K. Pratt,
*Digital image processing*, Wiley, 1978.Google Scholar - [3]M.D. Levine,
*Vision in Man and Machine*, McGraw-Hill, 1985.Google Scholar - [4]I.E. Abdou, W.K. Pratt, “Quantitative design and evaluation of enhancement/ thresholding edge detectors”,
*Proc. IEEE*, vol. 67, May 1979.Google Scholar - [5]D.H. Ballard, C.M. Brown,
*Computer vision*, Prentice-Hall, 1982.Google Scholar - [6]I. Pitas, A.N. Venetsanopoulos, “Edge detectors based on order statistics”,
*IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. PAMI-8, no. 4, pp. 538–550, July 1986.CrossRefGoogle Scholar - [7]X.Z. Sun, A.N. Venetsanopoulos, “Adaptive schemes for noise filtering and edge detection by the use of local statistics”,
*IEEE Transactions on Circuits and Systems*, vol. CAS-35, no. 1, pp. 57–69, Jan. 1988.Google Scholar - [8]A.C. Bovik, D.C. Munson, “Edge detection using median comparisons”,
*Computer Vision Graphics and Image Processing*, vol. 33, pp. 377–389, 1986.CrossRefGoogle Scholar - [9]J. Hajek, Z. Sidak,
*Theory of rank tests*, Academic Press, 1967.Google Scholar - [10]A.C. Bovik, T.S. Huang, D.C. Munson, “Nonparametric tests for edge detection in noise”,
*Pattern recognition*, vol. 19, no. 3, pp. 209–219, 1986.CrossRefGoogle Scholar - [11]J.S. Lee, “Digital image enhancement and noise filtering by local statistics”,
*IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. PAMI-2, no. 2, pp. 165–168, March 1980.Google Scholar - [12]J.S. Lee, “Refined filtering of image noise using local statistics”,
*Computer Graphics and Image processing*, vol. 15, pp. 380–389, 1981.CrossRefGoogle Scholar - [13]X.Z. Sun, A.N. Venetsanopoulos, “Adaptive schemes for noise filtering and edge detection by use of local statistics”,
*IEEE Transactions on Circuits and Systems*, vol. CAS-35, no. 1, pp. 57–69, Jan. 1988.Google Scholar - [14]I. Scollar, B. Weidner, T.S. Huang, “Image filtering using the interquartile distance”,
*Computer Vision*,*Graphics and Image processing*, vol. 25, pp. 236–251, 1984.CrossRefGoogle Scholar - [15]N.F. Nahi, A. Habibi, “Decision directed recursive image enhancement”,
*IEEE Transactions on Circuits and Systems*, vol. CAS-6, pp. 286–293, March 1975.Google Scholar - [16]I. Pitas, A.N. Venetsanopoulos, “Nonlinear order statistic filters for image filtering and edge detection”
*Signal Processing*, vol. 10, pp. 395–413, 1986.CrossRefGoogle Scholar - [17]R.L.J. Martens,
*Adaptive nonlinear order statistic filters: analysis and comparison*, M.A.Sc. Thesis, University of Toronto, 1988.Google Scholar - [18]A. Restrepo, A.C. Bovik, “Adaptive trimmed mean filters for image restoration”, IEEE
*Transactions on Acoustics*,*Speech and Signal Processing*, vol. 36, no. 8, pp. 1326–1337, 1988.MATHCrossRefGoogle Scholar - [19]R.V. Hogg, “More light on the kurtosis and related statistics”,
*Journal Amer. Statist. Assoc.*, vol. 67, pp. 422–424, 1972.MathSciNetMATHCrossRefGoogle Scholar - [20]R. Bernstein, “Adaptive nonlinear filters for simultaneous removal of different kinds of noise in images”,
*IEEE Transactions on Circuits and Systems*, vol. CAS-34, no. 11, pp. 1275–1291, Nov. 1987.Google Scholar - [21]T. Peli, J.S. Lim, “Adaptive filtering for image enhancement”,
*Journal Opt. Eng.*,*vol.*21, pp. 108–112, 1982.Google Scholar - [22]J.S. Lim, “Image enhancement”, in
*Digital image processing techniques*, M.P. Ekstrom editor, Academic Press, 1984.Google Scholar - [23]T. Alexander,
*Adaptive signal processing*, Springer Verlag, 1986.Google Scholar - [24]M. Bellanger,
*Adaptive digital filters and signal analysis*, Marcel Dekker, 1987.Google Scholar - [25]I. Pitas, A.N. Venetsanopoulos, “Adaptive filters based on order statistics”,
*IEEE Transactions on Acoustics*,*Speech and Signal Processing*,under review.Google Scholar - [26]R. Martens, A.N. Venetsanopoulos, “A comparison of adaptive nonlinear filters using different images”, Proc. IEEE Int. Conf. on Systems Engineering, Dayton, Ohio, 1989.Google Scholar
- [27]R. Ding, A.N. Venetsanopoulos, “Generalized homomorphic and adaptive order statistic filters for the removal of impulsive and signal dependent noise”,
*IEEE Transactions on Circuits and Systems*, vol. CAS-34, no. 8, pp. 948–955, Aug. 1987.Google Scholar - [28]T. Koh, E.J. Powers, “An adaptive nonlinear digital filter with lattice orthogonalization”,
*Proc. IEEE ICASSP-83*, pp. 37–40, 1983.Google Scholar - [29]H.H. Chiang, C.L. Nikias, A.N. Venetsanopoulos, “Efficient implementations of quadratic digital filters” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.ASSP-34, pp. 1511–1528, Dec. 1986.Google Scholar
- [30]D. Mansour, A.H. Gray, “Frequency domain non-linear adaptive filter”,
*1981 IEEE Int. Conf. Acoustics*,*Speech and Signal Processing*, pp.550553, 1981.Google Scholar - [31]C.E. Davila, A.J. Welch, H.G. Rylander, “A second-order adaptive Volterra filter with rapid convergence”, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.ASSP-34, pp. 1259–1263, Sept. 1987.Google Scholar
- [32]J.C. Stapleton, S.C. Bass, “Adaptive noise cancellation for a class of nonlinear, dynamic reference channels”,
*1984 Int. Symp. on Circuits and Systems*, pp. 268–271, 1984.Google Scholar - [33]G.L. Sicuranza, G. Ramponi, “A variable-step adaptation algorithm for memory-oriented Volterra filters”,
*IEEE Transactions on Acoustics*,*Speech*,*and Signal Processing*, vol.ASSP-35, pp. 1492–1494, Oct. 1987.Google Scholar - [34]G.L. Sicuranza, G. Ramponi, “Adaptive nonlinear digital filters using distributed arithmetic”, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.ASSP-34, pp. 518–526, June 1986.Google Scholar
- [35]Y. Lou, C.L. Nikias, A.N. Venetsanopoulos, “VLSI array processing structure of quadratic digital filters with LMS algorithm”,
*1987 IEEE Int. Conf. Acoustics*,*Speech and Signal Processing*, pp. 1394–1397, 1987.Google Scholar - [36]E.J. Thomas, “Some considerations on the application of the Volterra representation of nonlinear networks to adaptive echo cancelers”,
*Bell System Technical Journal*, vol. 50, pp. 2797–2805, Oct. 1971.Google Scholar - [37]H.H. Chiang, C.L. Nikias, A.N. Venetsanopoulos, “Reconfigurable systolic array implementation of quadratic digital filters”,
*IEEE Transactions on Circuits and Systems*, vol. CAS-33, pp. 845–848, Aug. 1986.Google Scholar - [38]G. L. Sicuranza, A. Bucconi, P. Mitri, “Adaptive echo cancellation with nonlinear digital filters”,
*1984 IEEE Int. Conf. Acoustics*,*Speech and Signal Processing*, pp.3.10.1–3. 10. 4, 1984.Google Scholar - [39]X.Y. Gao, W.M. Snelgrove, D.A. Johns, “Nonlinear IIR adaptive filtering using a bilinear structure”,
*1989 Int. Symp. on Circuits and Systems*, Portland Oregon, 1989.Google Scholar - [40]D.D. Falconer, “Adaptive equalization of channel nonlinearities in QAM data transmission systems”,
*Bell System Technical Journal*, vol. 57, pp. 2589–2611, Sept. 1978.MATHGoogle Scholar - [41]T. Koh, E.J. Powers, “Second-order Volterra filtering and its application to nonlinear system identification”,
*IEEE Transactions on Acoustics*,*Speech*,*and Signal Processing*, vol.ASSP-33, pp. 1445–1455, Dec. 1985.Google Scholar - [42]E. Biglieri, A. Gersho, R.D. Gitlin, T.L. Lim, “Adaptive cancellation of nonlinear intersymbol interference for voiceband data transmission”,
*IEEE J. Selected Areas in Communications*, vol.SAC-2, pp. 765–777, Sept. 1984.Google Scholar - [43]G.L. Sicuranza, “Nonlinear digital filter realization by distributed arithmetic”,
*IEEE Transactions on Acoustics*,*Speech*,*and Signal Processing*, vol. ASSP-33, pp. 939–945, Aug. 1985.Google Scholar - [44]O. Agazzi, D.G. Messerschmitt, D.A. Hodges, “Nonlinear echo cancellation of data signals”,
*IEEE Transactions on Communications*, vol. COM30, pp. 2421–2433, Nov. 1982.Google Scholar - [45]C.F.N. Cowan, P.F. Adams, “Nonlinear system modeling: concept and application”,
*1984 IEEE Int. Conf. Acoustics*,*Speech and Signal Processing*, Mar. 1984.Google Scholar - [46]H.H. Chiang, C.L. Nikias, A.N. Venetsanopoulos, “Efficient implementations of digital Volterra filters”,
*1986 IEEE Int. Conf. Acoustics*,*Speech and Signal Processing*, pp. 857–860, 1986.Google Scholar - [47]T. Koh, E.J. Powers, R.W. Miksad, F.J. Fischer, “Application of nonlinear digital filters to modeling low-frequency, nonlinear drift oscillations of moored vessels in random seas”,
*The 16th Annual Offshore Technology Conference*, pp. 309–314, Houston, Texas, May 1984.Google Scholar - [48]M.J. Coker and D.N. Simkins, “A nonlinear adaptive noise canceler”,
*1980 IEEE Int. Conf. Acoustics*,*Speech and Signal Processing*, pp.470473, 1980.Google Scholar - [49]X.Y. Gao,
*Adaptive Volterra filters: algorithms and applications*, Technical report, University of Toronto, 1988.Google Scholar - [50]F. Palmieri, C.G. Boncelet Jr., “A class of nonlinear adaptive filters”,
*Proc. IEEE International Conference on Acoustics*,*Speech and Signal Processing*, pp. 1483–1486, New York, 1988.Google Scholar - [51]I. Picas, A.N. Venetsanopoulos, “Adaptive L-filters”, Proc. European Conference on Circuit Theory and Design, Brighton, England, 1989.Google Scholar
- [52]F. Palmieri “A backpropagation algorithm for multilayer hybrid order statistics filters”,
*Proc. IEEE International Conference on Acoustics*,*Speech and Signal Processing*, Glasgow, Scotland, 1989.Google Scholar