This Article 
 Bibliographic References 
 Add to: 
A Study of Methods of Choosing the Smoothing Parameter in Image Restoration by Regularization
April 1991 (vol. 13 no. 4)
pp. 326-339

The method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness, with the tradeoff being determined by a scalar smoothing parameter. Various ways of choosing this parameter are discussed in the case of quadratic regularization criteria. They are compared algebraically, and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images.

[1] T. W. Anderson,The Statistical Analysis of Time Series. New York: Wiley, 1971.
[2] I. J. D. Craig and J. C. Brown,Inverse Problems in Astronomy: A Guide to Inversion Strategies for Remotely Sensed Data. Boston: Adam Hilger, 1986.
[3] I. J. D. Craig, K. G. McClements, A. M. Thompson, and J. C. Brown, "The numerical inversion of synchrotron spectra and the Crab Nebula as a synchrotron source,"Astron. Astrophys., vol. 149, pp. 171-178, 1985.
[4] T. Gasser, L. Sroka, and C. Jennen-Steinmetz, "Residual variance and residual pattern in nonlinear regression,"Biometrika, vol. 73, pp. 625-633, 1986.
[5] G.H. Golub, M. Heath, and G. Wahba, "Generalized cross-validation as a method for choosing a good ridge parameter,"Technometrics, vol. 21, pp. 215-223, 1979.
[6] R. M. Gray, "On the asymptotic eigenvalue distribution of Toeplitz matrices,"IEEE Trans. Inform. Theory, vol. IT-18, pp. 725-730, 1972.
[7] W. E. L. Grimson, "An implementation of a computational theory of visual surface interpolation,"Comput. vision, Graphics, Image Processing, vol. 21, pp. 215-223, 1983.
[8] C. W. Groetsch,The Theory of Tikhonov Regularisation for Fredholm Equations of the First Kind (Res. Notes in Math No.105). New York: Pitman, 1984.
[9] S.F. Gull and J. Skilling, "Maximum entropy method in image processing,"Proc. Inst. Elec. Eng. F., vol. 131, pp. 646-659, 1984.
[10] P. Hall and D. M. Titterington, "On some smoothing techniques used in image processing,"J. Roy. Statist. Soc. B, vol. 48, pp. 330-343, 1986.
[11] P. Hall and D. M. Titterington, "Common structure of techniques for choosing smoothing parameters in regression problems,"J. Roy. Statist. Soc. B, vol. 49, pp. 184-198, 1987.
[12] W. Härdle, P. Hall, and J. S. Marron, "How far are automatically chosen regression smoothing parameters from their optimum?"J. Amer. Statist. Assoc., vol. 83, pp. 86-101, 1988.
[13] A. K. Jain, "Fast inversion of banded Toeplitz matrices by circular decompositions,"IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP-26, pp. 121-126, 1978.
[14] N.L. Johnson and S. Kotz,Distribution in Statistics: Continuous Univariate Distributions-2, New York: Wiley, 1970.
[15] J. W. Kay, "On the choice of regularisation parameter in image restoration,"Springer Lecture Notes in Computer Science, vol. 301, pp. 587-596, 1988.
[16] A.L. MacKinnon, J. C. Brown, and J. Hayward, "Quantitative analysis of hard X-ray 'footprint flares' observed by the Solar Maximum Mission,"Solar Physics, vol. 99, pp. 231-262, 1985.
[17] J. Marroquin, S. Mitter, and T. Poggio, "Probabilistic solution of ill-posed problems in computational vision,"J. Amer. Statist. Assoc., vol. 82, pp. 76-89, 1987.
[18] J. Philip, "The most ill-posed non-negative kernels in discrete deconvolution,"Inverse Problems, vol. 3, pp. 309-328, 1987.
[19] D. L. Phillips, "A technique for the numerical solution of certain integral equations of the first kind,"J. Assoc. Comput. Mach., vol. 9, pp. 84-97, 1962.
[20] T. Poggio, V. Torre, and C. Koch, "Computational vision and regularization theory,"Nature, vol. 317, pp. 314-319, 1985.
[21] J. Rice, "Choice of smoothing parameter in deconvolution problems,"Contemporary Math., vol. 59, pp. 137-151, 1986.
[22] B. Shararay and D. J. Anderson, "Optimal estimation of contour properties by cross-validated regularization,"IEEE Trans. Pattern Anal. Machine Intell., vol. 11, pp. 600-610, 1989.
[23] A.M. Thompson, "On the use of quadratic regularisation within maximum entropy image restoration," inMaximum Entropy and Bayesian Methods, J. Skilling, Ed. New York: Kluwer, 1988, pp. 497-504.
[24] A.M. Thompson, J. W. Kay, and D. M. Titterington, "A cautionary note about crossvalidatory choice,"J. Statist. Comput. Simul., vol. 33, pp. 199-216, 1989.
[25] A. N. Tikhonov and V. Y. Arsenin,Solutions of Ill-Posed Problems. Washington, DC: Winston, 1977.
[26] D. M. Titterington, "General structure of regularisation procedures in image processing,"Astron. Astrophys., vol. 144, pp. 381-387, 1985.
[27] D. M. Titterington, "Common structure of smoothing techniques in statistics,"Int. Statist. Rev., vol. 53, pp. 141-170, 1985.
[28] D. M. Titterington, "Comments on a paper by F. O'Sullivan,"Statist. Sci., vol. 1, pp. 519-521, 1986.
[29] G. Wahba, "Constrained regularization for ill-posed linear operator equations, with applications in meteorology and medicine," inStatistical Theory and Related Topics III, vol. 2, S. S. Gupta and J. O. Berger, Eds. New York: Academic, 1982, pp. 383-418.
[30] G. Wahba, "Bayesian 'confidence intervals' for the cross-validated smoothing spline,"J. Roy. Statist. Soc. B., vol. 45, pp. 133-150, 1985.
[31] G. Wahba and S. Wold, "A completely automatic French curve: Fitting splines by cross-validation,"Commun. Statist., vol. 4, pp. 1-17, 1985.
[32] Y. Yasumoto and G. Medioni, "Robust estimation of three-dimensional motion parameters from a sequence of image frames using regularization,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, pp. 464-471, 1986.

Index Terms:
picture processing; image restoration; scalar smoothing parameter; quadratic regularization criteria; filtering and prediction theory; picture processing; statistical analysis
A.M. Thompson, J.C. Brown, J.W. Kay, D.M. Titterington, "A Study of Methods of Choosing the Smoothing Parameter in Image Restoration by Regularization," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 4, pp. 326-339, April 1991, doi:10.1109/34.88568
Usage of this product signifies your acceptance of the Terms of Use.