This Article
	Subscribers, please Login Purchase article: $19 PDF IEEE Xplore Subscribers RSS feed
	Share
	Email this Article to a friend
	Bibliographic References
	ASCII Text BibTex RefWorks Procite/RefMan
	Add to:
	Digg Furl Spurl Blink Simpy Google Del.icio.us Y!MyWeb
	Search
	Similar Articles Articles by B. Shahraray Articles by D.J. Anderson

Optimal Estimation of Contour Properties by Cross-Validated Regularization

June 1989 (vol. 11 no. 6)

pp. 600-610

[1] P. R. Adby and M. A. H. Dempster,Introduction to Optimization Theory. New York: Wiley, 1974.[2] R. S. Anderssen and P. Bloomfield, "A time series approach to numerical differentiation,"Technometrics, vol. 16, no. 1, pp. 69-75, Feb. 1974.[3] D. M. Bates and G. Wahba, "Computational methods for generalized cross-validation with large data sets," inTreatment of Integral Equations by Numerical Methods, C. T. H. Baker and G. F. Miller, Eds. New York: Academic, 1982.[4] D. M. Bates, M. J. Lindstrom, G. Wahba, and B. S. Yandel, "GCVPACK--Routines for generalized cross-validation," Dep. Statistics, Univ. Wisconsin, Tech. Rep. 775, Nov. 1985; also inCommun. Statist. Simul. Comput., vol. 16, pp. 263-297, 1987.[5] T. Boult, "What is regular in regularization," inProc. First Int. Conf. Computer Vision, London, 1987, pp. 457-462.[6] R. L. Brown, J. Durbin, and J. M. Evans, "Techniques of testing consistency of regression relationships over time,"J. Royal Stat. Assoc., vol. 70, pp. 70-79, 1975.[7] P. Craven and G. Wahba, "Smoothing noisy data with spline functions-Estimating the correct degree of smoothing by the method of generalized cross-validation,"Numerische Mathematik, no. 31, pp. 377-403, 1979.[8] J. Duchon, "Splines minimizing rotation-invariant semi-norms in Sobolev spaces," inConstructive Theory of Functions of Several Variables, W. Schempp and K. Zeller, Eds. 1976, pp. 85-100.[9] L. Elden, "Algorithms for the regularization of ill-conditioned least squares problems,"BIT, vol. 17, pp. 134-145, 1977.[10] L. Elden, "A note on the computation of generalized cross-validation function for ill-conditioned least squares problems,"BIT, vol. 24, pp. 467-472, 1984.[11] S. Geisser, "The predictive sample reuse method with applications,"J. Amer. Stat. Assoc., vol. 70, no. 350, pp. 320-328, 1975.[12] G. H. Golub, M. Heath, and G. Wahba, "Generalized cross-validation as a method for choosing a good ridge parameter,"Technometrics, vol. 21, no. 2, pp. 215-223, May 1979.[13] W. E. L. Grimson, "An implementation of a computational theory of visual surface interpolation,"Comput. Vision, Graphics, Image Processing, vol. 22, pp. 39-69, 1983.[14] W. J. Kennedy, Jr., and J. E. Gentle,Statistical Computing. New York: Dekker, 1980.[15] G. S. Kimeldorf and G. Wahba, "A correspondence between Bayesian estimation on stochastic processes and smoothing by splines,"Ann. Math. Stat., vol. 41, no. 2, pp. 495-502, 1970.[16] D. Lee and T. Pavlidis, "One-dimensional regularization with discontinuities," inProc. First Int. Conf. Computer Vision, London, 1987, pp. 573-577.[17] K. C. Li, "From Stein's unbiased estimate to the method of generalized cross validation,"Ann. Stat., vol. 13, no. 4, pp. 1352-1377, 1985.[18] A. A. Melkman and C. A. Micchelli, "Optimal estimation of linear operators in Hilbert spaces from inaccurate data,"SIAM J. Numer. Anal., vol. 16, pp. 87-105, 1979.[19] D. Nychka, G. Wahba, S. Goldfrab, and T. Pugh, "Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations on two-dimensional cross-sections,"J. Amer. Stat. Assoc., vol. 79, pp. 832-846, 1984.[20] T. Poggio and V. Torre, "Ill-posed problems and regularization analysis in early vision," inProc. DARPA Image Understanding Workshop, 1984, pp. 257-263.[21] T. Poggio, V. Torre, and C. Koch, "Computational vision and regularization theory,"Nature, vol. 317, pp. 314-319, 1985.[22] T. Poggio, H. Voorhees, and A. Yuille, "A regularized solution to edge detection," Tech. Rep. MA, Rep. AIM-833, MIT Artificial Intell. Lab., May 1985.[23] D. L. Ragozin, "Error bounds for derivative estimates based on spline smoothing of exact or noisy data,"J. Approximation Theory, no. 37, pp. 335-355, 1983.[24] C. M. Reinsch, "Smoothing by spline functions,"Numerische Mathematik, vol. 10, pp. 177-183, 1967.[25] I. J. Schoenberg, "Spline functions and tbe problem of graduation,"Proc. Nat. Acad. Sci., vol. 52, pp. 947-950, 1964.[26] B. Shahraray, "Measurement of boundary curvature of quantized shapes: A method based on smoothing spline approximation," Ph.D. dissertation, Dep. Elec. Eng. Comput. Sci., Univ. Michigan, Ann Arbor, 1985.[27] B. Shahraray and D. J. Anderson, "Optimal smoothing of digitized contours," inProc. IEEE Comput. Soc. Conf. Computer Vision and Pattern Recognition, Miami, FL, June 22-26, 1986, pp. 210-218.[28] J. H. Shiau, "Smoothing spline estimation of functions with discontinuities," Ph.D. dissertation, Dep. Stat., Univ. Wisconsin, Madison, 1985.[29] B. W. Silverman, "A fast and efficient cross-validation method for smoothing parameter choice in spline regression,"J. Amer. Stat. Assoc., vol. 79, pp. 584-589, 1984.[30] M. Stone, "Cross-validatory choice and assessment of statistical prediction,"J. Roy. Stat. Soc. Ser. B, vol. 36, no. 2, pp. 111-147, 1974.[31] D. Terzopoulos, "Regularization of inverse visual problems involving discontinuities,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, no. 4, pp. 413-424, July 1986.[32] A. N. Tikhonov, "Regularization of incorrectly posed problems,"Sov. Math. Dokl., vol. 4, pp. 1624-1627, 1963.[33] A. N. Tikhonov and V. Y. Arsenin,Solutions of Ill-Posed Problems. New York: Wiley, 1977.[34] G. Wahba and S. Wold, "A completely automatic french curve: Fitting splines by cross validation,"Commun. Stat., vol. 4, no. 1, pp. 1-17, 1975.[35] G. Wahba and S. Wold, "Periodic splines for spectral density estimation: The use of cross validation for determining the degree of smoothing,"Commun. Stat., vol. 4, no. 2, pp. 125-141, 1975.[36] G. Wahba, "A survey of some smoothing problems and the method of generalized cross-validation for solving them," inApplications of Statistics, P. R. Krishnaiah, Ed. Amsterdam, The Netherlands: North-Holland, 1977, pp. 507-523.[37] G. Wahba, "Ill-posed problems: Numerical and statistical methods for mildy, moderately and severely ill-posed problems with noisy data," Dep. Stat., Univ. Wisconsin, Madison, Tech. Rep. 595, 1980; also inProc. Int. Symp. Ill-Posed Problems, Newark, DE, Oct. 2-6, 1979.[38] G. Wahba, "Constrained regularization for ill-posed linear operator equations with applications in meteorology and medicine," inStatistical Decision Theory and Related Topics III, vol. 2, S. Gupta and J. O. Berger, Eds. New York: Academic, 1982, pp. 383-418.[39] G. Wahba and J. Wendelberger, "Some new mathematical methods for variational objective analysis using splines and cross-validation,"Monthly Weather Rev., vol. 108, pp. 36-57, 1980.[40] G. Wahba, "A comparison of GCV and GML for choosing the smoothing parameter in the generalized spline smoothing problem,"Ann. Stat., vol. 13, no. 4, pp. 1378-1402, 1985.[41] J. Wendelberger, "The computation of Laplacian smoothing splines with examples," Dep. Stat., Univ. Wisconsin, Madison, Tech. Rep. 648, 1981.[42] J. Wendelberger, "Smoothing noisy data with multi-dimensional splines and generalized cross-validation," Ph.D. dissertation, Dep. Stat., Univ. Wisconsin, Madison, 1983.[43] E. T. Whittaker, "On a new method of graduation,"Proc. Edinburgh Math. Soc., vol. 41, pp. 63-75, 1923.[44] 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.