Discontinuous regression surfaces fitting. (English) Zbl 0927.62041

Qiu, Peihua

Ann. Stat. 26, No. 6, 2218-2245 (1998).

Summary: We suggest a three-stage procedure to recover discontinuous regression surfaces when noisy data are present. In the first stage, jump candidate points are detected using a jump detection criterion. A local principal component line is then fitted through these points in a neighborhood of a design point. This line provides a first-order approximation to the true jump location curve in that neighborhood. In the third stage, observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point. If there are no jump candidate points in the neighborhood, then all observations in that neighborhood are used in the surface fitting.
If, however, the center of the neighborhood is on a jump location curve, only those observations on one side of the line are used. Thus blurring is automatically avoided around the jump locations. This methodology requires \(O(N(k^*)^2)\) computation, where \(N\) is the sample size and \(k^*\) is the window width. Its assumptions on the model are flexible. Some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths.

MSC:

62G08	Nonparametric regression and quantile regression
62G20	Asymptotic properties of nonparametric inference

Keywords:

discontinuous regression surfaces; image processing; jump detection criterion; jump location curves; least squares coefficients; principal component line; threshold value

Software:

KernSmooth

References:

[1]	BESAG, J. 1986. On the statistical analysis of dirty pictures with discussion. J. Roy. Statist. Soc. Ser. B 48 259 302. Z. JSTOR: · Zbl 0609.62150
[2]	BESAG, J., GREEN, P., HIGDON, D. and MENGERSEN, K. 1995. Bayesian computation and stochasZ. tic sy stems with discussions. Statist. Sci. 10 3 66. Z.
[3]	BHANDARKAR, S. M., ZHANG, Y. and POTTER, W. D. 1994. An edge detection technique using genetic algorithm-based optimization. Pattern Recognition 27, 1159 1180. Z.
[4]	BOW, S. T. 1992. Pattern Recognition and Image Preprocessing. Marcel Dekker, New York. Z.
[5]	BRACEWELL, R. N. 1995. Two-Dimensional Imaging. Prentice-Hall, Englewood Cliffs, NJ. Z. · Zbl 0867.68113
[6]	CHENG, K. F. and LIN, P. E. 1981. Nonparametric estimation of a regression function. Z. Wahrsch. Verw. Gebiete 57 223 233. Z. · Zbl 0518.62017 · doi:10.1111/j.1467-842X.1981.tb00776.x
[7]	CRESSIE, N. 1991. Statistics for Spatial Data. Wiley, New York. Z. · Zbl 0799.62002
[8]	EUBANK, R. L. and SPECKMAN, P. L. 1994. Nonparametric estimation of functions with jump Z discontinuities. In Change-Point Problems E. Carlstein, H. G. Muller and D. Sieg\". mund, eds. 130 144. IMS, Hay ward, CA. Z. · Zbl 1163.62313
[9]	FARIN, G. 1993. Curves and Surfaces for Computer Aided Geometric Design, 3rd ed. Academic Press, New York. Z.
[10]	GASSER, T. and MULLER, H. G. 1979. Kernel estimation of regression functions. Lecture Notes ïn Math. 757 23 68. Springer, Berlin. Z. · Zbl 0418.62033
[11]	GEMAN, S. and GEMAN, D. 1984. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans. Pattern Analy sis and Machine Intelligence 6 721 741. Z. · Zbl 0573.62030 · doi:10.1109/TPAMI.1984.4767596
[12]	GONZALEZ, R. C. and WOODS, R. E. 1992. Digital Image Processing. Addison-Wesley, Reading, MA.Z.
[13]	GREEN, P. J. 1990. Bayesian reconstruction from emission tomography data using a modified EM algorithm. IEEE Trans. Medical Imaging 9 84 93. Z.
[14]	HALL, P. and TITTERINGTON, M. 1992. Edge-preserving and peak-preserving smoothing. Technometrics 34 429 440. JSTOR:
[15]	KOROSTELEV, A. P. and TSy BAKOV, A. B. 1993. Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82. Springer, New York. Z. · Zbl 0833.62039
[16]	LI, S. Z. 1995. On discontinuity-adaptive smoothness priors in computer vision. IEEE Trans. Pattern Analy sis and Machine Intelligence 17 576 586. Z.
[17]	LOADER, C. R. 1996. Change point estimation using nonparametric regression. Ann. Statist. 24 1667 1678. Z. · Zbl 0867.62033 · doi:10.1214/aos/1032298290
[18]	MARR, D. and HILDRETH, E. 1980. Theory of edge detection. Proc. Roy. Soc. London Ser. A 207 187 217. Z.
[19]	MCDONALD, J. A. and OWEN, A. B. 1986. Smoothing with split linear fits. Technometrics 28 195 208. Z. JSTOR: · Zbl 0626.65010 · doi:10.2307/1269075
[20]	MULLER, H. G. 1992. Change-points in nonparametric regression analysis. Ann. Statist. 20 \" 737 761. Z. · Zbl 0783.62032 · doi:10.1214/aos/1176348654
[21]	MULLER, H. G. and SONG, K. S. 1994. Maximin estimation of multidimensional boundaries J. \" Multivariate Anal. 50 265 281. Z. O’SULLIVAN, F. 1995. A study of least squares and maximum likelihood for image reconstruction in positron emission tomography. Ann. Statist. 23 1267 1300. Z. O’SULLIVAN, F. and QIAN, M. 1994. A regularized contrast statistic for object boundary estimation implementation and statistical evaluation. IEEE Trans. Pattern Analy sis and Machine Intelligence 16 561 570. Z.
[22]	OWEN, A. 1984. A neighborhood-based classifier for Landsat data. Canad. J. Statist. 12 191 200. Z. JSTOR: · Zbl 0548.62041
[23]	QIU, P. 1994. Estimation of the number of jumps of the jump regression functions. Comm. Statist. Theory Methods 23 2141 2155. Z. · Zbl 0825.62205 · doi:10.1080/03610929408831378
[24]	QIU, P. 1997. Nonparametric estimation of jump surface. Sankhy a Ser. A 59 268 294. Z. · Zbl 0886.62046
[25]	QIU, P., ASANO, CHI. and LI, X. 1991. Estimation of jump regression functions. Bull. Inform. Cy bernet. 24 197 212.Z. · Zbl 0761.62044
[26]	QIU, P. and BHANDARKAR, S. M. 1996. An edge detection technique using local smoothing and statistical hy pothesis testing. Pattern Recognition Lett. 17 849 872. Z. QIU P. and YANDELL, B. 1997. Jump detection in regression surfaces. J. Comput. Graph. Statist. 6 332 354. Z. JSTOR:
[27]	QIU, P. and YANDELL, B. 1998. A local poly nomial jump detection algorithm in nonparametric regression. Technometrics 40 141 152. Z. JSTOR:
[28]	RICE, J. 1984. Boundary modification for kernel regression. Comm. Statist. Theory Methods 13 893 900. Z. · Zbl 0552.62022 · doi:10.1080/03610928408828728
[29]	ROSENFELD, A. and KAK, A. C. 1982. Digital Picture Processing 1, 2. Academic Press, New York. Z. · Zbl 0564.94001
[30]	RUDEMO, M. and STRy HN, H. 1994. Approximating the distribution of maximum likelihood contour estimators in two-region images. Scand. J. Statist. 21 41 55. Z. · Zbl 0804.62044
[31]	SHIAU, J. H., WAHBA, G. and JOHNSON, D. R. 1986. Partial spline models for the inclusion of tropopause and frontal boundary information in otherwise smooth twoand three-dimensional objective analysis. J. Atmospheric and Oceanic Technology 3 714 725. Z.
[32]	SINHA, S. S. and SCHUNK, B. G. 1992. A two-stage algorithm for discontinuity-preserving surface reconstruction. IEEE Trans. Pattern Analy sis and Machine Intelligence 14 36 55. Z.
[33]	STONE, C. J. 1982. Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040 1053. Z. · Zbl 0511.62048 · doi:10.1214/aos/1176345969
[34]	SWITZER, P. 1983. Some spatial statistics for the interpretation of satellite data. Bull. Internat. Statist. Inst. 50 962 972. Z.
[35]	SWITZER, P. 1986. Statistical image processing. In Quantitative Analy sis of Mineral and Energy Z. Resources C. F. Chung, A. G. Fabbri and R. Sinding-Larsen, eds. 271 282. Reidel, Dordrecht. Z.
[36]	SWITZER, P., KOWALIK, W. S. and Ly ON, R. J. P. 1982. A prior method for smoothing discriminant analysis classification maps. J. Internat. Assoc. Math. Geology 14 433 444. Z.
[37]	TITTERINGTON, D. M. 1985. Common structure of smoothing techniques in statistics. Internat. Statist. Rev. 53 141 170. JSTOR: · Zbl 0569.62026 · doi:10.2307/1402932
[38]	TORRE, V. and POGGIO, T. A. 1986. On edge detection. IEEE Trans. Pattern Analy sis and Machine Intelligence 8 147 163. Z.
[39]	WAND, M. P. and JONES, M. C. 1995. Kernel Smoothing. Chapman and Hall, London. Z. · Zbl 0854.62043
[40]	WANG, Y. 1995. Jump and sharp cusp detection by wavelets. Biometrika 82 385 397. Z. JSTOR: · Zbl 0824.62031 · doi:10.1093/biomet/82.2.385
[41]	WU, J. S. and CHU, C. K. 1993a. Kernel ty pe estimators of jump points and values of a regression function. Ann. Statist. 21 1545 1566. Z. WU. J. S. and CHU, C. K. 1993b. Nonparametric function estimation and bandwidth selection for discontinuous regression functions. Statist. Sinica 3 557 576. Z. · Zbl 0795.62043 · doi:10.1214/aos/1176349271
[42]	YI, J. H. and CHELBERG, D. M. 1995. Discontinuity-preserving and viewpoint invariant reconstruction of visible surfaces using a first order regularization. IEEE Trans. Pattern Analy sis and Machine Intelligence 17 624 629.
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