Abstract
A new local smoothing procedure is suggested for jump-preserving surface reconstruction from noisy data. In a neighborhood of a given point in the design space, a plane is fitted by local linear kernel smoothing, giving the conventional local linear kernel estimator of the surface at the point. The neighborhood is then divided into two parts by a line passing through the given point and perpendicular to the gradient direction of the fitted plane. In the two parts, two half planes are fitted, respectively, by local linear kernel smoothing, providing two one-sided estimators of the surface at the given point. Our surface reconstruction procedure then proceeds in the following two steps. First, the fitted surface is defined by one of the three estimators, i.e., the conventional estimator and the two one-sided estimators, depending on the weighted residual means of squares of the fitted planes. The fitted surface of this step preserves the jumps well, but it is a bit noisy, compared to the conventional local linear kernel estimator. Second, the estimated surface values at the original design points obtained in the first step are used as new data, and the above procedure is applied to this data in the same way except that one of the three estimators is selected based on their estimated variances. Theoretical justification and numerical examples show that the fitted surface of the second step preserves jumps well and also removes noise efficiently. Besides two window widths, this procedure does not introduce other parameters. Its surface estimator has an explicit formula. All these features make it convenient to use and simple to compute.
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References
Besag J. (1986). On the statistical analysis of dirty pictures (with discussion). Journal of the Royal Statistical Society, B, 48:259–302
Besag J., Green P., Higdon D., Mengersen K. (1995). Bayesian computation and stochastic systems (with discussion). Statistical Science 10:3–66
Carlstein E., Krishnamoorthy C.(1992). Boundary estimation. Journal of the American Statistical Association 87:430–438
Chu C.K., Glad I.K., Godtliebsen F., Marron J.S. (1998). Edge-preserving smoothers for image processing (with discussion). Journal of the American Statistical Association 93:526–556
Cleveland W.S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74:828–836
Fan J., Gijbels I. (1996). Local polynomial modelling and its applications. London, Chapman & Hall
Fessler J.A., Erdogan H., Wu W.B. (2000). Exact distribution of edge-preserving MAP estimators for linear signal models with Gaussian measurement noise. IEEE Transactions on Image Processing 9:1049–1055
Geman S., Geman D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6:721–741
Gijbels I., Lambert A., Qiu P. (2006). Edge-preserving image denoising and estimation of discontinuous surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 28:1075–1087
Gijbels I., Lambert A., Qiu P. (2007). Jump-preserving regression and smoothing using local linear fitting: a compromise. Annals of the Institute of Statistical Mathematics 59:235–272
Godtliebsen F., Sebastiani G. (1994). Statistical methods for noisy images with discontinuities. Journal of Applied Statistics 21:459–476
Gonzalez R.C., Woods R.E. (1992). Digital image processing. Reading, Addison-Wesley
Hall P., Molchanov (2003). Sequential methods for design-adaptive estimation of discontinuities in regression curves and surfaces. The Annals of Statistics 31:921–941
Hall P., Peng L., Rau C. (2001). Local likelihood tracking of fault lines and boundaries. Journal of the Royal Statistical Society, B 63:569–582
Hall P., Rau C. (2000). Tracking a smooth fault line in a response surface. The Annals of Statistics 28:713–733
Härdle W. (1990). Applied nonparametric regression. Oxford, Oxford University Press
Keeling S.L., Stollberger R. (2002). Nonlinear anisotropic diffusion filtering for multiscale edge enhancement. Inverse Problems 18:175–190
Li S.Z. (1995). Markov random field modeling in computer vision. New York, Springer
Marroquin J.L., Velasco F.A., Rivera M., Nakamura M. (2001). Gauss–Markov measure field models for low-level vision. IEEE Transactions on Pattern Analysis and Machine Intelligence 23:337–347
Müller H.G., Song K.S. (1994). Maximin estimation of multidimensional boundaries. Journal of the Multivariate Analysis 50:265–281
Nason G., Silverman B. (1994). The discrete wavelet transform in S. Journal of Computational and Graphical Statistics 3:163–191
O’Sullivan F., Qian M. (1994). A regularized contrast statistic for object boundary estimation—implementation and statistical evaluation. IEEE Transactions on Pattern Analysis and Machine Intelligence 16:561–570
Perona P., Malik J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12:629–639
Polzehl J., Spokoiny V.G. (2000). Adaptive weights smoothing with applications to image restoration. Journal of the Royal Statistical Society, B 62:335–354
Qiu P. (1997). Nonparametric estimation of jump surface. Sankhyā (A) 59:268–294
Qiu P. (1998). Discontinuous regression surfaces fitting. The Annals of Statistics 26:2218–2245
Qiu P. (2002). A nonparametric procedure to detect jumps in regression surfaces. Journal of Computational and Graphical Statistics 11:799–822
Qiu P. (2003). A jump-preserving curve fitting procedure based on local piecewise-linear kernel estimation. Journal of Nonparametric Statistics 15:437–453
Qiu P. (2004). The local piecewisely linear kernel smoothing procedure for fitting jump regression surfaces. Technometrics 46:87–98
Qiu P. (2005). Image processing and jump regression analysis. New York, Wiley
Qiu P. (2007). Jump surface estimation, edge detection, and image restoration. Journal of the American Statistical Association 102:745–756
Qiu P., Yandell B. (1997). Jump detection in regression surfaces. Journal of Computational and Graphical Statistics 6:332–354
Sebastiani G., Godtliebsen F. (1997). On the use of Gibbs priors for Bayesian image restoration. Signal Processing 56:111–118
Stone C. (1982). Optimal global rates of convergence for nonparametric regression. The Annals of Statistics 10:1040–1053
Sun J., Qiu P. (2007). Jump detection in regression surfaces using both first-order and second-order derivatives. Journal of Computational and Graphical Statistics 16:289–311
Titterington D.M. (1985). Common structure of smoothing techniques in statistics. International Statistical Review 53:141–170
Tomasi, C., Manduchi, R. (1998). Bilateral filtering for gray and color images. In: Proceedings of the 1998 IEEE international conference on computer vision (pp. 839–846). Bombay
Tukey J.W. (1977). Exploratory data analysis. Reading, MA: Addison-Wesley.
Weikert J., ter Haar Romeny B.M., Viergever M. (1998). Efficient and reliable schemes for nonlinear diffusion filtering. IEEE Transactions on Image Processing 7:398–410
Yi J.H., Chelberg D.M. (1995). Discontinuity-preserving and viewpoint invariant reconstruction of visible surfaces using a first order regularization. IEEE Transactions on Pattern Analysis and Machine Intelligence 17:624–629
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Qiu, P. Jump-preserving surface reconstruction from noisy data. Ann Inst Stat Math 61, 715–751 (2009). https://doi.org/10.1007/s10463-007-0166-9
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DOI: https://doi.org/10.1007/s10463-007-0166-9