×

Jump-preserving surface reconstruction from noisy data. (English) Zbl 1332.62369

Summary: 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.

MSC:

62M40 Random fields; image analysis
62G05 Nonparametric estimation
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
68U10 Computing methodologies for image processing

Software:

wavethresh; AWS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Besag J. (1986). On the statistical analysis of dirty pictures (with discussion). Journal of the Royal Statistical Society, B, 48:259–302 · Zbl 0609.62150
[2] Besag J., Green P., Higdon D., Mengersen K. (1995). Bayesian computation and stochastic systems (with discussion). Statistical Science 10:3–66 · Zbl 0955.62552 · doi:10.1214/ss/1177010123
[3] Carlstein E., Krishnamoorthy C.(1992). Boundary estimation. Journal of the American Statistical Association 87:430–438 · Zbl 0781.62043 · doi:10.2307/2290274
[4] 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 · Zbl 0954.62115 · doi:10.2307/2670100
[5] Cleveland W.S. (1979). Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74:828–836 · Zbl 0423.62029 · doi:10.2307/2286407
[6] Fan J., Gijbels I. (1996). Local polynomial modelling and its applications. London, Chapman & Hall · Zbl 0873.62037
[7] 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 · Zbl 0992.94002 · doi:10.1109/83.846247
[8] 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 · Zbl 0573.62030 · doi:10.1109/TPAMI.1984.4767596
[9] 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 · Zbl 05111825 · doi:10.1109/TPAMI.2006.140
[10] 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 · Zbl 1332.62132 · doi:10.1007/s10463-006-0045-9
[11] Godtliebsen F., Sebastiani G. (1994). Statistical methods for noisy images with discontinuities. Journal of Applied Statistics 21:459–476 · doi:10.1080/757584021
[12] Gonzalez R.C., Woods R.E. (1992). Digital image processing. Reading, Addison-Wesley
[13] Hall P., Molchanov (2003). Sequential methods for design-adaptive estimation of discontinuities in regression curves and surfaces. The Annals of Statistics 31:921–941 · Zbl 1028.62069 · doi:10.1214/aos/1056562467
[14] 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 · Zbl 1040.62043 · doi:10.1111/1467-9868.00299
[15] Hall P., Rau C. (2000). Tracking a smooth fault line in a response surface. The Annals of Statistics 28:713–733 · Zbl 1105.62327 · doi:10.1214/aos/1015951995
[16] Härdle W. (1990). Applied nonparametric regression. Oxford, Oxford University Press · Zbl 0714.62030
[17] Keeling S.L., Stollberger R. (2002). Nonlinear anisotropic diffusion filtering for multiscale edge enhancement. Inverse Problems 18:175–190 · Zbl 1006.94002 · doi:10.1088/0266-5611/18/1/312
[18] Li S.Z. (1995). Markov random field modeling in computer vision. New York, Springer
[19] 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 · Zbl 05111049 · doi:10.1109/34.917570
[20] Müller H.G., Song K.S. (1994). Maximin estimation of multidimensional boundaries. Journal of the Multivariate Analysis 50:265–281 · Zbl 0798.62053 · doi:10.1006/jmva.1994.1042
[21] Nason G., Silverman B. (1994). The discrete wavelet transform in S. Journal of Computational and Graphical Statistics 3:163–191 · doi:10.2307/1390667
[22] 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 · Zbl 05111680 · doi:10.1109/34.295901
[23] Perona P., Malik J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12:629–639 · Zbl 05111848 · doi:10.1109/34.56205
[24] Polzehl J., Spokoiny V.G. (2000). Adaptive weights smoothing with applications to image restoration. Journal of the Royal Statistical Society, B 62:335–354 · Zbl 04558575 · doi:10.1111/1467-9868.00235
[25] Qiu P. (1997). Nonparametric estimation of jump surface. Sankhyā (A) 59:268–294 · Zbl 0886.62046
[26] Qiu P. (1998). Discontinuous regression surfaces fitting. The Annals of Statistics 26:2218–2245 · Zbl 0927.62041 · doi:10.1214/aos/1024691468
[27] Qiu P. (2002). A nonparametric procedure to detect jumps in regression surfaces. Journal of Computational and Graphical Statistics 11:799–822 · doi:10.1198/106186002880
[28] Qiu P. (2003). A jump-preserving curve fitting procedure based on local piecewise-linear kernel estimation. Journal of Nonparametric Statistics 15:437–453 · Zbl 1054.62047 · doi:10.1080/10485250310001595083
[29] Qiu P. (2004). The local piecewisely linear kernel smoothing procedure for fitting jump regression surfaces. Technometrics 46:87–98 · doi:10.1198/004017004000000149
[30] Qiu P. (2005). Image processing and jump regression analysis. New York, Wiley · Zbl 1070.68146
[31] Qiu P. (2007). Jump surface estimation, edge detection, and image restoration. Journal of the American Statistical Association 102:745–756 · Zbl 1172.62326 · doi:10.1198/016214507000000301
[32] Qiu P., Yandell B. (1997). Jump detection in regression surfaces. Journal of Computational and Graphical Statistics 6:332–354 · doi:10.2307/1390737
[33] Sebastiani G., Godtliebsen F. (1997). On the use of Gibbs priors for Bayesian image restoration. Signal Processing 56:111–118 · Zbl 1008.68505 · doi:10.1016/S0165-1684(97)00002-9
[34] Stone C. (1982). Optimal global rates of convergence for nonparametric regression. The Annals of Statistics 10:1040–1053 · Zbl 0511.62048 · doi:10.1214/aos/1176345969
[35] 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 · doi:10.1198/106186007X204753
[36] Titterington D.M. (1985). Common structure of smoothing techniques in statistics. International Statistical Review 53:141–170 · Zbl 0569.62026 · doi:10.2307/1402932
[37] 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
[38] Tukey J.W. (1977). Exploratory data analysis. Reading, MA: Addison-Wesley. · Zbl 0409.62003
[39] 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 · doi:10.1109/83.661190
[40] 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 · Zbl 05111698 · doi:10.1109/34.387510
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.