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Edge-Preserving Image Denoising and Estimation of Discontinuous Surfaces
July 2006 (vol. 28 no. 7)
pp. 1075-1087
In this paper, we are interested in the problem of estimating a discontinuous surface from noisy data. A novel procedure for this problem is proposed based on local linear kernel smoothing, in which local neighborhoods are adapted to the local smoothness of the surface measured by the observed data. The procedure can therefore remove noise correctly in continuity regions of the surface and preserve discontinuities at the same time. Since an image can be regarded as a surface of the image intensity function and such a surface has discontinuities at the outlines of objects, this procedure can be applied directly to image denoising. Numerical studies show that it works well in applications, compared to some existing procedures.

[1] P. Qiu, Image Processing and Jump Regression Analysis. John Wiley and Sons, 2005.
[2] T. Poggio, “Early Vision: From Computational Structure to Algorithms and Parallel Hardware,” Computer Vision Graphics Image Processing, vol. 31, pp. 139-155, 1985.
[3] M.A. García, “Efficient Surface Reconstruction from Scattered Points through Geometric Data Fusion,” Proc. IEEE Int'l Conf. Multisensor Fusion and Integration for Intelligent Systems, pp. 559-566, 1994.
[4] R.C. Gonzalez and R.E. Woods, Digital Image Processing. Addison-Wesley, 1992.
[5] S. Geman and D. Geman, “Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, pp. 721-741, 1984.
[6] J. Besag, “On the Statistical Analysis of Dirty Pictures (with Discussion),” J. Royal Statistical Soc. (Series B), vol. 48, pp. 259-302, 1986.
[7] J. Besag, “Spatial Interaction and the Statistical Analysis of Lattice Systems (with Discussions),” J. Royal Statistical Soc. (Series B), vol. 36, pp. 192-236, 1974.
[8] J. Moussouris, “Gibbs and Markov Systems with Constraints,” J. Statistical Physics, vol. 10, pp. 11-33, 1974.
[9] F. Godtliebsen and G. Sebastiani, “Statistical Methods for Noisy Images with Discontinuities,” J. Applied Statistics, vol. 21, pp. 459-477, 1994.
[10] A. Blake and A. Zisserman, Visual Reconstruction. MIT Press, 1987.
[11] D. Geiger and F. Girosi, “Parallel and Deterministic Algorithms for MRFs: Surface Reconstruction,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, pp. 401-412, 1991.
[12] D. Geman and G. Reynolds, “Constrained Restoration and the Recovery of Distributions,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, pp. 367-383, 1992.
[13] J.A. Fessler, H. Erdogan, and W.B. Wu, “Exact Distribution of Edge-Preserving MAP Estimators for Linear Signal Models with Gaussian Measurement Noise,” IEEE Trans. Image Processing, vol. 9, pp. 1049-1055, 2000.
[14] J.L. Marroquin, F.A. Velasco, M. Rivera, and M. Nakamura, “Gauss-Markov Measure Field Models for Low-Level Vision,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 23, pp. 337-347, 2001.
[15] J.W. Tukey, Exploratory Data Analysis. Addison-Wesley, 1977.
[16] N.C. GallagherJr. and G.L. Wise, “A Theoretical Analysis of the Properties of Median Filtering,” IEEE Trans. Acoustics, Speech, and Signal Processing, vol. 29, pp. 1136-1141, 1981.
[17] Two-Dimensional Digital Signal Processing, T.S. Huang, ed. New York: Springer-Verlag, 1981.
[18] A.C. Bovik, T.S. Huang, and D.C. Munson, “The Effect of Median Filtering on Edge Estimation and Detection,” IEE Trans. Pattern Analysis and Machine Intelligence, vol. 9, pp. 181-194, 1987.
[19] D.R.K. Brownrigg, “The Weighted Median Filtering,” Comm. ACM, vol. 27, pp. 807-818, 1984.
[20] P. Haavisto, M. Gabbouj, and Y. Neuvo, “Median Based Idempotent Filters,” J. Circuits, Systems, and Computers, vol. 1, pp. 125-148, 1991.
[21] T. Sun, M. Gabbouj, and Y. Neuvo, “Center Weighted Median Filters: Some Properties and Their Applications in Image Processing,” Signal Processing, vol. 35, pp. 213-229, 1994.
[22] P. Saint-Marc, J. Chen, and G. Medioni, “Adaptive Smoothing: A General Tool for Early Vision,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, pp. 514-529, 1991.
[23] C. Tomasi and R. Manduchi, “Bilateral Filtering for Gray and Color Images,” Proc. 1998 IEEE Int'l Conf. Computer Vision, pp. 839-846, 1998.
[24] J. Koenderink, “The Structure of Images,” Biological Cybernetics, vol. 50, pp. 363-370, 1984.
[25] A. Hummel, “Representations Based on Zero-Crossings in Scale-Space,” Proc. IEEE Computer Vision and Pattern Recognition Conf., pp. 204-209, 1987.
[26] P. Perona and J. Malik, “Scale Space and Edge Detection Using Anisotropic Diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, pp. 629-639, 1990.
[27] D. Barash, “A Fundamental Relationship Between Bilateral Filtering, Adaptive Smoothing, and the Nonlinear Diffusion Equation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 24, pp. 844-847, 2002.
[28] D.L. Donoho and I.M. Johnstone, “Ideal Spatial Adaptation by Wavelet Shrinkage,” Biometrika, vol. 81, pp. 425-455, 1994.
[29] D.L. Donoho and X. Huo, “Beamlets and Multiscale Image Analysis,” Lecture Notes in Computational Science and Eng.: Multiscale and Multiresolution Methods, pp. 149-196, 2001.
[30] S.G. Chan, B. Yu, and M. Vetterli, “Spatially Adaptive Wavelet Thresholding with Context Modeling for Image Denoising,” IEEE Trans. Image Processing, vol. 9, pp. 1522-1531, 2000.
[31] M.A.T. Figueiredo and R.D. Nowak, “Wavelet-Based Image Estimation: An Empirical Bayes Approach Using Jeffrey's Noninformative Prior,” IEEE Trans. Image Processing, vol. 10, pp. 1322-1331, 2001.
[32] J. Portilla, V. Strela, M.J. Wainwright, and E.P. Simoncelli, “Image Denoising Using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Trans. Image Processing, vol. 12, pp. 1338-1351, 2003.
[33] G. Nason and B. Silverman, “The Discrete Wavelet Transform in S,” J. Computational and Graphical Statistics, vol. 3, pp. 163-191, 1994.
[34] S.Z. Li, “On Discontinuity-Adaptative Smoothness Prior in Computer Vision,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, pp. 576-586, 1995.
[35] M. Rivera and J.L. Marroquin, “Adaptive Rest Condition Potentials: First and Second Order Edge-Preserving Regularization,” J. Computer Vision and Image Understanding, vol. 88, pp. 76-93, 2002.
[36] M.J. Black and A. Rangarajan, “On the Unification of Line Processes, Outlier Rejection, and Robust Statistics with Applications in Early Vision,” Int'l J. Computer Vision, vol. 19, pp 57-91, 1996.
[37] S.S. Sinha and B.G. Schunck, “A Two-Stage Algorithm for Discontinuity-Preserving Surface Reconstruction,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, pp. 36-55, 1992.
[38] R.L. Stevenson and E.J. Delp, “Viewpoint Invariant Recovery of Visual Surfaces from Sparse Data,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 14, pp. 897-909, 1992.
[39] J.H. Yi and D.M. Chelberg, “Discontinuity-Preserving and Viewpoint Invariant Reconstruction of Visible Surfaces Using a First Order Regularization,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, pp. 624-629, 1995.
[40] P. Qiu, “Discontinuous Regression Surfaces Fitting,” The Annals of Statistics, vol. 26, pp. 2218-2245, 1998.
[41] C.-K. Chu, I.K. Glad, F. Godtliebsen, and J.S. Marron, “Edge Preserving Smoothers for Image Processing (with Discussion),” J. Am. Statistical Assoc., vol. 93, pp. 526-556, 1998.
[42] J. Polzehl and V.G. Spokoiny, “Adaptative Weights Smoothing with Applications to Image Restoration,” J. Royal Statistical Soc., vol. B-62, pp. 335-354, 2000.
[43] P. Qiu, “The Local Piecewisely Linear Kernel Smoothing Procedure for Fitting Jump Regression Surfaces,” Technometrics, vol. 46, pp. 87-98, 2004.
[44] J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications. New York: Chapman and Hall, 1996.
[45] I. Gijbels, A. Lambert, and P. Qiu, “Jump-Preserving Regression and Smoothing Using Local Linear Fitting: A Compromise,” The Annals of the Inst. Statistical Math., vol. 58, no. 3, 2006, to appear.
[46] G.Z. Yang, P. Burger, D.N. Firmin, and S.R. Underwood, “Structure Adaptative Anisotropic Image Filtering,” Image and Vision Computing, vol. 14, pp. 135-145, 1996.
[47] F. Chabat, G.Z. Yang, and D.M. Hansell, “A Corner Orientation Detector,” Image and Vision Computing, vol. 17, pp. 761-769, 1999.
[48] I. Gijbels, A. Lambert, and P. Qiu, “Edge-Preserving Image Denoising and Estimation of Discontinuous Surfaces,” technical report, Dept. of Math., Inst. de Statistique, Université Catholique de Louvain, http://www.stat.ucl.ac.be~alambert, 2005.
[49] T. Strohmer, “Computationally Attractive Reconstruction of Bandlimited Images from Irregular Samples,” IEEE Trans. Image Processing, vol. 6, pp. 540-548, 1997.
[50] M. Arigovindan, M. Sülhing, P. Hunziker, and M. Unser, “Variational Image Reconstruction from Arbitrarily Spaced Samples: A Fast Multiresolution Spline Solution,” IEEE Trans. Image Processing, vol. 14, pp 450-460, 2005.
[51] D. Comaniciu, “An Algorithm for Data-Driven Bandwidth Selection,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, pp. 281-283, 2003.
[52] D. Barash and D. Comaniciu, “A Common Framework for Nonlinear Diffusion, Adaptive Smoothing, Bilateral Filtering and Mean Shift,” IEEE Image and Vision Computing, vol. 22, pp 73-81, 2004.

Index Terms:
Corners, edges, jump-preserving estimation, local linear fit, noise, nonparametric regression, smoothing, surface fitting, weighted residual mean square.
Citation:
Ir?ne Gijbels, Alexandre Lambert, Peihua Qiu, "Edge-Preserving Image Denoising and Estimation of Discontinuous Surfaces," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 28, no. 7, pp. 1075-1087, July 2006, doi:10.1109/TPAMI.2006.140
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