Automated Smoothing of Image and Other Regularly Spaced Data

This paper is primarily motivated by the problem of automatically removing unwanted noise from high-dimensional remote sensing imagery. The initial step involves the transformation of the data to a space of intrinsically lower dimensionality and the smoothing of images in the new space. Different images require different amounts of smoothing. The signal (assumed to be mostly smooth with relatively few discontinuities) is estimated from the data using the method of generalized cross-validation. It is shown how the generalized cross-validated thin-plate smoothing spline with observations on a regular grid (in d-dimensions) is easily approximated and computed in the Fourier domain. Space domain approximations are also investigated. The technique is applied to some remote sensing data. >

[1]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[2]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[3]  B. Silverman,et al.  Spline Smoothing: The Equivalent Variable Kernel Method , 1984 .

[4]  G. Wahba.  Bayesian "Confidence Intervals" for the Cross-validated Smoothing Spline , 1983 .

[5]  M. Hutchinson,et al.  An efficient method for calculating smoothing splines using orthogonal transformations , 1986 .

[6]  A. Girard.  A fast ‘Monte-Carlo cross-validation’ procedure for large least squares problems with noisy data , 1989 .

[7]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[8]  B. Silverman,et al.  Some Aspects of the Spline Smoothing Approach to Non‐Parametric Regression Curve Fitting , 1985 .

[9]  J. B. Lee,et al.  Enhancement of high spectral resolution remote-sensing data by a noise-adjusted principal components transform , 1990 .

[10]  Biao Zhang,et al.  Kernel approximations for universal kriging predictors , 1993 .

[11]  J. Crank.  Tables of Integrals , 1962 .

[12]  M. Stein.  A kernel approximation to the kriging predictor of a spatial process , 1991 .

[13]  J. Wendelberger,et al.  The Computation of Laplacian Smoothing Splines with Examples. , 1981 .

[14]  R. Mersereau,et al.  Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration , 1990 .

[15]  Glenn Stone,et al.  Computation of Thin-Plate Splines , 1991, SIAM J. Sci. Comput..

[16]  D. Titterington,et al.  A cautionary note about crossvalidatory choice , 1989 .

[17]  D. M. Titterington,et al.  A Study of Methods of Choosing the Smoothing Parameter in Image Restoration by Regularization , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  R. Singleton.  An algorithm for computing the mixed radix fast Fourier transform , 1969 .

[19]  R. Bracewell.  The Fourier Transform and Its Applications , 1966 .

[20]  F. O’Sullivan.  Discretized Laplacian Smoothing by Fourier Methods , 1991 .

[21]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[22]  M. Buckley.  Fast computation of a discretized thin-plate smoothing spline for image data , 1994 .

[23]  G. Wahba.  Spline models for observational data , 1990 .

[24]  G. Wahba.  Smoothing noisy data with spline functions , 1975 .

[25]  M. Hutchinson.  A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines , 1989 .

[26]  O. Dubrule.  Two methods with different objectives: Splines and kriging , 1983 .

[27]  P. Switzer,et al.  A transformation for ordering multispectral data in terms of image quality with implications for noise removal , 1988 .