Fractional Laplacian pyramids

We provide an extension of the L2-spline pyramid (Unser et al., 1993) using polyharmonic splines. We analytically prove that the corresponding error pyramid behaves exactly as a multi-scale Laplace operator. We use the multiresolution properties of polyharmonic splines to derive an efficient, non-separable filterbank implementation. Finally, we illustrate the potentials of our pyramid by performing an estimation of the parameters of multivariate fractal processes.

[1]  Dimitri Van De Ville,et al.  Invariances, Laplacian-Like Wavelet Bases, and the Whitening of Fractal Processes , 2009, IEEE Transactions on Image Processing.

[2]  J. Duchon.  Spline minimizing rotation-invariant seminorms in Sobolev spaces , 1977 .

[3]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[4]  Edward H. Adelson,et al.  The Laplacian Pyramid as a Compact Image Code , 1983, IEEE Trans. Commun..

[5]  Jean Duchon,et al.  Splines minimizing rotation-invariant semi-norms in Sobolev spaces , 1976, Constructive Theory of Functions of Several Variables.

[6]  Thierry Blu,et al.  Fast Computation of Polyharmonic B-Spline Autocorrelation Filters , 2008, IEEE Signal Processing Letters.

[7]  S. Kay,et al.  Fractional Brownian Motion: A Maximum Likelihood Estimator and Its Application to Image Texture , 1986, IEEE Transactions on Medical Imaging.

[8]  S. Jaffard,et al.  Elliptic gaussian random processes , 1997 .

[9]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[10]  Michael Unser,et al.  The L2-Polynomial Spline Pyramid , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing, 2nd Edition , 1999 .

[12]  W. Madych,et al.  Polyharmonic cardinal splines , 1990 .

[13]  Christophe Rabut,et al.  Elementarym-harmonic cardinal B-splines , 1992, Numerical Algorithms.