Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Journal of Approximation Theory
On the self-similar nature of Ethernet traffic
ACM SIGCOMM Computer Communication Review - Special twenty-fifth anniversary issue. Highlights from 25 years of the Computer Communication Review
Fractional Splines and Wavelets
SIAM Review
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Self-Similarity: Part II—Optimal Estimation of Fractal Processes
IEEE Transactions on Signal Processing
Self-Similarity: Part I—Splines and Operators
IEEE Transactions on Signal Processing
Generalized sampling: a variational approach .I. Theory
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
IEEE Transactions on Information Theory
A wavelet-based joint estimator of the parameters of long-range dependence
IEEE Transactions on Information Theory
Isotropic polyharmonic B-splines: scaling functions and wavelets
IEEE Transactions on Image Processing
On Rate-Distortion Models for Natural Images and Wavelet Coding Performance
IEEE Transactions on Image Processing
Wavelet steerability and the higher-order Riesz transform
IEEE Transactions on Image Processing
ICIP'09 Proceedings of the 16th IEEE international conference on Image processing
Fractal modelling and analysis of flow-field images
ISBI'10 Proceedings of the 2010 IEEE international conference on Biomedical imaging: from nano to Macro
Left-inverses of fractional Laplacian and sparse stochastic processes
Advances in Computational Mathematics
Journal of Approximation Theory
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In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional Brownian motion (fBm) fields. In particular, using a rigorous and powerful distribution theoretic formulation, we extend previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator. We then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional Laplacian, which also turns out to be the whitening operator for fBm fields. This fact allows us to study the probabilistic properties of the wavelet transform coefficients of fBm-like processes, leading for instance to ways of estimating the Hurst exponent of a multiparameter process from its wavelet transform coefficients. We provide theoretical and experimental verification of these results. To complement the toolbox available for multiresolution processing of stochastic fractals, we also introduce an extended family of multidimensional multiresolution spaces for a large class of (separable and nonseparable) lattices of arbitrary dimensionality.