Linear inversion of ban limit reflection seismograms
SIAM Journal on Scientific and Statistical Computing
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Characterization of Signals from Multiscale Edges
IEEE Transactions on Pattern Analysis and Machine Intelligence
A family of polynomial spline wavelet transforms
Signal Processing
Fractional Splines and Wavelets
SIAM Review
Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)
Wavelet-based signal de-noising via simple singularities approximation
Signal Processing
Seismic deconvolution by atomic decomposition: A parametric approach with sparseness constraints
Integrated Computer-Aided Engineering
The fractional spline wavelet transform: definition end implementation
ICASSP '00 Proceedings of the Acoustics, Speech, and Signal Processing, 2000. on IEEE International Conference - Volume 01
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Construction of Hilbert transform pairs of wavelet bases and Gabor-like transforms
IEEE Transactions on Signal Processing
Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform
IEEE Transactions on Image Processing
Higher-order Riesz transforms and steerable wavelet frames
ICIP'09 Proceedings of the 16th IEEE international conference on Image processing
Wavelets, fractals, and radial basis functions
IEEE Transactions on Signal Processing
Self-Similarity: Part I—Splines and Operators
IEEE Transactions on Signal Processing
Wavelet footprints: theory, algorithms, and applications
IEEE Transactions on Signal Processing
Analysis of multiscale products for step detection and estimation
IEEE Transactions on Information Theory
Greed is good: algorithmic results for sparse approximation
IEEE Transactions on Information Theory
Image representations using multiscale differential operators
IEEE Transactions on Image Processing
Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid
IEEE Transactions on Image Processing
Hi-index | 35.69 |
We introduce a family of elementary singularities that are point-Hölder -regular. These singularities are self-similar and are the Green functions of fractional derivative operators; i.e., by suitable fractional differentiation, one retrieves a Dirac function at the exact location of the singularity. We propose to use fractional operator-like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain. We show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed-form expression, termed the analytical footprint. Practically, this means that the dictionary of wavelet footprints is embodied in a single analytical form. We show that the wavelet coefficients of the (nonredundant) decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity. We propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations.