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2 results on '"shift-invariant"'

1. Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces

Author

Stefan Ericsson and Niklas Grip

Subjects
Shift-invariant, Reproducing kernel, Pure mathematics, Irregular sampling, Deconvolution, Mathematical Analysis, Meyer wavelet, symbols.namesake, B-spline, Matematisk analys, Differentiable function, Informationsteknik - Signalbehandling, Mathematics, Basis (linear algebra), Information technology - Signal processing, Applied Mathematics, Mathematical analysis, Generating function, Shannon wavelet, Interpolating function, Shift-invariant space, Function (mathematics), Scaling function, Fourier transform, Bounded variation, symbols, Franklin, Analysis
Abstract

An important cornerstone of both wavelet and sampling theory is shift-invariant spaces, that is, spaces V spanned by a Riesz basis of integer-translates of a single function. Under some mild differentiability and decay assumptions on the Fourier transform of this function, we show that V also is generated by a function with Fourier transform φ ˆ ( ξ ) = ∫ ξ − π ξ + π g ( ν ) d ν for some g with ∫ R g ( ξ ) d ξ = 1 . We explain why analysis of this particular generating function can be more likely to provide large jitter bounds e such that any f ∈ V can be reconstructed from perturbed integer samples f ( k + e k ) whenever sup k ∈ Z | e k | ⩽ e . We use this natural deconvolution of φ ˆ ( ξ ) to further develop analysis techniques from a previous paper. Then we demonstrate the resulting analysis method on the class of spaces for which g has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions), on some particular choices of φ for which we know of no previously published bounds and finally, we use it to improve some previously known bounds for B-spline shift-invariant spaces.

Published
2011
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2. Multivariate dictionary learning and shift & 2D rotation invariant sparse coding

Author

David Mercier, Aurélien Mayoue, Quentin Barthélemy, Jerome Mars, Anthony Larue, SIGMAPHY (GIPSA-SIGMAPHY), Département Images et Signal (GIPSA-DIS), Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-GIPSA Pôle Sciences des Données (GIPSA-PSD), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Laboratoire Outils d'Analyse des Données (LOAD), Département Métrologie Instrumentation & Information (DM2I), Laboratoire d'Intégration des Systèmes et des Technologies (LIST), Université Paris-Saclay-Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Laboratoire d'Intégration des Systèmes et des Technologies (LIST), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Information, Modèles, Apprentissage [Gif-sur-Yvette] (LIMA), GIPSA - Signal Images Physique (GIPSA-SIGMAPHY), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Stendhal - Grenoble 3-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Intégration des Systèmes et des Technologies (LIST (CEA)), Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Laboratoire d'Intégration des Systèmes et des Technologies (LIST (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Laboratoire d'Intégration des Systèmes et des Technologies (LIST)

Subjects
Shift-invariant, Multivariate statistics, Sparse coding, 02 engineering and technology, 03 medical and health sciences, statistical analysis, [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], 0202 electrical engineering, electronic engineering, information engineering, Invariant (mathematics), signal processing, Mathematics, 030307 microscopy, 0303 health sciences, Approximation theory, Signal processing, K-SVD, Rotation-invariant, business.industry, Handwritten characters, Approximation algorithm, 020206 networking & telecommunications, Pattern recognition, artificial intelligence, Levenberg-Marquardt, Levenberg–Marquardt algorithm, machine learning, Multivariate DLA, artifical intelligence, Online learning, [INFO.INFO-IR]Computer Science [cs]/Information Retrieval [cs.IR], Multivariate OMP, Artificial intelligence, dictionary learning, business, Neural coding, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing
Abstract

International audience; In this article, we present a new tool for sparse coding : Multivariate DLA which empirically learns the characteristic patterns associated to a multivariate signals set. Once learned, Multivariate OMP approximates sparsely any signal of this considered set. These methods are specified to the 2D rotation-invariant case. Shift and rotation invariant cases induce a compact learned dictionary. Our methods are applied to 2D handwritten data in order to extract the elementary features of this signals set.

Published
2011

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