Search

Your search keyword '"shift-invariant"' showing total 6 results
Start Over Descriptor "shift-invariant" Topic mathematics
6 results on '"shift-invariant"'

1. A Factory of Fractional Derivatives

Author

Manuel D. Ortigueira

Subjects
shift-invariant, scale-invariant, nabla derivative, fractional q-derivative, fractional Hahn derivative, Mathematics, QA1-939
Abstract

This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn.

Published
2024
Full Text
View/download PDF

2. Algebraic Signal Processing Theory: Foundation and 1-D Time.

Author

Püschel, Markus and Moura, José M. F.

Subjects
*DIGITAL signal processing, *INFORMATION measurement, *SIGNAL theory, *VECTOR analysis, *BOUNDARY value problems, *FOURIER transforms, *FOURIER analysis, *MATHEMATICAL transformations, *MATHEMATICS
Abstract

This paper introduces a general and axiomatic approach to linear signal processing (SP) that we refer to as the algebraic signal processing theory (ASP). Basic to ASP is the linear signal model defined as a triple (A, M, Φ) where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively. The mapping Φ generalizes the concept of a z-transform to bijective linear mappings from a vector space of signal samples into the module M. Common concepts like filtering, spectrum, or Fourier transform have their equivalent counterparts in ASP. Once these concepts and their properties are defined and understood in the context of ASP, they remain true and apply to specific instantiations of the ASP signal model. For example, to develop signal processing theories for infinite and finite discrete time signals, for infinite or finite discrete space signals, or for multidimensional signals, we need only to instantiate the signal model to one that makes sense for that specific class of signals. Filtering, spectrum, Fourier transform, and other notions follow then from the corresponding ASP concepts. Similarly, common assumptions in SP translate into requirements on the ASP signal model. For example, shift-invariance is equivalent to A being commutative. For finite (duration) signals shift invariance then restricts A to polynomial algebras. We explain how to design signal models from the specification of a special filter, the shift. The paper illustrates the general ASP theory with the standard time shift, presenting a unique signal model for infinite time and several signal models for finite time. The latter models illustrate the role played by boundary conditions and recover the discrete Fourier transform (DFT) and its variants as associated Fourier transforms. Finally, ASP provides a systematic methodology to derive fast algorithms for linear transforms. This topic and the application of ASP to space dependent signals and to multidimensional signals are pursued in companion papers. [ABSTRACT FROM AUTHOR]

Published
2008
Full Text
View/download PDF

3. 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
Full Text
View/download PDF

4. Shift & 2D Rotation Invariant Sparse Coding for Multivariate Signals

Author

Anthony Larue, David Mercier, Quentin Barthélemy, Aurélien Mayoue, Jerome Mars, 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), 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), 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), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, 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)), 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 (CEA))

Subjects
Multivariate statistics, handwritten data, sparse coding, orthogonal matching pursuit, online learning, 02 engineering and technology, multivariate, multichannel, statistical analysis, [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], shift-invariant, 0202 electrical engineering, electronic engineering, information engineering, trajectory characters, Electrical and Electronic Engineering, rotation invariant, signal processing, Mathematics, Approximation theory, business.industry, Approximation algorithm, 020206 networking & telecommunications, Pattern recognition, Sparse approximation, Invariant (physics), Data structure, artificial intelligence, Matching pursuit, machine learning, Dictionary learning algorithm, 020201 artificial intelligence & image processing, Artificial intelligence, dictionary learning, Neural coding, business, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, [STAT.ME]Statistics [stat]/Methodology [stat.ME]
Abstract

International audience; Classical dictionary learning algorithms (DLA) allow unicomponent signals to be processed. Due to our interest in two-dimensional (2D) motion signals, we wanted to mix the two components to provide rotation invariance. So, multicomponent frameworks are examined here. In contrast to the well-known multichannel framework, a multivariate framework is first introduced as a tool to easily solve our problem and to preserve the data structure. Within this multivariate framework, we then present sparse coding methods: multivariate orthogonal matching pursuit (M-OMP), which provides sparse approximation for multivariate signals, and multivariate DLA (M-DLA), which empirically learns the characteristic patterns (or features) that are associated to a multivariate signals set, and combines shift-invariance and online learning. Once the multivariate dictionary is learned, any signal of this considered set can be approximated sparsely. This multivariate framework is introduced to simply present the 2D rotation invariant (2DRI) case. By studying 2D motions that are acquired in bivariate real signals, we want the decompositions to be independent of the orientation of the movement execution in the 2D space. The methods are thus specified for the 2DRI case to be robust to any rotation: 2DRI-OMP and 2DRI-DLA. Shift and rotation invariant cases induce a compact learned dictionary and provide robust decomposition. As validation, our methods are applied to 2D handwritten data to extract the elementary features of this signals set, and to provide rotation invariant decomposition.

Published
2012

5. 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

6. BUILDING INVARIANCE PROPERTIES FOR DICTIONARIES OF SAR IMAGE PATCHES

Author

Charles-Alban Deledalle, Sonia Tabti, Florence Tupin, Loïc Denis, TSI-TII radar, Laboratoire Traitement et Communication de l'Information (LTCI), Télécom ParisTech-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS)-Télécom ParisTech-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Hubert Curien [Saint Etienne] (LHC), Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-Institut d'Optique Graduate School (IOGS), Télécom ParisTech-Institut Mines-Télécom [Paris] (IMT)-Centre National de la Recherche Scientifique (CNRS), Télécom ParisTech, and CNRS/DGA

Subjects
Semantic interpretation, Noise reduction, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, 02 engineering and technology, Image (mathematics), Speckle pattern, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, shift-invariant, 0202 electrical engineering, electronic engineering, information engineering, Computer vision, Representation (mathematics), Mathematics, Structure (mathematical logic), business.industry, contrast-invariant, 020206 networking & telecommunications, Pattern recognition, Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing), SAR images, Computer Science::Graphics, Computer Science::Computer Vision and Pattern Recognition, Patches, 020201 artificial intelligence & image processing, Artificial intelligence, Affine transformation, business, [SPI.SIGNAL]Engineering Sciences [physics]/Signal and Image processing, dictionary
Abstract

Adding invariance properties to a dictionary-based model is a convenient way to reach a high representation capacity while maintaining a compact structure. Compact dictionaries of patches are desirable because they ease semantic interpretation of their elements (atoms) and offer robust decompositions even under strong speckle fluctuations. This paper describes how patches of a dictionary can be matched to a speckled image by accounting for unknown shifts and affine radiometric changes. This procedure is used to build dictionaries of patches specific to SAR images. The dictionaries can then be used for denoising or classification purposes.

Catalog

Books, media, physical & digital resources
See catalog results