Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval

Rebollo-Neira, Laura; Xu, Zhiqiang

doi:10.1016/j.sigpro.2010.02.004

Cited by 10 publications

(8 citation statements)

References 22 publications

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“…Now, we choose a parameter

such that

for some integer

. We define functions

and

which form a redundant dictionary [1] , [2] , [21] . Obviously,

corresponds to a basis.…”

Section: Wavelet Dictionariesmentioning

confidence: 99%

Construction of wavelet dictionaries for ECG modeling

Černá

Rebollo-Neira

2021

MethodsX

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Technical details, algorithms, and MATLAB implementation for a method advanced in the paper ``Wavelet Based Dictionaries for Dimensionality Reduction of ECG Signals'', are presented. This work aims to be the companion of that publication, in which an adaptive mathematical model for a given ECG record is proposed. The method comprises the following building blocks. Construction of a suitable redundant set, called 'dictionary', for decomposing an ECG signal as a superposition of elementary components, called 'atoms', selected from that dictionary. Implementation of the greedy strategy Optimized Orthogonal Matching Pursuit (OOMP) for selecting the atoms intervening in the signal decomposition. This paper gives the details of the algorithms for implementing stage (i), which is not fully elaborated in the previous publication. The proposed dictionaries are constructed from known wavelet families, but translating the prototypes with a shorter step than that corresponding to a wavelet basis. Stage (ii) is readily implementable by the available function OOMP. The use of the software and the power of the technique is illustrated by reducing the dimensionality of ECG records taken from the MIT-BIH Arrhythmia Database. The MATLAB software has been made publicly available on a dedicated website. We provide the explanations, algorithms and software for the construction of scaling functions and wavelet prototypes for 17 different wavelet families. The procedure is designed to allow for straightforward extension of the software by the inclusion of additional options for the wavelet families.

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“…Now, we choose a parameter

such that

for some integer

. We define functions

and

which form a redundant dictionary [1] , [2] , [21] . Obviously,

corresponds to a basis.…”

Section: Wavelet Dictionariesmentioning

confidence: 99%

Construction of wavelet dictionaries for ECG modeling

Černá

Rebollo-Neira

2021

MethodsX

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“…We next show that, without the condition of strongly linearly independent on some Ω 0 , it is possible that any 2s points are not unisolvent. We consider the sparse interpolation for B-spline basis which is useful in signal processing [23]. The first order B-spline on the interval [j, j + 1) is defined by Thus, B 1,j0 (x t ) = 0 for t = 1, .…”

Section: Hence the Solution Tomentioning

confidence: 99%

On Sparse Interpolation and the Design of Deterministic Interpolation Points

Xu¹,

Zhou²

2014

SIAM J. Sci. Comput.

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In this paper, we build up a framework for sparse interpolation. We first investigate the theoretical limit of the number of unisolvent points for sparse interpolation under a general setting and try to answer some basic questions of this topic. We also explore the relation between classical interpolation and sparse interpolation. We second consider the design of the interpolation points for the s-sparse functions in high dimensional Chebyshev bases, for which the possible applications include uncertainty quantification, numerically solving stochastic or parametric PDEs and compressed sensing. Unlike the traditional random sampling method, we present in this paper a deterministic method to produce the interpolation points, and show its performance with ℓ1 minimization by analyzing the mutual incoherence of the interpolation matrix. Numerical experiments show that the deterministic points have a similar performance with that of the random points.

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“…The other component of the proposed mixed dictionary, which allows for the representation of bright spots and edges, is inspired by a general result holding for continuous spline spaces. Namely, that spline spaces on a compact interval can be spanned by dictionaries of B-splines of broader support than the corresponding B-spline basis functions [26,27]. This may result in a very considerable gain in sparsity for functions well approximated in spline spaces.…”

Section: Redundant Discrete B-spline (Rdbs) Based Dictionariesmentioning

confidence: 99%

Sparse representation of astronomical images

Rebollo-Neira

Bowley

2013

J. Opt. Soc. Am. A

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Sparse representation of astronomical images is discussed. It is shown that a significant gain in sparsity is achieved when particular mixed dictionaries are used for approximating these types of images with greedy selection strategies. Experiments are conducted to confirm: i)Effectiveness at producing sparse representations. ii)Competitiveness, with respect to the time required to process large images. The latter is a consequence of the suitability of the proposed dictionaries for approximating images in partitions of small blocks. This feature makes it possible to apply the effective greedy selection technique Orthogonal Matching Pursuit, up to some block size. For blocks exceeding that size a refinement of the original Matching Pursuit approach is considered. The resulting method is termed Self Projected Matching Pursuit, because is shown to be effective for implementing, via Matching Pursuit itself, the optional back-projection intermediate steps in that approach.

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Sparse signal representation by adaptive non-uniform B-spline dictionaries on a compact interval

Cited by 10 publications

References 22 publications

Construction of wavelet dictionaries for ECG modeling

Construction of wavelet dictionaries for ECG modeling

On Sparse Interpolation and the Design of Deterministic Interpolation Points

Sparse representation of astronomical images

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