Prony's Method for Multivariate Signals

Peter, Thomas; Plonka, Gerlind; Schaback, Robert

doi:10.1002/pamm.201510322

Cited by 16 publications

(17 citation statements)

References 10 publications

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“…from as few function samples as possible. Until recently, algorithms to solve the problem required a number of samples of the order O(n d ) [16,18,24,26]…”

Section: Multidimensional Exponential Analysis the Problem Of D-dimementioning

confidence: 99%

“…In addition, the new technique does not suffer the well-known curse of dimensionality. A d-dimensional exponential analysis of an n-term model can now be carried out from a mere O((d + 1)n) regularly collected samples, which is substantially less than in other Prony-based methods [28,37,30,16,24,26], where the sample usage explodes exponentially. In [37] the entailed complexity of these numerical algorithms is improved by the use of a slicing technique.…”

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confidence: 99%

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Sparse Multidimensional Exponential Analysis with an Application to Radar Imaging

Cuyt¹,

Hou²,

Knaepkens³

et al. 2020

SIAM J. Sci. Comput.

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We present a d-dimensional exponential analysis algorithm that offers a range of advantages compared to other methods. The technique does not suffer the curse of dimensionality and only needs O((d + 1)n) samples for the analysis of an n-sparse expression. It does not require a prior estimate of the sparsity n of the d-variate exponential sum. The method can work with sub-Nyquist sampled data and offers a validation step, which is very useful in low SNR conditions. A favourable computation cost results from the fact that d independent smaller systems are solved instead of one large system incorporating all measurements simultaneously. So the method also lends itself easily to a parallel execution. Our motivation to develop the technique comes from 2D and 3D radar imaging and is therefore illustrated on such examples.

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“…from as few function samples as possible. Until recently, algorithms to solve the problem required a number of samples of the order O(n d ) [16,18,24,26]…”

Section: Multidimensional Exponential Analysis the Problem Of D-dimementioning

confidence: 99%

mentioning

confidence: 99%

Sparse Multidimensional Exponential Analysis with an Application to Radar Imaging

Cuyt¹,

Hou²,

Knaepkens³

et al. 2020

SIAM J. Sci. Comput.

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“…Next, it is natural to extend the techniques developed here to the multi-dimensional setting as multivariate signals are of high importance in many applications such as DNA sequencing and Mass Spectrometry. This could be investigated using for instance some extension of Prony's method to several dimensions such as in [37], [28] and [22].…”

Section: Future Directionsmentioning

confidence: 99%

Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

Chrétien

Tyagi

2020

J Fourier Anal Appl

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Consider L groups of point sources or spike trains, with the l th group represented by x l (t). For a function g : R → R, let g l (t) = g(t/µ l ) denote a point spread function with scale µ l > 0, and with µ 1 < · · · < µ L . With y(t) = L l=1 (g l x l )(t), our goal is to recover the source parameters given samples of y, or given the Fourier samples of y. This problem is a generalization of the usual super-resolution setup wherein L = 1; we call this the multi-kernel unmixing superresolution problem. Assuming access to Fourier samples of y, we derive an algorithm for this problem for estimating the source parameters of each group, along with precise non-asymptotic guarantees. Our approach involves estimating the group parameters sequentially in the order of increasing scale parameters, i.e., from group 1 to L. In particular, the estimation process at stage 1 ≤ l ≤ L involves (i) carefully sampling the tail of the Fourier transform of y, (ii) a deflation step wherein we subtract the contribution of the groups processed thus far from the obtained Fourier samples, and (iii) applying Moitra's modified Matrix Pencil method on a deconvolved version of the samples in (ii).

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“…In the 1dimensional case, the Prony-like exponential analysis methods, such as matrix pencil [7], ESPRIT [8], TLS-Prony [9] have all been successfully applied in solving many practical problems. At the same time, several multi-dimensional versions of these Prony-like methods have been developed, e.g., [10][11][12][13][14][15]. However, due to complexity issues, until recently these methods were not very suitable to serve as a general tool for higher-dimensional decomposition.…”

Section: Introductionmentioning

confidence: 99%

Decomposing Textures using Exponential Analysis

Hou

Cuyt

Lee

et al. 2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Decomposition is integral to most image processing algorithms and often required in texture analysis. We present a new approach using a recent 2-dimensional exponential analysis technique. Exponential analysis offers the advantage of sparsity in the model and continuity in the parameters. This results in a much more compact representation of textures when compared to traditional Fourier or wavelet transform techniques. Our experiments include synthetic as well as real texture images from standard benchmark datasets. The results outperform FFT in representing texture patterns with significantly fewer terms while retaining RMSE values after reconstruction. The underlying periodic complex exponential model works best for texture patterns that are homogeneous. We demonstrate the usefulness of the method in two common vision processing application examples, namely texture classification and defect detection.

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Prony's Method for Multivariate Signals

Cited by 16 publications

References 10 publications

Sparse Multidimensional Exponential Analysis with an Application to Radar Imaging

Sparse Multidimensional Exponential Analysis with an Application to Radar Imaging

Multi-kernel Unmixing and Super-Resolution Using the Modified Matrix Pencil Method

Decomposing Textures using Exponential Analysis

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