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Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions [chapter]

Lutz Kämmerer, Stefan Kunis, Ines Melzer, Daniel Potts, Toni Volkmer
2014 Lecture Notes in Computational Science and Engineering  
: We present stable and effective algorithms for the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on an index set I ⊂ Z d .  ...  Efficient algorithms like the fast Fourier transform (FFT) have to be customised to these thinner discretisations and we focus on two major topics regarding the Fourier analysis of high-dimensional functions  ...  As an extension to the reconstruction problem, we considered the efficient approximate reconstruction of a smooth function from subspaces of the Wiener algebra by a trigonometric polynomial (1) , which  ... 
doi:10.1007/978-3-319-08159-5_17 fatcat:nwbbldtm7vdl7hj3bmfw2rsm64

Compressed Sensing-Based Simultaneous Recovery of Magnitude and Phase MR Images via Dual Trigonometric Sparsity

Wei He, Xinwen Liu, Linman Zhao, Ran Li, Feng Liu
2021 IEEE Access  
The CS method requires a sparse representation of the original images, and it is observed the trigonometric functions of phase images in the Wavelet domain promote sparsity.  ...  The combination of the dual trigonometric functions captures a unique, faithful four-quadrant phase information, which also improves the reconstructed magnitude images through an alternating optimization  ...  As shown in Fig.1 , it is observed that the images computed by any trigonometric function of the phase p are generally sparse in the Wavelet domain.  ... 
doi:10.1109/access.2021.3062837 fatcat:vtxawubspja4niclojezihlsxm

Sparse high-dimensional FFT based on rank-1 lattice sampling

Daniel Potts, Toni Volkmer
2016 Applied and Computational Harmonic Analysis  
In this paper, we suggest approximate algorithms for the reconstruction of sparse high-dimensional trigonometric polynomials, where the support in frequency domain is unknown.  ...  When we restrict the search space in frequency domain to a full grid [−N, N ] d ∩ Z d of refinement N ∈ N and assume that the cardinality of the support of the trigonometric polynomial in frequency domain  ...  Example 3.12. s-sparse approximate reconstruction of a function using a single-step algorithm.  ... 
doi:10.1016/j.acha.2015.05.002 fatcat:d4rkz4sgyjghjkpvjf6ghp6fzy

High-dimensional sparse FFT based on sampling along multiple rank-1 lattices [article]

Lutz Kämmerer and Daniel Potts and Toni Volkmer
2017 arXiv   pre-print
The reconstruction of high-dimensional sparse signals is a challenging task in a wide range of applications.  ...  In this paper, both concepts - dimension-incremental reconstruction and multiple rank-1 lattices - are coupled, which yields a distinctly improved high-dimensional sparse fast Fourier transform.  ...  We approximate the function f by multivariate trigonometric polynomials p.  ... 
arXiv:1711.05152v1 fatcat:dsuqwqqhonb2jlggtbmkcm27ui

Reconstruction of sparse multiband wavelet signals from Fourier measurements

Yang Chen, Cheng Cheng, Qiyu Sun
2015 2015 International Conference on Sampling Theory and Applications (SampTA)  
In this paper, we consider the problem of reconstructing sparse multiband wavelet signals of finite levels from their samples in Fourier domain.  ...  We show that those sparse signals are determined by their Fourier measurements on a set of size proportional to their sparsity.  ...  This together with (III.6) provides a reconstruction of the function f J−1 − f J . Following similar steps, we can reconstruct functions f i−1 − f i by induction on i = J, J − 1, . . . , 1.  ... 
doi:10.1109/sampta.2015.7148854 fatcat:5ny4gv2jmbbhrjnmksd5ym2ui4

Sparse reconstruction of correlated multichannel activity

Sem Peelman, Joachim Van der Herten, Maarten De Vos, Wen-shin Lee, Sabine Van Huffel, Annie Cuyt
2013 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC)  
Secondly, to explore its use in the reconstruction of real-life signals.  ...  In general, these methods start by representing each sinusoidal component by means of two complex exponential functions, thereby doubling the number of unknown parameters.  ...  Sparse trigonometric interpolation In this section we apply a matrix pencil method to reconstruct a K-sparse trigonometric sum from 2K samples in the time domain.  ... 
doi:10.1109/embc.2013.6610396 pmid:24110583 dblp:conf/embc/PeelmanHVLHC13 fatcat:wxmqdjhamzhyned6e3tjhfenfa

Reconstruction of Sparse Wavelet Signals From Partial Fourier Measurements

Yang Chen, Cheng Cheng, Qiyu Sun
2015 IEEE Signal Processing Letters  
In this paper, we show that high-dimensional sparse wavelet signals of finite levels can be constructed from their partial Fourier measurements on a deterministic sampling set with cardinality about a  ...  From the proof of Theorem III.1, we have the following result on the reconstruction of an s-sparse trigonometric polynomial from its samples on a set of size 2s. IV.  ...  3(2t − 3/2)χ [1/2,1) (t) be wavelet functions.  ... 
doi:10.1109/lsp.2015.2478007 fatcat:ed3aoltgw5dzzij3uj2cb42xvm

Efficient Spectral Estimation by MUSIC and ESPRIT with Application to Sparse FFT

Daniel Potts, Manfred Tasche, Toni Volkmer
2016 Frontiers in Applied Mathematics and Statistics  
For a trigonometric polynomial of large sparsity, we present a new sparse fast Fourier transform by shifted sampling and using MUSIC resp.  ...  Later this technique is extended to a new reconstruction of a multivariate trigonometric polynomial of large sparsity for given (noisy) values sampled on a reconstructing rank-1 lattice.  ...  The results of this paper were first presented during the Dagstuhl Seminar 15251 on "Sparse modeling and multi-exponential analysis" (June 14 -19, 2015).  ... 
doi:10.3389/fams.2016.00001 fatcat:4znmafeag5gyjmetk3467463xi

A Deterministic Algorithm for Constructing Multiple Rank-1 Lattices of Near-Optimal Size [article]

Craig Gross, Mark A. Iwen, Lutz Kämmerer, Toni Volkmer
2020 arXiv   pre-print
accurate and efficient reconstruction of trigonometric polynomials with coefficients in I (and, therefore, for the approximation of multivariate periodic functions).  ...  Additionally, we present a second multiple rank-1 lattice construction algorithm which constructs lattices with even fewer sampling points at the cost of only being able to reconstruct exact trigonometric  ...  An Application to the Recovery of More General Functions As previously mentioned, we construct a cubature rule (1.4) that exactly reconstructs all Fourier coefficients of multivariate trigonometric polynomials  ... 
arXiv:2003.09753v1 fatcat:2veetkelofelhlqsbnie7l2mse

Grid-free compressive beamforming

Angeliki Xenaki, Peter Gerstoft
2015 Journal of the Acoustical Society of America  
The direction-of-arrival (DOA) estimation problem involves the localization of a few sources from a limited number of observations on an array of sensors, thus it can be formulated as a sparse signal reconstruction  ...  On a discrete angular grid, the CS reconstruction degrades due to basis mismatch when the DOAs do not coincide with the angular directions on the grid.  ...  (Color online) Grid-free sparse reconstruction.  ... 
doi:10.1121/1.4916269 pmid:25920844 fatcat:244xdnlqbfholnltqonkswxdsi

Stability results for random sampling of sparse trigonometric polynomials [article]

Holger Rauhut
2008 arXiv   pre-print
Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of non-zero coefficients, can be reconstructed exactly from a small number of random samples using  ...  For BP in addition, the stability result is extended to (non-sparse) trigonometric polynomials that can be well-approximated by sparse ones.  ...  Additionally, for BP we consider also the case that f is not sparse in a strict sense, but can be well approximated by a sparse trigonometric polynomial.  ... 
arXiv:math/0609630v3 fatcat:cgoddoituvby3ot2o4p36ep2hm

Random Sampling Using Shannon Interpolation and Poisson Summation Formulae [article]

Xiao Z. Wang, Wei E.I. Sha
2009 arXiv   pre-print
The numerical results for the trigonometric signal, the Gaussian-modulated sinusoidal pulse, and the square wave were demonstrated and discussed.  ...  The basic idea underlying the theory is that the sparse signals can be reconstructed from generally incomplete non-adaptive information.  ...  The function ) ( sin x c damps as the order of x 1 , thus (5) may play the most important role. However, ) ( sin x c function is not compact support.  ... 
arXiv:0909.2292v3 fatcat:h2al7m32ojaxrkind6amgcng3m

Response Properties of Single Neurons Predicted by Sparse Representation

Jiqian Liu, Chengbin Zeng, Liping Xiao
2015 International Journal of Machine Learning and Computing  
To make the test simpler, we use a group of two dimensional trigonometric functions and two Gaussian functions illustrated in Fig. 1 (b) as the basis set.  ...  The two dimensional trigonometric functions are given by ( , ) sin( ( cos sin ) ) 2 k u x y f y i x i          (2) where  is a scalar, f=1+0.7n, and ∆θ=π⁄6.  ... 
doi:10.7763/ijmlc.2015.v5.546 fatcat:p6zjitneofayjpjevmp6bt5e7q

Random Sampling of Sparse Trigonometric Polynomials II - Orthogonal Matching Pursuit versus Basis Pursuit [article]

Stefan Kunis, Holger Rauhut
2007 arXiv   pre-print
We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP  ...  While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous  ...  Its basic idea is that sparse or compressible signals can be reconstructed from vastly incomplete non-adaptive information.  ... 
arXiv:math/0604429v2 fatcat:jd7awfelfrafvo5vq7incopjfm

Approximation of multivariate periodic functions by trigonometric polynomials based on rank-1 lattice sampling

Lutz Kämmerer, Daniel Potts, Toni Volkmer
2015 Journal of Complexity  
Recently an algorithm for the trigonometric interpolation on generalized sparse grids for this class of functions was investigated in [12] .  ...  In this paper, we present algorithms for the approximation of multivariate periodic functions by trigonometric polynomials.  ...  Hamaekers [12] , where the authors used trigonometric interpolation based on generalized sparse grids, especially so-called energy norm based sparse grids [4, 5] , and developed the related hyperbolic  ... 
doi:10.1016/j.jco.2015.02.004 fatcat:dw7bnne3rrhgpewfcdrl3xpvaq
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