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On the Design of Deterministic Matrices for Fast Recovery of Fourier Compressible Functions [article]

J. Bailey and M. A. Iwen and C. V. Spencer
2011 arXiv   pre-print
We present a general class of compressed sensing matrices which are then demonstrated to have associated sublinear-time sparse approximation algorithms.  ...  Ultimately, these considerations improve previous sampling requirements for deterministic sparse Fourier transform methods.  ...  Introduction This paper considers methods for designing matrices which yield near-optimal nonlinear approximations to the Fourier transform of a given function, f : [0, 2π] → C.  ... 
arXiv:1105.6138v1 fatcat:cv3chsp47zc2fkerawlue3mdam

On the Design of Deterministic Matrices for Fast Recovery of Fourier Compressible Functions

J. Bailey, M. A. Iwen, C. V. Spencer
2012 SIAM Journal on Matrix Analysis and Applications  
We present a general class of compressed sensing matrices which are then demonstrated to have associated sublinear-time sparse approximation algorithms.  ...  Ultimately, these considerations improve previous sampling requirements for deterministic sparse Fourier transform methods.  ...  In this paper we will focus on constructing m × N compressed sensing matrices, M, for the Fourier recovery problem which meet the following four design requirements: 1.  ... 
doi:10.1137/110835864 fatcat:nsmf6mb6gbhxxi3vft3q5lkv34

Simple deterministically constructible RIP matrices with sublinear fourier sampling requirements

M. A. Iwen
2009 2009 43rd Annual Conference on Information Sciences and Systems  
As a consequence, we obtain small deterministic sample sets which are guaranteed to allow the recovery of near-optimal sparse Fourier representations for all periodic functions having an integrable second  ...  The Fourier sampling requirements obtained herein improve on previous deterministic Fourier sampling results in [1], [2] .  ...  In doing so we provide fast algorithms for approximating the product of our measurement matrices with the discrete Fourier transform of an array of samples from an arbitrary Fourier-compressible signal  ... 
doi:10.1109/ciss.2009.5054839 dblp:conf/ciss/Iwen09 fatcat:htqjg5ypbzaz3fyx7lwbuvnpkq

Learning to Observation Matrices of Compressive Sensing [chapter]

Guosheng Gu, Jie Ling
2012 Advances in Intelligent and Soft Computing  
In this paper, a binary sparse observation matrix for compressive sensing is deterministically constructed via a pseudo-random sequence generated by the sub-shift mapping of finite type on the chaotic  ...  Analysis and experimental results demonstrate the proposed matrix's simplification can be regarded as a reliable method and is usable in compressive sensing applications.  ...  Deterministic Observation Matrices For the sake of making Compressive Sensing applications more practicable, a mount of variable techniques also have been presented for constructing deterministic observation  ... 
doi:10.1007/978-3-642-30223-7_50 fatcat:egyzm54pjff7vpf4xoevodvk2y

State of the art and prospects of structured sensing matrices in compressed sensing

Kezhi Li, Shuang Cong
2015 Frontiers of Computer Science  
Based on the restricted isometry property and coherence, couples of existing structured sensing matrices are reviewed in this paper, which have special structures, high recovery performance, and many advantages  ...  The number of measurements and the universality of different structure matrices are compared.  ...  Here we focus on the problem of designing proper sensing matrices.  ... 
doi:10.1007/s11704-015-3326-8 fatcat:ftl6hdkjdvaqflxcjo2imcn26a

Compressive Sensing Algorithms for Signal Processing Applications: A Survey

Mohammed M. Abo-Zahhad, Aziza I. Hussein, Abdelfatah M. Mohamed
2015 International Journal of Communications, Network and System Sciences  
As a main feature of CS, efficient algorithms such as 1  -minimization can be used for recovery.  ...  The measurements are not point samples but more general linear functions of the signal.  ...  In this section, a survey of deterministic sensing matrices for compressive sensing is presented.  ... 
doi:10.4236/ijcns.2015.86021 fatcat:vx2l5d2tunhn3cbsdtfrfjaj2a

Compressive Sensing Algorithms for Signal Processing Applications: A Survey

Mohammed M. Abo-Zahhad, Aziza I. Hussein, Abdelfatah M. Mohamed
2015 International Journal of Communications, Network and System Sciences  
As a main feature of CS, efficient algorithms such as 1  -minimization can be used for recovery.  ...  The measurements are not point samples but more general linear functions of the signal.  ...  In this section, a survey of deterministic sensing matrices for compressive sensing is presented.  ... 
doi:10.4236/ijcns.2015.85021 fatcat:jklt3phauvafno3d2634pj55qi

Multiple Target Localization in WSNs Based on Compressive Sensing Using Deterministic Sensing Matrices

Thu L. N. Nguyen, Yoan Shin
2015 International Journal of Distributed Sensor Networks  
For this purpose, random sensing matrices have been studied, while a few researches on deterministic sensing matrices have been considered.  ...  This cooperation between the compressive sensing using deterministic sensing matrices and multiple target localization provides a new point of view in WSN localization.  ...  Most of early works on compressive sensing construct random matrices as sensing matrices, and some recent works have designed specific deterministic sensing matrices based on some given conditions.  ... 
doi:10.1155/2015/947016 fatcat:5xaxifvpnbc5dony6k4knpuzim

Chirp sensing codes: Deterministic compressed sensing measurements for fast recovery

Lorne Applebaum, Stephen D. Howard, Stephen Searle, Robert Calderbank
2009 Applied and Computational Harmonic Analysis  
Here we design deterministic measurements and an algorithm to accomplish signal recovery with computational efficiency. A measurement matrix is designed with chirp sequences forming the columns.  ...  This is done by bounding the eigenvalues of sub-matrices, as well as an empirical comparison with random projections.  ...  Acknowledgment The authors would like to thank Jarvis Haupt, Waheed Bajwa and Rob Nowak for their useful discussions on RIP recovery conditions.  ... 
doi:10.1016/j.acha.2008.08.002 fatcat:mhz3ajgv4raubcfzlnwxij3nhi

Deterministic Construction of Partial Fourier Compressed Sensing Matrices Via Cyclic Difference Sets [article]

Nam Yul Yu
2010 arXiv   pre-print
The restricted isometry property (RIP) is statistically studied for the deterministic matrix to guarantee the recovery of sparse signals.  ...  This paper studies a K × N partial Fourier measurement matrix for compressed sensing which is deterministically constructed via cyclic difference sets (CDS).  ...  recovery rates for partial Fourier matrices (K = 48 and N = 2257) Figure 4 4 compares the performance of our reconstruction algorithm with matching pursuit recovery for K = 49 and N = 197 in the presence  ... 
arXiv:1008.0885v2 fatcat:xdpbs7h62nds5pncz2gdztn45a

Sketching via hashing

Piotr Indyk
2013 Proceedings of the 32nd symposium on Principles of database systems - PODS '13  
Unfortunately, any operation on such matrices takes O(nm) time, which makes the recovery algorithms somewhat slow for high values of n. 3 It was observed in [CM06] (cf.  ...  The first algorithms of this type were designed for the Hadamard Transform, i.e., the Fourier transform over the Boolean cube [KM91, Lev93] (cf. [GL89, Gol99] ).  ... 
doi:10.1145/2463664.2465217 dblp:conf/pods/Indyk13 fatcat:rt4bws4lkfdulj43wa3xnqxt2y

Semi-deterministic Sparse Matrix for Low Complexity Compressive Sampling

2017 KSII Transactions on Internet and Information Systems  
This paper focuses on the enhancement of the practicability of the structurally random matrices and proposes a semi-deterministic sensing matrix called Partial Kronecker product of Identity and Hadamard  ...  The construction of completely random sensing matrices of Compressive Sensing requires a large number of random numbers while that of deterministic sensing operators often needs complex mathematical operations  ...  Introduction The design of the sensing matrix ∈ Φ ℝ M×N ( ) M N << is one of the three key problems of Compressive Sensing [1] (or Compressed Sampling, CS), since the sensing matrix determines whether  ... 
doi:10.3837/tiis.2017.05.009 fatcat:z3xatzs3wbfs3p5cdxenvuoauu

A Class of Deterministic Sensing Matrices and Their Application in Harmonic Detection [article]

Shan Huang, Hong Sun, Lei Yu, Haijian Zhang
2015 arXiv   pre-print
In this paper, a class of deterministic sensing matrices are constructed by selecting rows from Fourier matrices.  ...  These matrices have better performance in sparse recovery than random partial Fourier matrices.  ...  In this paper, we shall verify that a class of deterministic partial Fourier matrices can be used as sensing matrices and utilize these matrices to design a deterministic sampling method for harmonic detection  ... 
arXiv:1509.02628v1 fatcat:qmj7yskumzevthbi72y7b2orsq

Improved sparse fourier approximation results: faster implementations and stronger guarantees

Ben Segal, M. A. Iwen
2012 Numerical Algorithms  
In particular, we prove the existence of sublinear-time Las Vegas Fourier Transforms which improve on the recent deterministic Fourier approximation results of [21, 22] for Fourier compressible functions  ...  In addition to our empirical evaluation, we also consider the existence of sublinear-time Fourier approximation methods with deterministic approximation guarantees for functions whose sequences of Fourier  ...  Acknowledgments We would like to thank Martin Strauss for answering questions concerning Theorem 1.  ... 
doi:10.1007/s11075-012-9621-7 fatcat:neq3qvwchzadlnyxhfpyqmsq5i

An Improved Toeplitz Measurement Matrix for Compressive Sensing

Xu Su, Yin Hongpeng, Chai Yi, Xiong Yushu, Tan Xue
2014 International Journal of Distributed Sensor Networks  
A proper measurement matrix for compressive sensing is significance in above processions. In most compressive sensing frameworks, random measurement matrix is employed.  ...  Compressive sensing (CS) takes advantage of the signal's sparseness in some domain, allowing the entire signal to be efficiently acquired and reconstructed from relatively few measurements.  ...  The project is CSTC, 2010BB2065. This Project is granted financial support from China Postdoctoral Science Foundation (2012M521676).  ... 
doi:10.1155/2014/846757 fatcat:j7czktgsp5f4darwcs44bpcbqe
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