A deterministic sparse FFT for functions with structured Fourier sparsity

Bittens, Sina; Zhang, Ruochuan; Iwen, Mark A.

doi:10.1007/s10444-018-9626-4

Cited by 24 publications

(39 citation statements)

References 45 publications

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“…Since all matrices occurring in (21) are invertible, we derive z (j+1) (0) as in (16). Then y (j+1) is given as in (17) and (18) and symmetry (5). Note that if L (j) = 2 j , then z (j) = y (j) and z…”

Section: A Dft Procedures For Case A: One-block Supportmentioning

confidence: 99%

Sparse fast DCT for vectors with one-block support

Bittens

Plonka

2018

Numer Algor

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In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II that reconstructs the input vector x ∈ R N , N = 2 J−1 , with short support of length m from its discrete cosine transformThe resulting algorithm has a runtime of O m log m log 2N m and requires O m log 2N m samples of x II .In order to derive this algorithm we also develop a new fast and deterministic inverse FFT algorithm that constructs the input vector y ∈ R 2N with reflected block support of block length m from y with the same runtime and sampling complexities as our DCT algorithm.

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Section: A Dft Procedures For Case A: One-block Supportmentioning

confidence: 99%

Sparse fast DCT for vectors with one-block support

Bittens

Plonka

2018

Numer Algor

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“…(2) F f =:f 4 Of course deterministic algorithms with error guarantees of the type of (1) do exist for more restricted classes of periodic functions f . See, e.g., [2,3,23] for some examples. These include USSFT methods developed for periodic functions with structured Fourier support [2] which are of use for, among other things, the fast approximation of functions which exhibit sparsity with respect to other bounded orthonormal basis functions [13].…”

Section: Thenmentioning

confidence: 99%

“…In addition, the method used to develop this new deterministic DSFT algorithm is general enough that it can be applied to any fast and noise robust USSFT method of the type mentioned above (be it deterministic, or randomized) in order to yield a new fast and robust DSFT algorithm. As a result, we are also able to use the fastest of the currently existing USSFT methods [15,17,19,25,21,5,3] in order to create new publicly available DSFT implementations herein which are both faster and more robust to noise than currently existing noise robust DSFT methods for large N .…”

Section: Introductionmentioning

confidence: 99%

A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees

Merhi

Zhang

Iwen

et al. 2018

J Fourier Anal Appl

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In this paper we consider Sparse Fourier Transform (SFT) algorithms for approximately computing the best s-term approximation of the Discrete Fourier Transform (DFT)f ∈ C N of any given input vector f ∈ C N in just (s log N ) O(1) -time using only a similarly small number of entries of f . In particular, we present a deterministic SFT algorithm which is guaranteed to always recover a near best s-term approximation of the DFT of any given input vector f ∈ C N in O s 2 log 11 2 (N )time. Unlike previous deterministic results of this kind, our deterministic result holds for both arbitrary vectors f ∈ C N and vector lengths N . In addition to these deterministic SFT results, we also develop several new publicly available randomized SFT implementations for approximately computingf from f using the same general techniques. The best of these new implementations is shown to outperform existing discrete sparse Fourier transform methods with respect to both runtime and noise robustness for large vector lengths N . 2 ∩Z, by sampling its associated trigonometric polynomial f (x) = ω∈Bf ω e iωx

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“…Parts of this thesis have already been published in our papers [Bit17c,BP18c,BP18a,BZI19]. I significantly contributed to the publications [BZI19,BP18c,BP18a], which constitute Chapters 3, 5 and 6, and I am the corresponding author for all three.…”

Section: Please Notementioning

confidence: 93%

Sparse Fast Trigonometric Transforms

Vanessa¹

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This thesis is the result of the work I did in the last three years, during which I had the opportunity to work at the Institute for Numerical and Applied Mathematics at the University of Göttingen.First of all I owe my deepest thanks to my adviser Gerlind Plonka-Hoch for introducing me to the world of signal and image processing. I could not imagine a more interesting and inspiring topic of research. Furthermore, I would like to thank her, and my coadviser Russell Luke, for their constantly open doors and their continued support during all stages of my PhD. My sincere thanks also go to the co-referees of my thesis, Daniel Potts and Felix Krahmer. I am particularly grateful to Felix Krahmer for inviting me to Munich in May 2016 to meet Mark Iwen. The collaboration initiated by that visit and finalized during a two-week research stay at Michigan State University in February 2017 resulted in Chapter 3 of my thesis. For the invitation to MSU and also for his support during our collaboration I would like to thank Mark Iwen as well.Furthermore, I gratefully acknowledge the financial support of the German Research Foundation (DFG) in the framework of the Research Training Group (RTG) 2088 "Discovering structure in complex data: Statistics meets Optimization and Inverse Problems". For the whole duration of my PhD I was either a member or an associated member of this RTG, enabling me to attend many conferences and workshops for broadening my scientific horizon, but also giving me the opportunity to improve my soft skills.I am particularly indebted to my wonderful colleagues, former colleagues and friends from the "Mathematical Signal and Image Processing" group for the incredible working environment and their emotional support. I would especially like to thank Hanna Knirsch, Inge Keller and Markus Petz for proofreading this thesis.I am eternally grateful to my family and all my friends for their unconditional support, advice and patience with me during the past three years. This thesis would not exist without you. Finally, I want to express my deepest gratitude to Daniel H.

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A deterministic sparse FFT for functions with structured Fourier sparsity

Cited by 24 publications

References 45 publications

Sparse fast DCT for vectors with one-block support

Sparse fast DCT for vectors with one-block support

A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees

Sparse Fast Trigonometric Transforms

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