Real sparse fast DCT for vectors with short support

Bittens, Sina; Plonka, Gerlind

doi:10.1016/j.laa.2019.08.006

Cited by 5 publications

(19 citation statements)

References 12 publications

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“…we can also restrict the matrix F 2 j W 2 j ∈ C 2 j ×2 j in the linear system (5) to its M j columns with indices n r . Finally, it suffices to restrict the system in (5) to M j linear independent rows, and x (j+1) 0 can still be uniquely computed. Therefore a restriction A (j) ∈ C Mj ×Mj of the product F 2 j W 2 j can be chosen as…”

Section: Multi-scale Sparse Sublinear Fft Algorithm From [16]mentioning

confidence: 99%

Deterministic Sparse Sublinear FFT with Improved Numerical Stability

Plonka

Wulffen

2021

Results Math

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In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (Numer Algorithms 78:133–159, 2018. 10.1007/s11075-017-0370-5) for fast reconstruction of M-sparse vectors $${\mathbf{x}}$$ x of length $$N= 2^J$$ N = 2 J , where we assume that all components of the discrete Fourier transform $$\hat{\mathbf{x}}= {\mathbf{F}}_{N} {\mathbf{x}}$$ x ^ = F N x are available. The sparsity of $${\mathbf{x}}$$ x needs not to be known a priori, but is determined by the algorithm. If the sparsity M is larger than $$2^{J/2}$$ 2 J / 2 , then the algorithm turns into a usual FFT algorithm with runtime $${\mathcal O}(N \log N)$$ O ( N log N ) . For $$M^{2} < N$$ M 2 < N , the runtime of the algorithm is $${\mathcal O}(M^2 \, \log N)$$ O ( M 2 log N ) . The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $$M>20$$ M > 20 because of numerical instabilities, the modified algorithm is still numerically stable for $$M=200$$ M = 200 .

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Section: Multi-scale Sparse Sublinear Fft Algorithm From [16]mentioning

confidence: 99%

Deterministic Sparse Sublinear FFT with Improved Numerical Stability

Plonka

Wulffen

2021

Results Math

Self Cite

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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: 95%

“…, n 2 , have to be zero. This motivates the notion of odd Vandermonde matrices introduced in [BP18a].…”

Section: Vandermonde Matrices and Chebyshev Polynomialsmentioning

confidence: 97%

“…The following chapter is based on our preprint [BP18a], and coincides in parts identically with said publication.…”

Section: Sparse Fast Idct-ii For Vectors With One-block Supportmentioning

confidence: 99%

“…Definition 6.4 (Reflected Periodization ((5) in [BP18a])) Let N = 2 J with J ∈ N and x ∈ R N . We set x [J] := x.…”

Section: Short Support and Reflected Periodizationsmentioning

confidence: 99%

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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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