arXiv subjects

Sina Bittens

Publications and source records attributed to Sina Bittens.

3 recordsLinked to original sources

arXiv2018

Real Sparse Fast DCT for Vectors with Short Support

In this paper we present a new fast and deterministic algorithm for the inverse discrete cosine transform of type II for reconstructing the input vector $\mathbf x\in\mathbb R^N$, $N=2^J$, with short support of length $m$ from its discrete cosine transform $\mathbf x^{\widehat{\mathrm{II}}}=C^{\mathrm{II}}_N\mathbf x$ if an upper bound $M\geq m$ is known. The resulting algorithm only uses real arithmetic, has a runtime of $\mathcal{O}\left(M\log M+m\log_2\frac{N}{M}\right)$ and requires $\mathcal{O}\left(M+m\log_2\frac{N}{M}\right)$ samples of $\mathbf x^{\widehat{\mathrm{II}}}$. For $m,M\rightarrow N$ the runtime and sampling requirements approach those of a regular IDCT-II for vectors with full support. The algorithm presented hereafter does not employ inverse FFT algorithms to recover $\mathbf x$.

math.NA↗

arXiv2018

Sparse Fast DCT for Vectors with One-block Support

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 $\mathbf{x}\in\mathbb{R}^{N}$, $N=2^{J-1}$, with short support of length $m$ from its discrete cosine transform $\mathbf{x}^{\widehat{\mathrm{II}}}=\mathbf{C}_N^{\mathrm{II}}\mathbf{x}$. The resulting algorithm has a runtime of $\mathcal{O}\left(m\log m\log \frac{2N}{m}\right)$ and requires $\mathcal{O}\left(m\log \frac{2N}{m}\right)$ samples of $\mathbf{x}^{\widehat{\mathrm{II}}}$. In order to derive this algorithm we also develop a new fast and deterministic inverse FFT algorithm that constructs the input vector $\mathbf{y}\in\mathbb{R}^{2N}$ with reflected block support of block length $m$ from $\widehat{\mathbf{y}}$ with the same runtime and sampling complexities as our DCT algorithm.

math.NA↗

arXiv2017

A Deterministic Sparse FFT for Functions with Structured Fourier Sparsity

In this paper a deterministic sparse Fourier transform algorithm is presented which breaks the quadratic-in-sparsity runtime bottleneck for a large class of periodic functions exhibiting structured frequency support. These functions include, e.g., the oft-considered set of block frequency sparse functions of the form $$f(x) = \sum^{n}_{j=1} \sum^{B-1}_{k=0} c_{ω_j + k} e^{i(ω_j + k)x},~~\{ ω_1, \dots, ω_n \} \subset \left(-\left\lceil \frac{N}{2}\right\rceil, \left\lfloor \frac{N}{2}\right\rfloor\right]\cap\mathbb{Z}$$ as a simple subclass. Theoretical error bounds in combination with numerical experiments demonstrate that the newly proposed algorithms are both fast and robust to noise. In particular, they outperform standard sparse Fourier transforms in the rapid recovery of block frequency sparse functions of the type above.

math.NA↗