Real Sparse Fast DCT for Vectors with Short Support

TL;DR

This paper introduces a fast, deterministic algorithm for inverse DCT-II that efficiently reconstructs vectors with short support from their DCT coefficients, using real arithmetic and optimized runtime.

Contribution

It presents a novel inverse DCT-II algorithm tailored for vectors with short support, avoiding inverse FFT and reducing computational complexity.

Findings

Runtime approaches that of full support IDCT for large support sizes

Requires fewer samples than traditional methods for sparse vectors

Operates solely with real arithmetic

Abstract

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 $\mathbf x$.