Fast DCT+: A Family of Fast Transforms Based on Rank-One Updates of the Path Graph

Samuel Fernández-Menduiña, Eduardo Pavez, Antonio Ortega
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Abstract

This paper develops fast graph Fourier transform (GFT) algorithms with O(n log n) runtime complexity for rank-one updates of the path graph. We first show that several commonly-used audio and video coding transforms belong to this class of GFTs, which we denote by DCT+. Next, starting from an arbitrary generalized graph Laplacian and using rank-one perturbation theory, we provide a factorization for the GFT after perturbation. This factorization is our central result and reveals a progressive structure: we first apply the unperturbed Laplacian's GFT and then multiply the result by a Cauchy matrix. By specializing this decomposition to path graphs and exploiting the properties of Cauchy matrices, we show that Fast DCT+ algorithms exist. We also demonstrate that progressivity can speed up computations in applications involving multiple transforms related by rank-one perturbations (e.g., video coding) when combined with pruning strategies. Our results can be extended to other graphs and rank-k perturbations. Runtime analyses show that Fast DCT+ provides computational gains over the naive method for graph sizes larger than 64, with runtime approximately equal to that of 8 DCTs.
快速 DCT+:基于路径图一级更新的快速变换系列
本文针对路径图的秩一更新,开发了运行复杂度为 O(nlog n) 的快速图傅立叶变换(GFT)算法。我们首先证明,几种常用的音频和视频编码变换属于这一类 GFT,我们将其称为 DCT+。接下来,我们从任意广义图拉普拉斯开始,利用秩一扰动理论,提供了扰动后 GFT 的因式分解。这种因式分解是我们的核心成果,揭示了一种渐进结构:我们首先应用未扰动拉普拉斯的 GFT,然后将结果乘以考奇矩阵。通过将这种分解特殊化为路径图并利用考奇矩阵的特性,我们证明了快速 DCT+ 算法的存在。我们还证明,当结合剪枝策略时,渐进性可以加快涉及秩一扰动相关多变换(如视频编码)的应用中的计算速度。我们的结果可以扩展到其他图和阶一扰动。运行时间分析表明,当图的大小大于 64 时,快速 DCT+ 比传统方法带来了计算上的优势,其运行时间约等于 8 个 DCT 的运行时间。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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