基于Kaleidoscope变换的离散傅立叶空间分形采样模式
Bespoke Fractal Sampling Patterns for Discrete Fourier Space via the Kaleidoscope Transform
摘要：
采样策略对于稀疏成像方法，特别是那些采用离散傅里叶变换（DFT）的方法是重要的。混沌感测是一种这样的方法，其采用确定性的分形采样结合有限的迭代重建方案来从有限的样本形成图像。使用完全由DFT空间中的周期线构造的采样模式，发现混沌感知优于用于磁共振成像的传统压缩感知;然而，仅提出了一种这样的采样模式，并且未证明其分形性质的原因。本文通过引入一种新的图像变换--万花筒变换（Kaleidoscope transform），形式化和扩展了下采样和图像拼接的概念，（1）证明了模运算中的乘法与下采样之间的基本关系，（2）对DFT中采样模式的分形性质提供了严格的数学解释，（3）对DFT中的采样模式的分形性质给出了严格的数学解释。以及（3）利用这种理解来开发具有可定制属性的2D DFT的新颖分形采样模式的集合。设计定制的分形采样模式的能力扩展了DFT在混沌成像中的实用性，并且可以形成定制的混沌感测方法的基础，其中分形采样与成像任务相匹配以改善重建。
英文摘要：
Sampling strategies are important for sparse imaging methodologies, especially those employing the discrete Fourier transform (DFT). Chaotic sensing is one such methodology that employs deterministic, fractal sampling in conjunction with finite, iterative reconstruction schemes to form an image from limited samples. Using a sampling pattern constructed entirely from periodic lines in DFT space, chaotic sensing was found to outperform traditional compressed sensing for magnetic resonance imaging; however, only one such sampling pattern was presented and the reason for its fractal nature was not proven. Through the introduction of a novel image transform known as the kaleidoscope transform, which formalises and extends upon the concept of downsampling and concatenating an image with itself, this paper: (1) demonstrates a fundamental relationship between multiplication in modular arithmetic and downsampling; (2) provides a rigorous mathematical explanation for the fractal nature of the sampling pattern in the DFT; and (3) leverages this understanding to develop a collection of novel fractal sampling patterns for the 2D DFT with customisable properties. The ability to design tailor-made fractal sampling patterns expands the utility of the DFT in chaotic imaging and may form the basis for a bespoke chaotic sensing methodology, in which the fractal sampling matches the imaging task for improved reconstruction.
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