Bespoke Fractal Sampling Patterns for Discrete Fourier Space via the Kaleidoscope Transform

TL;DR

This paper introduces the kaleidoscope transform to explain and generate custom fractal sampling patterns in the discrete Fourier space, enhancing chaotic sensing techniques for improved sparse imaging.

Contribution

It provides a mathematical foundation linking modular arithmetic to fractal sampling patterns and develops new customizable fractal sampling patterns for the 2D DFT.

Findings

Demonstrates the relationship between modular multiplication and downsampling.

Provides a rigorous explanation for the fractal nature of DFT sampling patterns.

Develops a collection of novel, customizable fractal sampling patterns.

Abstract

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.