# The Discrete Fourier Transform for Golden Angle Linogram Sampling

## TL;DR

This paper introduces the Golden Angle Linogram Fourier Domain (GALFD) and a fast evaluation method, GALE, enabling efficient and accurate approximation of the DTFT at GALFD points, with applications in imaging like MRI.

## Contribution

The paper presents GALFD and GALE, novel tools that improve the efficiency and flexibility of DTFT estimation in imaging applications, especially MRI.

## Key findings

- GALE computes DTFT accurately with low floating point operations.
- GALFD allows incremental data collection, beneficial for MRI.
- The methods extend to arbitrary radial patterns.

## Abstract

Estimation of the Discrete-Time Fourier Transform (DTFT) at points of a finite domain arises in many imaging applications. A new approach to this task, the Golden Angle Linogram Fourier Domain (GALFD), is presented, together with a computationally fast and accurate tool, named Golden Angle Linogram Evaluation (GALE), for approximating the DTFT at points of a GALFD. A GALFD resembles a Linogram Fourier Domain (LFD), which is efficient and accurate. A limitation of linograms is that embedding an LFD into a larger one requires many extra points, at least doubling the domain's cardinality. The GALFD, on the other hand, allows for incremental inclusion of relatively few data points. Approximation error bounds and floating point operations counts are presented to show that GALE computes accurately and efficiently the DTFT at the points of a GALFD. The ability to extend the data collection in small increments is beneficial in applications such as Magnetic Resonance Imaging. Experiments for simulated and for real-world data are presented to substantiate the theoretical claims. The mathematical analysis, algorithms, and software developed in the paper are equally suitable to other angular distributions of rays and therefore we bring the benefits of linograms to arbitrary radial patterns.