# Linear Predictability in MRI Reconstruction: Leveraging Shift-Invariant Fourier Structure for Faster and Better Imaging

## TL;DR

This paper reviews how leveraging shift-invariant Fourier structure and linear predictability in MRI can enable faster imaging with fewer samples, unifying classical constraints without strong assumptions.

## Contribution

It provides an overview of linear predictability methods in MRI, highlighting their role in reducing sampling rates and unifying various reconstruction constraints.

## Key findings

- Linear predictability allows MRI data to be sampled below Nyquist rate.
- Linear predictive methods are widely used in clinical MRI.
- Unifies classical MRI constraints without strong assumptions.

## Abstract

Over the past several decades, many different types of computational imaging approaches have been proposed for improving MRI. In this paper, we provide an overview of methods that assume that MRI Fourier data is linearly predictable. Linear prediction is well known in signal processing and may be most recognizable for its usefulness in speech processing and spectrum estimation applications. In MRI, linear predictability implies that data can be sampled below the conventional Nyquist rate, since unmeasured data may be imputed as a shift-invariant linear combination of measured samples. Linear predictive methods include some of the earliest methods in the computational MRI reconstruction field, some of the most widely utilized computational MRI methods in modern clinical practice, and some of the most flexible and versatile modern computational imaging approaches that are enabling unprecedented new styles of data acquisition. In addition, the concept of linear predictability can be used to unify a number of more classical MRI reconstruction constraints, but without needing to make the strong assumptions of classical constrained reconstruction methods.