High-Dimensional Confidence Regions in Sparse MRI

TL;DR

This paper extends high-dimensional confidence region methods to MRI, enabling pixel-wise confidence intervals using debiased LASSO with complex Fourier measurements, requiring specific data quantities for accuracy.

Contribution

It adapts the debiased LASSO approach for MRI's complex Fourier measurements, providing a framework for pixel-wise confidence intervals in high-dimensional imaging.

Findings

Confidence intervals can be constructed for MRI pixels.

Sufficient data size is characterized by specific logarithmic conditions.

Method extends high-dimensional inference to complex Fourier measurement systems.

Abstract

One of the most promising solutions for uncertainty quantification in high-dimensional statistics is the debiased LASSO that relies on unconstrained $\ell_1$-minimization. The initial works focused on real Gaussian designs as a toy model for this problem. However, in medical imaging applications, such as compressive sensing for MRI, the measurement system is represented by a (subsampled) complex Fourier matrix. The purpose of this work is to extend the method to the MRI case in order to construct confidence intervals for each pixel of an MR image. We show that a sufficient amount of data is $n \gtrsim \max\{ s_0\log^2 s_0\log p, s_0 \log^2 p \}$.