Density Evolution on a Class of Smeared Random Graphs: A Theoretical Framework for Fast MRI

TL;DR

This paper develops a theoretical framework to analyze iterative decoding on a new class of structured random graphs called the smearing ensemble, with applications to fast MRI signal reconstruction.

Contribution

It introduces the smearing ensemble of bipartite graphs, derives exact density evolution recurrences despite small cycles, and provides explicit MRI reconstruction thresholds.

Findings

Exact density evolution for smearing ensemble with smear-length two.

Lower bounds on performance for larger smearing classes.

MRI system architecture with explicit Fourier sampling thresholds.

Abstract

We introduce a new ensemble of random bipartite graphs, which we term the `smearing ensemble', where each left node is connected to some number of consecutive right nodes. Such graphs arise naturally in the recovery of sparse wavelet coefficients when signal acquisition is in the Fourier domain, such as in magnetic resonance imaging (MRI). Graphs from this ensemble exhibit small, structured cycles with high probability, rendering current techniques for determining iterative decoding thresholds inapplicable. In this paper, we develop a theoretical platform to analyze and evaluate the effects of smearing-based structure. Despite the existence of these small cycles, we derive exact density evolution recurrences for iterative decoding on graphs with smear-length two. Further, we give lower bounds on the performance of a much larger class from the smearing ensemble, and provide numerical experiments showing tight agreement between empirical thresholds and those determined by our bounds. Finally, we describe a system architecture to recover sparse wavelet representations in the MRI setting, giving explicit thresholds on the minimum number of Fourier samples needing to be acquired for the $1$-stage Haar wavelet setting. In particular, we show that $K$-sparse $1$-stage Haar wavelet coefficients of an $n$-dimensional signal can be recovered using $2.63K$ Fourier domain samples asymptotically using $\mathcal{O}(K\log{K})$ operations.