Function-Valued Mappings, Total Variation and Compressed Sensing for diffusion MRI

Abstract

Being the only imaging modality capable of delineating the anatomical structure of the white matter, diffusion magnetic resonance imaging (dMRI) is currently believed to provide a long-awaited means for early diagnosis of various neurological conditions as well as for interrogating the brain connectivity. Despite substantial advances in practical use of dMRI, a solid mathematical platform for modelling and treating dMRI signals still seems to be missing. Accordingly, in this paper, we show how a Hilbert space of $\mathbb{L}^2$-valued mappings $u: X \to \mathbb{L}^2({\mathbb{S}^2})$, with X being a subset of ℝ³ and $\mathbb{L}^2({\mathbb{S}^2})$ being the set of squared-integrable functions supported on the unit sphere ${\mathbb{S}^2}$, provides a natural setting for a specific example of dMRI, known as high-angular resolution diffusion imaging. The proposed formalism is also shown to provide a basis for image processing schemes such as total variation minimization. Finally, we discuss a way to amalgamate the proposed models with the tools of compressed sensing to achieve a close-to-perfect recovery of diffusion signals from a minimal number of their discrete measurements. The main outcomes of this paper are supported by a series of experimental results.