Linear transforms for Fourier data on the sphere: Application to high angular resolution diffusion MRI of the brain

Abstract

This paper presents a novel family of linear transforms that can be applied to data collected from the surface of a 2-sphere in three-dimensional Fourier space. This family of transforms generalizes the previously-proposed Funk-Radon Transform (FRT), which was originally developed for estimating the orientations of white matter fibers in the central nervous system from diffusion magnetic resonance imaging data. The new family of transforms is characterized theoretically, and efficient numerical implementations of the transforms are presented for the case when the measured data is represented in a basis of spherical harmonics. After these general discussions, attention is focused on a particular new transform from this family that we name the Funk-Radon and Cosine Transform (FRACT). Based on theoretical arguments, it is expected that FRACT-based analysis should yield significantly better orientation information (e.g., improved accuracy and higher angular resolution) than FRT-based analysis, while maintaining the strong characterizability and computational efficiency of the FRT. Simulations are used to confirm these theoretical characteristics, and the practical significance of the proposed approach is illustrated with real diffusion weighted MRI brain data. These experiments demonstrate that, in addition to having strong theoretical characteristics, the proposed approach can outperform existing state-of-the-art orientation estimation methods with respect to measures such as angular resolution and robustness to noise and modeling errors.

Quantitative Comparison of Reconstruction Methods for Intra-Voxel Fiber Recovery From Diffusion MRI

“…Lately, the advent of compressed sensing [31], [32] inspired new approaches to solve the reconstruction problem whose main goal is to drastically reduce the acquisition time [33]- [39]. This list is not exhaustive and many other algorithms exist in the literature; see [40]- [42] for a more comprehensive list of available techniques. All these methods differ in a great deal of aspects, like for instance the acquisition scheme adopted (number and position of the diffusion gradients), the targeted feature of the diffusion process to estimate, e.g.…”

Fiber up‐sampling and quality assessment of tractograms – towards quantitative brain connectivity

“…A Fiber Assignment by Continuous Tracking (FACT) inspired deterministic algorithm generalized to the Orientation Distribution Function (ODF) was used in the tractography step. The ODF was reconstructed using the FRACT method (Haldar & Leahy, 2013). The tracking direction was selected according to the local diffusion maximum of the ODF.…”

Greedy algorithms for nonnegativity-constrained simultaneous sparse recovery

“…(Bottom Row) ODFs, computed using the Funk-Radon and Cosine Transform [51], corresponding to the diffusion basis functions from the top row. The locations of the largest maximum of each ODF indicate the primary orientations of the corresponding fiber bundle.…”