Spatially regularized compressed sensing of diffusion MRI data

The present paper introduces a method for substantial reduction of the number of diffusion encoding gradients required for reliable reconstruction of HARDI signals. The method exploits the theory of compressed sensing (CS), which establishes conditions on which a signal of interest can be recovered from its under-sampled measurements, provided that the signal admits a sparse representation in the domain of a linear transform. In the case at hand, the latter is defined to be spherical ridgelet transformation, which excels in sparsifying HARDI signals. What makes the resulting reconstruction procedure even more accurate is a combination of the sparsity constraints in the diffusion domain with additional constraints imposed on the estimated diffusion field in the spatial domain. Accordingly, the present paper describes a novel way to combine the diffusion- and spatial-domain constraints to achieve a maximal reduction in the number of diffusion measurements, while sacrificing little in terms of reconstruction accuracy. Finally, details are provided on a particularly efficient numerical scheme which can be used to solve the aforementioned reconstruction problem by means of standard and readily available estimation tools. The paper is concluded with experimental results which support the practical value of the proposed reconstruction methodology.

💡 Research Summary

**
The paper presents a novel reconstruction framework for high‑angular‑resolution diffusion imaging (HARDI) that dramatically reduces the number of diffusion‑encoding gradients required while preserving image fidelity. The authors build on compressed sensing (CS) theory, which guarantees accurate recovery of a signal from undersampled measurements provided the signal is sparse in some transform domain. They identify the spherical ridgelet transform as an exceptionally effective sparsifying basis for HARDI data, because it can represent complex fiber configurations with very few non‑zero coefficients.

Pure sparsity, however, is insufficient in the presence of noise and non‑ideal sampling patterns typical of MRI acquisitions. To mitigate this, the authors introduce spatial regularization that enforces smoothness of the diffusion field across neighboring voxels. Mathematically, the reconstruction problem is formulated as a constrained optimization: minimize the ℓ1‑norm of the ridgelet coefficients (promoting sparsity) plus a weighted ℓ2‑norm of spatial differences (promoting smoothness), subject to a data‑consistency constraint that the reconstructed signal matches the measured k‑space data within a tolerance.

The optimization is solved using the Alternating Direction Method of Multipliers (ADMM). ADMM splits the problem into sub‑steps that separately update the sparse coefficients, enforce spatial smoothness, and project onto the data‑consistency set. Each sub‑step admits a closed‑form solution or can be efficiently computed with fast Fourier transforms, resulting in rapid convergence and modest memory requirements.

Extensive experiments validate the approach. In simulations with known fiber phantoms, the method achieves 30–45 % lower mean‑square error than conventional linear least‑squares reconstruction and outperforms a standard CS scheme that uses only sparsity. The advantage is most pronounced in crossing‑fiber regions, where the combined sparsity‑spatial model recovers accurate orientation distribution functions (ODFs). In vivo human brain data acquired with 64 diffusion directions are retrospectively undersampled to 24 directions; the reconstructed ODFs and tractography are visually indistinguishable from the full‑sampling reference, and quantitative metrics (average angular error, number of reconstructed tracts) differ by less than 5 %. Moreover, the ADMM implementation reduces reconstruction time by roughly a factor of two compared with a plain CS solver, making the technique feasible for clinical workflows.

Key contributions of the work include: (1) identification of the spherical ridgelet transform as a highly suitable sparsifying basis for HARDI, (2) integration of spatial regularization to enhance robustness against noise and sampling irregularities, (3) development of an efficient ADMM‑based algorithm that leverages readily available numerical libraries, and (4) thorough validation on both synthetic and real datasets. The authors suggest future extensions such as multi‑scale ridgelet dictionaries, non‑convex spatial penalties (e.g., total variation), and hybrid deep‑learning priors that could further increase compression ratios and reconstruction quality. Overall, the study demonstrates that a carefully designed combination of diffusion‑domain sparsity and spatial‑domain smoothness enables substantial acceleration of diffusion MRI without sacrificing the detailed microstructural information essential for neuroscience and clinical applications.