Joint spatial-angular sparse coding for dMRI with separable dictionaries

Abstract

Diffusion MRI (dMRI) provides the ability to reconstruct neuronal fibers in the brain, in vivo, by measuring water diffusion along angular gradient directions in q-space. High angular resolution diffusion imaging (HARDI) can produce better estimates of fiber orientation than the popularly used diffusion tensor imaging, but the high number of samples needed to estimate diffusivity requires longer patient scan times. To accelerate dMRI, compressed sensing (CS) has been utilized by exploiting a sparse dictionary representation of the data, discovered through sparse coding. The sparser the representation, the fewer samples are needed to reconstruct a high resolution signal with limited information loss, and so an important area of research has focused on finding the sparsest possible representation of dMRI. Current reconstruction methods however, rely on an angular representation per voxel with added spatial regularization, and so, for non-zero signals, one is required to have at least one non-zero coefficient per voxel. This means that the global level of sparsity must be greater than the number of voxels. In contrast, we propose a joint spatial-angular representation of dMRI that will allow us to achieve levels of global sparsity that are below the number of voxels. A major challenge, however, is the computational complexity of solving a global sparse coding problem over large-scale dMRI. In this work, we present novel adaptations of popular sparse coding algorithms that become better suited for solving large-scale problems by exploiting spatial-angular separability. Our experiments show that our method achieves significantly sparser representations of HARDI than is possible by the state of the art.

(k, q)-Compressed Sensing for dMRI with Joint Spatial-Angular Sparsity Prior

“…Therefore, in our experiments we vary the level of λ and select the value that leads to a minimal reconstruction error. The efficiency of Kron-SFISTA over the traditional SFISTA can be viewed in the same vein as for Kron-FISTA analyzed in [11].…”

“…In this paper, we present a new (k, q)-CS framework that subsamples jointly in (k, q)-space while analogously imposing sparsity in the joint spatial-angular domain. Building upon the recent findings of [10,11] which show increased levels of dMRI sparsity using joint spatial-angular sparse coding, our proposed (k, q)-CS has the potential to further accelerate dMRI than prior methods by exploiting this underlying sparse representation. Our main objective in this paper is to evaluate the advantages of imposing sparsity in the joint spatial-angular domain versus previous formulations that involve separate spatial and angular sparsity terms.…”

“…To overcome this difficulty, we propose to impose additional structure on Φ. Following [10,11] one can choose Φ to be separable over the spatial and angular domains resulting in the Kronecker dictionary Φ = Γ ⊗ Ψ , where Ψ and Γ are, as above, spatial and angular dictionaries, respectively. Then, we can rewrite (7) in an equivalent matrix form as:…”

Harmonization of diffusion MRI datasets with adaptive dictionary learning

“…As updating α needs an optimization scheme, this can be done independently for each α n using coordinate descent (Friedman et al, 2010). For updating D, we use the parameter-free closed form update from Mairal et al (2010, Algorithm 2), which only requires storing intermediary matrices of the previous iteration using α and X n to update D. Building dictionaries for the task at hand has been used previously in the context of diffusion MRI for denoising (Gramfort et al, 2014;St-Jean et al, 2016) and compressed sensing (Gramfort et al, 2014;Merlet et al, 2013;Schwab et al, 2018) amongst other tasks. Note that it is also possible to design dictionaries based on products of fixed basis or adding additional constraints such as positivity or spatial consistency to Eq.…”

“…Nevertheless, our method can still be used on such datasets, but would not be aware of the relationship between DWIs beyond the angular domain. Other approaches to build the dictionary could be used to inform the algorithm and potentially increase performance on such datasets by explicitly modeling the spatial and angular relationship (Schwab et al, 2018) or using an adaptive weighting considering the b-values in the angular domain (Duits et al, 2019) amongst other possible strategies. Modeling explicitly the angular part of the signal could also be used to sample new gradients directions directly, an aspect we covered in the original CDMRI challenge by using the spherical harmonics basis (Descoteaux et al, 2007).…”