Reconstruction algorithms

Summary of the reconstruction algorithms evaluated in this study. The methods have been grouped into four different classes, depending on the sampling scheme adopted: DTI-like (violet), SPARSE-like (green), HARDI-like (yellow-red) and DSI-like (blue). The colors used here will be used coherently in every plot to identify each algorithm:

Method	Acquisition	Brief description of the algorithm	Spatial filter
DTI	one-shell b = 1000 6 samples	Diffusion Tensor Imaging (DTI). The estimated directions were directly recovered in the optimization procedure.	none
DTIneigh	one-shell b = 1000 12 samples	Contextual enhancing of spherical diffusion functions for improving DTI reconstructions, based on the notion of context to extrapolate additional information about the underlying fiber structure. Maxima detection was performed using a finite difference algorithm on a discrete grid (tessellated icosahedron) on the reconstructed profiles. If a mesh point was above the threshold and it was above all its neighbors, the point was considered a maxima. This threshold was chosen as 0.4 in order to avoid selecting small peaks that may appear due to noise.	regularization
L2-L1-DL	one-shell b = 2000 15 samples	A parametric dictionary is learnt from a set of training diffusion data, providing a highly sparse representation of the diffusion signal and enabling to analytically recover important diffusion features as the ODF with a small number of measurements. The fibre orientation estimation was performed via an analytical extraction of ODF maxima using a polynomial approach presented in reference [4] of the corresponding abstract.	none
L1-L1	one-shell b = 750 24 samples	Sparse reconstruction using the l1 norm both in the data fidelity term and the sparsity regularization with a Gaussian diffusion profile dictionary. Peaks were detected by a finite difference over the first level of neighbors of the tesselation used in the challenge (724 directions) and a threshold of 0.5 was used on the peaks. ODFs were first min-max normalized.	none
L2-L1-TV	one-shell b = 1200 30 samples	Sparse reconstruction with combined q-space and k-space with Total Variation regularization in the spatial domain. The fiber orientation were estimated by searching for ODF maxima in a neighbourhood defined by a cone of 10 degrees, after thresholding ODF amplitudes smaller than 0.3×ODFmax, where ODFmax is the amplitude of the global maximum.	regularization
L2-L2	one-shell b = 3333 37 samples	Multi-tensor fitting guided by ODF estimation, to obtain the true diffusivities corresponding to each different fiber population in the voxel. In the estimation process, the number, orientations and relative intensities of fiber populations were assumed to be known and equal to the values determined from the ODF maxima. All local maxima were obtained by comparing the ODF amplitudes between each point in the grid and its nearest neighbors within and interval of 15 degrees. The largest three local maxima were preserved if their amplitudes were above the value 0.4×ODFmax, where ODFmax is the amplitude of the global maximum. Second, to refine the spatial positions of the resulting maxima, all neighbors around each maximum were used to fit an ellipsoid centered at the origin. The position of the principal direction of each ellipsoid was used to specify the local fiber orientation.	smoothing
NN-L2	one-shell b = 1500 48 samples	Diffusion Basis Functions + selective spatial smoothing based on FA and diffusion orientation similarity. Each atom of the diffusion dictionary has associated a discrete diffusion orientation. From the atoms computed as the solution, the method clusters close orientations in order to provide a continuos solution. The number of clusters and their centroids are reported as the number of diffusion compartments and the diffusion directions, respectively. Diffusion orientations with a size compartment smaller than 30% of the biggest size compartment within the voxel were removed.	smoothing
L2-L1-TGV	multi-shell b = 1500/2500 64 samples	l1-based constrained spherical deconvolution with Total Generalized Variation regularization optimized by a first-order primal-dual approach. On the discretized sphere all local maxima of the ODF are extracted. Then, around a small neighborhood (all neighbors according to a Delaunay triangulation) a quadratic form is fitted and its principal direction is designated to be the underlying fiber direction. There is one exclusion criteria: if the principal direction of the quadratic form is not within the triangles formed by the neighborhood of the local maximum, the direction is discarded.	regularization
DOT	one-shell b = 3000 60 samples	Diffusion Orientation Transform (DOT) with the following parameters: Spherical Harmonics (SH) order lmax = 8 and EAP evaluated at radius R = 15 μm. All local maxima were obtained by comparing the ODF amplitudes between each point in the grid and its nearest neighbors within and interval of 15 degrees. The largest three local maxima were preserved if their amplitudes were above the value 0.4×ODFmax, where ODFmax is the amplitude of the global maximum. Second, to refine the spatial positions of the resulting maxima, all neighbors around each maximum were used to fit an ellipsoid centered at the origin. The position of the principal direction of each ellipsoid was used to specify the local fiber orientation.	none
QBI	one-shell b = 3000 60 samples	Analytical Q-Ball Imaging (QBI) with the following parameters: SH order lmax = 6 and regularization parameter λ = 0.006. Peaks were detected by a finite difference over the first level of neighbors of the tesselation used in the challenge (724 directions) and a threshold of 0.5 was used on the peaks. ODFs were first min-max normalized.	none
QBISWT	one-shell b = 3000 60 samples	Q-Ball imaging with Spherical Wavelet Transform with the following parameters: SH order lmax = 8. Peaks were detected by a finite difference over the first level of neighbors of the tesselation used in the challenge (724 directions) and a threshold of 0.5 was used on the peaks. ODFs were first min-max normalized.	none
QBICSA	one-shell b = 3000 60 samples	Q-Ball imaging in Constant Solid Angle with the following parameters: SH order lmax = 6 and regularization parameter λ = 0.006. Peaks were detected by a finite difference over the first level of neighbors of the tesselation used in the challenge (724 directions) and a threshold of 0.5 was used on the peaks. ODFs were first min-max normalized.	none
CSD	one-shell b = 3000 60 samples	Non-negativity constrained super-resolved spherical deconvolution (CSD) with the following parameters: SH order lmax = 8 and diffusion profile of [1.7, 0.3, 0.3]×10−3 mm2/s. Peaks were detected by a finite difference over the first level of neighbors of the tesselation used in the challenge (724 directions) and a threshold of 0.5 was used on the peaks. ODFs were first min-max normalized.	none
SDT	one-shell b = 3000 60 samples	Sharpening Deconvolution Transform with the following parameters: SH order lmax = 6 and diffusion profile of [1.7, 0.3, 0.3]×10−3 mm2/s. Peaks were detected by a finite difference over the first level of neighbors of the tesselation used in the challenge (724 directions) and a threshold of 0.5 was used on the peaks. ODFs were first min-max normalized.	none
STD	one-shell b = 3000 64 samples	Spherical deconvolution via Symmetric Tensor Decomposition. The estimated directions were directly recovered in the optimization procedure.	none
SPIRAL	spiral b = 1200 82 samples	Gradient directions were set on a spherical spiral curve resulting in voxel-wise diffusion signal periodicity. The periodic pattern of the signal in the gradient direction domain was exploited for de-noising and fiber orientation estimation. Periodic spiral sampling scheme led to acquiring a periodic signal in each voxel (combination of all the samples). The location of extrema was used to find the main orientation of the fibres. No threshold has been used. Given our assumption about the number of fibres (m=2), the extrema represent the plane that contains the main orientation of the fibres.	none
DSI	cartesian b <= 8000 257 samples	Diffusion Spectrum Imaging (DSI) with the following parameters: zero-padded 35×35×35 grid for the EAP reconstruction, Hanning filtering before discrete Fourier transformation and ODF integration over the full radius range. All local maxima were obtained by comparing the ODF amplitudes between each point in the grid and its nearest neighbors within and interval of 15 degrees. The largest three local maxima were preserved if their amplitudes were above the value 0.4×ODFmax, where ODFmax is the amplitude of the global maximum. Second, to refine the spatial positions of the resulting maxima, all neighbors around each maximum were used to fit an ellipsoid centered at the origin. The position of the principal direction of each ellipsoid was used to specify the local fiber orientation.	none
DSILR	cartesian b <= 8000 257 samples	General framework based on Lucy-Richardson deconvolution to rectify the EAP reconstructed with DSI, taking into account the blurring introduced in the Fourier inversion due to the discrete and finite support of q-space measurements. All local maxima were obtained by comparing the ODF amplitudes between each point in the grid and its nearest neighbors within and interval of 15 degrees. The largest three local maxima were preserved if their amplitudes were above the value 0.4×ODFmax, where ODFmax is the amplitude of the global maximum. Second, to refine the spatial positions of the resulting maxima, all neighbors around each maximum were used to fit an ellipsoid centered at the origin. The position of the principal direction of each ellipsoid was used to specify the local fiber orientation.	none
DSISWT	cartesian b <= 8000 257 samples	Multi-resolution approach based on 3D Stationary Wavelet Transform (3D-SWT) to de-noise the propagator reconstructed with DSI. Fiber orientations were obtained by means of finite differences over the first level of neighbors of the 724 directions tesselation and discarding those with an amplitude below 0.4×ODFmax, where ODFmax is the amplitude of the global maximum.	none
GQI2	cartesian b <= 8000 257 samples	Extension of the Generalized Q-sampling Imaging to analytically reconstruct the ODF with solid angle consideration. As it is expressed as a direct linear transform of the raw signal, it is faster to compute than DSI. Local maxima were detected by searching for maxima on the ODF using the triangularization of the unit spehre as given in the challenge as support. A threshold of 0.5 was used to filter out spurious peaks.	none