Equivariant Spherical Deconvolution: Learning Sparse Orientation Distribution Functions from Spherical Data
EEquivariant Spherical Deconvolution: LearningSparse Orientation Distribution Functions fromSpherical Data
Axel Elaldi (cid:63) , Neel Dey , Heejong Kim , and Guido Gerig Department of Computer Science and Engineering, New York University, USA { axel.elaldi, neel.dey, heejong.kim, gerig } @nyu.edu Abstract.
We present a rotation-equivariant unsupervised learning frame-work for the sparse deconvolution of non-negative scalar fields defined onthe unit sphere. Spherical signals with multiple peaks naturally arise inDiffusion MRI (dMRI), where each voxel consists of one or more signalsources corresponding to anisotropic tissue structure such as white mat-ter. Due to spatial and spectral partial voluming, clinically-feasible dMRIstruggles to resolve crossing-fiber white matter configurations, leading toextensive development in spherical deconvolution methodology to recoverunderlying fiber directions. However, these methods are typically linearand struggle with small crossing-angles and partial volume fraction es-timation. In this work, we improve on current methodologies by nonlin-early estimating fiber structures via unsupervised spherical convolutionalnetworks with guaranteed equivariance to spherical rotation. Experimen-tally, we first validate our proposition via extensive single and multi-shellsynthetic benchmarks demonstrating competitive performance againstcommon baselines. We then show improved downstream performance onfiber tractography measures on the Tractometer benchmark dataset. Fi-nally, we show downstream improvements in terms of tractography andpartial volume estimation on a multi-shell dataset of human subjects.
Diffusion-weighted MRI (dMRI) measures voxel-wise molecular diffusivity andenables the in vivo investigation of tissue microstructure via analysis of whitematter fiber configurations and tractography. Localized profiles of water diffu-sion can be constructed via multiple directional magnetic excitations, with eachexcitation direction corresponding to an image volume. Due to partial voluming,voxels with two or more crossing fibers require increased directional sampling[19] for reliable resolution of multiple fiber directions. However, higher numbersof directions (also referred to as diffusion gradients) lead to clinically infeasiblescanning times. Consequently, a series of reconstruction models [4] characterizingvoxel-specific diffusivity and enabling fewer gradients have been proposed. (cid:63)
These authors contributed equally. a r X i v : . [ ee ss . I V ] F e b A. Elaldi, et al.
Fig. 1: Equivariant Spherical Deconvolution (ESD) can outperform conventionalmethods (CSD [6]) on the separation of crossing-fibers with small angles (here33 ◦ on the left and 45 ◦ on the right) across a wide range of diffusion gradients.In particular, these reconstruction models seek to estimate a fiber orienta-tion distribution function (fODF) [17]: a function on the unit sphere S providingfiber orientation/direction and intensity. The fODF can be obtained via spheri-cal deconvolution of the dMRI signal with a tissue response function (analogousto a point spread function for planar images). The fODF model represents tis-sue micro-structure as a sparse non-negative signal, giving higher precision tofiber estimation. The constrained spherical deconvolution (CSD) model [17] hasbeen extended to handle multiple tissue types (e.g., white and grey matter, cere-brospinal fluid) with multiple excitation shells (MSMT-CSD) [6]. Recent worksaim to recover a sparser fODF, either via dictionary-learning [1] or by using spe-cialized basis functions [20]. However, these methods may fail to recover difficultmicro-structures such as crossing-fibers with small crossing angles.Emerging literature demonstrates the utility of deep networks towards learn-ing fODFs. Patel et al. use an autoencoder pretrained for fODF reconstruction asa regularizer for the MSMT-CSD optimization problem [13]. Nath et al. train aregression network on ground truth fiber orientations acquired via ex-vivo con-focal microscopy images of animal histology sections co-registered with dMRIvolumes [12]. Such an approach is typically impracticable due to the need for exvivo histological training data. More recently, a series of work [8,7,10,15] pro-poses to train supervised deep regression networks directly on pairs of inputdMRI signals and their corresponding MSMT-CSD model fits. We argue thatsuch approaches are inherently limited by the quality of the MSMT-CSD solutionand show that the underlying deconvolution itself can be improved via an un-supervised deep learning approach. Moreover, none of these learning approaches(with the exception of [15]) are equivariant to spherical rotation.In these inverse problems aiming to recover the fODF model, we argue thatas the DWI signal lives on the unit sphere, planar convolutional layers maynot have the appropriate inductive bias. Standard convolutional layers are con-structed for equivariance to planar translation, whereas the analogous operationfor spherical signals is rotation. Designing the appropriate form of equivariancefor a given network and task is key, as it enables a higher degree of weight shar-ing, parameter efficiency, and generalization to poses not seen in training data quivariant Spherical Deconvolution 3 Fig. 2:
Framework overview . The raw DWI signal is interpolated onto a spheri-cal Healpix grid [5] and fed into a rotation-equivariant spherical U-Net which pre-dicts sparse fiber orientation distribution functions. The architecture is trainedunder a regularized reconstruction objective.[2]. Fortunately, rotation-equivariant spherical convolutional layers for data on S have been proposed [3,14] with natural applicability to dMRI data.In this work, we tackle sparse unsupervised fODF estimation via rotation-equivariant spherical convolutional networks. This reformulation is trained undera regularized reconstruction objective and allows for the nonlinear estimation ofsparse non-negative fiber structures with incorporation of relevant symmetriesand leads to improved performance on a variety of fODF estimation settingsfor both single-shell and multi-shell data. When ground truth is available viabenchmark datasets, we obtain more accurate fiber detection and downstreamfiber tractography. For real-world application to humans without ground truth,we show downstream improvements in terms of tractography and partial volumeestimation on a real-world multi-shell dataset. Our learning framework is flexibleand amenable to various regularizers and inductive priors and is applicable togeneric spherical deconvolution tasks. Our code is publicly available at https://github.com/AxelElaldi/equivariant-spherical-deconvolution . The dMRI signal is a function S : S → R B , where B is the number of gradients strengths/sampling shells, with the fODF given as F : S → R such that S = A ( F ), where A is the spherical convolution of thefODF with a response function (RF) R : S → R B . The RF can be seen as thedMRI signal containing only one fiber in the y -axis direction. The spherical con-volution is defined as S ( p ) = ( R ∗ F )( p ) = (cid:82) S R ( P − p q ) F ( q ) dq , where p, q ∈ S are spherical coordinates and P p is the rotation associated to the spherical anglesof coordinate p . We note that the fODF is voxel-dependent and shell-independent A. Elaldi, et al.
Fig. 3: Qualitative synthetic (Sec. 3.1) results showing fODF estimation on 128-gradient 2-fiber samples with red arrows representing ground truth fibers andthe heatmap showing model prediction. Row 1: CSD [6], Row 2: ESD (ours).with antipodal symmetry and that the RF is voxel-independent, shell-dependent,and rotationally symmetric about the y -axis. Typically, convolution between two S signals yields a signal on SO (3) (the group of 3D rotations) as the rotationmatrix P p ∈ SO (3) is a 3D rotation [3]. Specific to spherical deconvolution, asthe RF is symmetric about the y -axis, all rotations which differ only by rota-tion around the y -axis give the same convolution result. Therefore, P p can beexpressed with two angles and the output of the convolution lives on S . Spherical Harmonics.
We utilize the orthonormal Spherical Harmonics (SH)basis { Y ml } l ∈ N ,m ∈{− l,...,l } : S → R to express square-integrable f : S → R as f ( p ) = (cid:80) ∞ l =0 (cid:80) lm = − l f l,m Y ml ( p ) , where p ∈ S , { f l,m } l ∈ N ,m ∈{− l,...,l } ∈ R arethe spherical harmonic coefficients (SHC) of f . We assume f to be bandwidthlimited such that 0 ≤ l ≤ l max . As even degree SH functions are antipo-dally symmetric and odd degree SH functions are antipodally anti-symmetric,the odd degree SHC of both RF and fODF are null. Moreover, the m or-der SH functions are azimuthal symmetric only for m = 0. Thus, RF hasonly 0-order SHC. Therefore, the dMRI signal S b for the b -th shell is S b ( p ) = (cid:80) l max l =0 (cid:80) lm = − l (cid:113) π l +1 r b l f l,m Y m l ( p ) for l ∈ { , ..., l max } , m ∈ {− l, ..., l } where p ∈ S , L = (2 l max + 1)( l max + 1) is the number of coefficients, and { r b l, } and { f l,m } are the SHC of the RF and fODF, respectively. Matrix Formulation.
Let the dMRI signal be sampled over B shells for V voxels. For a specific b -shell, a set of n b gradient directions { ( θ bi , φ bi ) } ≤ i ≤ n b ischosen, where θ bi and φ bi are angular coordinates of the i th gradient direction.This set gives n b values { S b,vi } ≤ i ≤ n b for each voxel v , where S b,vi = S b,v ( θ bi , φ bi ).Let S b ∈ M V,n b ( R ) be the sampling of the v th voxel of the b -shell, with the v th row being { S b,vi } ≤ i ≤ n . Let Y b ∈ M L,n ( R ) be the SH sampled on the i th gradient of the b -shell { ( θ bi , φ bi ) } with its i th column being { Y m l ( θ bi , φ bi ) } l,m . Let F ∈ M V,L ( R ) be the matrix of the fODF SH coefficients with the v th row be-ing the coefficients { f v l,m } l,m of the v th voxel. Finally, let R b ∈ M L,L ( R ) be a quivariant Spherical Deconvolution 5 Fig. 4: Synthetic results relative to the number of gradients on 3 dMRI settings interms of Success Rate, Angular Error, Over-estimation, and Under-estimation.Arrows indicate whether lower or higher is better for a given score.diagonal matrix, with diagonal elements (cid:113) π l +1 r b l in blocks of length 4 l + 1 for l ∈ { , ..., l max } . The diffusion signal can now be written as S b = FR b Y b , with FR b giving the SHC of S and Y b transforming SHC into spatial data. Multi-Tissue decomposition.
So far, formalism has been presented for voxelswith a single tissue type. In reality, brain tissue comprises multiple components,e.g., white matter (WM), grey matter (GM), and cerebrospinal fluid (CSF). Tothis end, [6] presents a diffusion signal decomposition between WM/GM/CSFsuch that S ( p ) = S wm ( p ) + S gm + S csf . GM and CSF are assumed to haveisotropic diffusion limiting their spherical harmonic bandwidth to l max = 0.Thus, S b = F wm R bwm Y bwm + ( F gm R bgm + F csf R bcsf ) Y biso , where F gm , F csd ∈M V, ( R ), R gm , R csd ∈ M , ( R ), and Y iso ∈ M ,n ( R ). Simplifying notation, wefinally aim to solve the spherical deconvolution optimization problem to find SHcoefficients of the V fODFs. Deconvolution methods typically assume the SHcoefficients of the RF R to be known. Thus, the estimated fODF is: ˆF = argmin F , FY ≥ || S − FRY || + λReg ( F ) (1)where Reg is a sparsity regularizer on the fODF and
F Y ≥ A. Elaldi, et al.
Method ↑ ) IB ( ↓ ) VC ( ↑ ) IC ( ↓ ) NC ( ↓ ) CSD 1
117 34 .
79 65 .
07 0 .
123 48 .
06 51 .
82 0 .
111 48 .
80 51 .
06 0 . .
81 53 .
06 0 . .
12 34 .
86 0 . .
22 34 .
77 0 . Table 1: Downstream post-deconvolution tractography results on Tractometer interms of Valid/Invalid Bundles, Valid/Invalid Connections, and No Connections.Arrows indicate whether lower or higher is better for a given score.
Here, we outline how to take an unsupervised deep spherical network towardssparse non-negative fODF estimation, with an overview shown in Figure 2 andapplicability to single-shell single-tissue (SSST), single-shell multi-tissue (SSMT),and multi-shell multi-tissue (MSMT) deconvolution.
Graph convolution
We utilize the rotation-equivariant graph convolution de-veloped in [14] due to its improved time complexity over harmonic methods suchas [3]. The sphere is discretized into a graph G , such that f : S → R B is sampledon the N vertices of G such that the signal becomes a matrix f ∈ M N,B ( R ).A graph convolution can now be written as h ( L ) f = (cid:80) Pi =0 w i L i f , where theconvolutional filter is fully described by weights { w i } , and L = D − A is thegraph Laplacian with degree matrix D and adjacency matrix A . This graphconvolution can be made approximately rotation equivariant following [14] byfixing the edge weights of the graph and the discretization of the sphere. We useexponential weighting d i,j = e − || xi − xj || ρ if i and j are neighbors and d i,j = 0otherwise, where i and j are vertex indices of the graph, x i is the coordinate ofthe i -th vertex, and ρ is the average distance between two neighbors. We usehierarchical Healpix sampling [5] of the sphere to construct our graph. Spherical harmonics resampling.
Real-world DWI acquisition protocols sam-ple diffusion signals over a few dozen to at most a few hundred points with thesepoints not corresponding to Healpix sampling. Therefore, to construct the de-convolution network input, we resample the diffusion signal onto the Healpixgrid using spherical harmonics interpolation as illustrated in Figure 2. We thusobtain S input ∈ M V,B,N ( R ), where V is the number of voxels in a batch, B is the number of shells and N is the number of vertices. In the case the sam-pling and the frequency bandwidth of the DWI signal are not consistent, thisstep introduces a non-rotation equivariant operation. However, the model is still quivariant Spherical Deconvolution 7 Fig. 5: Tractometer partial volume fractions estimated from CSD (row 1) andESD (row 2), for the WM compartment (col. 1-2) and the isotropic GM andCSF compartments (col. 3-4). ESD returns more accurate localized tissue maps.equivariant to the set of rotations that permute the sampling gradients.
Sparsity.
Typically CSD methods represent the fODF with spherical harmonicsup to degree 8, implying that the fODF representation cannot approximate aDirac-like function and thus two close fibers cannot be distinguished. We main-tain the spherical harmonic basis, but increase their degree to 20 for betterrepresentation with the ability to separate small crossing-fibers. To ensure thesparsity of the predicted fODF, we forgo L regularization as in other sparse re-construction work [9] and assume that the fODF follows a heavy-tailed Cauchydistribution and regularize towards it as Reg ( F ) = (cid:80) Ni =1 log (1 + f i σ c ), where N is the number of points we estimate the fODF on, f i is the fODF value on the i th sphere pixel and σ c controls the sparsity level of the fODF. Learning Framework.
An overview of the overall network architecture isdescribed in Fig. 2. Response functions for the tissue compartments are pre-calculated with
MRTrix3 [18]. We input the Healpix-resampled signal into arotation-equivariant graph convolutional network, following a U-net style archi-tecture. We use max pooling and unpooling over the Healpix grid for down andupsampling, with batch normalization and ReLU nonlinearities following everyconvolution except for the last layer. For the final pre-fODF layer, we use theSoftplus activation for MSMT and ReLU for SSST for increased stability.The fODF outputs are T signals O ∈ M V,T,N ( R ) each corresponding to atissue compartment. The SHCs of the WM fODF are the even SHC degrees ofthe first output signal. For the isotropic GM and CSF, we take the maximumvalues of the second and third output signal and use them as the GM and CSFfODF SHC. Finally, the predicted fODF is convolved with the RF to reconstruct A. Elaldi, et al. the signal S , training the network under a regularized reconstruction objective asbelow. For simplicity, we omit the indices for the multiple tissue compartments. L ( f ODF ) = || S − f ODF ∗ RF || + λ N (cid:88) i =1 log (1 + f ODF i σ c ) + || f ODF fODF < || where the first term represents signal reconstruction, the second correspondsto sparsity as described before, and the third term encouraging non-negativity.As we use a ReLU or Softplus nonlinearity on the network output, the initiallyestimated fODF is entirely non-negative. However, as we use only the even-orderSHC of the fODF for convolution with the RF for reconstruction (the fODF issymmetric), eliminating the odd-order SHC may introduce negative values to thefODF which we suppress using an L regularizer on only its negative elements f ODF fODF < . We use a batch size of 32 and Adam for minimization with a 10 − step size, step-decayed on loss plateau. Depending on the dataset, all networksare found to rapidly converge within 20-30 epochs. We benchmark our methodologies across diverse datasets and deconvolution set-tings including: (1) synthetic data generated from a noisy multi-tensor modelwhere we evaluate SSST, SSMT, and MSMT; (2) the Tractometer dMRI bench-mark for SSST and SSMT; (3) an in-vivo human multi-shell dataset (MSMT).We compare our methods against the state-of-the-art CSD implementationsavailable in
MRTrix3 [18]. We note that ground truth fODFs are not available forTractometer and the human dataset, motivating our use of surrogate evaluationsin terms of downstream utility via tractography and partial volume estimation.Finally, for completeness, we test whether feature engineering by way of con-catenating CSD deconvolutions to the network input (denoted as ESD++CSD)would provide performance improvements.
We use the multi-tissue diffusion model from [6] to generate a multi-shell multi-tissue dataset. Representative response functions are estimated froma human subject (subject 1, center 1 from [16]) using
MRTrix . We add Riciannoise to the simulated signal corresponding to real MR noise. To assess therobustness of the method against the number of gradients and shells, we gen-erate five single-shell datasets with b -value 3000 s/mm and { , , , , } diffusion gradients per method. We also generate five multi-shell datasets ( b = { , , } s/mm ) corresponding to { , , , , } diffusion gradi-ents per shell. Finally, we generate a single b = 0 signal. We simulate 1 e { e , e , e } for training, validation and testing. quivariant Spherical Deconvolution 9 Fig. 6: Post-deconvolution tractography for Tractometer (rows 1-2; single-shell)and a human subject (rows 3-4; multi-shell). For both datasets, ESD (rows 2 and4) demonstrates clearer streamlines with lower noise as opposed to CSD (rows1 and 3). Readers are encouraged to zoom-in for visual inspection.
Evaluation Scores.
As ground truth microstructure is known, performanceevaluation is performed via five scores: (1) The success rate measures the abilityto correctly estimate the number and location of white matter fibers via the num-ber of voxels corrected processed. A voxel is successfully processed by the modelif each ground truth fiber can be match to a predicted fiber and the number ofpredicted fibers is the same as ground truth fibers. Following [1], a ground truthfiber is matched to a predicted fiber if it is no further than 25 degrees away; (2)The angular error measures angular distance between a ground truth fiber andthe closest predicted fiber; (3) The overestimatation error estimates the numberof predicted fibers outside of the 25-degree cone of the ground truth fibers; (4)The underestimation error measures the number of ground-truth fibers withoutpredicted fibers in their 25-degree cone; (5) Finally, we measure the estimatedfractional volume of each tissue component (a probability mass function) pervoxel in terms of the KL-divergence to the ground truth. For scores 1-4, we usethe peak detection algorithm from [1] to predict fiber directions, where the am-
Number of GradientsMethod
CSD 2 0 . . .
39 0 .
39 0 . .
39 0 .
61 0 . . . . . . . . CSD 3 0 .
54 0 .
44 0 .
38 0 .
36 0 . .
09 0 .
07 0 .
06 0 .
06 0 . ESD++CSD 3 0 . .
07 0 .
06 0 . . Method S. 1 S. 2 S. 3
CSD 3 .
46 3 .
10 3 . .
23 0 . . ESD++CSD .
83 0 . . Table 2: Partial Volume Fraction estimation in terms of KL divergence to groundtruth.
Left: synthetic performance relative to the number of gradients.
Right:
Performance on a multi-shell Human dataset for individual subjects (S1-3).plitude threshold selection is done on each validation set.
Results.
Evaluation scores are presented in Fig. 4. Our models consistently im-prove success rate, angular error and under-estimation over all gradients, shells,and number of tissue decompositions, save for 8 gradients. These results sug-gest that sharp fODFs returned by our methods allow for better localizationand detection of fibers, decreased undetected fibers, while reducing the numberof spurious fibers detected. Fig. 3 shows the better capacity of ESD to detectsmall-angle crossing fibers over CSD. Table 2 demonstrates better partial vol-ume fraction estimation with our models, leading to better tissue componentestimation and better localization of white matter fibers.
To assess downstream deconvolution utility, we utilize the ISMRM2015 Tractometer challenge [11] which provides a realistic single-shell multi-tissue brain phantom with 25 ground truth fiber-bundles, 32 diffusion gradientsand one b0 image. We apply basic dMRI motion-correction and use probabilis-tic tractography algorithm from
DiPy on the deconvolved fODF to estimate thefiber tracks. For tractography, we use the brain mask as the seed region, withdensity 1, we use a maximum angle of 75 and a stopping criterion threshold of0 .
25 on the FA map and delete short tracks from the output.
Evaluation Scores.
We follow Tractometer evaluation scores using Valid Bun-dles ( VB /True Positives), Invalid Bundles ( IB /False Positives), Valid Connec-tions ( VC /fraction of predicted streamlines part of a valid bundle), Invalid Con-nections ( IC /fraction of predicted streamlines part of an invalid bundle), andNo Connections ( NC /fraction of predicted streamlines not part of any bundle). Results.
Tractometer results are shown in Table 1, where our proposed modelsincrease the fraction of valid connections while decreasing the fraction of invalid quivariant Spherical Deconvolution 11 and non-connected streamlines. This suggests that our models allow for more ac-curate downstream fiber tracking. While the proposed model does not impact onthe number of valid bundles, it halves the number of invalid bundles in a 2-tissuedecomposition setting. We show the partial volume maps on that specific settingon Fig. 5, where ESD shows a clear separation between the two compartmentswhile CSD overestimates the white matter volume fraction in the isotropic partof the phantom. A qualitative view of the tractography in the 2-tissue decompo-sition setting is presented in figure 6. We see that the streamlines are less noisywith ESD than CSD with better trajectory coherence.
Here, we use the preprocessed multishell human dataset from [16]. The datasethas three different subjects and we use scans of all three subjects from the firstcenter. The protocol has three b -values { , , } each with 98 gradientsand 27 B Evaluation Scores.
Assessing deconvolution performance on a real brain isnon-trivial due to the lack of micro-structural ground-truth. Therefore, we fol-low a downstream utility based evaluation similar to Section 3.2. We estimate theground-truth partial volume from the T1 images of each subject using
FSL FAST [21]. We also analyze qualitative tractograms, computed with the same proba-bilistic tractography algorithm as the tractometer dataset.
Results.
Quantitative results are shown in Table 2. The KL-divergence is con-sistently improved for the three subjects, suggesting a better estimation of theoverall partial volume fractions. Moreover, we show qualitative tractographyviews of the three subjects in the figure 6. Again, with ESD, we observe lessnoise and more streamline consistency.
We present an unsupervised rotation-equivariant spherical CNN for the nonlin-ear estimation of the fiber orientation distribution function from diffusion MRIsignals. With a Cauchy distribution prior on the fODF intensity and increasedspherical harmonics order, we obtain sharp and accurate localizations. Our modelis flexible and can be used in a wide range of settings, including generic natu-ral image spherical deconvolution. It can work with or without multi-shell dataand with an arbitrary number of DWI gradients. Our experiments demonstrateimproved micro and macro-structure estimation over state-of-the-art deconvolu-tion frameworks in terms of the detection of smaller crossing-angle fibers, betterfiber localization, and better partial-volume estimation. Finally, improved localperformances have a positive impact on global white matter fiber tracking interms of noise and streamline coherence.
Acknowledgements
The authors were supported by NIH grants 1R01DA038215-01A1, R01-HD055741-12, 1R01HD088125-01A1, 1R01MH118362-01, R01ES032294, R01MH122447, and1R34DA050287.
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