HARDI denoising with mean-curvature enhancement PDE on SE(3)
Etienne St-Onge1, Stephan Meesters2, Erick J Bekkers2, Maxime Descoteaux1, and Remco Duits2

1SCIL, Université de Sherbrooke, Sherbrooke, QC, Canada, 2CASA, Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands


We present a new High Angular Resolution Diffusion-weighted Imaging (HARDI) mean-curvature enhancement (MCE) on the homogeneous space of positions and orientations, embedded in the rigid body motion group SE(3). Its potential for crossing-preserving enhancement of fiber orientation distribution (FOD) fields is demonstrated. Compared to previous partial differential equation (PDE) enhancements on SE(3), denoising FOD fields with MCE better preserve boundaries, resulting in a lower overall angular error.


Anisotropic filtering was first introduced to reduce noise from images without blurring sharp edges1,2. These edge-preserving smoothing PDE encourage the conductivity along contours, increasing spatial consistency, while still preserving angular features (boundaries, corners, contour curves). As an alternative to these methods, curvature based filtering was proposed to reduce noise in images and discrete surfaces3,4,5.

For diffusion-weighted MRI (DW-MRI), numerous filtering methods have been proposed to increase the local coherence of Diffusion Tensor Image (DTI)6,7,8. Since, DTI is unable fully represent the diffusion signal in crossing regions, known to occur in most of the white matter [9,10], advanced methods were proposed to increase spatial continuity of multi-tensor fields11,12,13 or FOD fields14,15. Multiple approaches for HARDI denoising have also been compared by Daducci et al16.

Previous works exposed the advantage of PDE on SE(3) for crossing-preserving filtering17. This representation enables the processing of spatial and angular components together, since the underlying metric combines both, i.e. high diffusivity in a certain orientation should persist in the spatial domain along this orientation. Elaborations of these concepts on $$$\mathbb{R}^3 \rtimes \mathbb{S}^2$$$ led to contextual enhancement (CE) and Perona-Malik enhancement (PME) for 2D or 3D orientation score images and DW-MRI18,19.


In this work, we adapted an anisotropic filtering approach for HARDI, to enhance the alignment of elongated structures while maintaining crossing angles. This enhancement follows previous research on mean-curvature ($$$\kappa$$$) enhancement on 2D images ($$$I(\mathbf{x}), \, \mathbf{x} \in \mathbb{R}^2$$$) or manifold2,3,4,5:$$\frac{\partial J(\mathbf{x},t)}{\partial t}= \|\nabla J(\mathbf{x},t)\| \kappa(\mathbf{x},t)= \|\nabla J(\mathbf{x},t)\| \text{div} \left(\frac{\nabla J(\mathbf{x},t)}{\|\nabla J(\mathbf{x},t)\|} \right)\\
J(\mathbf{x}, 0)= I(\mathbf{x}), \quad t>0$$

For HARDI, the image domain ($$$\mathbb{R}^3$$$) was extended with a sphere ($$$\mathbb{S}^2$$$), considering that FODs generate a spatial field of spherical function ($$$U(\mathbf{x},\mathbf{n}),\, (\mathbf{x},\mathbf{n}) \in \mathbb{R}^3 \times \mathbb{S}^2$$$). Hence, the MCE equation was adapted to the Lie group SE(3) (or rather Lie group quotient $$$\mathbb{R}^3 \rtimes \mathbb{S}^2$$$) with the use of Left-invariant derivative19,20,21:
$$\frac{\partial W(\mathbf{x},\mathbf{n},t)}{\partial t}= \|\nabla_{sr} W(\mathbf{x},\mathbf{n}, t) \| \left(D_{\mathbf{x}} (\mathbf{n} \cdot \nabla_{\mathbb{R}^3})\left(\frac{(\mathbf{n} \cdot \nabla_{\mathbb{R}^3}) W(\mathbf{x},\mathbf{n},t)}{\|\nabla_{sr} W(\mathbf{x},\mathbf{n},t)\|}\right) + D_{\mathbf{n}}\, \textrm{div}_{\mathbb{S}^2}\left(\frac{\nabla_{\mathbb{S}^2} W(\mathbf{x},\mathbf{n},t)}{\|\nabla_{sr} W(\mathbf{x},\mathbf{n},t)\|}\right) \right)\\
\|\nabla_{sr} W(\mathbf{x},\mathbf{n}, t) \| = \sqrt{\frac{D_{\mathbf{x}}}{D_{\mathbf{n}}}\|(\mathbf{n} \cdot \nabla_{\mathbb{R}^3}) W(\mathbf{x},\mathbf{n},t)\|^2 + \|\nabla_{\mathbb{S}^2} W(\mathbf{x},\mathbf{n},t)\|^2}\\
W(\mathbf{x},\mathbf{n}, 0)= U(\mathbf{x},\mathbf{n}), \quad t>0$$


To evaluate various SE(3) filtering techniques, we used the ISBI-HARDI challenge synthetic phantom22 at SNR10 and computed the FOD field with constrained spherical deconvolution (CSD)23 (Figure #1). To analyze results, the noisy FOD field was filtered with different parameters on 40 iterations ($$$t=4, dt=0.1$$$): angular conductivity $$$D_{\mathbf{n}} = 0.001, 0.002, 0.004$$$ and PME sensitivity $$$K = 0.1, 0.2, 0.4, 0.8$$$. A constant rescaling of both $$$D_{\mathbf{x}}$$$ and $$$D_{\mathbf{n}}$$$ correspond to a scaling of the diffusion time ($$$t, dt$$$), therefore we kept the spatial conductivity $$$D_{\mathbf{x}} = 1$$$ fixed.

Quantitatively, the L1 and L2 absolute error, defined as the distance to the noiseless FOD, was computed at each timestep, for all SE(3) enhancements. A second version of both L1 and L2 error was included to assess only the orientational error, where at each position the FOD was normalized.

Qualitatively, a standard in-vivo HARDI acquisition (49 directions at b-value 1000) was included to visually compare results.


Figure #2 illustrates filtering results over the synthetic phantom, comparing the proposed CME to the PME. Since there was no perceptual difference between the CE and PME, only one of them is displayed. The L1 error progression for all methods are shown in Figures #3-4, the L2 error followed the same trend. The resulting enhanced FOD, over a standard in-vivo dataset, can be observed in Figure #5. Only PM results with sensitivity $$$K = 0.2, 0.4$$$ are presented as they showed best performance compared to $$$K = 0.1, 0.8$$$.


As illustrated in Figure #2, all methods are able to remove spurious peaks, and preserve crossings. Contextual enhancement and Perona-Malik methods reduce noise from FOD field within few iterations, but afterward tend to blur edges ($$$t > 2$$$), causing the error to increase (see Figures #3-4). The proposed MCE is able to further decrease the error before over-smoothing. In addition, the reduced normalized error, combined with the smaller slope, (Figure #4) makes the proposed filtering very appealing when the FOD field is only used as an angular probability, e.g. for tractography. In Figure #5, sharper and more consistent FOD is obtained with MCE.


Mean-curvature enhancement, with SE(3) framework, can denoise HARDI both spatially and angularly. The proposed method is particularly good to recover the angular distribution of FODs. Parameters and diffusion time can be optimized to maximize FOD denoising while preserving sharp edges. In future work, it would be interesting to adapt the MCE for contextual interpolation/upsampling or to regularize the CSD process.


We would like to thank Dr. A. Roebroeck from the Faculty of Psychology & Neuroscience, Maastricht University, for providing us with the in-vivo data. Supported by grants from Université de Sherbrooke Research Chair, NSERC Discovery Grant, European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013), ERC grant Lie Analysis (agr. nr. 335555).


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Figure #1. Illustration of the synthetic phantom multi-tensor field (ground truth). The error is computed inside the red mask and the visualization in the blue rectangle. Top-right) Initial noiseless FOD, bottom-right) FOD at SNR10.

Figure #2. HARDI enhancement results, at time t=1,4: top) Perona-malik, bottom) Mean-curvature. All methods are able to recover the crossing and to remove spurious peaks.

Figure #3. Assessment of the filtering error, starting from the synthetic data at SNR10. The error computed from the L1 distance to the ground truth FOD (noiseless phantom), with and without FOD normalization. For every angular conductivity Dn the proposed MCE (in blue) is able to outperform other methods.

Figure #4. Table of the smallest error achieved, over all time, for each enhancement. For every distance to the ground truth (row), cells were color coded from green to white; from the smallest (best) to the largest (worst) minimum reached. For all angular conductivity Dn, the proposed MCE outperform other methods.

Figure #5. SE(3) denoising of the in-vivo acquisition (coronal cross section): a) Initial raw FOD, b) PME (K=0.4, Dn=0.001) and c) SE(3) MCE (Dn=0.001).

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)