NN

    neural network
CNN

    convolutional neural network
GCN

    graph convolutional network
SCNN

    spherical convolutional neural network (CNN)
dMRI

    diffusion magnetic resonance imaging
RCNN

    recurrent CNN
PCConv

    parametric continuous convolution
PCCNN

    parametric continuous CNN
DTI

    diffusion tensor imaging
FBA

    fixel-based analysis
NODDI

    neurite orientation dispersion and density imaging
MLP

    multi-layer perceptron
AFD

    apparent fibre density
AE

    absolute error
MAE

    mean absolute error
FA

    fractional anisotropy
ReLU

    rectified linear unit
HCP

    WU-Minn Human Connectome Project
MPEAE

    mean principal eigenvector angular error
FOD

    fibre orientation distribution
ACC

    angular correlation coefficient
CSD

    constrained spherical deconvolution
WM

    white matter
SS3T

    Single-Shell 3-Tissue
MSMT

    Multi-Shell Multi-Tissue
OD

    orientation dispersion index
ViT

    vision transformer
SH

    spherical harmonic

# Spatio-Angular Convolutions for Super-resolution in Diffusion MRI

Matthew Lyon
University of Manchester
<EMAIL_ADDRESS>
&Paul Armitage
University of Sheffield
<EMAIL_ADDRESS>
&Mauricio A Álvarez
University of Manchester
<EMAIL_ADDRESS>

###### Abstract

Diffusion MRI (dMRI) is a widely used imaging modality, but requires long
scanning times to acquire high resolution datasets. By leveraging the unique
geometry present within this domain, we present a novel approach to dMRI
angular super-resolution that extends upon the parametric continuous
convolution (PCConv) framework. We introduce several additions to the
operation including a Fourier feature mapping, global coordinates, and domain
specific context. Using this framework, we build a fully parametric continuous
convolution network (PCCNN) and compare against existing models. We
demonstrate the PCCNN performs competitively while using significantly less
parameters. Moreover, we show that this formulation generalises well to
clinically relevant downstream analyses such as fixel-based analysis, and
neurite orientation dispersion and density imaging.

##  1 Introduction

dmri is a clinical imaging modality that is routinely used to diagnose white
matter pathology within the human brain. Advancements in downstream analyses
using diffusion magnetic resonance imaging (dMRI) data continuously expand the
diagnostic capabilities of the imaging modality, but often require higher
resolution data than is clinically feasible [24]. It has been shown, however,
that deep learning can be used to obtain high resolution images whilst only
acquiring low resolution data [22, 35, 32].

dmri data can be represented using a non-Euclidean geometry. Specifically, a
dMRI dataset consists of a series of three-dimensional volumes that measure
water diffusivity in a given direction. Diffusivity directions, quantified by
a 3D vector known as a ‘b-vector’, are sparsely sampled from a unit sphere,
whilst each three-dimensional volume is regularly sampled in a discrete grid.
Despite this relatively unique geometry, most dMRI deep learning methods use
operations grounded in a Euclidean framework, such as CNNs [34, 35, 32].

This presents an opportunity as typical CNN architectures do not fully utilise
the geometric properties present in dMRI data. For example, implicit within
the formulation of the CNN is the assumption that data are densely and
regularly sampled in a discrete manner. This is only true when considering the
three spatial dimensions of dMRI data, whilst the other dimensions would be
more suited using approaches like graph convolutional networks [37], spherical
CNNs [5], and point cloud CNNs [19]. This domain would therefore benefit from
a framework that convolves over densely sampled grid data and sparsely sampled
spherical data simultaneously.

A promising candidate for such a framework is the parametric continuous
convolution (PCConv) introduced by Wang et al. [31], which extended the
discrete convolution into the continuous domain. This was achieved via an
inner parameterised hypernetwork [14] responsible for generating convolutional
weights. Here, the convolutional weights were not learnt directly but were
instead sampled from the hypernetwork given kernel coordinates as input. This
operation could therefore be used to model arbitrary geometries.

In this work, we build upon the PCConv operation to convolve across both
spatial and angular dimensions of dMRI data via a new, flexible, coordinate
embedding. We subsequently create a deep parametric continuous  (PCCNN) that
is well-suited for inference within this domain. Furthermore, we develop
additional modifications to the PCCNN by including supplemental prior
information in the coordinate embedding. These modifications include the
application of Fourier feature mappings, the inclusion of prior domain
information, and the incorporation of global coordinate components.

We demonstrate the effectiveness and flexibility of this approach through the
task of angular super-resolution of dMRI data in various sampling schemes,
including single-shell and multi-shell data [16]. To assess the performance of
our method, we compare results across clinically relevant downstream analysis
tasks such as fixel-based analysis (FBA) [25], and neurite orientation
dispersion and density imaging (NODDI) [36]. We show that the PCCNN performs
competitively against a similar recurrent  (RCNN) network trained on the same
data whilst using approximately a tenth of the number of parameters.
Additionally, we demonstrate the generalisability of the network by comparing
it against models trained specifically for downstream analysis tasks.

##  2 Methods

This section presents related work within dMRI super-resolution, focusing on
previous methods that infer raw dMRI data, and those that infer high angular
resolution data within FBA and NODDI anlayses. Later we introduce parametric
continuous convolutions, and extend the framework into the dMRI data domain.
We highlight our key contributions to the PCConv operation, then finally
provide implementation details used to perform training and inference within
the task of dMRI super-resolution.

###  2.1 Related Work

The task of inferring high angular resolution data from low angular resolution
dMRI acquisitions can broadly be split into two categories: methods that
super-resolve raw dMRI data, and those that super-resolve downstream analyses.
The former allows for more flexible use of the data and analysis method, but
the unconstrained nature of the task makes inference more challenging. The
latter constrains the task by leveraging assumptions made within the analysis
method, however the mapping from raw dMRI data to downstream analysis is in
general not invertible, therefore the super-resolved data cannot be used to
derive other downstream analysis metrics.

Notable works in angular super-resolution of raw dMRI data include Yin et al.
[34], whereby a preliminary study constructed a 1D CNN autoencoder that only
convolved across the angular dimension, disregarding possible spatial
correlations within the data. Lyon et al. [22] developed a 3D RCNN that
benefited from parameter sharing in the spatial dimensions, but required
angular dimension shuffling to overcome sequence order bias. Both models
concatenated the b-vectors as channels within the input, therefore treating
the b-vectors as additional features, rather than coordinates of a manifold
for which the data lies on.

Whilst there is a myriad of downstream analyses for dMRI, this work focuses on
two commonly used methods: FBA [25] and NODDI [36]. FBA relies on segmenting
fibre orientation distributions [16] into ‘fixels’ or fibres within a voxel.
Relevant works in FOD super-resolution include Lucena et al. [21] and Zeng et
al. [35]. Both of these studies regressed low angular resolution derived FODs
with their high resolution counterparts through 3D CNNs. NODDI estimates
micro-structural complexity of the underlying brain tissue through modelling
axon and dendrite populations within a voxel. Approaches to infer high angular
resolution derived NODDI parameters from lower resolution raw dMRI data
include Golkov et al. [13], Ye et al. [32], and Ye et al. [33]. These studies
use various sub-sampling schemes and network architectures, including a 3D CNN
within Ye et al. [32].

###  2.2 Parametric Continuous Convolutions

A discrete convolution within the context of deep learning is defined as
follows

$h[n]=(f\ast g)[n]=\sum_{m=-M}^{M}f[m]g[n-m],$

where the data function $f:\mathcal{N}\rightarrow\mathbb{R}$ and kernel
$g:\mathcal{G}\rightarrow\mathbb{R}$ are defined over the support domain for
the finite integer set $\mathcal{N}=\mathbb{Z}^{D}$ and
$\mathcal{G}=\\{-M,-M+1,\ldots,M-1,M\\}^{D}$ respectively. Conversely, a
continuous convolution is defined as

$h(\mathbf{y}_{j})=(f\ast
g)(\mathbf{y}_{j})=\int^{\infty}_{-\infty}f(\mathbf{x})g(\mathbf{y}_{j}-\mathbf{x})d\mathbf{x},$

where both $f$ and $g$ are continuous functions over the support domain
$\mathcal{N}=\mathcal{G}=\mathbb{R}^{D}$. Typically in deep learning, this
integration is analytically intractable but can be approximated as in Wang et
al. [31] using the following form:

$h(\mathbf{y}_{j})=\int^{\infty}_{-\infty}f(\mathbf{x})g(\mathbf{y}_{j}-\mathbf{x})d\mathbf{x}\approx\frac{1}{N}\sum^{N}_{i=1}f(\mathbf{x}_{i})g(\mathbf{y}_{j}-\mathbf{x}_{i}),$
(1)

where a finite number of input points $\mathbf{x}_{i}$ are sampled from the
support domain for each output point $\mathbf{y}_{j}$. In the parametric
continuous convolution framework, we choose to use a multi-layer perceptron
(MLP) as the kernel function
$g(\mathbf{e};\bm{\theta})=\operatorname{MLP}(\mathbf{e};\bm{\theta})$, where
$\mathbf{e}$ is a coordinate embedding that depends on $\mathbf{x}_{i}$ and
$\mathbf{y}_{j}$, which is described in detail in Section 2.3. The kernel
function $g(\mathbf{e};\bm{\theta}):\mathbb{R}^{D}\rightarrow\mathbb{R}$ spans
the full continuous domain while being parameterised by a finite number of
parameters $\bm{\theta}$. In contrast to standard convolutional operations,
the weights of the PCConv are not learnt directly, and are instead sampled
from $g(\mathbf{e};\bm{\theta})$. As the kernel function is itself a fully
differentiable function, it can be trained through back-propagation.

In typical convolutional layers the set of output points form a subset of the
input points set, provided there is no padding. However, within the parametric
continuous framework, these two sets can be disjoint. This is key within the
task of dMRI angular super-resolution, where the angular component of
$\mathbf{y}_{j}$, i.e. the target b-vectors, is not contained within the set
of input b-vectors.

Each PCConv layer requires three inputs to produce the output features
$h_{k,j}(\mathbf{y}_{j})$: the input features $f_{c,i}(\mathbf{x}_{i})$, the
set of input points $\mathbf{x}$, and the set of output points $\mathbf{y}$.
Within each layer, following on from the right-hand side of Equation (1), the
PCConv operation is defined as

$h_{k,j}(\mathbf{y}_{j})=\sum^{C}_{c=1}\sum^{N}_{i=1}f_{c,i}(\mathbf{x}_{i})g_{c,k}(\mathbf{e};\bm{\theta}).$
(2)

Here, $C$ and $K$ denote the input and output feature dimensions respectively.
Within each layer the number of input points $N$ is fixed, therefore the
kernel function $g$ will encapsulate the $\frac{1}{N}$ factor in Equation (1)
during training.

###  2.3 Convolutions in Q-Space

Figure 1:  Visualisation of dMRI parametric continuous convolution. Points
$\mathbf{x}_{i}$ and $\mathbf{y}_{j}$ are of the form
$(u,v,w,\rho,\vartheta,\varphi)^{\top}$ with spatial components $(u,v,w)$ and
angular components $(\rho,\vartheta,\varphi)$. Left: The raw dMRI data
$f(\mathbf{x})$. Points $\mathbf{x}_{4}$ and $\mathbf{x}_{6}$ are located in
the same region on the q-space sphere and thus have the same angular
components. Similarly, $\mathbf{x}_{6}$ and $\mathbf{x}_{20}$ have the same
spatial coordinates but different angular components. Middle: the parametric
continuous kernel, whose geometry is defined by the coordinate embedding
$\mathbf{e}$. Right: Sampled points from q-space are pointwise multiplied with
the kernel and summed over to produce one output feature $h(\mathbf{y}_{j})$.

We define q-space $\mathbb{Q}$ as the joint spatial-angular domain within dMRI
where $\mathbb{Q}=\\{(u,v,w,\rho,\vartheta,\varphi)^{\top}\mid
u,v,w,\rho\in\mathbb{R},0\leq\vartheta<2\pi,0\leq\varphi\leq\pi\\}$ and
$\mathbf{x}_{j},\mathbf{y}_{i}\in\mathbb{Q}$. Here, $(u,v,w)^{\top}$ denote
the spatial components, and $(\rho,\vartheta,\varphi)^{\top}$ denote the
angular components. Specifically, $\vartheta$ and $\varphi$ refer to the
azimuthal and polar angles, respectively, which form the set of points on the
surface of a sphere of radius $\rho$. In this context, $\rho$ is the diffusion
weighting or b-value. Figure 1 visualises convolutions across q-space through
Equation (2). To perform convolutions using q-space, we first calculate a
coordinate embedding $\mathbf{e}\in\mathbb{R}^{2LE}$ via

$\mathbf{e}=[\bm{\gamma}(p_{1}),\bm{\gamma}(p_{2}),\cdots,\bm{\gamma}(p_{E})]^{\top},$
(3)

where

$\bm{\gamma}(p_{m})=[\sin{(2^{0}\pi p_{m})},\cos{(2^{0}\pi
p_{m})},\cdots,\sin{(2^{L-1}\pi p_{m})},\cos{(2^{L-1}\pi p_{m})}].$

Here $p_{m}$ is the $m$th component of the coordinate vector
$\mathbf{p}\in\mathbb{R}^{E}$ given by
$\mathbf{p}=[u_{i}-u_{j},v_{i}-v_{j},w_{i}-w_{j},\rho_{i}-\rho_{j},d_{r}]$. We
have opted to enforce rotational invariance across the sphere by omitting
$\vartheta$ and $\varphi$ directly, and instead use
$p_{5}=d_{r}(\vartheta_{j},\varphi_{j},\vartheta_{i},\varphi_{i})$ where
$d_{r}(\vartheta_{j},\varphi_{j},\vartheta_{i},\varphi_{i})=\sin(\vartheta_{j})\sin(\vartheta_{i})+\cos(\vartheta_{j})\cos(\vartheta_{i})\cos(\varphi_{j}-\varphi_{i})$.
The function $\gamma:\mathbb{R}\rightarrow\mathbb{R}^{2L}$ is a Fourier
feature mapping motivated by Tancik et al. [28] and Mildenhall et al. [23]
which is used to improve the hypernetwork’s ability to learn high-frequency
functions. In practice, $L$ is a hyperparameter of the PCConv layer. Figure 2
provides examples of weights sampled from a trained PCConv layer with varying
$d_{r}$. The channels selected within the Figure demonstrate a variety of
functions across the sphere, highlighting the network’s ability to learn both
high and low frequency kernels.

An important aspect of the discrete convolution is its finite support domain
$M$, thus allowing the network to share parameters across the image domain.
Within the PCConv framework this is achieved by limiting the number of points
to convolve across, given some constraint. When convolving over q-space, the
spatial dimensions follow the same constraint as in discrete convolutions.
That is, for each dimension given a kernel size $n$, only the $n$ closest
points are selected. As the angular components represent sparsely sampled data
across a sphere, a kernel size $n$ selects the $n$ closest points, as
determined by $d_{r}$. We refer to the angular kernel size as $k_{q}$ from
this point onward. There is an additional constraint with the angular
components that only points within a fixed radius $d_{\mathrm{max}}$, such
that $d_{r}\leq d_{\mathrm{max}}$, are included. To achieve this, a binary
mask is applied after the pointwise multiplication of the kernel weight with
the input feature.

$g_{7,1}(\mathbf{e};\bm{\theta})$$g_{10,1}(\mathbf{e};\bm{\theta})$$g_{17,3}(\mathbf{e};\bm{\theta})$$g_{24,10}(\mathbf{e};\bm{\theta})$$g_{36,10}(\mathbf{e};\bm{\theta})$
Figure 2:  Visualisation of kernel weights sampled from
$g_{c,k}(\mathbf{e};\bm{\theta})$ within a PCConv layer, where $c$ and $k$
index the input and output feature dimensions respectively. To generate the
spheres, $d_{r}$ is varied whilst all other components of $\mathbf{p}$ are
kept constant.

The PCConv is a flexible operation that enables convolutions over non-
Euclidean spaces. Additionally, the framework allows for kernel weights to be
dependent upon coordinates that are not defined in terms of the relative
distance between an input and output point. In this work, we additionally
introduce a variant of the PCCNN that uses one such encoding. Denoted as
‘PCCNN-Bv’, this model uses
$\mathbf{p}=[u_{i}-u_{j},v_{i}-v_{j},w_{i}-w_{j},\rho_{i},\rho_{j},d_{r}]$.
The inclusion of this model is motivated by the non-linear relationship
between different shells within multi-shell data, which may not be
appropriately captured by the relative difference of b-values.

###  2.4 Global Information

Discrete convolutional layers are computed over a limited kernel size, and
therefore have a limited field of view. This affords the convolutional layer
computational efficiency, as the kernel size is typically much smaller than
the image. However, this comes at the cost of losing global information.
Within dMRI, structures within the brain are highly spatially dependent and
follow a similar pattern in each individual. Non-local coordinates, and
therefore global context, can be incorporated within the PCConv layer through
the coordinate embedding. For example, the relative position of the kernel
within an image could be incorporated into the coordinate embedding. This idea
is inspired by how vit [10] actively extract global information through
correlations between image patches. The availability of global context allows
vision transformers to show competitive performance in image recognition
tasks, where accurate synthesis of semantic meaning more heavily relies on
non-local information [17].

In clinical practice, neuroimaging data are routinely aligned to a reference
space, such as Talairach space [2]. As such, coordinates within reference
spaces are correlated across different subjects, and therefore can be used to
provide context for the location of a voxel within the brain. In this work,
after normalising the coordinates with respect to the subject brain mask size,
we use the centroid of the training patches, as discussed in Section 2.7, as
additional coordinates within $\mathbf{p}$. Denoted as an ‘-Sp’ suffix, both
‘PCCNN-Sp’ and ‘PCCNN-Bv-Sp’ models within Section 3 have this modification.

###  2.5 Factorised Convolutions

dMRI is routinely stored in a dense four-dimensional format, with three
spatial dimensions and one angular dimension whose coordinates are defined by
$\mathbf{b}$. Performing a convolution in this high-dimensional space can be
computationally expensive. We mitigate this cost by using factorised
convolutions [27] within the PCConv framework. For example, in the two-
dimensional case, an $(n\times m)$ kernel would be factorised into two
sequential convolutions of size $(n\times 1)$ and $(1\times m)$. As in the
non-factorised case, the PCConv layer’s weights are sampled from one
hypernetwork $g(\mathbf{e};\bm{\theta})$. Following the approach used in Wang
et al. [31], we further factorise the kernel weight tensor from
$W\in\mathbb{R}^{N\times O\times C\times K}$ into two sequential operations:
an input convolution $W_{\mathrm{in}}=\mathbb{R}^{N\times O\times C}$ followed
by a forward projection matrix $W_{\mathrm{out}}=\mathbb{R}^{C\times K}$,
where $O$ is the number of output points.

###  2.6 Implementation

Since PCConv layers are capable of convolving over both spatial and angular
dimensions, the PCCNN models are constructed exclusively of PCConv layers and
non-linear activation functions. In practice, this involves a sequence of
spatially pointwise $(1\times 1\times 1\times k_{q})$ PCConv layers, followed
by blocks of parallel convolutions consisting of spatially pointwise and
$(3\times 3\times 3\times k_{q})$ factorised convolutions. The hyperparameters
for each PCCNN, including the number of PCConv layers, can be found in Figure
4 in the Appendix. Within this work, each PCConv layer has its own independent
hypernetwork. This configuration allows for significant weight sharing
capabilities, as the number of parameters for each hypernetwork does not
depend on the kernel size or dimensionality. Each PCConv layer or residual
PCConv block is followed by a rectified linear unit (ReLU), excluding the
final layer. Each hypernetwork is composed of two dense layers, each followed
by a leaky ReLU with a negative slope of $0.1$, and a final dense layer with
output size $1$ and no subsequent activation.

###  2.7 Data and Training

Given the high dimensionality of dMRI data, using complete acquisitions as
individual training examples is prohibitively expensive in terms of memory. To
mitigate this issue, training sets were derived from input image patches
$\mathbf{X}_{\mathrm{in}}\in\mathbb{R}^{10\times 10\times 10\times
q_{\mathrm{in}}}$, and target image patches
$\mathbf{X}_{\mathrm{out}}\in\mathbb{R}^{10\times 10\times 10\times
q_{\mathrm{out}}}$ where $q_{\mathrm{in}}$ and $q_{\mathrm{out}}$ denote the
number of input and target b-vectors respectively. During training, to sample
the angular dimension of the input sets, an initial b-vector $\mathbf{q}_{0}$
was randomly selected. The next $n$, up to $q_{\mathrm{in}}$, b-vectors
$\mathbf{q}$ were chosen using
$\operatorname*{arg\,max}_{j}(\min_{i}(d_{r}(\mathbf{q}_{i},\mathbf{q}_{j}))),\forall
i,j\in\\{0,1,...,n-1\\}$. Similarly, this process was repeated to sample the
output set up to size $q_{\mathrm{out}}$.

To reduce inter-subject variability, each dMRI dataset was normalised such
that 99% of the data lay between $[-1,1]$. Initially, a normalisation range of
$[0,1]$ was used; however, this resulted in training instabilities within the
hypernetwork. This was likely due to a greater mismatch between the data
distribution and the weight initialisation scheme within the hypernetwork,
which followed
$w\sim\mathcal{U}(-\sqrt{\frac{1}{k_{\mathrm{in}}}},\sqrt{\frac{1}{k_{\mathrm{in}}}})$,
where $k_{\mathrm{in}}$ was the number of input features in a given layer.
dMRI data from the WU-Minn Human Connectome Project (HCP) [30] were used for
training, validation and testing. This data were initially processed with the
standard HCP processing pipeline [12], as well as denoised using the
‘patch2self’ algorithm [11]. Models were trained on twenty-seven subjects from
the HCP dataset, while three subjects were used for validation during
development. Hyperparameters for the PCCNN were selected through a random grid
search with RayTune [18].

Each acquisition within the HCP dataset had shells of
$b_{\mathrm{full}}=\\{1000,2000,3000\\}$ $\mathrm{s/mm^{2}}$, each with $90$
directions. Training examples contained randomly selected shells
$b_{\mathrm{in}},b_{\mathrm{out}}\sim U(b_{\mathrm{full}})$. This sampling
scheme trained each network on single-shell and multi-shell examples
simultaneously. The input angular dimension size was determined via
$q_{\mathrm{in}}\sim
U(q_{\mathrm{sample}}),q_{\mathrm{sample}}=\\{6,7,...,19,20\\}$. This approach
allowed each network to learn the data distribution for a range of angular
input sizes. To implement this, training examples had a fixed
$q_{\mathrm{in}}=20$, with zero-filled entries for input size less than 20.
Models were trained using 4 NVIDIA A100’s with a batch size of 16, and an
$\ell_{1}$ loss function, for 400,000 iterations using AdamW [20]. Each PCCNN
model took approximately a week to train. Details of pre-processing and
training for other models can be found in the Appendix. The model weights used
for test inference in Section 3 were the best performing in the validation
dataset.

##  3 Experiments and Results

Table 1:  Total number of parameters, in millions, of different deep learning approaches.  Model | Parameters
---|---
SR-q-DL [32] | 0.52
PCCNN | 0.77
PCCNN-Bv | 0.79
PCCNN-Sp | 0.83
PCCNN-Bv-Sp | 0.85
RCNN [22] | 6.82
FOD-Net [35] | 48.17

We evaluated the performance in dMRI angular super-resolution using eight,
previously unseen subjects within the HCP dataset. Our results included four
variants of the PCCNN, as listed in Table 1. As dMRI data is most commonly
used for downstream analyses, we compared the results in commonly used
techniques for both single-shell and multi-shell experiments. Our results
included other methods where appropriate, such as spherical harmonic (SH)
interpolation for single-shell experiments, FOD-Net [35] for multi-shell FBA,
and SR-q-DL [32] for multi-shell NODDI. FOD-Net is a 3D CNN that maps low
resolution FOD data to high resolution FOD data. SR-q-DL is a 3D CNN that maps
low resolution dMRI data to high resolution NODDI data. RCNN is a 3D recurrent
CNN that maps low resolution dMRI data to high resolution dMRI data. When
applicable, we also compared against low angular resolution data. All error
metrics were calculated with respect to the full angular resolution HCP
dataset.

The reported error values consist of the mean and standard deviation, with
distribution statistics generated from the mean performance per test subject.
Only voxels within the brain are included in the calculations, as determined
by a brain mask segmentation. The best mean value in a given category is
denoted in bold.

###  3.1 Single-shell dMRI

Within single-shell inference, we evaluated performance across two analyses.
Firstly, on raw dMRI data, of which any downstream analysis task would be
derived from. We compared performance across all three shells available within
the HCP dataset, as well as in three q-space sub-sampling regimes, as outlined
in Table 2. Secondly, we compared performance within FBA, where metrics were
derived from the inferred dMRI data. For this, we considered the $b=1000$
$\mathrm{s}/\mathrm{mm}^{2}$ shell at three sub-sampling regimes.

####  3.1.1 dMRI Analysis

Table 2 presents the absolute error (AE) of the inferred dMRI data as compared
to the high resolution ground truth. For the $b=2000\mathrm{s/mm^{2}}$
results, see Table 6 within the Appendix. The PCCNN-Bv-Sp model had the lowest
AE in five out of the nine shell and sub-sampling combinations and had the
overall best performance when averaged across all data. Despite having
approximately a tenth of the number of parameters of the RCNN [22] model, the
PCCNN models had lower mean errors across all data regimes except
$q_{\mathrm{in}}=6$ in shells $b=\\{2000,3000\\}$
$\mathrm{s}/\mathrm{mm}^{2}$.

Table 2:  ae of dMRI intensity in eight subjects with data derived from
various models. Models use single-shell data with angular dimension size
$q_{\mathrm{in}}$ as input, and produce inferred data with angular dimension
size $90-q_{\mathrm{in}}$ from the same shell. b-values are quoted in units of
$\mathrm{s/mm^{2}}$.

$q_{\mathrm{in}}=6$ $q_{\mathrm{in}}=10$ $q_{\mathrm{in}}=20$ $b=1000$
$b=3000$ $b=1000$ $b=3000$ $b=1000$ $b=3000$ SH Interpolation $84.85\pm 5.47$
$106.40\pm 4.29$ $47.25\pm 2.85$ $46.14\pm 2.02$ $45.07\pm 3.10$ $42.27\pm
1.86$ RCNN $74.01\pm 4.03$ $\bm{52.11\pm 1.97}$ $56.01\pm 3.50$ $38.68\pm
1.45$ $53.89\pm 3.92$ $34.31\pm 1.39$ PCCNN $74.55\pm 3.69$ $57.99\pm 2.60$
$48.41\pm 2.98$ $34.99\pm 1.63$ $44.03\pm 3.09$ $24.86\pm 1.23$ PCCNN-Bv
$\bm{73.55\pm 3.63}$ $56.71\pm 2.56$ $48.40\pm 2.89$ $34.18\pm 1.65$ $45.32\pm
3.13$ $24.77\pm 1.20$ PCCNN-Sp $75.01\pm 3.64$ $57.41\pm 2.57$ $47.79\pm 2.99$
$34.16\pm 1.49$ $\bm{41.88\pm 2.76}$ $23.88\pm 0.98$ PCCNN-Bv-Sp $74.87\pm
3.51$ $56.78\pm 2.48$ $\bm{46.92\pm 2.53}$ $\bm{33.53\pm 1.47}$ $42.52\pm
2.57$ $\bm{23.70\pm 0.98}$

####  3.1.2 Fixel Analysis

fba is an analysis method used to estimate the intra-voxel fibre populations.
To generate fixel data [8], response functions were first estimated in an
unsupervised manner [7], and Single-Shell 3-Tissue (SS3T) constrained
spherical deconvolution (CSD) [6] was performed to obtain white matter (WM)
FODs. WM FODs were then normalised [9], and finally segmented into fixels.
This workflow used the MRtrix3 [29] package. FOD data is stored as a finite SH
expansion, and as such, was compared using the angular correlation coefficient
(ACC) [1], given by Equation (4). To compare FBA data, the AE in the apparent
fibre density (AFD) within each voxel was calculated by taking the absolute
difference between the magnitudes of each inferred fibre population with
respect to the ground truth values.

Table 3 summarises the performance in FBA in the low resolution baseline and
model based reconstructions compared to the high resolution ground truth.
Here, all PCCNN models performed better than the low resolution baseline in
all three sampling schemes in FOD ACC, and had lower AFD AE in the two lowest
sampling schemes. The PCCNN models performed competitively against the RCNN.
However, in the smallest sub-sampling scheme $q_{\mathrm{in}}=6$, or
equivalently $6.6\%$ of the original number of voxels, the RCNN performed
slightly better. Whilst the low resolution data had lower AFD AE at
$q_{\mathrm{in}}=20$, the PCCNN models had, in the worse case, a relative
increased mean error of approximately $6\%$, compared to $35\%$ increase in
the RCNN model. Additionally, the PCCNN models had greater ACC compared to the
low resolution baseline across all sub-sampling schemes, suggesting that their
use in probabilistic tractography [26] and connectomics [3] would be
beneficial.

Table 3:  Comparison of metrics in fixel-based analysis for eight subjects
with metrics derived from both input data and inferred data using various
models. Models use single-shell data ($b=1000\mathrm{s/mm^{2}}$) with angular
dimension size $q_{\mathrm{in}}$ as input, and produce
$b=1000\mathrm{s/mm^{2}}$ inferred data with angular dimension size
$90-q_{\mathrm{in}}$. _Lowres_ denotes metrics derived from single-shell input
only. fod angular correlation coefficient (ACC) is defined by Equation (4),
whilst apparent fibre density (AFD) absolute error (AE) is calculated using
the magnitudes of all fixels within a voxel. $\uparrow$ denotes that a higher
value is better.

$q_{\mathrm{in}}=6$ $q_{\mathrm{in}}=10$ $q_{\mathrm{in}}=20$ FOD ACC
$\uparrow$ AFD AE $\downarrow$ FOD ACC $\uparrow$ AFD AE $\downarrow$ FOD ACC
$\uparrow$ AFD AE $\downarrow$ Lowres $0.699\pm 0.007$ $0.133\pm 0.011$
$0.793\pm 0.009$ $0.085\pm 0.006$ $0.850\pm 0.007$ $\bm{0.048\pm 0.003}$ SH
Interpolation $0.702\pm 0.010$ $0.180\pm 0.014$ $0.792\pm 0.008$ $0.090\pm
0.005$ $0.833\pm 0.006$ $0.078\pm 0.004$ RCNN $\bm{0.708\pm 0.005}$ $0.081\pm
0.006$ $0.784\pm 0.007$ $0.071\pm 0.005$ $0.823\pm 0.005$ $0.065\pm 0.004$
PCCNN $0.705\pm 0.007$ $0.082\pm 0.006$ $\bm{0.806\pm 0.008}$ $0.063\pm 0.004$
$\bm{0.856\pm 0.006}$ $0.049\pm 0.003$ PCCNN-Bv $0.708\pm 0.006$ $\bm{0.079\pm
0.005}$ $0.802\pm 0.006$ $0.063\pm 0.004$ $0.853\pm 0.006$ $0.051\pm 0.003$
PCCNN-Sp $0.699\pm 0.007$ $0.082\pm 0.005$ $0.803\pm 0.008$ $\bm{0.062\pm
0.004}$ $0.855\pm 0.006$ $0.048\pm 0.003$ PCCNN-Bv-Sp $0.704\pm 0.007$
$0.081\pm 0.005$ $0.802\pm 0.007$ $0.062\pm 0.004$ $0.852\pm 0.006$ $0.049\pm
0.003$

###  3.2 Multi-shell dMRI

To evaluate the performance of multi-shell inference, we used a varying
$q_{\mathrm{in}}$ number of $b=1000\mathrm{s/mm^{2}}$ volumes as input, and
inferred the remaining known volumes from all three shells as output. The
output volumes set contained the output volumes from single-shell inference,
in addition to the other two shells. The $q_{\mathrm{in}}=6$, $10$, and $20$
sub-sampling schemes correspond to an under-sampling rate of $2.2\%$, $3.7\%$,
and $7.4\%$ respectively. Outlined in the sections below are the performances
of the models evaluated within FBA and NODDI metrics.

####  3.2.1 Fixel Analysis

Fixel data were generated in the same manner as outlined in Section 3.1.2,
except where the Multi-Shell Multi-Tissue (MSMT) algorithm [16] was used in
place of SS3T. Table 4 outlines the performance within multi-shell inference.
In contrast to single-shell inference, here FOD ACC and AFD AE for PCCNN
models were lower than the baseline in all sub-sampling schemes. The minimum
difference in AFD AE, as compared to baseline, was $13.8\%$ at
$q_{\mathrm{in}}=20$, whilst the maximum difference was $43.04\%$ at
$q_{\mathrm{in}}=6$. The FOD-Net model had the best performance overall in
this experiment, obtaining the best error rate in three out of the six
metrics. Despite having approximately 50-fold less parameters, the PCCNN
models still perform competitively, and boasts the highest FOD ACC in
$q_{\mathrm{in}}=\\{10,20\\}$.

Table 4:  Comparison of metrics in fixel-based analysis for eight subjects
with metrics derived from both input data and inferred data using various
models. Models use single-shell data ($b=1000\mathrm{s/mm^{2}}$) with angular
dimension size $q_{\mathrm{in}}$ as input, and produce $90-q_{\mathrm{in}}$
$b=1000\mathrm{s/mm^{2}}$, $90$ $b=2000\mathrm{s/mm^{2}}$, and $90$
$b=3000\mathrm{s/mm^{2}}$ inferred volumes. _Lowres_ denotes metrics derived
from single-shell input only. fod angular correlation coefficient (ACC) is
defined by Equation (4), whilst apparent fibre density (AFD) absolute error
(AE) is calculated using the magnitudes of all fixels within a voxel.
$\uparrow$ denotes that a higher value is better.

$q_{\mathrm{in}}=6$ $q_{\mathrm{in}}=10$ $q_{\mathrm{in}}=20$ FOD ACC
$\uparrow$ AFD AE $\downarrow$ FOD ACC $\uparrow$ AFD AE $\downarrow$ FOD ACC
$\uparrow$ AFD AE $\downarrow$ Lowres $0.653\pm 0.008$ $0.157\pm 0.014$
$0.724\pm 0.008$ $0.119\pm 0.012$ $0.757\pm 0.010$ $0.086\pm 0.011$ FOD-Net
$\bm{0.743\pm 0.008}$ $0.087\pm 0.005$ $0.767\pm 0.006$ $\bm{0.072\pm 0.004}$
$0.776\pm 0.008$ $\bm{0.062\pm 0.004}$ RCNN $0.685\pm 0.010$ $\bm{0.087\pm
0.006}$ $0.749\pm 0.009$ $0.080\pm 0.006$ $0.765\pm 0.010$ $0.079\pm 0.006$
PCCNN $0.658\pm 0.009$ $0.090\pm 0.005$ $0.753\pm 0.008$ $0.077\pm 0.005$
$0.792\pm 0.009$ $0.068\pm 0.006$ PCCNN-Bv $0.681\pm 0.010$ $0.091\pm 0.005$
$\bm{0.770\pm 0.011}$ $0.080\pm 0.006$ $\bm{0.807\pm 0.011}$ $0.074\pm 0.006$
PCCNN-Sp $0.670\pm 0.013$ $0.092\pm 0.006$ $0.761\pm 0.013$ $0.075\pm 0.006$
$0.791\pm 0.014$ $0.070\pm 0.007$ PCCNN-Bv-Sp $0.675\pm 0.013$ $0.089\pm
0.005$ $0.766\pm 0.014$ $0.075\pm 0.006$ $0.798\pm 0.015$ $0.067\pm 0.006$

####  3.2.2 NODDI Analysis

NODDI metrics were generated from input and inferred output volumes using the
cuDIMOT [15] package. The full angular resolution derived NODDI metrics were
used as ground truth. Data for $q_{\mathrm{in}}=\\{6,20\\}$ are presented in
Table 5, whilst $q_{\mathrm{in}}=10$ can be found in Table 7 within the
Appendix. Input, or low resolution, data were not included in this analysis as
multi-shell data is required to reconstruct NODDI parameters. The parameters
$f_{\mathrm{intra}}$ and $f_{\mathrm{iso}}$ refer to the estimated volume
fraction for intra-cellular and cerebrospinal fluid respectively, whilst
orientation dispersion index (OD) is an estimation of the dispersion of white
matter fibre bundles.

Table 5:  ae of orientation dispersion index (OD), $f_{\mathrm{intra}}$, and
$f_{\mathrm{iso}}$ derived from neurite orientation dispersion and density
imaging using input data and inferred data from various models. Models use
single-shell data ($b=1000\mathrm{s/mm^{2}}$) with angular dimension size
$q_{\mathrm{in}}$ as input, and produce $90-q_{\mathrm{in}}$
$b=1000\mathrm{s/mm^{2}}$, $90$ $b=2000\mathrm{s/mm^{2}}$, and $90$
$b=3000\mathrm{s/mm^{2}}$ inferred volumes.

$q_{\mathrm{in}}=6$ $q_{\mathrm{in}}=20$ OD $f_{\mathrm{iso}}$
$f_{\mathrm{intra}}$ OD $f_{\mathrm{iso}}$ $f_{\mathrm{intra}}$ SR-q-DL
$0.095\pm 0.004$ $0.162\pm 0.015$ $0.132\pm 0.010$ $0.089\pm 0.003$ $0.162\pm
0.015$ $0.129\pm 0.010$ RCNN $\bm{0.074\pm 0.007}$ $0.030\pm 0.003$ $0.052\pm
0.005$ $0.045\pm 0.005$ $0.025\pm 0.003$ $0.048\pm 0.006$ PCCNN $0.084\pm
0.006$ $0.030\pm 0.003$ $0.054\pm 0.006$ $0.039\pm 0.006$ $0.024\pm 0.003$
$\bm{0.045\pm 0.004}$ PCCNN-Bv $0.075\pm 0.006$ $0.026\pm 0.003$ $0.050\pm
0.006$ $\bm{0.036\pm 0.004}$ $0.023\pm 0.003$ $0.045\pm 0.006$ PCCNN-Sp
$0.080\pm 0.008$ $0.026\pm 0.004$ $\bm{0.050\pm 0.010}$ $0.038\pm 0.006$
$0.025\pm 0.004$ $0.051\pm 0.012$ PCCNN-Bv-Sp $0.083\pm 0.008$ $\bm{0.025\pm
0.003}$ $0.050\pm 0.012$ $0.040\pm 0.007$ $\bm{0.023\pm 0.004}$ $0.047\pm
0.012$

In this experiment, the PCCNN models perform competitively, yielding the
lowest error in $f_{\mathrm{iso}}$ and $f_{\mathrm{intra}}$ across all three
sub-sampling rates. The relative error in OD across different models mirrors
that of the FOD ACC found in Table 4, suggesting that the trained models are
able to infer data that generalise across different downstream analyses. The
worst performing model across all sub-sampling schemes was the SR-q-DL model,
as is demonstrated qualitatively by Figure 3. Whilst this model could benefit
from the more constrained task of inferring NODDI data directly, the lack of
geometric prior information, such as b-vector coordinates, suggests that these
additions within the PCCNN models were important to its relatively high
performance in this task.

(a) High Resolution

(b) SR-q-DL

(c) RCNN

(d) PCCNN-Bv

Figure 3: Axial slice of orientation dispersion index (OD) within one subject
across different models. Models use single-shell data
($b=1000\mathrm{s/mm^{2}}$) with angular dimension size $q_{\mathrm{in}}=10$
as input, and produce $80$ $b=1000\mathrm{s/mm^{2}}$, $90$
$b=2000\mathrm{s/mm^{2}}$, and $90$ $b=3000\mathrm{s/mm^{2}}$ inferred
volumes.

##  4 Conclusion

The results of the experiments show that the proposed PCConv operation can be
used to build a flexible low parameter network to effectively infer high
angular resolution raw dMRI data. It does this through appropriate
incorporation of task specific geometric properties, as well as suitable
parameter sharing via a hypernetwork. We show that inference of raw dMRI data
can be used in downstream analysis with moderately high fidelity, and
therefore demonstrates the generalisability of the model.

All three of the PCCNN models with additional modifications (PCCNN-Bv, PCCNN-
Sp, PCCNN-Bv-Sp) performed greater or equal to the standard PCCNN in
approximately 47% of the experiments presented in this work. Additionally, at
least one of the modified PCCNN’s outperformed the standard PCCNN in
approximately 87% of the experiments. Overall, this suggests that the
inclusion of additional domain specific context through the coordinate
embedding was beneficial to the models performance.

Modern deep learning frameworks rely on highly optimised subroutines to
perform standard discrete convolutions [4]. Subsequently, despite having a
lower number of parameters, the PCCNNs inference and train times were longer
than the other methods presented in this work. This is because the PCConv uses
a bespoke implementation that does not benefit from the aforementioned
subroutines. This highlights the need for future research into more efficient
implementations of the PCConv operation.

In summary, our work further explores the integration of geometric priors into
the distinct geometry of dMRI data, and in doing so provides a flexible
framework to incorporate arbitrary geometries in different domains and problem
formulations.

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##  Appendix A Appendix

###  A.1 Model Hyperparameters

Figure 4:  PCCNN model architecture. Angular kernel size is
$k_{q}=q_{\mathrm{in}}=20$. $C_{1}$ and $C_{3}$ denote the number of output
channels for a $(1\times 1\times 1\times 20)$ and $(3\times 3\times 3\times
20)$ PCConv kernel respectively. $W_{h}$ denotes the size of the hidden units
within each PCConv’s hypernetwork. $L$ is a hyperparameter of the coordinate
embedding $\mathbf{e}_{s}$ as shown in Equation (3).

###  A.2 FOD-Net Data and Training

In this work we modify the training scheme used within Zeng et al. [35] to
more closely match the PCCNN training methodology. Specifically, each training
example input was a low angular resolution FOD volume generated from single-
shell dMRI data with b-value $b_{\mathrm{in}}$ and angular dimension
$q_{\mathrm{in}}$ determined via

$q_{\mathrm{in}}\sim
U(q_{\mathrm{sample}}),q_{\mathrm{sample}}=\\{6,10,20\\},b_{\mathrm{in}}\sim
U(b_{\mathrm{sample}}),b_{\mathrm{sample}}=\\{1000,2000,3000\\}.$

Training parameters such as number of iterations, batch size, learning rate,
hardware used, and optimisation method remain consistent with those detailed
in Section 2.7. Model training took approximately 6 hours to complete.

###  A.3 SR-q-DL Data and Training

The data sampling used to train the SR-q-DL for this work differs from that
outlined in Ye et al. [32]. As in Section A.2, this was done to match the
PCCNN training scheme. Here, each training example input was a low angular
resolution single-shell dMRI dataset with b-value $b_{\mathrm{in}}$ and
angular dimension $q_{\mathrm{in}}$ determined via

$q_{\mathrm{in}}\sim
U(q_{\mathrm{sample}}),q_{\mathrm{sample}}=\\{6,7,...,19,20\\},b_{\mathrm{in}}\sim
U(b_{\mathrm{sample}}),b_{\mathrm{sample}}=\\{1000,2000,3000\\}.$

Training parameters such as number of iterations, batch size, learning rate,
hardware used, and optimisation method remain consistent with those detailed
in Section 2.7. Model training took approximately 30 minutes to complete.

###  A.4 Additional Single-shell dMRI Analysis

Table 6:  ae of dMRI intensity in eight subjects with data derived from
various models. Models use single-shell $b=2000\mathrm{s/mm^{2}}$ data with
angular dimension size $q_{\mathrm{in}}$ as input, and produce inferred
$b=2000\mathrm{s/mm^{2}}$ data with angular dimension size
$90-q_{\mathrm{in}}$.

$q_{\mathrm{in}}=6$ $q_{\mathrm{in}}=10$ $q_{\mathrm{in}}=20$ SH Interpolation
$103.53\pm 4.64$ $45.63\pm 2.21$ $43.67\pm 2.25$ RCNN $\bm{61.60\pm 2.43}$
$42.79\pm 1.75$ $39.14\pm 1.93$ PCCNN $65.72\pm 2.83$ $37.93\pm 1.93$
$30.21\pm 1.81$ PCCNN-Bv $64.63\pm 2.74$ $36.98\pm 1.94$ $30.84\pm 1.93$
PCCNN-Sp $65.74\pm 2.78$ $36.82\pm 1.77$ $28.39\pm 1.49$ PCCNN-Bv-Sp $65.33\pm
2.71$ $\bm{36.18\pm 1.66}$ $\bm{28.32\pm 1.48}$

###  A.5 Angular Correlation Coefficient

The angular correlation coefficient (ACC) is given by

$\mathrm{ACC}(\alpha,\beta)=\frac{\sum^{8}_{l=0}\sum^{l}_{m=-l}\alpha_{l,m}\beta_{l,m}^{*}}{\sqrt{\sum_{l=0}^{8}\sum_{m=-l}^{l}\alpha_{l,m}^{2}}\sqrt{\sum_{l=0}^{8}\sum_{m=-l}^{l}\beta_{l,m}^{2}}},$
(4)

where $\alpha_{l,m}$ and $\beta_{l,m}$ denote the SH coefficients for two FODs
with degree $l$ and order $m$.

###  A.6 Additional NODDI Analysis

Table 7:  ae of orientation dispersion index (OD), $f_{\mathrm{intra}}$, and
$f_{\mathrm{iso}}$ derived from neurite orientation dispersion and density
imaging using both input data and inferred data from various models. Models
use single-shell data ($b=1000\mathrm{s/mm^{2}}$) with angular dimension size
$q_{\mathrm{in}}=10$ as input, and produce $80$ $b=1000\mathrm{s/mm^{2}}$,
$90$ $b=2000\mathrm{s/mm^{2}}$, and $90$ $b=3000\mathrm{s/mm^{2}}$ inferred
volumes.

OD $f_{\mathrm{iso}}$ $f_{\mathrm{intra}}$ SR-q-DL $0.089\pm 0.004$ $0.162\pm
0.015$ $0.128\pm 0.010$ RCNN $0.054\pm 0.006$ $0.026\pm 0.003$ $0.048\pm
0.004$ PCCNN $0.048\pm 0.006$ $0.027\pm 0.003$ $0.049\pm 0.006$ PCCNN-Bv
$\bm{0.044\pm 0.005}$ $0.025\pm 0.003$ $\bm{0.048\pm 0.006}$ PCCNN-Sp
$0.048\pm 0.007$ $0.025\pm 0.004$ $0.048\pm 0.010$ PCCNN-Bv-Sp $0.050\pm
0.008$ $\bm{0.024\pm 0.004}$ $0.049\pm 0.012$

  *[CNN]: convolutional neural network
  *[dMRI]: diffusion magnetic resonance imaging
  *[CNNs]: convolutional neural network
  *[PCConv]: parametric continuous convolution
  *[]:
  *[PCCNN]: parametric continuous
  *[FBA]: fixel-based analysis
  *[NODDI]: neurite orientation dispersion and density imaging
  *[RCNN]: recurrent
  *[FOD]: fibre orientation distribution
  *[FODs]: fibre orientation distribution
  *[MLP]: multi-layer perceptron
  *[ReLU]: rectified linear unit
  *[HCP]: WU-Minn Human Connectome Project
  *[SH]: spherical harmonic
  *[AE]: absolute error
  *[SS3T]: Single-Shell 3-Tissue
  *[CSD]: constrained spherical deconvolution
  *[WM]: white matter
  *[ACC]: angular correlation coefficient
  *[AFD]: apparent fibre density
  *[MSMT]: Multi-Shell Multi-Tissue
  *[OD]: orientation dispersion index
  *[PCCNNs]: parametric continuous