11institutetext: Department of Biomedical Engineering, King’s College London.
11email: {mohamed.suliman, logan.williams, abdulah.fawaz,
<EMAIL_ADDRESS>
# A Deep-Discrete Learning Framework for Spherical Surface Registration

Mohamed A. Suliman  Logan Z. J. Williams  Abdulah Fawaz  Emma C. Robinson

###### Abstract

Cortical surface registration is a fundamental tool for neuroimaging analysis
that has been shown to improve the alignment of functional regions relative to
volumetric approaches. Classically, image registration is performed by
optimizing a complex objective similarity function, leading to long run times.
This contributes to a convention for aligning all data to a global average
reference frame that poorly reflects the underlying cortical heterogeneity. In
this paper, we propose a novel unsupervised learning-based framework that
converts registration to a multi-label classification problem, where each
point in a low-resolution control grid deforms to one of fixed, finite number
of endpoints. This is learned using a spherical geometric deep learning
architecture, in an end-to-end unsupervised way, with regularization imposed
using a deep Conditional Random Field (CRF). Experiments show that our
proposed framework performs competitively, in terms of similarity and areal
distortion, relative to the most popular classical surface registration
algorithms and generates smoother deformations than other learning-based
surface registration methods, even in subjects with atypical cortical
morphology.

###### Keywords:

Deep learning unsupervised learning cortical surface registration conditional
random fields

##  1 Introduction

The human cerebral cortex is highly convoluted structure, with complex
patterns of functional organisation that vary considerably across individuals
[1, 13]. Image registration is an important tool that supports comparison of
brain images, through mapping of all data to a global average space. However,
the degree of variation of cortical morphology and topography across
individuals generates considerable uncertainty with regards to the optimal
mapping between brains.

Recently, cortical surface registration algorithms [11, 26, 23, 22] have led
to improvements in the precision with which it is possible to compare features
on the cortical surface through learning mappings that regularize deformations
with respect to displacements along with the cortical sheet. Increasingly,
these methods support multimodal registration [23, 22, 27], allowing improved
evaluation of cortical functional areas [5, 13]. However, increasing evidence
suggests that there is a limit to these improvements since cortical
topographies vary in ways that break the diffeomorphic assumptions of
classical registration algorithms [13].

To this end, learning-based registration algorithms [6, 16] present an
attractive framework for exploring these problems, on the grounds that they
train fast, unsupervised deformation frameworks that can learn to adapt to
sub-populations in the data [6]. Previous learning-based registration
frameworks have predominately been generated for 2D or 3D Euclidean domains
such as brain volumes [3, 6, 7, 8], lung CT [15, 16, 12], and histology [4,
24, 20]. However, increasing efforts have been made to adapt convolutional
networks to non-Euclidean domains [19, 21, 28], resulting in the development
of tools for learning-based registration of surfaces and point clouds [2, 25,
28]. Most notably, the recent S3Reg framework [27] proposes the first
learning-based registration framework for cortical surfaces, leveraging the
Spherical U-Net algorithm [28] to learn multi-resolution hexagonal filters
across regular subdivisions of an icosphere.

One limitation of the Spherical U-Net is that its hexagonal convolutions do
not generate a rotationally equivariant solution since the lack of a global
coordinate system on a sphere causes filter orientation to flip at the poles.
When learning registrations for S3Reg, this generates distortions that must be
corrected through averaging warps learned across multiple rotated orientations
of the sphere. Recent work showed that contrary to Spherical U-Net, MoNet
convolutions (learned from a mixture of Gaussian kernels) could indeed be
trained to be rotationally equivariant [9]. Therefore, we develop a new
framework for spherical cortical registration based on MoNet, which also takes
inspiration from deep-discrete registration frameworks [15, 16] designed to
learn larger deformations than deep regression frameworks. We hypothesize that
this could improve the generalization of our framework to brains with atypical
topographies.

Contributions In this paper, we propose the first deep-discrete framework for
cortical surface registration. We validate on the alignment of cortical folds
and compare the proposed network against state-of-the-art classical [11, 26,
23, 22] and learning-based [27] methods. Specific focus is placed on
evaluating the smoothness of the generated warps and the capacity of the
network to generalize to atypical cortical morphologies.

##  2 Method

The proposed method combines ideas from the discrete frameworks of [16, 23,
22] to propose a Deep-Discrete spherical Registration (DDR) network for
alignment of cortical surface features. The full DDR architecture consists of
three networks that compose global rigid rotations and non-linear warps
(learned in a coarse-to-fine fashion). The objective is to learn a spatial
transformation $\hbox{\boldmath$\Phi$}:M\to F$, that aligns cortical features
on a moving mesh ($M$) to those of a fixed mesh ($F$) by optimizing a
dissimilarity metric $\mathcal{L}$ of the form:

$\hat{\hbox{\boldmath$\theta$}}=\operatorname*{arg\,min}_{\hbox{\boldmath$\theta$}}\mathcal{L}\left(\hbox{\boldmath$\Phi$}_{\hbox{\boldmath$\theta$}};F,M\right)+\Sigma\left(\hbox{\boldmath$\Phi$}_{\hbox{\boldmath$\theta$}}\right),$
(1)

where $\theta$ are transformation parameters that parametrize $\Phi$, while
$\Sigma\left(\cdot\right)$ is a regularization function that imposes
smoothness on $\Phi$.

Let $f\left(\cdot\right)$ represents our deep learning network architecture,
with $\eta$ being a set of learnable parameters; then, our deep learning image
registration problem may be represented as

$\hbox{\boldmath$\theta$}=f_{\hbox{\boldmath$\eta$}}\left(F,M\right).$ (2)

In all cases, data is presented to the network as concatenated cortical metric
maps of $F$ and $M$, defined on a sphere $S^{2}$ that is parametrized by a
sixth-order icosphere (that has 40962 vertices). The base architecture of each
network is a MoNet U-Net.

###  2.1 Rotation Architecture

Registration is learned in 3 stages, with the first network learning global
rotations. To obtain the rotation matrix between $M$ and $F$, we use a MoNet
U-Net to estimate the 12 parameters of the rotation matrix. With conventional
rotation estimation methods, such as quaternions and Euler angles, being shown
to be discontinuous in the real Euclidean spaces of four or fewer dimensions
[30], we apply the continuous 6D representation formula of the 3D rotations
estimation proposed in [30]. The network optimization is driven using an
unsupervised mean-squared-error loss. The rotationally aligned subjects are
then passed to the next registration stage.

###  2.2 Deep-Discrete Networks

Deformable registration is subsequently learned using two deep-discrete
networks, which learn optimal displacements as a classification problem,
regularized by a deep CRF. Let $\\{{\bf c}_{i}\\}_{i=1}^{N_{c}}\in G\subset
S^{2}$ be a set of $N_{c}$ control points, on the input image sphere,
generated from a low-resolution icosphere (bottom left of Fig. 1), and let
$\\{{\bf l}_{i}\\}_{i=1}^{N_{l}}\in S^{2}$ represent a set of label points,
defined around each control point ${\bf c}_{i}$, from a high-order icosphere
(bottom left of Fig. 1). Then, the objective of DDR is to learn to predict the
optimal label (displacement) for each control point to ensure features of the
fixed and moving mesh optimally overlap. Importantly unlike classical discrete
frameworks [23, 22], for which run-time is linked to the label dimensionality,
DDR is far less constrained by the extent of the label space.

Figure 1: DDR network architecture. The dimensions in red represent the input
and the output dimensions at different levels on the network, while blue boxes
represent features in the spherical space. $N_{d}$ is the number of data
vertices, $N_{c}$ is the total number of control points, $N_{l}$ is the number
of labels around each control point, and $f_{in}$ is the number of features in
each subject.

Fig. 1 shows an overview of our proposed approach. The first part of the
network takes $M$ and $F$ concatenated to a single input, then passes this
through a U-Net followed by a feedforward spherical network (FSN) to output
$Q=\text{Softmax}\left(U\right)\in\mathbb{R}^{N_{c}\times N_{l}}$: softmax
probabilities for each label. The second part of the network is a CFR-RNN
network, which imposes smoothness by encouraging neighboring control points to
take similar labels.

####  2.2.1 Classifier Architecture

The surface U-Net implements a symmetric encoder-decoder architecture with six
resolution stages $i$, each defined at a different icosphere order. The number
of icosphere vertices $V_{i}$ at each stage is related to the previous one
through $V_{i+1}=\left(V_{i}+6\right)/4$. MoNet convolutional filters [19]
with a kernel size of 3, spherical polar pseudo-coordinates, and mean
aggregation operators are applied to learn $C_{i}$ features at each level. In
our network, we set $C_{i+1}=2C_{i}$ for all $i$. Each convolution is followed
by a LeakyReLU activation with parameter 0.2. Downsampling and upsampling are
performed using the hexagonal mean pooling and transpose convolution
operations proposed by [28].

The learned features at the final stage of the decoder, which is of
dimensionality of $N_{d}\times C_{1}$, are then passed to a feedforward
spherical network that decompresses channel dimensions over $r$ convolutional
layers, each learning $\bar{C}_{i}$ features, to return an output of
resolution $N_{d}\times N_{l}$ (a label prediction for each location in the
input mesh); then regularized by downsampling label predictions $U$ to the
desired control point resolution of $N_{c}\times N_{l}$. Once optimal labels
are determined from the softmax operation $Q$, we deform our control grid
accordingly using the spherical coordinates of the labels, i.e., ${\bf
l}_{i}$.

####  2.2.2 CRF-RNN Network:

On its own, the classifier is of limited use since cortical registration is an
ill-posed problem with many possible solutions. Therefore, smoothness is
imposed through a CRF-RNN, adapted from [29], that optimizes the following CRF
energy cost function:

$E=\sum_{i}Q_{\left({\bf c}_{i},{\bf l}_{i}\right)}+\sum_{i\neq
j}\varphi\left({\bf l}_{{\bf c}_{i}},{\bf l}_{{\bf c}_{j}}\right).$ (3)

Here, $Q_{\left({\bf c}_{i},{\bf l}_{i}\right)}$ represents the cost
(likelihood) of deforming ${\bf c}_{i}$ to ${\bf l}_{i}$ while

$\varphi\left({\bf l}_{{\bf c}_{i}},{\bf l}_{{\bf c}_{j}}\right)=\mu\left({\bf
l}_{i},{\bf l}_{j}\right)K_{G}\left({\bf l}_{{\bf c}_{i}},{\bf l}_{{\bf
c}_{j}}\right),$ (4)

measures the pairwise cost of deforming ${\bf c}_{i}$ and ${\bf c}_{j}$ to the
label points ${\bf l}_{i}$ and ${\bf l}_{j}$, respectively. Moreover, $\mu$ is
a learnable label compatibility function that captures correspondences between
different pairs of label points (i.e., penalizing the assignment of different
labels to different control points with similar properties), and $K_{G}$ is a
Gaussian kernel of the form [18, 29]

$K_{G}\left({\bf l}_{{\bf c}_{i}},{\bf l}_{{\bf
c}_{j}}\right)=\omega\left({\bf c}_{i},{\bf
c}_{j}\right)\exp\left(-\frac{1}{2\gamma^{2}}\left(\left({\bf l}_{{\bf
c}_{i}}-{\bf l}_{{\bf c}_{j}}\right)\hbox{\boldmath$\Lambda$}\left({\bf
l}_{{\bf c}_{i}}-{\bf l}_{{\bf c}_{j}}\right)\right)\right),$ (5)

with ${\bf l}_{{\bf c}_{i}}$ being the spatial location of the deformed ${\bf
c}_{i}$, $\omega$ being learnable filter weights, $\Lambda$ being symmetric,
positive-definite, kernel characterization matrix, and $\gamma$ being a kernel
parameter. Energy (3) is optimized using the Recurrent Neural Network (RNN)
implementation of [29], which is based on learning multiple iterations of a
mean-field CRF.

####  2.2.3 Final Deformed Grid and the Loss Function:

The updated deformed control grid from the CRF-RNN network is upsampled to the
input data icosphere order using barycentric interpolation. Then, the moving
image features are resampled to the deformed data grid and compared to those
of the fixed image. Optimization is driven using an unsupervised loss function
(2) that is a sum of the mean-squared error and the cross-correlation. To
allow for more user control over the balance between accurate alignment and
smooth deformation, we add a diffusion regularization penalty on the spatial
gradients of the $\Phi$ in the form
$\lambda\left(\left|\bigtriangledown\hbox{\boldmath$\Phi$}_{{\bf
x}}\right|+\left|\bigtriangledown\hbox{\boldmath$\Phi$}_{{\bf
y}}\right|+\left|\bigtriangledown\hbox{\boldmath$\Phi$}_{{\bf
z}}\right|\right)$. The hexagonal filter is applied to compute
$\bigtriangledown$.

####  2.2.4 Coarse to Fine Registration:

As with surface registration methods, we perform multi-stage registration in
the form of coarse-to-fine using two DDR networks. The first network is
trained to align image features using a coarse grid of control points. The
deformed grid is then upsampled to the higher level using the hexagonal
upsampling method and passed to the second network that uses a higher
resolution control grid.

##  3 Experiments

To validate our proposed framework, we conduct a series of experiments that
register individual cortical surfaces to an atlas. In each case, the proposed
framework was compared against classical surface registration methods:
FreeSurfer [11], Spherical Demons [26], and Multimodal Surface Matching (MSM)
[23, 22], as well as the learning-based method S3Reg [27].

####  3.0.1 Datasets and Preprocessing:

Experiments were performed using cortical surface data collected as a part of
the adult Human Connectome Project (HCP) [14] resampled to a regular icosphere
of order 6 using barycentric interpolation. A total number of 1110 brain MRI
scans were used in the experiments with 888-111-111 split for train-
validation-test and batch sizes of 1 for all. For simplicity, registration was
driven solely using sulcal depth as a feature, i.e., $C_{in}=2$; however, we
point out that the network can be straightforwardly adapted to accept multi-
channel features.

####  3.0.2 Implementation and Training:

MoNet filters are implemented using the PyTorch Geometric library [10], while
optimization is performed using ADAM [17] with a learning rate of $10^{-3}$.
During the coarse registration stage, we use $N_{c}=162$ vertices formed from
an icosphere of order 2 with a 26.7 mm distance between adjacent points.
$N_{l}=600$ labels are generated from an icosphere of order 5 with 3.7 mm
neighboring distances. Finally, we set $C_{1}=32,\gamma=0.7,\lambda=1.5,r=1,$
and $\bar{C}_{1}=600$. For the fine registration, a total of $N_{c}=2542$
points, formed using the vertices of an icosphere of order 4 with 6.9 mm
neighboring distances, are applied. $N_{l}=1000$ label vertices are generated
from an icosphere of order 8 with 0.4 mm neighboring distances. Finally, we
let $C_{1}=2,\gamma=0.2,\lambda=0.6,r=5$, and set
$[\bar{C}_{i}]_{i=1}^{5}=[8,16,64,128,1000]$. All networks are trained on 100
epochs and we report the network performance with the best validation score.

####  3.0.3 Baseline Registration Methods:

We validate against the official implementations of Spherical Demons
(SD)111https://github.com/ThomasYeoLab/CBIG, MSM Pair 222Available through
FSLv6.0, MSM Strain333https://github.com/ecr05/MSM$\\_$HOCR, and the Spherical
U-Net multi-stage registration algorithm
(S3Reg)444https://github.com/zhaofenqiang/SphericalUNetPackage. Freesurfer is
run as part of the HCP Pipeline, so we obtain its result from there.

To achieve fair comparison across all methods, we ran parameter optimization
and report performance across all runs. For SD, we run an additional 11
experiments by setting the number of smooth iterations to take
$\left[1,5,10\right]$ and the smoothing variance $\sigma_{x}$ to be
$\left[1,2,6,10\right]$. For MSM Pair, we run 22 experiments modifying the
regularization penalty $\lambda\in\left[0.0001,0.2\right]$, while for MSM
Strain, we run 22 experiments with $\lambda\in\left[0.0001,0.9\right]$. For
S3Reg, we ran 7 more experiments by varying smoothness regularization at
different registration levels to take the sets $[2,5,6,8]$, $[2,10,12,20]$,
$[2,10,12,14]$, $[2,5,12,16]$, $[2,10,6,8]$, $[2,10,12,8]$, and $[2,5,6,16]$;
in each case training for 100 epochs and reporting the performance of the
network with the best validation score. As FreeSurfer is not tunable by the
user, we only report its results with default parameterization. Note that all
these methods register two surfaces at 4 levels of icosphere subdivisions
(coarse to fine), with S3Reg having an additional spherical transform network
that ensures a diffeomorphic registration.

####  3.0.4 Evaluation Metrics:

All methods are compared in terms of cross-correlation (CC) similarity, areal
strain ($J$), shape distortion $R$, and run time. Here
$J=\lambda_{1}/\lambda_{2}$ and $R=\lambda_{1}/\lambda_{2}$, where
$\lambda_{1}$ and $\lambda_{2}$ represent the eigenvalues of the local
deformation gradient $F_{pqr}$ estimated from the deformation of each
triangular face, defined by vertices: $p,q,r$. $\log_{2}J$ is equivalent to
areal distortion.

##  4 Results

Fig. 2 plots the similarity performances of the different runs of all methods
versus the 95th percentile of the absolute value of $J$. At each similarity
level, DDR returns distortions within the range of the best classical methods
(SD and MSM Strain) and reduced extremes of strain $J$ relative to that of
S3Reg.

This trend is repeated across the full distributions of $|J|$ and $|R|$ for
each comparable run. Fig. 3 plots the full histogram distributions for $J$ and
$R$ across all runs that return a CC value of approximately 0.88. Most of the
areal distortions of SD and DDR at this CC level are around zero. However,
S3Reg, MSM Pair, and Freesurfer all lead to extreme distortions across
subjects, with an average of $150$ vertices at each subject returning $|J|$
above $2$ in S3Reg compared to $0.567$ for DDR.

Figure 2: Similarity performances of all methods vs. the 95th percentile of
the areal distortion at multiple regularization levels across runs.

(a) Strain $J$.

(b) Strain $R$.

Figure 3: Histogram plots comparing methods distortions $J$ and $R$ across all
test subjects for runs with CC $\sim 0.88$.

Table. 1 further reports summary statistics for these histogram distributions
in terms of the mean, max 95th, and 98th percentiles of $|J|$ and the mean CC
across all runs for these parameters. Again, DDR performs comparably to SD and
MSM Strain, whereas S3Reg and MSM Pair provide the worst performance. These
values reflect the best performance of S3Reg (in terms of distortion) across
all runs. In terms of average run time, for a PC with NVIDIA Titan RTX 24GB
GPU and Intel Core i9-9820X 3.30 GHz CPU, DDR has the least GPU and CPU times
across all methods.

Table 1: Distortions measures and average runtime for different methods obtained for CC $\sim 0.88$; top: classical methods, bottom: learning-based methods. Methods |  CC Similarity | Areal Distortion | Shape Distortion | Avg. Time
---|---|---|---|---
Mean | Max | 95% | 98% | Mean | Max | CPU | GPU
| Freesurfer
---
0.75 | 0.34 | 11.73 | 0.82 | 1.00 | 0.63 | 6.77 | 30 min | -
| MSM Pair
---
0.877 | 0.41 | 9.17 | 1.24 | 1.76 | 0.62 | 9.05 | 13 min | -
| MSM Strain
---
0.880 | 0.27 | 1.06 | 0.70 | 0.66 | 0.64 | 1.93 | 1 hour | -
| SD
---
0.875 | 0.18 | 2.00 | 0.50 | 0.65 | 0.24 | 1.98 | 1 min | -
| S3Reg
---
0.875 | 0.266 | 22.22 | 0.82 | 1.16 | 0.51 | 21.65 | 8.8 sec | 8.0 sec
| DDR
---
0.878 | 0.19 | 2.66 | 0.53 | 0.71 | 0.26 | 3.14 | 7.7 sec | 2.3 sec

Fig. 4 presents a qualitative comparison of the alignment quality for 2
subjects across all methods, together with the strain $J$ metrics across the
surface. Results show that DDR, SD, and MSM Strain generate comparable
alignment for reduced distortion relative to S3Reg and MSM Pair that produce
alignments that result in peaks of high distortions.

In Fig. 5, we provide qualitative comparisons of the methods registration
performance with the template on subjects with atypical cortical folding
patterns at CC of $\sim 0.88$ (except for FreeSurfer, which has $0.75$ CC).
Moreover, we also provide the associated strain $J$ and strain $R$ of these
methods. The figure shows clearly that DDR, SD, and MSM Strain all produce
good comparable alignments with reduced distortions, while MSM Pair and S3Reg
provide alignments with peaks of high distortions.

Finally, we plot in Fig. 6 the histograms for $J$ and $R$ across all test
subjects for methods that return a CC value of approximately $0.90$ (i.e., SD
$=0.901$ CC, DDR $=0.895$ CC, and S3Reg $=0.897$ CC). Fig. 6 shows that most
SD and DDR distortions at this CC level are around zero, while S3Reg leads to
excess distortions.

Figure 4: Qualitative comparisons of sulcal depth alignment for 2 subjects.
Results reflect distortion values presented in Fig. 3 and Table. 1. Columns 1
and 3 show registered sulcal depth maps; columns 2 and 4 reflect strain $J$
across the surface. Cyan areas of the distortion maps highlight significant
distortions.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: (a), (c), (d) Qualitative comparisons of methods’ performance on
subjects with atypical cortical folding patterns at CC of $\sim 0.88$. (b),
(d), (f) Associated Strain $J$ (top) and Strain $R$ (bottom).

(a) Strain $J$.

(b) Strain $R$.

Figure 6: Histogram plots comparing distortions across all test subjects for
methods with CC $\sim 0.90$.

##  5 Conclusions

In this work, we propose the first deep-discrete registration (DDR) framework
for cortical surface alignment that aligns two surfaces by deforming a set of
control points on the surface into a finite set of possible deformations. Our
results show that DDR outperforms other deep learning-based cortical surface
registration frameworks (S3Reg) in terms of similarity and distortions
measures and provides a competitive performance to state-of-the-art
conventional surface registration methods. Future work will extend DDR to
multimodal alignment with high dimensional feature sets and address
topographical variation.

## References

  * [1] Amunts, K., Malikovic, A., Mohlberg, H., Schormann, T., Zilles, K.: Brodmann’s areas 17 and 18 brought into stereotaxic space—where and how variable? Neuroimage 11(1), 66–84 (2000)
  * [2] Aoki, Y., Goforth, H., Srivatsan, R.A., Lucey, S.: Pointnetlk: Robust & efficient point cloud registration using pointnet. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 7163–7172 (2019)
  * [3] Balakrishnan, G., Zhao, A., Sabuncu, M.R., Guttag, J., Dalca, A.V.: Voxelmorph: a learning framework for deformable medical image registration. IEEE transactions on medical imaging 38(8), 1788–1800 (2019)
  * [4] Borovec, J., Kybic, J., Arganda-Carreras, I., Sorokin, D.V., Bueno, G., Khvostikov, A.V., Bakas, S., Eric, I., Chang, C., Heldmann, S., et al.: Anhir: automatic non-rigid histological image registration challenge. IEEE transactions on medical imaging 39(10), 3042–3052 (2020)
  * [5] Coalson, T.S., Van Essen, D.C., Glasser, M.F.: The impact of traditional neuroimaging methods on the spatial localization of cortical areas. Proceedings of the National Academy of Sciences 115(27), E6356–E6365 (2018)
  * [6] Dalca, A., Rakic, M., Guttag, J., Sabuncu, M.: Learning conditional deformable templates with convolutional networks. Advances in neural information processing systems 32 (2019)
  * [7] De Vos, B.D., Berendsen, F.F., Viergever, M.A., Sokooti, H., Staring, M., Išgum, I.: A deep learning framework for unsupervised affine and deformable image registration. Medical image analysis 52, 128–143 (2019)
  * [8] Fan, J., Cao, X., Xue, Z., Yap, P.T., Shen, D.: Adversarial similarity network for evaluating image alignment in deep learning based registration. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 739–746. Springer (2018)
  * [9] Fawaz, A., Williams, L.Z., Alansary, A., Bass, C., Gopinath, K., da Silva, M., Dahan, S., Adamson, C., Alexander, B., Thompson, D., et al.: Benchmarking geometric deep learning for cortical segmentation and neurodevelopmental phenotype prediction. bioRxiv (2021)
  * [10] Fey, M., Lenssen, J.E.: Fast graph representation learning with PyTorch Geometric. In: ICLR Workshop on Representation Learning on Graphs and Manifolds (2019)
  * [11] Fischl, B., Sereno, M.I., Tootell, R.B., Dale, A.M.: High-resolution intersubject averaging and a coordinate system for the cortical surface. Human brain mapping 8(4), 272–284 (1999)
  * [12] Fu, Y., Lei, Y., Wang, T., Higgins, K., Bradley, J.D., Curran, W.J., Liu, T., Yang, X.: Lungregnet: An unsupervised deformable image registration method for 4d-ct lung. Medical physics 47(4), 1763–1774 (2020)
  * [13] Glasser, M.F., Coalson, T.S., Robinson, E.C., Hacker, C.D., Harwell, J., Yacoub, E., Ugurbil, K., Andersson, J., Beckmann, C.F., Jenkinson, M., et al.: A multi-modal parcellation of human cerebral cortex. Nature 536(7615), 171–178 (2016)
  * [14] Glasser, M.F., Sotiropoulos, S.N., Wilson, J.A., Coalson, T.S., Fischl, B., Andersson, J.L., Xu, J., Jbabdi, S., Webster, M., Polimeni, J.R., et al.: The minimal preprocessing pipelines for the human connectome project. Neuroimage 80, 105–124 (2013)
  * [15] Heinrich, M.P.: Closing the gap between deep and conventional image registration using probabilistic dense displacement networks. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 50–58. Springer (2019)
  * [16] Heinrich, M.P., Hansen, L.: Highly accurate and memory efficient unsupervised learning-based discrete ct registration using 2.5 d displacement search. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 190–200. Springer (2020)
  * [17] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980 (2014)
  * [18] Krähenbühl, P., Koltun, V.: Efficient inference in fully connected crfs with gaussian edge potentials. Advances in neural information processing systems 24 (2011)
  * [19] Monti, F., Boscaini, D., Masci, J., Rodola, E., Svoboda, J., Bronstein, M.M.: Geometric deep learning on graphs and manifolds using mixture model cnns. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 5115–5124 (2017)
  * [20] Pielawski, N., Wetzer, E., Öfverstedt, J., Lu, J., Wählby, C., Lindblad, J., Sladoje, N.: Comir: Contrastive multimodal image representation for registration. Advances in neural information processing systems 33, 18433–18444 (2020)
  * [21] Qi, C.R., Su, H., Mo, K., Guibas, L.J.: Pointnet: Deep learning on point sets for 3d classification and segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 652–660 (2017)
  * [22] Robinson, E.C., Garcia, K., Glasser, M.F., Chen, Z., Coalson, T.S., Makropoulos, A., Bozek, J., Wright, R., Schuh, A., Webster, M., et al.: Multimodal surface matching with higher-order smoothness constraints. Neuroimage 167, 453–465 (2018)
  * [23] Robinson, E.C., Jbabdi, S., Glasser, M.F., Andersson, J., Burgess, G.C., Harms, M.P., Smith, S.M., Van Essen, D.C., Jenkinson, M.: Msm: a new flexible framework for multimodal surface matching. Neuroimage 100, 414–426 (2014)
  * [24] Shao, W., Banh, L., Kunder, C.A., Fan, R.E., Soerensen, S.J., Wang, J.B., Teslovich, N.C., Madhuripan, N., Jawahar, A., Ghanouni, P., et al.: Prosregnet: A deep learning framework for registration of mri and histopathology images of the prostate. Medical image analysis 68, 101919 (2021)
  * [25] Wang, Y., Solomon, J.M.: Deep closest point: Learning representations for point cloud registration. In: Proceedings of the IEEE/CVF International Conference on Computer Vision. pp. 3523–3532 (2019)
  * [26] Yeo, B.T., Sabuncu, M.R., Vercauteren, T., Ayache, N., Fischl, B., Golland, P.: Spherical demons: fast diffeomorphic landmark-free surface registration. IEEE transactions on medical imaging 29(3), 650–668 (2009)
  * [27] Zhao, F., Wu, Z., Wang, F., Lin, W., Xia, S., Shen, D., Wang, L., Li, G.: S3reg: superfast spherical surface registration based on deep learning. IEEE Transactions on Medical Imaging 40(8), 1964–1976 (2021)
  * [28] Zhao, F., Xia, S., Wu, Z., Duan, D., Wang, L., Lin, W., Gilmore, J.H., Shen, D., Li, G.: Spherical u-net on cortical surfaces: methods and applications. In: International Conference on Information Processing in Medical Imaging. pp. 855–866. Springer (2019)
  * [29] Zheng, S., Jayasumana, S., Romera-Paredes, B., Vineet, V., Su, Z., Du, D., Huang, C., Torr, P.H.: Conditional random fields as recurrent neural networks. In: Proceedings of the IEEE international conference on computer vision. pp. 1529–1537 (2015)
  * [30] Zhou, Y., Barnes, C., Lu, J., Yang, J., Li, H.: On the continuity of rotation representations in neural networks. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 5745–5753 (2019)