Diffeomorphic Template Transformers
Authors: Tycho F.A. van der Ouderaa
Ivana Isgum
Wouter B. Veldhuis
Bob D. De Vos
Pim Moeskops
Venue:  (2021)

Abstract: In this paper we propose a spatial transformer network where the spatial transformations are limited to the group of diffeomorphisms. Diffeomorphic transformations are a kind of homeomorphism, which by definition preserve topology, a compelling property in certain applications.
We apply this diffemorphic spatial transformer to model the output of a neural network as a topology preserving mapping of a prior shape. By carefully choosing the prior shape we can enforce properties on the output of the network without requiring any changes to the loss function, such as smooth boundaries and a hard constraint on the number of connected components.
The diffeomorphic transformer networks outperform their non-diffeomorphic precursors when applied to learn data invariances in classification tasks. On a breast tissue segmentation task, we show that the approach is robust and flexible enough to deform simple artificial priors, such as Gaussian-shaped prior energies, into high-quality predictive probability densities. In addition to desirable topological properties, the segmentation maps have competitive quantitative fidelity compared to those obtained by direct estimation (i.e. plain U-Net).

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DIFFEOMORPHIC TEMPLATE TRANSFORMERS
Anonymous authors Paper under double-blind review
ABSTRACT
In this paper we propose a spatial transformer network where the spatial transforma- tions are limited to the group of diffeomorphisms. Diffeomorphic transformations are kind homeomorphism, which by definition preserve topology, a compelling property in certain applications. We apply this diffemorphic spatial transformer to model the output of a neural network as a topology preserving mapping of a prior shape. By carefully choosing the prior shape we can enforce properties the output of the network without requiring any changes to the loss function, such as smooth boundaries and hard constraint on the number of connected components. The diffeomorphic transformer networks outperform their non- non-diffcomorphic pre- cursors when applied to learn data invariances in classification tasks. On breast tissue segmentation task, we show that the approach is robust and flexible enough to deform simple artificial priors, such as Gaussian-shapp -hapee prior energies, into high-quality predictive probability densities. In addition to desirable topological properties, the segmentation maps have competitive quantitative fidelity compared to those obtained by direct estimation (i.e. plain U-Net).
INTRODUCTION
The success of Convolutional Neural Networks (CNNs) in many modeling tasks is often attributed to their depth and inductive bias. An important inductive bias in CNNs is spatial symmetry (c.g. translational equivariance) which are embedded in the architecture through Altemattvely, spatial transformers constrain networks through predicted spatial affine or thin hhin-plte-- spline transformations. this work, we investigate special type of spatial transformer, where the transformations are limited to flexible diffécomophisms. Diffeomorphisms belong to the group of homeomorphisms that preserve topology by design, and thereby guarantee that relations between structures remain, i.c. connected breginregions to stay connected.
constraints.
We propose to use such diffeomorphic spatial transformer in a template transformer setting (Lee et al. 2019), where a prior shape is deformed to the output of the model. Here neural network is used to predict the deformation of the shape, rather than the output itself. By introducing a diffeomorphic mapping of prior shape, and carefully choosing properties of the prior shape, we can enforce desirable properties on the output, such as a smooth decision boundary or a constraint on the number of connected components.
To obtain flexible diffeomorphic transformations, we use a technique known as scaling-ande -an-squurinn which has been successfully applied in the context of image registration in prior work (Dalca et al. 2018), but has received relatively little attention in other areas in machine learning. In an attempt to increase flexibility of the flow, we try approximate time-dependent parameterisation using Baker-Cample -aapsell-Hausdorff (BCH) formula, rather than a stationary field. Hereby, diffeomorphic constraints are directly built into the architecture itself, not requiring any changes to the loss function.
Experimentally, we first validate the diffeomorphic spatial transformer to learn data-invarte invariances in a MNIST handwritten digits classification task, as proposed by (Jaderberg et af.) 2015) to evaluate the original spatial transformer. The results show that better results can be achieved by employing diffeomorphic transformations. Additionally, we explore the use of diffeomorphic mappings in spatial template transformer set-up for 3D medical breast tissue segmentation. We find that the diffeomorphic spatial transformer able to deform simple prior shapes, such as normally distributed energy, into high-quality predictive probability densities. We are successful in limiting the number of connected components in the output and achieve competitive performance measured by quantitative metrics compared to direct estimation of class probabilities.

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2 RELATED WORK
Spatial Transformers were introduced by Jaderberrg et al. (2015) as a learnable module that deform an input image, and can be incorporated into CNNs for various tasks. Spatial Transformer Networks (STNs), the module is used to learn data invariances in order to do better in image classification tasks. The work focuses on simple linear transformations (e.g. translations, rotations, affine) but also allows for more flexible mappings such as thin plate spline (TPS) transformations. The use of spatial transformations template transformer setting was first proposed by Lee et al. (2019), but does not use diffeomorphisms and requires defining a discrete image as shape prior.
In the field of image registration, diffeomorphisms have been actively studied and have been suc- cesfully applied in variety of methods including LDDMM by |Beg et al.] (2005). Diffeomorphic Demons by Vercauteren et al. (2009), and SyN by Avants et al. (2008). More recently, efforts have been made to fuse such diffeomorphic image registration approaches with neural networks (Dalca et al. 2018)) Haskins et al. (2020)) is well known that although these models mathematically describe diffeomorphisms, transformations
not always diffeomorphic; in practice and negative Jacobian determinants can still occur due to approximation errors. To reduce such errors, additional
regularisation is often applied (Bro-Nielsen and Gramkow (1996), Ashburner 20077) Dalca et al. (2018). ), but typically requries careful tuning.
Image registration has also been applied to perform segmentation by deforming basis template commonly referred to as an 'atlas' onto a target image (Rohlfing et al. (2005), Fortunati et al. (2013)) for instance by combining (e.g. averaging) manually labelled training annotations (Gee et 1993).
There have been some studies that investigated how to obtain smoother segmentation boundaries in neural-based image registration. For instance, Monteiro et al. (2020) proposed model spatial correlation by modeling joint distributions over entire label maps, in contrast to pixel-wise estimates. In other work, post-procsssing steps have been applied in order to smooth predictions or to enforce topological constraints (Chlebus et al. (2018). Jafari et al. (2016)).
There have been some studies that try to enforce more consistent topology during training of neural network, but often use soft constraint that required alteration of the loss function, such as in Hu ett al. (001)) and GAN-based approaches which addition require a separately trained discriminator
model Sekuboyina al. (2018).
Lastly, there have been some studies in which diffeomorphisms in context of spatial transformer networks were investigated. In Skafte Detlefsen et al. (2018), subsequent layers of spatial transformer layers with piece-wise affine transformations (PCAB) were used to construct a diffeomorphic neural network, but requires tessellation strategy (Freifeld et al. (2015), Freifeld et al.) (2017)). In Deep Diffcomorphic Normalizing Flows (Salman et al. (2018) a neural network is used to predict diffeomorphic transformations normalizing flow but to obtain more expressive posteriors for variational inference.
3 DIFFEOMORPHIC SPATIAL TRANSFORMERS
The Spatial Transformer is a learnable module which explicitly allows for spatial manipulation of data within a neural network. The module takes an input feature map U passed through learnable function which regresses the transformation parameters 0. A spatial grid G over the output is transformed to an output grid (G), which applied the input to produce the output O. In the original spatial transformer, # could represent arbitrary parameterised mappings such as simple rotation, translation or aftine transformation matrices. We propose flexible transformations in the group of diffeomorphisms To DD which preserve topology, by continuity and continuity of the inverse.
In Section 4 we will describe how we can use a diffeomorphic spatial transformer to warp shape prior, as illustrated in Figure [ in a template transformer setting illustrated in Figure?)
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Diffeomorphic Transformation Let define the diffeomorphic mapping v/ using an ordinary differential equation (ODE):
0el v( (x))
(1)
Ot
where e is a stationary velocity field, the Id the identity transformation and is time. By integrating over unit time we obtain th the time flow of the stationary velocity field
The most basic way to solve an ordinary differential equation from some initial point æy is Euler's method, in which the trajectory is approximated by taking small discrete steps and adding the difference to the running approximation in time. The method straightforward implement, but may take many steps to converge to good approximations. In this work, we will use a technique known as scaling-ande and-squaring (Moler and Van Loan 2003), which allows for fast exponentiation of stationary velocity fields and thus the solution to the ODE defined in Equation I
To solve the ODE from Equation with a stationary velocity field v and the solution is the matrix exponential á exp(v), we is the scaling-and- and-squaring method (Moler and Van Loan (003). Arsigny et al. (0066). The method is very similar to Euler's ' method, but is typically more efficient by exploiting the relation exp(v) = exp(v/2f)² with T E N together with the fact that exp(v) can be well approximated by Padé or Taylor approximation near the origin (i.e. for small |]e|]). The main idea is to pick certain step size T such that ||u|l/2" 0.5 and divide the diagonal values in u by the power integral 2 to obtain the approximation for exp(v/2") z Id+w/22 and then squaring (self-composing)) it times to find obtain approximate solution for exp(v).
Scaling-and-Squaring
Algorithm 1: Approximating e exp(v) using scaling-and-squaring
Result: exp(v) T + ceig (max(1e]) + du u/2"
for f = to T do
Φ. Φ. 6t-1
end
such The approach can efficiently be implemented in existing numerical differentiation frameworks as PyTorch or Tensorflow by element-wi wise dividing the vector components velocity field v by 2 and then self-composing the resulting field 2 times using the linear grid sampling operation defined in Section 3
Spatial Sampling To perform a spatial transformation on the input feature map, a sampler takes set of sampling points To(G), along with an input feature map U = / with input image I to produce output O. In case of template transformer, explained in Section the input feature map would be concatenation U = I o S of input image with some prior shape S.
We follow the general sampling framework described Jaderberg et al. (2015). defined for arbitrary sampling kernels of which the (sub)gradients can be computed, and the 3D trilinear interpolation in particular:
D Σ
0
Iodd maax(0,11 1 1y5 - -hl) (0, |af w|) max(0, 1112f - -d)
(2)
This procedure should be differentiable with respect to both the sampling grid coordinates and the input feature map by using partial (sub)gradients, allowing it to be used in conjunction with backpropagation.
For experiments, we utilized the ..rri d_sampl function PyTorch 1.6 to perform grid sampling.
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(5)
Outel Stage (0)
Dapei Stage (1)
g
6
On of
Figure I: Illustration of diffeomorphism integrated over in time applied to shape prior. RIGHT: Target annotation. ToP: Class probabilities over voxel location. BOTTOM: Thresholded class probabilities (p 0.5) laid over the input.
Caater--Cmmell-Haussoon formula Instead of parameterising our flow by single stationary velocity field, we might also think of piece-wise dimedeependent sequence vector fields. By pa- rameterising the deformation performance by sequentially modeling larger movements first and detailed refinements thereafter. Composing multiple diffeomorphic transformations will also yield a diffeomorphic transformation, as the space of diffeomorphie transformations D an algebraic group that is closed under the compo- sition operation. The scaling-and-squaree and-squ algorithm offers efficient way to find diffcomorphic transformations from a stationary vector field, but can not straightforwardly be applied to such time-dependent parameterisations. To address this, for simplicity of notation, to form the Lie exponential mapping:
-eeendenn sequence of velocities we hope improve predictive
combine two timepoints, A and B
exp(Z) 7)) =exp(A) exp (B)
(3)
and apply the Caker-CCambelll-Haasdooo (BCH) formula up to a certain order to approximate
1 [A + (4)
Z (ch
(A, A+ B 2 [].
[4, [A.
is the Lie bracket. We apply the formula approximate the logarithm of matrix expo- nentials of two noncommutative velocity fields Z = log(cxp( cxp(B)) and then use scaling-and- squaring one time find the exponential exp(Z).
where [
Binary Tree Composition Naive composition of the T diffcomorphic transformations would result ((((((á) oΦ22) oØ3) )0))) (45)) ....)...ooT--)) o To reduce possible interpolation errors in the resampling from growing as result of such repetitive composing, we compose the field using a binary tree scheme & (((Ø) Φ22) o (d3 o (44) (...o (ØT-1 (T))). Treating the composition scheme as a tree structure, the depth now scales in order of complexity O(T) compared to (log(T)) when using naive composition, reducing the maximum number of times an BCH approximation is repetitively applied to single timepoint.
into long chain of composition operations
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Input
Diffeomorphic Transformation
Neural Network
Integrate Velocities
1
V
- TΦ (G)
Prior Shape
Output
Target
L
Resampler
Figure Illustration of Diffeomorphic Spatial Template Transformer: neural network predicts set of velocity fields V f(T,S) from an input image and prior shape S. The fields are integrated diffeomorphic transformation grid (G), which transforms the prior shape into an output O, while preserving topology. The model is trained end-to-end that output O matches target Y.
DIFFEOMORPHIC TEMPLATE TRANSFORMER
Now that we have defined how obtain a flexible diffeomorphic spatial transformer, we will investigate its use in a template transformer setting, We define the output of segmentation model diffeomorphic transformation of prior shape, based on input image and the prior shape. By carefully choosing the prior shape and its properties, we obtain explicit control over the properties of the output, such as the number of connected components.
Let the input of the model f be feature map U R'*xxDX2C = IoS be concatenation of an input image E To. xXxxx>DxxC and an prior shape E HR#xWDDCC along the channel dimension, with width W, height H, depth D and channels C' that outputs set of T velocity fields
= f(U) = {ur)²=1 where the fields 2 E | x3x are concatenated along the channel dimension of the output. Then we compute the diffcomorphic transformation & = =II exp(v:) by approximating the product of matrix exponentials discussed in Section 3 Lastly, we subsample the pixels in the original prior shape S using the diffcomorphic grid to obtain the output of the model
To (S) as explained Section 3 The resulting model is illustrated in Figure 2
4.1
PRIOR SHAPE
The template transformer can in principle any prior shape, such discrete image by averaging annotations (or "atlas" (Gee et al. 1993) (Cabezas et al. (011)). But, by carefully choosing prior shape and by continuity of the diffeomorphic transformation, connected component and smooth boundaries on the model output. In this paper we aim to keep the prior shape simple more general form emphasise the expressivity of the diffeomorphism in our experiments. We choose an analytical shape prior inspired by the generalised multivariate Gaussian, and define the probability of voxel at location æ belonging to the main class by
enforce properties such single
pl:; μc, E, 3) exp [-((x
1)) -
(5)
where μ,, Σ and 3 directly influence the mean, (co)variances and kurtosis of the prior shape in the spatial domain, and can be kept fixed or trained as part of the model parameters (sce Section 5.4). The log(2) factor ensures that the decision boundary (p=0.5) is independent from 3.
EXPERIMENTS AND RESULTS
The diffeomorphic spatial transformer is evaluated two tasks: classification task using handwritten MNIST dataset and a medical 3D breast tissue segmentation problem in the template transformer setting. In both settings, its performance is compared with its non-diffcomorphic counterparts. For the segmentation, we additionally analyse the effect of training different shape prior parameters in Section 5.4
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Table 1: Quantitative evaluation of diffeomorphic spatial transformer on MNIST classification task when learning to augment the input with various types of Spatial Transformer Networks (STNs).
Model
Type
Parumeter Count
Accuracy
(hoseline)
CNN TPS STN (aderberg 2015
27100 20056
96,00 0.46
0.05 00000
CNN 1PS-STN
x 3-grgid
27100 27506
95.92 4: 0.40
0.05 0.003
CNN TPS-STN
5 E-arid
27100 29140
96.63 0.332
0.04 0.002
CNN 195-SIN
12x 17-ard
27100 41278
96.49 9-0.0
0.05 i 0.006
CNN- TP5-STN
16 x 16-grit
27100+
55220
96.12 + 0.43
0.07 + 0.010
CNN Difrecmopphe SIN + [[ools penaty (Ouns)
28 x Mb-grat
21100
25953
96.45 11 0:48
0.31 a 0,009
CNN difteemoopphc
integaatio)
28 and
27100
29652
CNN Difieemoppiie SIN (Ours)
28 x 30-grd
21100 29002
97.34
5.1 IMPLEMENTATION
For the MNIST classification experiments, adapt existing spatial transformer network imple- mentation on Githut and added transformations using diffcomorphic vector fields. We train for 10 epochs with batch size of 64 and a learning rate of 0.001 (31 0.9, .3 0.999) without any further learning rate decay.
For the 3D breast tissue experiments, trained for 40k iterations using Adam (Kingma and Ba 2014) with a batch size of I and a learning rate of 0.0002 (3) = 0.9, 32 - 0.999) decayed with cosine annealing (Loshchilov and Hutter 2016). The input was normalised using 1-99 percentile normalisation [Patrice et al. (2018) and training samples consist of randomly sampled 128 x 128 x 64 patches. For the neural network, a 3D U-Net (Ronneberger et al.) 2015), 4 times spatial down- and up-sampling using linear interpolation (Odena et al.] 2016). instance normalisation (Ulyanov et al. 2016) and the Leaky ReLU (slope 0.2) activation functions. Lastly, an hyperbolic tangent scaled with = 256 limits the vector components of the vector fields to the range (-a, a.. The network uses a single input channel and 3 x T output channels, with T and a 2nd order BCH approximation. We train the model using standard cross entropy loss.
The prior shape was initialized as a centred Gaussian with diagonal co- corvariance matrix with diagonal components set at half the volume size of the corresponding dimension. Aside from trying optimise these values as part of the model parameters (see Section we did not perform hyper-parameter tuning on these values. All experiments were performed single NVIDIA GeForce RTX 2080 Ti.
5.2 MNIST EXPERIMENTS
In this experiment, analogous to the one described in Jaderberrg et al. (2015), we take a simple CNN classifier with and without an additional spatial transformers added to the beginning of the network and train it end-to-end. We train and evaluate the model on the well-known MNIST dataset (LeCun) jet al. 2010)) comprising 60000 training and 10000 testing images with size of 28 x 28 pixels that contain hand-drawn numbers in the range O to 10, with images randomly rotated to angle uniformly sampled from the range [-90, 90] degrees. The idea is that the spatial transformer can learn invariances in the data (e.g. translation) and thereby aid the classifier. The spatial transformer networks were designed in such a way that they have approximately the same parameter count, and the same classifier model was used in all cases. The experiment was repeated 20 times and standard deviations were reported. For fair comparison, we did not tune hyper-parameters in favor of the diffeomorphic-STr
In Table [ the results between different types of spatial transformers are shown. We find that our diffeomorphic spatial transformer network results in highest predictive accuracy, compared to non-diffeomorphhic and more course thin-plate-spline (TPS) spatial transformers. The TPS models generate inherently smoother fields, and are therefore less prone to folding resulting in fewer negative Jacobian determinants. On the other hand, coarser TPS grids have less fiexibility, which would make them unsuitable for application in complex anatomical segmentation tasks. We do observe that integration in our diffeomorphic model helps lowering the number of negative Jacobian determinants when compared to the same model without integration. In addition, we performed an experiment with an added regularisation term to the loss penalizing the spatial gradient of the field A|| //VVll2 22, where = 10 controls the amount of regularisation. We find that this helps to limit the number of negative Jacobian determinants, but also negatively impacts overall accuracy.
Used TPS-STN implementation:

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Table Quantitative evaluation of breast tissue segmentation on breast tissue segmentation dataset for different methods, indicating whether method uses template, preserves topology by-design, and comparison of loss, Sørensen -iie coefficient, Hausdorff distance, ratio of negative Jacobian determinants and average number of connected components on the validation set.
Template
Topulagy
Presetves
Dice
Housdorff Distance
58 [44]
Counectel Compusants
Net (direct estimation)
0.846 0.24
12.22
15.48
51.79
Spatial Tomphit Trandorner LLe b019 Shape Prior (ours)
0,877 0.31
12.58 12.11
0.43 0:01
19.68
Diffoomompiic Speiial Templin-rrmetmmin tranetmeerrrammm Shhpe Price (ours)
0.801 0.208
10.10 6.51
0.30 + 0.009
$1.20
Diffeomophiic Sputil Template Transfomer (ours) Trriteed Shape Prior (ours)
0.822 0.249
7,39
0,35 1.01
4.5
5.3 SEGMENTATION EXPERIMENTS
To assess the applicability of diffeomorphic spatial transformer in a template transformer setting, we compare our differentiable spatial transformer with and without trained shape prior with direct esti- mation (i.e. plain U-Net) on a breast tissue segmentation task. We also evaluate a non-diffeomorphic spatial transformer, as is done in Lee et al. (2019), but applied it in combination with our shape prior as it was not obvious to us how to create a discrete 3D template from our 2D annotations.
The dataset comprises 20 training volumes and 20 evaluation volumes of dynamic contrast enhance- ment series of subjects with extremely dense breast tissue (Volpara Density Grade 4). Each series contains DCE-MRI images (384 384 x 60 voxels with spacing 0.97 x 0.97 3.00 mm resampled to 2.5mm²) acquired on 3.OT Achieva or Ingenia Philips system in the axial plane with bilateral anatomic coverage. A randomly selected axial 2D slice was annotated to be used for training and evaluation labels. All annotations make up single connected component.
Performance was measured using the well-known Sprensen- coefficient (Fy-score) (Dice (1945), Sorensen ) and Hausdorff distance (Hausdorff) 1978) metrics. In addition, measure the percentage of negative Jacobian determinants of the approximated flow, a well-known metric for deformation quality in image registration that measures the amount of folding. Lastly, we evaluate whether the connected components in the thresholded output (p > 0.5) is close to 1, as should be the case without approximation errors. The HD and CC metrics on this medical imaging tasks are particularly important indicating high-quality and robust results.
In Table 2 comparison of spatial template transformer with fixed shape prior, a diffeomorphic spatial transformer with fixed shape prior and a diffcomorphic spatial transformer with trained shape prior (trained mean μa, diagonal covariance diag(22 (E) and B) with direct estimation (i.e. plain U-Net) is shown. We find that all template transformer models perform better in terms of Hausdorff distance. The diffeomorphic spatial template transformers perform worse in terms of Sørensen coefficient, but in combination with trained shape prior are able to reduce the number of connected components and Hausdorff distance. Lastly, we observe negative Jacobian determinants as result of approximation errors in all template transformers, but to lower degree in the diffeomorphic models.
5.4 ANALYSIS OF PRIOR SHAPPS
In this section we empirically assess the impact of different prior shapes, with fixed or varying μ., Σ and [3 parameters. In Table 3 Sörensen-Dice coefficient, Hausdorff distance, ratio of negative Jacobians % [&o < 0 and average number of connected components are reported for different combinations of trained prior shape parameters.
Table Ablation study where different parameters for the prior shapes are kept fixed or learnt.
Trained prior shape
Dice Score
Hausdorff Distance
% [Jo] <
Connected Components
Fixed Prior Shape
0.803 +:2 0.298
10.10 + 6.51
0.39 = 0.009
31,20
Train mean y
0.820 + 0.262
11.30 + 10.65
0.22 = 0.013
4.38 :: 4,87
Train scale n (X d²
0.807 + 0.297
9.92 = 12.17
0.36 006=0.000 0.023
61.16 4 45.1
Train (iag(E))
0.821 0.284
8.66 5.57
0.47 = 0.007
27.47
11.5
Train μ o
0.821 +: 0.252
10.10 +:: 7.84
0.25 0.012
5.23 4 5.48
Train diag x
0.790 + 0.268
12.58 + 11.64
0.22 = 0.020
5.02 = 5.34
Train diag (E) +
0.822 +:: 0.249
9.81 +: 7.39
0.15 = 0.012
4.75 4.37
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We find that learning parameters of the shape prior positively contributes to performance and helps to reduce the number of negative Jacobian determinants, most notably for learnt position μu. The result suggests that, in case of more complex shape priors such as segmentation atlas, the model could benefit from deforming the prior shape with some linear transformations (e.g. translation or affine) before being warped by the diffeomorphic transformation predicted by the network.
DISCUSSION AND CONCLUSION
We have presented a special type of spatial transformer where the spatial transformations re- stricted to the group of diffeomorphisms. Diffeomorphic deformations are topology preserving by continuity and continuity of their inverse. which can be compelling property when designing deep learning architectures. We show how expressive diffeomorphic mappings can be obtained by time-dependent parameterisation of multiple vector fields utilizing the Baker- Seeeer-Cmmmblll-Haae Hausdorff formula in combination with an efficient integration method known scaling-and saalin-and-squaring.. By building these constraints directly into the architecture itself, no changes to the loss function are re- quired. In addition, we propose to use the diffeomorphic spatial transformer in template transformer setting, constraining the output a neural segmentation model as analytical prior shape. Hereby, we show that the diffeomorphic transform enforces smooth boundaries and explicit control over the topology of the output such as its number of connected components.
mapping of an
The diffeomorphic spatial transformer outperforms the original spatial transformer network when used to learn intaiinnariances on MNIST. In a template transformer set-up, we found that a neural network predicting a diffeomorphic mapping of prior shape offers flexible way insert knowledge about the structure of the output without having to alter the loss function or optimisation scheme. We were able to warp shape priors into high-quality segmentations in a medical 3D breast tissue segmentation task, resulting in lower number of connected components and obtain higher performance in terms of Hausdorff distance but lower in terms of Dice Score compared with direct estimation (i.e. plain U-Net).
To show the effectiveness of the approach, we used a general and simple dlussian-shappl shape prior as template. Interestingly. the method is flexible enough to find diffeomorphic mappings from such simple shapes into high-quality posteriors. We expect that designing shape priors, specifically tailored to task (e.g. an atlas or average segmentation) might achieve even better results. It would be interesting to explore applicability on more complex anatomical structures, such as in coronary artery tree segmentation (Lee et al. (2019)).
piece-wise constant time-dependent parameterisation performed slightly better than modeling a stationary velocity field. This surprised us, because for every diffeomorphic mapping generated by a picce-wise constant demendependnnt field there also exists a single (stationary) vector field that describes the same diffeomorphism. We hypothesise that directly optimising this stationary velocity field is harder and that the uneedeeendeett parameterisation aids the optimisation process by allowing the network to model larger and more detailed deformations separately.
We used the BCH-formula to integrate a piece-wise time-dependent velocity field using the scaling- and-squaring method. It would be interesting see how this method relates to other ODE solvers that are capable of integrating time-dependent velocity fields, such as those proposed in Chen et al. (2018). However, we were unable to use these solvers under available hardware constraints.
In some cases, negative Jacobian determinants are still present in the obtained flows. This is likely caused by spatial discretisation and interpolation during resampling operations. Future research could assess whether the approach could benefit from regularisations of the vector fields (Ashburner) 2007) inverse consistency (Christensen 1999) or better interpolation methods.
To conclude, we show that diffeomorphic spatial transformations can successfully be applied to preserve topology neural networks, and for template transformer networks in particular. We have provided several insights on how to incorporate diffeomorphisms in neural network architectures for classification and segmentation. We expect that these insights can aid in tailoring neural network architectures to specific structure and geometry in data.
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ACKNOWLEDGEMENTS
Anonymised
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Under review as a conference paper at ICLR 2021
A
BAKER- SEEEER-CAMBELLHHASSDOOO FORMULA ORDERS TO 27
we denote the Baker-Campbell-Haauddrrff formula as
20 (X,
(Xx, Y) (x)(exx(())xx((()
can simplify (Van-Brumt Brunt and Visser (2016) the lower order terms to to
(X,,Y XY
(X, 1 (XY - -YX)
xX,, = (X 2XXX - -X-- +YX - XYX + +2X)
12
xX,Y)) (x2y² - 2XYXY + 2YXXX Y²X²)
24
2x(XY 1
(-X* +6XYXYX + +XX X y + -YXXXY -
6!
26 (x,, = 2 6!
2-2X"X² XY 6XYXXXY XY
Higher order 27 and 2g terms were not used this study, but can be found the Supplementary Material of Van-Brunt and Visser (0066).
B TIMING MEASUREMENTS
A comparison of performance in terms of inference time be found in Table Average inference time was calculated over 20 full 3d volumes on the breast tissue segmentation task on the U-Net model baseline, non-diffeomorphic model (without field-integration) and diffeomorphic models with stationary and denendeenddnnt vector field parameterisations. Due to integration procedures, the average inference time the diffeomorphic template transformer models is slightly slower (s 10%) compared to the U-Net baseline, but well within practical bounds.
Table 4: Average inference time in seconds (s) measured over 20 full 3d volumes on the U-Net baseline, non-diffcoomophic template transformer, diffeomorphic template transformer with stationary field and diffeomorphic template transformer with deeedeepndentt field.
Method
Average inference time over 20 full 3d volumes
U-Net
1.03
U-Net non-diffeomorphic field
1.06
U-Net stationary diffeomorphic field
1.17
U-Net diffeomorphic field (ours)
1.19
12

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Under review as a conference paper at ICLR 2021
C EXAMPLE VARIATIONS IN SHAPE PRIOR PARAMETERS
To illustrate how variations the parameters μ., a, Σ and / spatially change the shape prior, we plot the probability (white: p 0 and purple 1) for each voxel p(æ; μ. E/o1, 3) with different parameter values both smooth (top) and thresholded (bottom):
p-0.3
P-1
B-2
P-5
5-800
Figure Shape prior under different values for B.
u-1200.
Figure 4: Shape prior under different values for μ..
Figure 5: Shape prior under different values for
10.
50% 10. 0, 11
10, 20.2000.. 01
1005. 10. 0.
(300%.
10.0, II
0. -11
Figure 6: Shape prior under different values for Σ.
13


Reviews:
[{"review_id": "YY1YBzcEb_x", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper propose a novel method to incorporate shape prior in neural networks based on Diffeomorphic transformation. This is useful as by design it preserves certain desirable properties of output such as smooth boundaries and connected components which are of interest in medical imaging applications.   The method is validated on Mnist for data invariance and a medical imaging task for segmentation.\n\nNovelty: The idea of  incorporating shape prior information into neural network based image segmentation is inspired by Lee et al. This method shows  how to use a diffeomorphic spatial transformer to warp a shape prior where warping is based on time-dependent parameterisation of multiple vector fields utilizing the Baker-Campbell-Hausdorff formula.\n\nClarity and Presentation: The paper is overall well written and motivated. Minor comment: As an outsider, I found the argument on negative Jacobian determinants hard to follow and it comes up several times. So it seems quite important.\n\nExperimental Validation: It is partially on weaker side in my opinion.  e.g. Lee et al. experiments were shown on segmentation of coronary lumen structures. Is there a good reason to instead choose breast tissue segmentation task only and not show any experiments on the former one? similarly, to prove data invariance, STN is validated on several benchmarks whereas this method only uses one, MNIST.", "extra_scores": {}, "extra_text": {}}, {"review_id": "PLFWabOlZS", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Authors present a spatial transformer layer that models\ndiffeomorphisms. Modelling diffeormophisms with neural networks is not very\nnew. Prior work successfully utilized stationary vector field prediction and\nfast solvers through scaling-and-squaring techniques. These prior work have been\nused to tackle registration and segmentation problems. The technical novelty\nhere is to use time-dependent vector fields for modeling\ndiffeomorphisms. Experiments on MNIST classification, using the proposed spatial\ntransformer as a layer, and breast tissue segmentation, using the proposed\ntransformer as a single network to deform a template, are presented and compared\nwith conventional spatial transformer layers using thin-plate-splines as\ntransformation models.\n\nWhile the introduction of time-dependency is interesting, more from a technical\nperspective than a conceptual perspective, I believe there are several aspects\nof the paper that needs improvement. \n\n1. Since the main innovation is integration of time-dependency in modelling\nvector fields with neural networks, I suggest directly comparing time-dependent\nwith stationary vector field approaches.\na. Experimental comparison can focus on the differences between stationary and\npiece-wise stationary vector field modelling. I could not see this comparison in\nthe experiments, only a single mention in the conclusion stating better\nperformance of the proposed model.\nb. Analysis of computational time would be interesting. Predicting and\nintegrating a stationary vector field will be faster I assume. But by how much?\n\n2. MNIST classification experiments raise some concerns:\na. The increase in classification accuracy of the proposed model compared to\nCNN + Field-STN is small. This difference may as well be due to randomness of\nthe optimization. This result is not motivating for the proposed\nmethod. Achieving lower number of negative Jacobian determinants is interesting\nbut its value is questionable.\nb. What is CNN + Field-STN? This is not defined.\n\n3. Segmentation results raise the following concerns:\na. The method proposed here does not seem to be better than the method\nproposed in Lee et al. 2019. The lower HD distance can be motivating but value\nof this may be better justified. If the proposed innovation was substantial,\nlower accuracy would have been completely fine. However, since the innovation is\nsubtle, such a lower accuracy lowers the enthusiasm for the paper.\nb. The proposed method achieves lower number of connected components but the\nvalue of this is not very well motivated from an application\nperspective. Providing such a motivation would be helpful.\n\n4. I suggest editing section 3 to present the proposed method much more\nclearly. The method only becomes clear in Section 4. \n", "extra_scores": {}, "extra_text": {}}, {"review_id": "KWotcs72mb", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This submission proposes a diffeomorphic spatial transform network, which considers the specific transformation, i.e., diffeomorphism, in the data. The research topic is quite interesting, but the submission is at the preliminary stage and needs more work before publishing.\n\n1) The writing of the paper could be improved in terms of clarification. Most of the time, the authors present the solutions without explaining the motivations or underlying intuitions. For instance, why do we have to have the time-dependent vector field? Why do we need to use the prior shape? \n\n2) The novelty of the proposed method is limited. The main contribution of this paper is the use of BCH formula to approximate the piece-wise time-dependent sequence of the vector field. Very likely, this approximation is not very efficient, and we also don't know its approximation accuracy. \n\n3) The performance improvement is limited. In the MNIST experimental results, the accuracy improvement is subtle. Compared to CNN+Field-STN, the CNN+Diffeomorhpic-STN has a slightly increased mean accuracy with a larger standard deviation and a slightly decreased number of negative Jacobians with also a larger standard deviation. For the breast tissue segmentation task, if we consider the transformer only (without considering the shape prior), the Dice Score performance of the proposed method is downgraded. \n\n4) The efficiency of the proposed method is questionable. If we use this diffeomorphic transform network as a plugin module in another network, will it become the computational bottleneck of the whole model, since the estimated spatiotemporal vector field is really high-dimensional?\n\n5) Minor question: Is the identity map missing in the Algirhtm 1? phi_0 = Id + v/2^T?", "extra_scores": {}, "extra_text": {}}, {"review_id": "IIgwyLqmmXh", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "Summary:\n\nThe authors present a modification to spatial transformer networks that restricts the transformations to the group of diffeomorphisms. When combined with shape priors, this imposes topological constraints on the mappings produced by the network. These are important considerations in applications such as segmentation tasks where we expect there to be constraints on, for example, the number of connected components. The authors demonstrate the effectiveness of their approach in MNIST experiments and a breast tissue segmentation task. \n\nStrengths:\n1) The paper is clear and well written, and positions its contributions well in relation to existing research.\n\n2) The method is well motivated and the procedure is reasonable. The paper tackles an important consideration when applying neural networks to segmentation tasks, as constraining to the group of diffeomorphisms imposes smoothness on the transformations that ensures the outputs share topological properties with our prior expectations (e.g. number of connected components, smoothness of boundaries).\n\n3) The authors use the scaling-and-squaring to solve the ODEs that describe the diffeomorphic transformations, which I have not seen used in other ML studies.\n\nWeaknesses:\n\n1) Section 3, which describes the components of the method, can be expanded (there is space for this at the end of page 2) and reorganised to make it clearer. Specifically:\n\n\ta) the last two lines of paragraph one of page 4, starting with ‘The scaling-and-squaring...’, say this algorithm is not amenable to time-dependent parametrisations as laid out by the ODE in eq. 1. It is not immediately clear to me as to what the A and B refer to in this situation, are these the separate time steps t?\n\n\tb) I would like to see more discussion about the binary tree assumption on the composition of the diffeomorphisms. I can understand why this is a useful assumption with regards to complexity and numerical stability, but I do not have good intuition about the implications of this assumption on the transformations the model outputs.\n\nMy suspicion is the combination of these two assumptions explain some of the behaviours seen in the results. \n\n2) The results section could also be expanded for a bit more clarity. My main concerns are around the metric that measures the percent of the Jacobian determinants that are less than zero. Is this computed from all observations or just on a hold out test set? Why is it the case that the supposedly non-diffeomorphic models in Table 1 perform better on this metric than the diffeomorphic models?\n\nWhy is there such a large discrepancy in the size of the standard deviations in Table 2?\n\nI do not think the results are particularly convincing in terms of improvements in the quality or the accuracy of the methods compared with, but the point of the paper stands if the behaviour of this metric is properly discussed. \n\n\nReasons for score: \nI vote for accepting the paper. On the whole, it is a solid paper with an interesting and novel contribution, but I think it is hampered by the lack of clarity in the areas listed above. I am willing to revise my assessment up if these concerns are addressed. \n\nQuestions for the rebuttal period:\n\nPlease refer to the questions in the weaknesses section. \n\n\n---------------------------------------------------------------------------------------------------------------\nUpdate after discussion:\nAs stated in my initial review, I wanted to see an in depth discussion of what the implications of the binary tree assumption are for the types of transformations that are produced and the impact on the metric using the Jacobian. If these concerns are addressed, I think the paper will be greatly strengthened in the future. ", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper has initially received mixed reviews, with two favorable and two unfavorable reviews. Several serious issues have been raised, in particular on experiments and validation; limited novelty; limited performance improvement; on the preliminary stage of the paper, in particular presentation and writing, and on the justification of key choices.

The authors provided responses to some of these issues, but in the discussion phase the reviewers (and the AC) judged the the responses did not sufficiently address the weaknesses of the paper, in particular:
- The experimental setup does not assess the key innovation of the paper. 
- Several contributions claimed by the authors (in the paper and in the response) are judged not to be novel.
- Concerns regarding the metric chosen to measure smoothness.
and other issues.

The reviewers and AC agreed, that the paper has potential and merits, but that at at this point it is not yet ready for publication.
Decision: Reject