Stefan Sommer

Stefan Sommer, Mads Nielsen, Sune Darkner et al.

To achieve sparse parametrizations that allows intuitive analysis, we aim to represent deformation with a basis containing interpretable elements, and we wish to use elements that have the description capacity to represent the deformation compactly. To accomplish this, we introduce in this paper higher-order momentum distributions in the LDDMM registration framework. While the zeroth order moments previously used in LDDMM only describe local displacement, the first-order momenta that are proposed here represent a basis that allows local description of affine transformations and subsequent compact description of non-translational movement in a globally non-rigid deformation. The resulting representation contains directly interpretable information from both mathematical and modeling perspectives. We develop the mathematical construction of the registration framework with higher-order momenta, we show the implications for sparse image registration and deformation description, and we provide examples of how the parametrization enables registration with a very low number of parameters. The capacity and interpretability of the parametrization using higher-order momenta lead to natural modeling of articulated movement, and the method promises to be useful for quantifying ventricle expansion and progressing atrophy during Alzheimer's disease.

Gefan Yang, Stefan Sommer

We propose a novel denoising diffusion generative model for predicting nonlinear fluid fields named FluidDiff. By performing a diffusion process, the model is able to learn a complex representation of the high-dimensional dynamic system, and then Langevin sampling is used to generate predictions for the flow state under specified initial conditions. The model is trained with finite, discrete fluid simulation data. We demonstrate that our model has the capacity to model the distribution of simulated training data and that it gives accurate predictions on the test data. Without encoded prior knowledge of the underlying physical system, it shares competitive performance with other deep learning models for fluid prediction, which is promising for investigation on new computational fluid dynamics methods.

Francesco Santini, Jakob Wasserthal, Abramo Agosti et al.

Purpose: To present and evaluate Dafne (deep anatomical federated network), a freely available decentralized, collaborative deep learning system for the semantic segmentation of radiological images through federated incremental learning. Materials and Methods: Dafne is free software with a client-server architecture. The client side is an advanced user interface that applies the deep learning models stored on the server to the user's data and allows the user to check and refine the prediction. Incremental learning is then performed at the client's side and sent back to the server, where it is integrated into the root model. Dafne was evaluated locally, by assessing the performance gain across model generations on 38 MRI datasets of the lower legs, and through the analysis of real-world usage statistics (n = 639 use-cases). Results: Dafne demonstrated a statistically improvement in the accuracy of semantic segmentation over time (average increase of the Dice Similarity Coefficient by 0.007 points/generation on the local validation set, p < 0.001). Qualitatively, the models showed enhanced performance on various radiologic image types, including those not present in the initial training sets, indicating good model generalizability. Conclusion: Dafne showed improvement in segmentation quality over time, demonstrating potential for learning and generalization.

Morten Akhøj, James Benn, Erlend Grong et al.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank $k$ tangent subbundle on $\mathbb{R}^d$, $k<d$, which we call a principal subbundle. This determines a sub-Riemannian metric on $\mathbb{R}^d$. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold $M$, construction of a representation of the point-cloud in $\mathbb{R}^k$, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

Elizabeth L. Baker, Moritz Schauer, Stefan Sommer

We propose a new algorithm for learning bridged diffusion processes using score-matching methods. Our method relies on reversing the dynamics of the forward process and using this to learn a score function, which, via Doob's $h$-transform, yields a bridged diffusion process; that is, a process conditioned on an endpoint. In contrast to prior methods, we learn the score term $\nabla_x \log p(t, x; T, y)$ directly, for given $t, y$, completely avoiding first learning a time-reversal. We compare the performance of our algorithm with existing methods and see that it outperforms using the (learned) time-reversals to learn the score term. The code can be found at https://github.com/libbylbaker/forward_bridge.

Elizabeth Baker, Thomas Besnier, Stefan Sommer

Modelling randomness in shape data, for example, the evolution of shapes of organisms in biology, requires stochastic models of shapes. This paper presents a new stochastic shape model based on a description of shapes as functions in a Sobolev space. Using an explicit orthonormal basis as a reference frame for the noise, the model is independent of the parameterisation of the mesh. We define the stochastic model, explore its properties, and illustrate examples of stochastic shape evolutions using the resulting numerical framework.

Lili Bao, Jiahao Lu, Shihui Ying et al.

In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a non-differentiable kernel. This allows to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion.

Du Nguyen, Stefan Sommer

We express parallel transport for several common matrix Lie groups with a family of pseudo-Riemannian metrics in terms of matrix exponential and exponential actions. The metrics are constructed from a deformation of a bi-invariant metric and are naturally reductive. There is a similar picture for homogeneous spaces when taking quotients satisfying a general condition. In particular, for a Stiefel manifold of orthogonal matrices of size $n\times d$, we give an expression for parallel transport along a geodesic from time zero to $t$, that could be computed with time complexity of $O(n d^2)$ for small $t$, and of $O(td^3)$ for large $t$, contributing a step in a long-standing open problem in matrix manifolds. A similar result holds for {\it flag manifolds} with the canonical metric. We also show the parallel transport formulas for the {\it general linear group} and the {\it special orthogonal group} under these metrics.

Lili Bao, Bin Xiao, Shihui Ying et al.

In this paper, we propose a novel mathematical framework for piecewise diffeomorphic image registration that involves discontinuous sliding motion using a diffeomorphism groupoid and algebroid approach. The traditional Large Deformation Diffeomorphic Metric Mapping (LDDMM) registration method builds on Lie groups, which assume continuity and smoothness in velocity fields, limiting its applicability in handling discontinuous sliding motion. To overcome this limitation, we extend the diffeomorphism Lie groups to a framework of discontinuous diffeomorphism Lie groupoids, allowing for discontinuities along sliding boundaries while maintaining diffeomorphism within homogeneous regions. We provide a rigorous analysis of the associated mathematical structures, including Lie algebroids and their duals, and derive specific Euler-Arnold equations to govern optimal flows for discontinuous deformations. Some numerical tests are performed to validate the efficiency of the proposed approach.

Stefan Sommer, Gefan Yang, Elizabeth Louise Baker

Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of shape spaces, the mathematical construction of suitable stochastic shape processes is far from immediate. We define and formalize properties that stochastic shape processes should ideally satisfy to be compatible with the shape structure, and we link this to Kunita flows that, when acting on shape spaces, induce stochastic processes that satisfy these criteria by their construction. We couple this with a survey of other relevant shape stochastic processes and show how bridge sampling techniques can be used to condition shape stochastic processes on observed data thereby allowing for statistical inference of parameters of the stochastic dynamics.

Gefan Yang, Frank van der Meulen, Stefan Sommer

Inference in non-linear continuous stochastic processes on trees is challenging, particularly when observations are sparse (leaf-only) and the topology is complex. Exact smoothing via Doob's $h$-transform is intractable for general non-linear dynamics, while particle-based methods degrade in high dimensions. We propose Neural Backward Filtering Forward Guiding (NBFFG), a unified framework for both discrete transitions and continuous diffusions. Our method constructs a variational posterior by leveraging an auxiliary linear-Gaussian process. This auxiliary process yields a closed-form backward filter that serves as a ``guide'', steering the generative path toward high-likelihood regions. We then learn a neural residual--parameterized as a normalizing flow or a controlled SDE--to capture the non-linear discrepancies. This formulation allows for an unbiased path-wise subsampling scheme, reducing the training complexity from tree-size dependent to path-length dependent. Empirical results show that NBFFG outperforms baselines on synthetic benchmarks, and we demonstrate the method on a high-dimensional inference task in phylogenetic analysis with reconstruction of ancestral butterfly wing shapes.

Elizabeth Louise Baker, Gefan Yang, Michael L. Severinsen et al.

Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.

Erlend Grong, Karen Habermann, Stefan Sommer

Simulation of conditioned diffusion processes is an essential tool in inference for stochastic processes, data imputation, generative modelling, and geometric statistics. Whilst simulating diffusion bridge processes is already difficult on Euclidean spaces, when considering diffusion processes on Riemannian manifolds the geometry brings in further complications. In even higher generality, advancing from Riemannian to sub-Riemannian geometries introduces hypoellipticity, and the possibility of finding appropriate explicit approximations for the score of the diffusion process is removed. We handle these challenges and construct a method for bridge simulation on sub-Riemannian manifolds by demonstrating how recent progress in machine learning can be modified to allow for training of score approximators on sub-Riemannian manifolds. Since gradients dependent on the horizontal distribution, we generalise the usual notion of denoising loss to work with non-holonomic frames using a stochastic Taylor expansion, and we demonstrate the resulting scheme both explicitly on the Heisenberg group and more generally using adapted coordinates. We perform numerical experiments exemplifying samples from the bridge process on the Heisenberg group and the concentration of this process for small time.

Nicklas Boserup, Gefan Yang, Michael Lind Severinsen et al.

We introduce a methodology for performing parameter inference in high-dimensional, non-linear diffusion processes. We illustrate its applicability for obtaining insights into the evolution of and relationships between species, including ancestral state reconstruction. Estimation is performed by utilising score matching to approximate diffusion bridges, which are subsequently used in an importance sampler to estimate log-likelihoods. The entire setup is differentiable, allowing gradient ascent on approximated log-likelihoods. This allows both parameter inference and diffusion mean estimation. This novel, numerically stable, score matching-based parameter inference framework is presented and demonstrated on biological two- and three-dimensional morphometry data.

Gefan Yang, Frank van der Meulen, Stefan Sommer

We propose a novel method for simulating conditioned diffusion processes (diffusion bridges) in Euclidean spaces. By training a neural network to approximate bridge dynamics, our approach eliminates the need for computationally intensive Markov Chain Monte Carlo (MCMC) methods or score modeling. Compared to existing methods, it offers greater robustness across various diffusion specifications and conditioning scenarios. This applies in particular to rare events and multimodal distributions, which pose challenges for score-learning- and MCMC-based approaches. We introduce a flexible variational family, partially specified by a neural network, for approximating the diffusion bridge path measure. Once trained, it enables efficient sampling of independent bridges at a cost comparable to sampling the unconditioned (forward) process.

Du Nguyen, Stefan Sommer

We specify the conditions when a manifold M embedded in an inner product space E is an invariant manifold of a stochastic differential equation (SDE) on E, linking it with the notion of second-order differential operators on M. When M is given a Riemannian metric, we derive a simple formula for the Laplace-Beltrami operator in terms of the gradient and Hessian on E and construct the Riemannian Brownian motions on M as solutions of conservative Stratonovich and Ito SDEs on E. We derive explicitly the SDE for Brownian motions on several important manifolds in applications, including left-invariant matrix Lie groups using embedded coordinates. Numerically, we propose three simulation schemes to solve SDEs on manifolds. In addition to the stochastic projection method, to simulate Riemannian Brownian motions, we construct a second-order tangent retraction of the Levi-Civita connection using a given E-tubular retraction. We also propose the retractive Euler-Maruyama method to solve a SDE, taking into account the second-order term of a tangent retraction. We provide software to implement the methods in the paper, including Brownian motions of the manifolds discussed. We verify numerically that on several compact Riemannian manifolds, the long-term limit of Brownian simulation converges to the uniform distributions, suggesting a method to sample Riemannian uniform distributions

Liwei Hu, Wenyong Wang, Yu Xiang et al.

We propose to incorporate Riemannian geometric features from the geometry of aircraft wing surfaces in the prediction of coefficient of pressure (CP) on the aircraft wing. Contrary to existing approaches that treat the wing surface as a flat object, we represent the wing as a piecewise smooth manifold and calculate a set of Riemannian geometric features (Riemannian metric, connection, and curvature) over points of the wing. Combining these features in neighborhoods of points on the wing with coordinates and flight conditions gives inputs to a deep learning model that predicts CP distributions. Experimental results show that the method with incorporation of Riemannian geometric features, compared to state-of-the-art Deep Attention Network (DAN), reduces the predicted mean square error (MSE) of CP by an average of 15.00% for the DLR-F11 aircraft test set.

Alexander Christgau, Alexis Arnaudon, Stefan Sommer

Models of stochastic image deformation allow study of time-continuous stochastic effects transforming images by deforming the image domain. Applications include longitudinal medical image analysis with both population trends and random subject specific variation. Focusing on a stochastic extension of the LDDMM models with evolutions governed by a stochastic EPDiff equation, we use moment approximations of the corresponding Itô diffusion to construct estimators for statistical inference in the full stochastic model. We show that this approach, when efficiently implemented with automatic differentiation tools, can successfully estimate parameters encoding the spatial correlation of the noise fields on the image.

Alexander Barbie, Wilhelm Hasselbring, Niklas Pech et al.

In this decade, the amount of (industrial) Internet of Things devices will increase tremendously. Today, there exist no common standards for interconnection, observation, or the monitoring of these devices. In context of the German "Industrie 4.0" strategy the Reference Architectural Model Industry 4.0 (RAMI 4.0) was introduced to connect different aspects of this rapid development. The idea is to let different stakeholders of these products speak and understand the same terminology. In this paper, we present an approach using Digital Twins to prototype different layers along the axis of the RAMI 4.0, by the example of an autonomous ocean observation system developed in the project ARCHES.

Alexander Barbie, Niklas Pech, Wilhelm Hasselbring et al.

During the research cruise AL547 with RV ALKOR (October 20-31, 2020), a collaborative underwater network of ocean observation systems was deployed in Boknis Eck (SW Baltic Sea, German exclusive economic zone (EEZ)) in the context of the project ARCHES (Autonomous Robotic Networks to Help Modern Societies). This network was realized via a Digital Twin Prototype approach. During that period different scenarios were executed to demonstrate the feasibility of Digital Twins in an extreme environment such as underwater. One of the scenarios showed the collaboration of stage IV Digital Twins with their physical counterparts on the seafloor. This way, we address the research question, whether Digital Twins represent a feasible approach to operate mobile ad hoc networks for ocean and coastal observation.

Jakob Stolberg-Larsen, Stefan Sommer

Generative neural networks have a well recognized ability to estimate underlying manifold structure of high dimensional data. However, if a single latent space is used, it is not possible to faithfully represent a manifold with topology different from Euclidean space. In this work we define the general class of Atlas Generative Models (AGMs), models with hybrid discrete-continuous latent space that estimate an atlas on the underlying data manifold together with a partition of unity on the data space. We identify existing examples of models from various popular generative paradigms that fit into this class. Due to the atlas interpretation, ideas from non-linear latent space analysis and statistics, e.g. geodesic interpolation, which has previously only been investigated for models with simply connected latent spaces, may be extended to the entire class of AGMs in a natural way. We exemplify this by generalizing an algorithm for graph based geodesic interpolation to the setting of AGMs, and verify its performance experimentally.

Stefan Sommer, Alex Bronstein

We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convolution layers by sampling manifold valued random variables that center around a weighted diffusion mean. Both methods are inspired by stochastics on manifolds and geometric statistics, and provide examples of how stochastic methods -- here horizontal frame bundle flows and non-linear bridge sampling schemes, can be used in geometric deep learning. We outline the theoretical foundation of the two methods, discuss their relation to Euclidean deep networks and existing methodology in geometric deep learning, and establish important properties of the proposed constructions.

Line Kühnel, Alexis Arnaudon, Tom Fletcher et al.

Modelling deformation of anatomical objects observed in medical images can help describe disease progression patterns and variations in anatomy across populations. We apply a stochastic generalisation of the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework to model differences in the evolution of anatomical objects detected in populations of image data. The computational challenges that are prevalent even in the deterministic LDDMM setting are handled by extending the FLASH LDDMM representation to the stochastic setting keeping a finite discretisation of the infinite dimensional space of image deformations. In this computationally efficient setting, we perform estimation to infer parameters for noise correlations and local variability in datasets of images. Fundamental for the optimisation procedure is using the finite dimensional Fourier representation to derive approximations of the evolution of moments for the stochastic warps. Particularly, the first moment allows us to infer deformation mean trajectories. The second moment encodes variation around the mean, and thus provides information on the noise correlation. We show on simulated datasets of 2D MR brain images that the estimation algorithm can successfully recover parameters of the stochastic model.

Sune Darkner, Stefan Sommer, Andreas Schuhmacher et al.

Humans are highly dependent on the ability to process audio in order to interact through conversation and navigate from sound. For this, the shape of the ear acts as a mechanical audio filter. The anatomy of the outer human ear canal to approximately 15-20 mm beyond the Tragus is well described because of its importance for customized hearing aid production. This is however not the case for the part of the ear canal that is embedded in the skull, until the typanic membrane. Due to the sensitivity of the outer ear, this part, referred to as the bony part, has only been described in a few population studies and only ex-vivo. We present a study of the entire ear canal including the bony part and the tympanic membrane. We form an average ear canal from a number of MRI scans using standard image registration methods. We show that the obtained representation is realistic in the sense that it has acoustical properties almost identical to a real ear.

Mauricio Orbes Arteaga, Lauge Sørensen, M. Jorge Cardoso et al.

For proper generalization performance of convolutional neural networks (CNNs) in medical image segmentation, the learnt features should be invariant under particular non-linear shape variations of the input. To induce invariance in CNNs to such transformations, we propose Probabilistic Augmentation of Data using Diffeomorphic Image Transformation (PADDIT) -- a systematic framework for generating realistic transformations that can be used to augment data for training CNNs. We show that CNNs trained with PADDIT outperforms CNNs trained without augmentation and with generic augmentation in segmenting white matter hyperintensities from T1 and FLAIR brain MRI scans.

Line Kuhnel, Tom Fletcher, Sarang Joshi et al.

Given data, deep generative models, such as variational autoencoders (VAE) and generative adversarial networks (GAN), train a lower dimensional latent representation of the data space. The linear Euclidean geometry of data space pulls back to a nonlinear Riemannian geometry on the latent space. The latent space thus provides a low-dimensional nonlinear representation of data and classical linear statistical techniques are no longer applicable. In this paper we show how statistics of data in their latent space representation can be performed using techniques from the field of nonlinear manifold statistics. Nonlinear manifold statistics provide generalizations of Euclidean statistical notions including means, principal component analysis, and maximum likelihood fits of parametric probability distributions. We develop new techniques for maximum likelihood inference in latent space, and adress the computational complexity of using geometric algorithms with high-dimensional data by training a separate neural network to approximate the Riemannian metric and cometric tensor capturing the shape of the learned data manifold.

Alexis Arnaudon, Darryl Holm, Stefan Sommer

Matching of images and analysis of shape differences is traditionally pursued by energy minimization of paths of deformations acting to match the shape objects. In the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, iterative gradient descents on the matching functional lead to matching algorithms informally known as Beg algorithms. When stochasticity is introduced to model stochastic variability of shapes and to provide more realistic models of observed shape data, the corresponding matching problem can be solved with a stochastic Beg algorithm, similar to the finite temperature string method used in rare event sampling. In this paper, we apply a stochastic model compatible with the geometry of the LDDMM framework to obtain a stochastic model of images and we derive the stochastic version of the Beg algorithm which we compare with the string method and an expectation-maximization optimization of posterior likelihoods. The algorithm and its use for statistical inference is tested on stochastic LDDMM landmarks and images.

Stefan Sommer

We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covariant tensors to allow global transport of principal eigenvectors, and the model is thereby an example of how principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. We show how curvature implies non-integrability of the equivalent of Euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction of such subspaces. We describe estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties of the model on embedded surfaces.

Alexis Arnaudon, Darryl Holm, Stefan Sommer

In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincaré theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms a template image that is undergoing its own internal dynamics as it deforms. This type of deformation allows more freedom for image matching and has analogies with complex fluids when the template properties are regarded as order parameters (coset spaces of broken symmetries). The first stochastic perturbation we consider corresponds to uncertainty due to random errors in the reconstruction of the deformation map from its vector field. We also consider a second stochastic perturbation, which compounds the uncertainty in of the deformation map with the uncertainty in the reconstruction of the template position from its velocity field. We apply this general geometric theory to several classical examples, including landmarks, images, and closed curves, and we discuss its use for functional data analysis.

Stefan Sommer, Alexis Arnaudon, Line Kuhnel et al.

We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood.

Akshay Pai, Stefan Sommer, Lars Lau Raket et al.

Template estimation plays a crucial role in computational anatomy since it provides reference frames for performing statistical analysis of the underlying anatomical population variability. While building models for template estimation, variability in sites and image acquisition protocols need to be accounted for. To account for such variability, we propose a generative template estimation model that makes simultaneous inference of both bias fields in individual images, deformations for image registration, and variance hyperparameters. In contrast, existing maximum a posterori based methods need to rely on either bias-invariant similarity measures or robust image normalization. Results on synthetic and real brain MRI images demonstrate the capability of the model to capture heterogeneity in intensities and provide a reliable template estimation from registration.

Alexis Arnaudon, Darryl D. Holm, Stefan Sommer

We introduce a stochastic model of diffeomorphisms, whose action on a variety of data types descends to stochastic evolution of shapes, images and landmarks. The stochasticity is introduced in the vector field which transports the data in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for shape analysis and image registration. The stochasticity thereby models errors or uncertainties of the flow in following the prescribed deformation velocity. The approach is illustrated in the example of finite dimensional landmark manifolds, whose stochastic evolution is studied both via the Fokker-Planck equation and by numerical simulations. We derive two approaches for inferring parameters of the stochastic model from landmark configurations observed at discrete time points. The first of the two approaches matches moments of the Fokker-Planck equation to sample moments of the data, while the second approach employs an Expectation-Maximisation based algorithm using a Monte Carlo bridge sampling scheme to optimise the data likelihood. We derive and numerically test the ability of the two approaches to infer the spatial correlation length of the underlying noise.

Alexis Arnaudon, Darryl D. Holm, Akshay Pai et al.

In the study of shapes of human organs using computational anatomy, variations are found to arise from inter-subject anatomical differences, disease-specific effects, and measurement noise. This paper introduces a stochastic model for incorporating random variations into the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. By accounting for randomness in a particular setup which is crafted to fit the geometrical properties of LDDMM, we formulate the template estimation problem for landmarks with noise and give two methods for efficiently estimating the parameters of the noise fields from a prescribed data set. One method directly approximates the time evolution of the variance of each landmark by a finite set of differential equations, and the other is based on an Expectation-Maximisation algorithm. In the second method, the evaluation of the data likelihood is achieved without registering the landmarks, by applying bridge sampling using a stochastically perturbed version of the large deformation gradient flow algorithm. The method and the estimation algorithms are experimentally validated on synthetic examples and shape data of human corpora callosa.

Line Kühnel, Stefan Sommer, Akshay Pai et al.

This paper introduces a class of mixed-effects models for joint modeling of spatially correlated intensity variation and warping variation in 2D images. Spatially correlated intensity variation and warp variation are modeled as random effects, resulting in a nonlinear mixed-effects model that enables simultaneous estimation of template and model parameters by optimization of the likelihood function. We propose an algorithm for fitting the model which alternates estimation of variance parameters and image registration. This approach avoids the potential estimation bias in the template estimate that arises when treating registration as a preprocessing step. We apply the model to datasets of facial images and 2D brain magnetic resonance images to illustrate the simultaneous estimation and prediction of intensity and warp effects.

Stefan Sommer, Henry O. Jacobs

We survey the role of symmetry in diffeomorphic registration of landmarks, curves, surfaces, images and higher-order data. The infinite dimensional problem of finding correspondences between objects can for a range of concrete data types be reduced resulting in compact representations of shape and spatial structure. This reduction is possible because the available data is incomplete in encoding the full deformation model. Using reduction by symmetry, we describe the reduced models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. Symmetry also arises in reduction to the Lie algebra using particle relabeling symmetry allowing the equations of motion to be written purely in terms of Eulerian velocity field. Reduction by symmetry has recently been applied for jet-matching and higher-order discrete approximations of the image matching problem. We outline these constructions and further cases where reduction by symmetry promises new approaches to registration of complex data types.

Henry O. Jacobs, Stefan Sommer

We discretize a cost functional for image registration problems by deriving Taylor expansions for the matching term. Minima of the discretized cost functionals can be computed with no spatial discretization error, and the optimal solutions are equivalent to minimal energy curves in the space of $k$-jets. We show that the solutions convergence to optimal solutions of the original cost functional as the number of particles increases with a convergence rate of $O(h^{d+k})$ where $h$ is a resolution parameter. The effect of this approach over traditional particle methods is illustrated on synthetic examples and real images.