The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Michael Miller - One of the best experts on this subject based on the ideXlab platform.
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hierarchical Computational Anatomy unifying the molecular to tissue continuum via measure representations of the brain
bioRxiv, 2021Co-Authors: Michael Miller, Daniel J Tward, Alain TrouveAbstract:ABSTRACT This paper presents a unified representation of the brain based on mathematical functional measures integrating the molecular and cellular scale descriptions with continuum tissue scale descriptions. We present a fine-to-coarse recipe for traversing the brain as a hierarchy of measures projecting functional description into stable empirical probability laws that unifies scale-space aggregation. The representation uses measure norms for mapping the brain across scales from different measurement technologies. Brainspace is constructed as a metric space with metric comparison between brains provided by a hierarchy of Hamiltonian geodesic flows of diffeomorphisms connecting the molecular and continuum tissue scales. The diffeomorphisms act on the brain measures via the 3D varifold action representing “copy and paste” so that basic particle quantities that are conserved biologically are combined with greater multiplicity and not geometrically distorted. Two applications are examined, the first histological and tissue scale data in the human brain for studying Alzheimer’s disease, and the second the RNA and cell signatures of dense spatial transcriptomics mapped to the meso-scales of brain atlases. The representation unifies the classical formalism of Computational Anatomy for representing continuum tissue scale with non-classical generalized functions appropriate for molecular particle scales.
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coarse to fine hamiltonian dynamics of hierarchical flows in Computational Anatomy
Computer Vision and Pattern Recognition, 2020Co-Authors: Michael Miller, Daniel J Tward, Alain TrouveAbstract:We present here the Hamiltonian control equations for hierarchical diffeomorphic flows of particles. We define the controls to be a series of multi-scale vector fields, each with their own reproducing kernel Hilbert space norm. The hierarchical control is connected across scale through successive refinements that refine as they ascend the hierarchy with commensurately higher bandwidth Green's kernels. Interestingly the geodesic equations do not separate, with fine scale motions determined by all of the particle information simultaneously, from coarse to fine. Additionally, the hierarchical conservation law is derived, defining the geodesics and demonstrating the constancy of the Hamiltonian. We show results on one simulated example and one example from histological images of an Alzheimer's disease brain. We introduce the varifold action to transport the weights of micro-scale particles for mapping to sub millimeter scale cortical folds.
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expanding the Computational Anatomy gateway from clinical imaging to basic neuroscience research
Proceedings of the Practice and Experience in Advanced Research Computing on Rise of the Machines (learning), 2019Co-Authors: Daniel J Tward, Anthony Kolasny, Fatima Khan, Juan C Troncoso, Michael MillerAbstract:The Computational Anatomy Gateway, powered largely by the Comet (San Diego Super-computer Center) and Stampede (Texas Advanced Computing Center) clusters through XSEDE, provides software as a service tools for atlas based analysis of human brain magnetic resonance images. This includes deformable registration, automatic labeling of tissue types, and morphometric analysis. Our goal is to extend these services to the broader neuroscience community, accommodating multiple model organisms and imaging modalities, as well as low quality or missing data. We developed a new approach to multimodality registration: by predicting one modality from another, we can replace ad hoc image similarity metrics (such as mutual information or normalized cross correlation) with a log likelihood under a noise model. This statistical approach enables us to account for missing data using the Expectation Maximization algorithm. For portability and scalability we have implemented this algorithm in tensorflow. For accessibility we have compiled and many working examples for multiple model organisms, imaging systems, and missing tissue or image anomaly situations. These examples are made easily usable in the form of Jupyter notebooks, and made publicly available through github. This framework will significantly reduce the barrier to entry for basic neuroscientists, enabling the community to benefit from atlas based Computational image analysis techniques.
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on variational solutions for whole brain serial section histology using a sobolev prior in the Computational Anatomy random orbit model
PLOS Computational Biology, 2018Co-Authors: Daniel J Tward, Partha P Mitra, Michael MillerAbstract:This paper presents a variational framework for dense diffeomorphic atlas-mapping onto high-throughput histology stacks at the 20 μm meso-scale. The observed sections are modelled as Gaussian random fields conditioned on a sequence of unknown section by section rigid motions and unknown diffeomorphic transformation of a three-dimensional atlas. To regularize over the high-dimensionality of our parameter space (which is a product space of the rigid motion dimensions and the diffeomorphism dimensions), the histology stacks are modelled as arising from a first order Sobolev space smoothness prior. We show that the joint maximum a-posteriori, penalized-likelihood estimator of our high dimensional parameter space emerges as a joint optimization interleaving rigid motion estimation for histology restacking and large deformation diffeomorphic metric mapping to atlas coordinates. We show that joint optimization in this parameter space solves the classical curvature non-identifiability of the histology stacking problem. The algorithms are demonstrated on a collection of whole-brain histological image stacks from the Mouse Brain Architecture Project.
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Computational Anatomy and diffeomorphometry a dynamical systems model of neuroAnatomy in the soft condensed matter continuum
Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 2018Co-Authors: Michael Miller, Daniel J Tward, Sylvain Arguillere, Laurent YounesAbstract:The nonlinear systems models of Computational Anatomy that have emerged over the past several decades are a synthesis of three significant areas of Computational science and biological modeling. First is the algebraic model of biological shape as a Riemannian orbit, a set of objects under diffeomorphic action. Second is the embedding of anatomical shapes into the soft condensed matter physics continuum via the extension of the Euler equations to geodesic, smooth flows with inverses, encoding divergence for the compressibility of atrophy and expansion of growth. Third, is making human shape and form a metrizable space via geodesic connections of coordinate systems. These three themes place our formalism into the modern data science world of personalized medicine supporting inference of high-dimensional anatomical phenotypes for studying neurodegeneration and neurodevelopment. The dynamical systems model of growth and atrophy that emerges is one which is organized in terms of forces, accelerations, velocities, and displacements, with the associated Hamiltonian momentum and the diffeomorphic flow acting as the state, and the smooth vector field the control. The forces that enter the model derive from external measurements through which the dynamical system must flow, and the internal potential energies of structures making up the soft condensed matter. We examine numerous examples on growth and atrophy. This article is categorized under: Analytical and Computational Methods > Computational Methods Laboratory Methods and Technologies > Imaging Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.
Alain Trouve - One of the best experts on this subject based on the ideXlab platform.
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hierarchical Computational Anatomy unifying the molecular to tissue continuum via measure representations of the brain
bioRxiv, 2021Co-Authors: Michael Miller, Daniel J Tward, Alain TrouveAbstract:ABSTRACT This paper presents a unified representation of the brain based on mathematical functional measures integrating the molecular and cellular scale descriptions with continuum tissue scale descriptions. We present a fine-to-coarse recipe for traversing the brain as a hierarchy of measures projecting functional description into stable empirical probability laws that unifies scale-space aggregation. The representation uses measure norms for mapping the brain across scales from different measurement technologies. Brainspace is constructed as a metric space with metric comparison between brains provided by a hierarchy of Hamiltonian geodesic flows of diffeomorphisms connecting the molecular and continuum tissue scales. The diffeomorphisms act on the brain measures via the 3D varifold action representing “copy and paste” so that basic particle quantities that are conserved biologically are combined with greater multiplicity and not geometrically distorted. Two applications are examined, the first histological and tissue scale data in the human brain for studying Alzheimer’s disease, and the second the RNA and cell signatures of dense spatial transcriptomics mapped to the meso-scales of brain atlases. The representation unifies the classical formalism of Computational Anatomy for representing continuum tissue scale with non-classical generalized functions appropriate for molecular particle scales.
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coarse to fine hamiltonian dynamics of hierarchical flows in Computational Anatomy
Computer Vision and Pattern Recognition, 2020Co-Authors: Michael Miller, Daniel J Tward, Alain TrouveAbstract:We present here the Hamiltonian control equations for hierarchical diffeomorphic flows of particles. We define the controls to be a series of multi-scale vector fields, each with their own reproducing kernel Hilbert space norm. The hierarchical control is connected across scale through successive refinements that refine as they ascend the hierarchy with commensurately higher bandwidth Green's kernels. Interestingly the geodesic equations do not separate, with fine scale motions determined by all of the particle information simultaneously, from coarse to fine. Additionally, the hierarchical conservation law is derived, defining the geodesics and demonstrating the constancy of the Hamiltonian. We show results on one simulated example and one example from histological images of an Alzheimer's disease brain. We introduce the varifold action to transport the weights of micro-scale particles for mapping to sub millimeter scale cortical folds.
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distortion minimizing geodesic subspaces in shape spaces and Computational Anatomy
European Congress on Computational Methods in Applied Sciences and Engineering, 2017Co-Authors: Benjamin Charlier, Jean Feydy, David W Jacobs, Alain TrouveAbstract:The estimation of finite dimensional nonlinear submanifold representing shape samples is of paramount importance in many applications. The Distortion Minimizing Geodesic Submanifold (DMGS) approach allows to select the most accurate submanifolds in term of distortion under a dimensionality constraint for shape spaces. We show that the computation of DMGS is widely compatible with the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework and the varifold distortion for application to Computational Anatomy. It allows the estimation of finite dimensional geodesic submanifolds in the difficult situation where we do not assume any one to one correspondance between shapes (parametrisation invariance). Unlike regular Tangent PCA, the computation of DMGS does not need to deal with the classical balance between the deformation cost from the template to target and the resulting distortion. On the contrary, the greedy minimization of the distortion under dimensionality constraints, hiding the deformation metric in the exponential map, suggests a new way to select between alternative metrics and shape spaces under the unifying point of view of the dimension/distortion curves in the spirit of the rate/distortion curves in information theory. Proof of concept on 2D and 3D experiments are discussed.
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template estimation in Computational Anatomy frechet means in top and quotient spaces are not consistent
Siam Journal on Imaging Sciences, 2017Co-Authors: Loic Devilliers, Alain Trouve, Stephanie Allassonniere, Xavier PennecAbstract:In this article, we study the consistency of the template estimation with the Frechet mean in quotient spaces. The Frechet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.
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template estimation in Computational Anatomy fr e chet means in top and quotient spaces are not consistent
arXiv: Statistics Theory, 2016Co-Authors: Loic Devilliers, Alain Trouve, Stephanie Allassonniere, Xavier PennecAbstract:In this article, we study the consistency of the template estimation with the Frechet mean in quotient spaces. The Frechet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.
Laurent Younes - One of the best experts on this subject based on the ideXlab platform.
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Computational Anatomy and diffeomorphometry a dynamical systems model of neuroAnatomy in the soft condensed matter continuum
Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 2018Co-Authors: Michael Miller, Daniel J Tward, Sylvain Arguillere, Laurent YounesAbstract:The nonlinear systems models of Computational Anatomy that have emerged over the past several decades are a synthesis of three significant areas of Computational science and biological modeling. First is the algebraic model of biological shape as a Riemannian orbit, a set of objects under diffeomorphic action. Second is the embedding of anatomical shapes into the soft condensed matter physics continuum via the extension of the Euler equations to geodesic, smooth flows with inverses, encoding divergence for the compressibility of atrophy and expansion of growth. Third, is making human shape and form a metrizable space via geodesic connections of coordinate systems. These three themes place our formalism into the modern data science world of personalized medicine supporting inference of high-dimensional anatomical phenotypes for studying neurodegeneration and neurodevelopment. The dynamical systems model of growth and atrophy that emerges is one which is organized in terms of forces, accelerations, velocities, and displacements, with the associated Hamiltonian momentum and the diffeomorphic flow acting as the state, and the smooth vector field the control. The forces that enter the model derive from external measurements through which the dynamical system must flow, and the internal potential energies of structures making up the soft condensed matter. We examine numerous examples on growth and atrophy. This article is categorized under: Analytical and Computational Methods > Computational Methods Laboratory Methods and Technologies > Imaging Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.
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hamiltonian systems and optimal control in Computational Anatomy 100 years since d arcy thompson
Annual Review of Biomedical Engineering, 2015Co-Authors: Michael Miller, Alain Trouve, Laurent YounesAbstract:The Computational Anatomy project is the morphome-scale study of shape and form, which we model as an orbit under diffeomorphic group action. Metric comparison calculates the geodesic length of the diffeomorphic flow connecting one form to another. Geodesic connection provides a positioning system for coordinatizing the forms and positioning their associated functional information. This article reviews progress since the Euler-Lagrange characterization of the geodesics a decade ago. Geodesic positioning is posed as a series of problems in Hamiltonian control, which emphasize the key reduction from the Eulerian momentum with dimension of the flow of the group, to the parametric coordinates appropriate to the dimension of the submanifolds being positioned. The Hamiltonian viewpoint provides important extensions of the core setting to new, object-informed positioning systems. Several submanifold mapping problems are discussed as they apply to metamorphosis, multiple shape spaces, and longitudinal time series...
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metamorphosis of images in reproducing kernel hilbert spaces
arXiv: Optimization and Control, 2014Co-Authors: Casey L Richardson, Laurent YounesAbstract:Metamorphosis is a method for diffeomorphic matching of shapes, with many potential applications for anatomical shape comparison in medical imagery, a problem which is central to the field of Computational Anatomy. An important tool for the practical application of metamorphosis is a numerical method based on shooting from the initial momentum, as this would enable the use of statistical methods based on this momentum, as well as the estimation of templates from hyper-templates using morphing. In this paper we introduce a shooting method, in the particular case of morphing images that lie in a reproducing kernel Hilbert space (RKHS). We derive the relevant shooting equations from a Lagrangian frame of reference, present the details of the numerical approach, and illustrate the method through morphing of some simple images.
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Computational Anatomy gateway leveraging xsede Computational resources for shape analysis
Extreme Science and Engineering Discovery Environment, 2014Co-Authors: Saurabh Jain, Michael Miller, Tilak J Ratnanather, Daniel J Tward, David S Lee, Anthony Kolasny, Timothy Brown, Laurent YounesAbstract:Computational Anatomy (CA) is a discipline focused on the quantitative analysis of the variability in biological shape. The Large Deformation Diffeomorphic Metric Mapping (LDDMM) is the key algorithm which assigns computable descriptors of anatomical shapes and a metric distance between shapes. This is achieved by describing populations of anatomical shapes as a group of diffeomorphic transformations applied to a template, and using a metric on the space of diffeomorphisms. LDDMM is being used extensively in the neuroimaging (www.mristudio.org) and cardiovascular imaging (www.cvrgrid.org) communities. There are two major components involved in shape analysis using this paradigm. First is the estimation of the template, and second is calculating the diffeomorphisms mapping the template to each subject in the population. Template estimation is a Computationally expensive problem, which involves an iterative process, where each iteration calculates one diffeomorphism for each target. These can be calculated in parallel and independently of each other, and XSEDE is providing the resources, in particular those provided by the cluster Stampede, that make these computations for large populations possible. Mappings from the estimated template to each subject can also be run in parallel. In addition, the use of NVIDIA Tesla GPUs available on Stampede present the possibility of speeding up certain convolution-like calculations which lend themselves well to the General Purpose GPU computation model. We are also exploring the use of the available Xeon Phi Co-processors to increase the efficiency of our codes. This will have a huge impact on both the neuroimaging and cardiac imaging communities as we bring these shape analysis tools online for use by these communities through our webservice (www.mricloud.org), with the XSEDE Computational Anatomy Gateway providing the resources to handle the Computational demands for large populations.
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evolutions equations in Computational Anatomy
NeuroImage, 2009Co-Authors: Laurent Younes, Felipe Arrate, Michael MillerAbstract:One of the main purposes in Computational Anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Over the last decade, our group has progressively developed several approaches for this problem, all related to the Riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act. Several important shape evolution equations that are now used routinely in applications have emerged over time. Our goal in this paper is to provide an overview of these equations, placing them in their theoretical context, and giving examples of applications in which they can be used. We introduce the required theoretical background before discussing several classes of equations of increasingly complexity. These equations include energy minimizing evolutions deriving from Riemannian gradient descent, geodesics, parallel transport and Jacobi fields.
Xavier Pennec - One of the best experts on this subject based on the ideXlab platform.
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statistical analysis of organs shapes and deformations the riemannian and the affine settings in Computational Anatomy
2020Co-Authors: Xavier PennecAbstract:Computational Anatomy is an emerging discipline at the interface of geometry, statistics and medicine that aims at analyzing and modeling the biological variability of organs' shapes at the population level. Shapes are equivalence classes of images, surfaces or deformations of a template under rigid body (or more general) transformations. Thus, they belong to non-linear manifolds. In order to deal with multiple samples in non-linear spaces, a consistent statistical framework on Riemannian manifolds has been designed over the last decade. We detail in this chapter the extension of this framework to Lie groups endowed with the affine symmetric connection, a more invariant (and thus more consistent) but non-metric structure on transformation groups. This theory provides strong theoretical bases for the use of one-parameter subgroups and diffeomorphisms parametrized by stationary velocity fields (SVF), for which efficient image registration methods like log-Demons have been developed with a great success from the practical point of view. One can further reduce the complexity with locally affine transformations , leading to parametric diffeomorphisms of low dimension encoding the major shape variability. We illustrate the methodology with the modeling of the evolution of the brain with Alzheimer's disease and the analysis of the cardiac motion from MRI sequences of images.
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multimodal brain image analysis and mathematical foundations of Computational Anatomy
2019Co-Authors: Heng Huang, Xavier Pennec, Mads Nielsen, Tom Fletcher, Stanley Durrleman, Carl-fredrik Westin, Sarang Joshi, Paul M. Thompson, Li Shen, Stefan SommerAbstract:This book constitutes the refereed joint proceedings of the 4th International Workshop on Multimodal Brain Image Analysis, MBAI 2019, and the 7th International Workshop on Mathematical Foundations of Computational Anatomy, MFCA 2019, held in conjunction with the 22nd International Conference on Medical Imaging and Computer-Assisted Intervention, MICCAI 2019, in Shenzhen, China, in October 2019. The 16 full papers presented at MBAI 2019 and the 7 full papers presented at MFCA 2019 were carefully reviewed and selected. The MBAI papers intend to move forward the state of the art in multimodal brain image analysis, in terms of analysis methodologies, algorithms, software systems, validation approaches, benchmark datasets, neuroscience, and clinical applications. The MFCA papers are devoted to statistical and geometrical methods for modeling the variability of biological shapes. The goal is to foster the interactions between the mathematical community around shapes and the MICCAI community around Computational Anatomy applications.
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graphs in biomedical image analysis Computational Anatomy and imaging genetics
2017Co-Authors: Jorge M Cardoso, Xavier Pennec, Sarang Joshi, Tal Arbel, Enzo Ferrante, Adrian V. Dalca, Sarah Parisot, Nematollah K. Batmanghelich, Aristeidis Sotiras, Mads NielsenAbstract:This book constitutes the refereed joint proceedings of the First International Workshop on Graphs in Biomedical Image Analysis, GRAIL 2017, the 6th International Workshop on Mathematical Foundations of Computational Anatomy, MFCA 2017, and the Third International Workshop on Imaging Genetics, MICGen 2017, held in conjunction with the 20th International Conference on Medical Imaging and Computer-Assisted Intervention, MICCAI 2017, in Quebec City, QC, Canada, in September 2017.
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template estimation in Computational Anatomy frechet means in top and quotient spaces are not consistent
Siam Journal on Imaging Sciences, 2017Co-Authors: Loic Devilliers, Alain Trouve, Stephanie Allassonniere, Xavier PennecAbstract:In this article, we study the consistency of the template estimation with the Frechet mean in quotient spaces. The Frechet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.
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template estimation in Computational Anatomy fr e chet means in top and quotient spaces are not consistent
arXiv: Statistics Theory, 2016Co-Authors: Loic Devilliers, Alain Trouve, Stephanie Allassonniere, Xavier PennecAbstract:In this article, we study the consistency of the template estimation with the Frechet mean in quotient spaces. The Frechet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.
Darryl D Holm - One of the best experts on this subject based on the ideXlab platform.
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a stochastic large deformation model for Computational Anatomy
arXiv: Computer Vision and Pattern Recognition, 2016Co-Authors: Alexis Arnaudon, Darryl D Holm, Akshay Pai, Stefan SommerAbstract: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.
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geometry of image registration the diffeomorphism group and momentum maps
arXiv: Differential Geometry, 2015Co-Authors: Martins Bruveris, Darryl D HolmAbstract:These lecture notes explain the geometry and discuss some of the analytical questions underlying image registration within the framework of large deformation diffeomorphic metric mapping (LDDMM) used in Computational Anatomy.
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inexact trajectory planning and inverse problems in the hamilton pontryagin framework
arXiv: Dynamical Systems, 2013Co-Authors: Christopher L Burnett, Darryl D Holm, David M MeierAbstract:We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in Computational Anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler-Lagrange equations is obtained from a higher-order Hamilton-Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre-Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton-Pontryagin principle and preserves the geometric properties of the continuous-time solution.
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interaction dynamics of singular wave fronts
arXiv: Pattern Formation and Solitons, 2013Co-Authors: Darryl D Holm, Martin F StaleyAbstract:Some of the most impressive singular wave fronts seen in Nature are the transbasin oceanic internal waves, which may be observed from the Space Shuttle as they propagate and interact with each other, for example, in the South China Sea. The characteristic feature of these strongly nonlinear wavefronts is that they reconnect when two of them collide transversely. We derive the EPDiff equation, and use it to model this phenomenon as elastic collisions between singular wave fronts (solitons) whose momentum is distributed along curves moving in the plane. Numerical methods for EPDiff based on compatible differencing algorithms (CDAs) are used for simulating these collisions among curves. The numerical results show the same nonlinear behavior of wavefront reconnections as that observed for internal waves in the South China Sea. We generalize the singular solutions of EPDiff for other applications, in Computational Anatomy and in imaging science, where the singular wavefronts are evolving image outlines, whose momentum may be distributed on surfaces moving though space in three dimensions. The key idea is always momentum exchange during collisions of the wavefronts. A suite of 2d and 3d numerical simulations provide collision rules for the wavefront reconnection phenomenon in a variety of scenarios.
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the momentum map representation of images
Journal of Nonlinear Science, 2011Co-Authors: Martins Bruveris, Darryl D Holm, Francois Gaybalmaz, Tudor S RatiuAbstract:This paper discusses the mathematical framework for designing methods of Large Deformation Diffeomorphic Matching (LDM) for image registration in Computational Anatomy. After reviewing the geometrical framework of LDM image registration methods, we prove a theorem showing that these methods may be designed by using the actions of diffeomorphisms on the image data structure to define their associated momentum representations as (cotangent-lift) momentum maps. To illustrate its use, the momentum map theorem is shown to recover the known algorithms for matching landmarks, scalar images, and vector fields. After briefly discussing the use of this approach for diffusion tensor (DT) images, we explain how to use momentum maps in the design of registration algorithms for more general data structures. For example, we extend our methods to determine the corresponding momentum map for registration using semidirect product groups, for the purpose of matching images at two different length scales. Finally, we discuss the use of momentum maps in the design of image registration algorithms when the image data is defined on manifolds instead of vector spaces.