| 1. |
- Balehowsky, Tracey, et al.
(author)
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Shape analysis via gradient flows on diffeomorphism groups
- 2023
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In: Nonlinearity. - : IOP Publishing. - 1361-6544 .- 0951-7715. ; 36:2, s. 862-877
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Journal article (peer-reviewed)abstract
- We study a Riemannian gradient flow on Sobolev diffeomorphisms for the problem of image registration. The energy functional quantifies the effect of transforming a template to a target, while also penalizing non-isometric deformations. The main result is well-posedness of the flow. We also give a geometric description of the gradient in terms of the momentum map.
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| 2. |
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| 3. |
- Bauer, M., et al.
(author)
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Diffeomorphic density matching by optimal information transport
- 2015
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In: SIAM Journal on Imaging Sciences. - : Society for Industrial & Applied Mathematics (SIAM). - 1936-4954. ; 8:3, s. 1718-1751
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Journal article (peer-reviewed)abstract
- We address the following problem: given two smooth densities on a manifold, find an optimal diffeomorphism that transforms one density into the other. Our framework builds on connections between the Fisher–Rao information metric on the space of probability densities and right-invariant metrics on the infinite-dimensional manifold of diffeomorphisms. This optimal information transport, and modifications thereof, allow us to construct numerical algorithms for density matching. The algorithms are inherently more efficient than those based on optimal mass transport or diffeomorphic registration. Our methods have applications in medical image registration, texture mapping, image morphing, nonuniform random sampling, and mesh adaptivity. Some of these applications are illustrated in examples.
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| 4. |
- Bauer, M., et al.
(author)
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Diffeomorphic density registration
- 2019
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In: Riemannian Geometric Statistics in Medical Image Analysis. - : Elsevier. - 9780128147269 ; , s. 577-603
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Book chapter (other academic/artistic)abstract
- In this book chapter we study the Riemannian geometry of the density registration problem: Given two densities (not necessarily probability densities) defined on a smooth finite-dimensional manifold find a diffeomorphism which transforms one to the other. This problem is motivated by the medical imaging application of tracking organ motion due to respiration in thoracic CT imaging, where the fundamental physical property of conservation of mass naturally leads to modeling CT attenuation as a density. We will study the intimate link between the Riemannian metrics on the space of diffeomorphisms and those on the space of densities. We finally develop novel computationally efficient algorithms and demonstrate their applicability for registering thoracic respiratory correlated CT imaging.
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| 5. |
- Bauer, Martin, et al.
(author)
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Diffeomorphic random sampling using optimal information transport
- 2017
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In: International Conference on Geometric Science of Information. GSI 2017: Geometric Science of Information. Lecture Notes in Computer Science.. - Cham : Springer. - 0302-9743 .- 1611-3349. - 9783319684444
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Conference paper (peer-reviewed)abstract
- © 2017, Springer International Publishing AG. In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)—an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge–Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.
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| 6. |
- Bauer, M., et al.
(author)
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On geodesic completeness for Riemannian metrics on smooth probability densities
- 2017
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In: Calculus of Variations and Partial Differential Equations. - : Springer Science and Business Media LLC. - 0944-2669 .- 1432-0835. ; 56:4
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Journal article (peer-reviewed)abstract
- The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are the Otto metric, yielding the L-2-Wasserstein distance of optimal mass transport, and the Fisher-Rao metric, predominant in the theory of information geometry. On the space of smooth probability densities, none of these Riemannian metrics are geodesically complete-a property desirable for example in imaging applications. That is, the existence interval for solutions to the geodesic flow equations cannot be extended to the whole real line. Here we study a class of Hamilton-Jacobi-like partial differential equations arising as geodesic flow equations for higher-order Sobolev type metrics on the space of smooth probability densities. We give order conditions for global existence and uniqueness, thereby providing geodesic completeness. The system we study is an interesting example of a flow equation with loss of derivatives, which is well-posed in the smooth category, yet non-parabolic and fully non-linear. On a more general note, the paper establishes a link between geometric analysis on the space of probability densities and analysis of Euler-Arnold equations in topological hydrodynamics.
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| 7. |
- Bauer, M., et al.
(author)
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Semi-invariant Riemannian metrics in hydrodynamics
- 2020
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In: Calculus of Variations and Partial Differential Equations. - : Springer Science and Business Media LLC. - 0944-2669 .- 1432-0835. ; 59:2
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Journal article (peer-reviewed)abstract
- Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa-Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev Hk-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid's internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted.
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| 8. |
- Benn, J., et al.
(author)
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Currents and Finite Elements as Tools for Shape Space
- 2019
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In: Journal of Mathematical Imaging and Vision. - : Springer Science and Business Media LLC. - 0924-9907 .- 1573-7683. ; 61:8, s. 1197-1220
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Journal article (peer-reviewed)abstract
- The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper, we study a general representation of shapes as currents, which are based on linear spaces and are suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element-based discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples.
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| 9. |
- Bogfjellmo, Geir, et al.
(author)
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A Numerical Algorithm for C-2-Splines on Symmetric Spaces
- 2018
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In: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 56:4, s. 2623-2647
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Journal article (peer-reviewed)abstract
- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized Bézier curves. To construct $C^2$-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for $C^2$-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the $n$-sphere, complex projective space, and the real Grassmannian.
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| 10. |
- Bogfjellmo, Geir, 1987, et al.
(author)
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A Numerical Algorithm for C-2-Splines on Symmetric Spaces
- 2018
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In: SIAM Journal on Numerical Analysis. - : Siam Publications. - 1095-7170 .- 0036-1429. ; 56:4, s. 2623-2647
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Journal article (peer-reviewed)abstract
- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.
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