Geometry of Image Registration: The Diffeomorphism Group and Momentum Maps

Bruveris, Martins; Holm, Darryl D.

doi:10.1007/978-1-4939-2441-7_2

Cited by 24 publications

(53 citation statements)

References 80 publications

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“…It is being applied in computational anatomy with images being MRI and CT scans to study the connections between anatomical shape and physiological function. See [23] for an overview of image registration within the LDDMM framework.…”

Section: Pattern Theorymentioning

confidence: 99%

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Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

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Section: Pattern Theorymentioning

confidence: 99%

“…2] that the minimization problems (23) and (26) are equivalent and that the induced distance functions are equal: Theorem 9.2 (Prop. 2, [87]) Let v be a minimizer of the energy functional (22).…”

Section: The Space Of Landmarksmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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“…A useful case is when V is a reproducing kernel Hilbert space (RKHS) with a symmetric and positive-definite reproducing kernel. Then V is an admissible Hilbert space [11]. In the rest of the paper, the space of vector fields is selected as an admissible RKHS for the advantages of sufficient smoothness and fast computability [17].…”

Section: The Lddmm Frameworkmentioning

confidence: 99%

“…The approach taken in this paper belongs to the second category and the motion model makes use of diffeomorphic deformations. The latter are provided by the large deformation diffeomorphic metric mapping (LDDMM) framework, which is a well-developed framework for diffeomorphic image registration (see, for instance, [53,25,37,5,30,60,54,55,11,12]). Diffeomorphic deformations based on LDDMM were used in [32] for joint image reconstruction and motion estimation in 4D computed tomography (CT), which is based on the growth model of LDDMM [30].…”

Section: Introductionmentioning

confidence: 99%

A New Variational Model for Joint Image Reconstruction and Motion Estimation in Spatiotemporal Imaging

Chen¹,

Gris²,

Öktem³

2019

SIAM J. Imaging Sci.

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We propose a new variational model for joint image reconstruction and motion estimation in spatiotemporal imaging, which is investigated along a general framework that we present with shape theory. This model consists of two components, one for conducting modified static image reconstruction, and the other performs sequentially indirect image registration. For the latter, we generalize the large deformation diffeomorphic metric mapping framework into the sequentially indirect registration setting. The proposed model is compared theoretically against alternative approaches (optical flow based model and diffeomorphic motion models), and we demonstrate that the proposed model has desirable properties in terms of the optimal solution. The theoretical derivations and efficient algorithms are also presented for a time-discretized scenario of the proposed model, which show that the optimal solution of the time-discretized version is consistent with that of the time-continuous one, and most of the computational components is the easy-implemented linearized deformation. The complexity of the algorithm is analyzed as well. This work is concluded by some numerical examples in 2D space + time tomography with very sparse and/or highly noisy data.

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“…Диффеоморфизмы не позволяют изменить топологию вдоль геодезических траекторий. Неточный вид диффеоморфизмов [6,7] обеспечи-вает механизм, который позволяет при эволюции геодезических отклоняться от точных деформаций. В задаче неточного сравнения минимизируемый функционал содержит член, который оценивает точность попадания точек…”

Section: Introductionunclassified