Geometric modeling in shape space

Kilian, Martin; Mitra, Niloy J.; Pottmann, Helmut

doi:10.1145/1239451.1239515

Cited by 30 publications

(37 citation statements)

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“…Among all existing approaches, a number of different concepts of a shape are employed, including landmark vectors [11,1], planar curves [12,13,14], surfaces in R 3 [4,15,16], boundary contours of objects [7,17,18], multiphase objects [19] as well as the morphologies of images [20].…”

Section: A Review Of Different Shape Space Conceptsmentioning

confidence: 99%

Variational Methods in Shape Analysis

Rumpf¹,

Wirth²

2015

Handbook of Mathematical Methods in Imaging

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The analysis of shapes as elements in a frequently infinite-dimensional space of shapes has attracted increasing attention over the last decade. There are pioneering contributions in the theoretical foundation of shape space as a Riemannian manifold as well as path-breaking applications to quantitative shape comparison, shape recognition, and shape statistics. The aim of this chapter is to adopt a primarily physical perspective on the space of shapes and to relate this to the prevailing geometric perspective. Indeed, we here consider shapes given as boundary contours of volumetric objects, which consist either of a viscous fluid or an elastic solid.In the first case, shapes are transformed into each other via viscous transport of fluid material, and the flow naturally generates a connecting path in the space of shapes. The viscous dissipation rate-the rate at which energy is converted into heat due to friction-can be defined as a metric on an associated Riemannian manifold. Hence, via the computation of shortest transport paths one defines a distance measure between shapes.In the second case, shapes are transformed via elastic deformations, where the associated elastic energy only depends on the final state of the deformation and not on the path along which the deformation is generated. The minimal elastic energy required to deform an object into another one can be considered as a dissimilarity measure between the corresponding shapes.In what follows we discuss and extensively compare the path-based and the state-based approach. As applications of the elastic shape model we consider shape averages and a principal component analysis of shapes. The viscous flow model is used to exemplarily cluster 2D and 3D shapes and to construct a flow type nonlinear interpolation scheme. Furthermore, we show how to approximate the viscous, path-based approach with a time-discrete sequence of state-based variational problems. A review of different shape space conceptsThe structure of shape spaces and statistical analyses of shapes have been examined in various settings, and applications range from the computation of priors for segmentation [1,2,3] and shape classification [4,5,6,7] to the construction of standardized anatomical atlases [8,9,10]. Among all existing approaches, a number of different concepts of a shape are employed, including landmark vectors [11,1], planar curves [12,13,14], surfaces in R 3 [4,15,16], boundary contours of

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Section: A Review Of Different Shape Space Conceptsmentioning

confidence: 99%

Variational Methods in Shape Analysis

Rumpf¹,

Wirth²

2015

Handbook of Mathematical Methods in Imaging

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“…Suppose two different deformation fields as X and Y , the distance between them are measured by X, Y in shape space. Some geometric structures have been established to provide the Isometric and Rigid metrics based on the Riemannian metric definition [7]. Here, we propose a quasi-conformal metric within this structure.…”

Section: Quasi-conformal Deformation Estimationmentioning

confidence: 99%

“…Likewise, all the segments are transferred so that the final transferred deformationÝ is achieved [7].…”

Section: Deformation Transfer and Dimension Reductionmentioning

confidence: 99%

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Comparative Analysis of Quasi-Conformal Deformations in Shape Space

Taimouri

Hua

2010

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2010

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Abstract.A novel approach based on the shape space concept is proposed to classify quasi-conformal deformations of 3D models. A new metric on the quotient space of meshes is introduced to capture changes of the curvature at each vertex of a simplicial complex during deformation. Then, the deformation curve is obtained by calculating the geodesic curve connecting two shapes in the shape space manifold. In order to compare the deformations, the deformation curves are first transferred to the same part of shape space. And then, the Multi-Dimensional Scaling method is employed to eliminate the redundant dimensions facilitating easy comparison of the deformations. To evaluate our method, some synthetic datasets and 23 datasets of gated images of the left heart ventricle during one heartbeat have been examined. Our experiments show that the algorithm can effectively classify normal and abnormal left heart ventricle deformations in shape space.

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“…It is this latter application that inspired the present work. Some alternatives to ARAP are As-Isometric-As-Possible (AIAP) methods that preserve distances within the mesh [39,40], the As-Similar-As-Possible method (ASAP) which enforces the preservation of the mesh tetrahedrals [41], path deformation techniques based on physics [42] or possible dynamics [43], and interpolation methods based on specific surface representations [44][45][46][47][48]. The reader is referred to [49] for a survey of mesh deformation techniques.…”

Section: Introductionmentioning

confidence: 99%

As-Rigid-As-Possible molecular interpolation paths

Nguyen

Jaillet

Redon

2017

J Comput Aided Mol Des

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This paper proposes a new method to generate interpolation paths between two given molecular conformations. It relies on the As-Rigid-AsPossible (ARAP) paradigm used in Computer Graphics to manipulate complex meshes while preserving their essential structural characteristics. The adaptation of ARAP approaches to the case of molecular systems is presented in this contribution. Experiments conducted on a large set of benchmarks show how such a strategy can efficiently compute relevant interpolation paths with large conformational rearrangements.

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Geometric modeling in shape space

Cited by 30 publications

References 0 publications

Variational Methods in Shape Analysis

Variational Methods in Shape Analysis

Comparative Analysis of Quasi-Conformal Deformations in Shape Space

As-Rigid-As-Possible molecular interpolation paths

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