Constrained Diffeomorphic Shape Evolution

Younes, Laurent

doi:10.1007/s10208-011-9108-2

Cited by 22 publications

(23 citation statements)

References 28 publications

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“…A second natural requirement would be for the deformations to be locally more complex than local translations similarly to [41,8,35] and in addition to allow the geometrical supports of the deformation (for instance control points) to be dierent from the shape data as in the frameworks presented in [15,8,35]. A last interesting quality would be to dene the possible deformation patterns as priors as 3 in the models of [8,35,50]. The deformation model presented in this article aims at combining all these features.…”

Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

“…A way to tackle this problem is then not only to combine local deformation generators into a single vector eld, but also to transport the local generators with the deformation that results from the integration of the velocity elds. Such a framework is introduced in [50] where local constraints depending on the shape allow a particular control on the structure of vector elds. Besides, this structure evolves along the resulting ow and induces a sub-Riemannian structure on the shape space.…”

Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

“…From what precedes, a natural deformation model should allow the denition of a metric on shape spaces similarly to the frameworks of the LDDMM [33,49,51] and dieons [50]. A second natural requirement would be for the deformations to be locally more complex than local translations similarly to [41,8,35] and in addition to allow the geometrical supports of the deformation (for instance control points) to be dierent from the shape data as in the frameworks presented in [15,8,35].…”

Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

“…This desirable consistency property leads to consider the geometrical descriptors as a shape on which dieomorphisms act. Hence, following [50], the positioning of the geometrical descriptors will be represented as a shape in a shape space.…”

Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

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A Sub-Riemannian Modular Framework for Diffeomorphism-Based Analysis of Shape Ensembles

Gris¹,

Durrleman²,

Trouvé³

2018

SIAM J. Imaging Sci.

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Abstract. Deformations, and dieormophisms in particular, have played a tremendous role in the eld of statistical shape analysis, as a proxy to measure and interpret dierences between similar objects but with dierent shapes. Dieomorphisms usually result from the integration of a ow of regular velocity elds, whose parameters have not enabled so far a full control of the local behaviour of the deformation.In this work, we propose a new mathematical and computational framework, in which these velocity elds are constrained to be built through a combination of local deformation modules with few degrees of freedom. Deformation modules contribute to the global velocity eld, and interact with it during integration so that the local modules are transported by the global dieomorphic deformation under construction. Such modular dieomorphisms are used to deform shapes and to provide the shape space with a sub-Riemannian metric.We then derive a method to estimate a Fréchet mean from a series of observations, and to decompose the variations in shape observed in the training samples into a set of elementary deformation modules encoding distinctive and interpretable aspects of the shape variability. We show how this approach brings new solutions to long lasting problems in the elds of computer vision and medical image analysis. For instance, the easy implementation of priors in the type of deformations oers a direct control to favor one solution over another in situations where multiple solutions may t the observations equally well. It allows also the joint optimisation of a linear and a non-linear deformation between shapes, the linear transform simply being a particular type of modules.The proposed approach generalizes previous methods for constructing dieomorphisms and opens up new perspectives in the eld of statistical shape analysis.

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Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

Section: A Sub-riemannian Modular Framework For Diffeomorphism Based mentioning

confidence: 99%

See 2 more Smart Citations

A Sub-Riemannian Modular Framework for Diffeomorphism-Based Analysis of Shape Ensembles

Gris¹,

Durrleman²,

Trouvé³

2018

SIAM J. Imaging Sci.

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“…For instance, for cortical surfaces with different sulci topography, one can prefer to favour lateral displacement over the creation of new sulci. Large deformations are commonly obtained through the integration of a vector field [4,10,13,18] and a natural route is to introduce the constraints in the vector fields instead of the final diffeomorphism [19]. The vector field could be restricted, via a finite dimensional control variable, to a state dependent finite dimensional subspace generated by a finite basis and conceptualized in structures called hereafter deformation modules.…”

mentioning

confidence: 99%

A Sub-Riemannian Modular Approach for Diffeomorphic Deformations

Gris

Durrleman

Trouvé

2015

Lecture Notes in Computer Science

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Abstract. We develop a generic framework to build large deformations from a combination of base modules. These modules constitute a dynamical dictionary to describe transformations. The method, built on a coherent sub-Riemannian framework, defines a metric on modular deformations and characterises optimal deformations as geodesics for this metric. We will present a generic way to build local affine transformations as deformation modules, and display examples. IntroductionA central aspect of Computational Anatomy is the comparison of different shapes, which are encoded as meshes or images. A common approach is the study of deformations matching one shape onto another, so that the differences between the two shapes are encoded by the deformation parameters [9,11,17]. In order to study differences between subjects on a particular structure, it should be useful to constrain locally the deformation, to favour realistic anatomic deformations, or to introduce some anatomical priors. For instance, for cortical surfaces with different sulci topography, one can prefer to favour lateral displacement over the creation of new sulci. Large deformations are commonly obtained through the integration of a vector field [4,10,13,18] and a natural route is to introduce the constraints in the vector fields instead of the final diffeomorphism [19]. The vector field could be restricted, via a finite dimensional control variable, to a state dependent finite dimensional subspace generated by a finite basis and conceptualized in structures called hereafter deformation modules. Deformation modules should create interpretable deformations, and several modules should be allowed to combine to form more complex compound deformation modules in the spirit of Grenander's Pattern Theory [7].Preliminary instantiations of the concept of deformation modules can be found in several early works. In the poly-affine framework [3,14,20], deformations are created by the integration of a poly-affine stationary vector field. This vector field is a sum of few local affine transformations, which are then easily interpretable and share some features of deformation modules even if regions of each affine component are not updated during the deformation. In the LDDMM framework, a Riemannian structure is defined on the group of diffeomorphisms and optimal matchings are geodesics for this metric [2,10]. Several discretization schemes based on landmarks induce examples of finite dimensional state dependent representations of the velocity fields updated along the deformation and could be rephrased inside our definition of deformation modules. Note that the discretization scheme are thought of as approximations of the unconstrained nonparametrized infinite dimensional case. In a recently developed sparse-LDDMM framework [6,12,15], the vector field is constrained to be a sum of a fixed number of local translations, carried by control points, with a more clearer focus on finite-dimensionality and local interpretability. Extension to locally more complex transformation a...

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