A sub-Riemannian modular approach for diffeomorphic deformations

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14320

Résumé

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.

A sub-Riemannian modular approach for diffeomorphic deformations

Collection

application/pdf A sub-Riemannian modular approach for diffeomorphic deformations Barbara Gris, Stanley Durrleman, Alain Trouvé

Média

Voir la vidéo

Métriques

176
4
1.17 Mo
 application/pdf
bitcache://dc3bf0e8b62ece74dd07f88eea66b86df67e5ea2

Licence

Creative Commons Attribution-ShareAlike 4.0 International

Sponsors

Organisateurs

logo_see.gif
logocampusparissaclay.png

Sponsors

entropy1-01.png
springer-logo.png
lncs_logo.png
Séminaire Léon Brillouin Logo
logothales.jpg
smai.png
logo_cnrs_2.jpg
gdr-isis.png
logo_gdr-mia.png
logo_x.jpeg
logo-lix.png
logorioniledefrance.jpg
isc-pif_logo.png
logo_telecom_paristech.png
csdcunitwinlogo.jpg
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/11784/14320</identifier><creators><creator><creatorName>Stanley Durrleman</creatorName></creator><creator><creatorName>Alain Trouvé</creatorName></creator><creator><creatorName>Barbara Gris</creatorName></creator></creators><titles>
            <title>A sub-Riemannian modular approach for diffeomorphic deformations</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Sun 8 Nov 2015</date>
	    <date dateType="Updated">Wed 31 Aug 2016</date>
            <date dateType="Submitted">Mon 15 Oct 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">dc3bf0e8b62ece74dd07f88eea66b86df67e5ea2</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>24712</version>
        <descriptions>
            <description descriptionType="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.

</description>
        </descriptions>
    </resource>
.

A sub-Riemannian modular approach for diffeomorphic deformations GSI 2015 Barbara Gris Advisors: Alain Trouvé (CMLA) and Stanley Durrleman (ICM) gris@cmla.ens-cachan.fr October 30, 2015 1 Introduction 2 Deformation modules Definition and first examples Modular large deformations Combining deformation modules 3 Numerical results Sommaire 1 Introduction 2 Deformation modules Definition and first examples Modular large deformations Combining deformation modules 3 Numerical results Introduction "Is it possible to mechanize human intuitive understanding of biological pictures that typically exhibit a lot of variability but also possess characteristic structure ?" Ulf Grenander Hands : a Pattern Theoric Study of Biological Shapes, 1991 Introduction Structure in data Introduction Structure in data ˙ϕt = vt ◦ ϕt , ϕt=0 = Id Introduction Structure in data Structure in deformations Introduction Structure in data Structure in deformations Type of vector fields Previous works locally affine deformations Poly-affine [C. Seiler , X. Pennec, and M. Reyes. Capturing the multiscale anatomical shape variability with polyaffine transformation trees. Medical image analysis, 2012] Previous works locally affine deformations Poly-affine [C. Seiler , X. Pennec, and M. Reyes. Capturing the multiscale anatomical shape variability with polyaffine transformation trees. Medical image analysis, 2012] v(x) = i wi (x)Ai (x) Previous works locally affine deformations Poly-affine [C. Seiler , X. Pennec, and M. Reyes. Capturing the multiscale anatomical shape variability with polyaffine transformation trees. Medical image analysis, 2012] v(x) = i wi (x)Ai (x) Deformation structure does not evolve with the flow Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Deformation structure imposed by shapes and action of vector fields Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Deformation structure imposed by shapes and action of vector fields Previous works : Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Deformation structure imposed by shapes and action of vector fields Previous works : LDDMM [M. I. Miller, L. Younes, and A. Trouvé. Diffeomorphometry and geodesic positioning systems for human anatomy, 2014] Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Deformation structure imposed by shapes and action of vector fields Previous works : LDDMM [M. I. Miller, L. Younes, and A. Trouvé. Diffeomorphometry and geodesic positioning systems for human anatomy, 2014] Higher-order momentum [S. Sommer M. Nielsen, F. Lauze, and X. Pennec. Higher-order momentum distributions and locally affie lddmm registration. SIAM Journal on Imaging Sciences, 2013] Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Deformation structure imposed by shapes and action of vector fields Previous works : LDDMM [M. I. Miller, L. Younes, and A. Trouvé. Diffeomorphometry and geodesic positioning systems for human anatomy, 2014] Higher-order momentum [S. Sommer M. Nielsen, F. Lauze, and X. Pennec. Higher-order momentum distributions and locally affie lddmm registration. SIAM Journal on Imaging Sciences, 2013] Sparse LDDMM [S. Durrleman, M. Prastawa, G. Gerig, and S. Joshi. Optimal data-driven sparse parameterization of diffeomorphisms for population analysis. In Information Processing in Medical Imaging , pages 123-134. Springer, 2011] Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Deformation structure imposed by shapes and action of vector fields Previous works : LDDMM [M. I. Miller, L. Younes, and A. Trouvé. Diffeomorphometry and geodesic positioning systems for human anatomy, 2014] Higher-order momentum [S. Sommer M. Nielsen, F. Lauze, and X. Pennec. Higher-order momentum distributions and locally affie lddmm registration. SIAM Journal on Imaging Sciences, 2013] Sparse LDDMM [S. Durrleman, M. Prastawa, G. Gerig, and S. Joshi. Optimal data-driven sparse parameterization of diffeomorphisms for population analysis. In Information Processing in Medical Imaging , pages 123-134. Springer, 2011] Deformation structure evolves with flow Previous works Shape space (S. Arguillère) Shape space [S. Arguillere. Géométrie sous-riemannienne en dimension infinie et applications à l’analyse mathématique des formes . PhD thesis, 2014.]: Deformation structure imposed by shapes and action of vector fields Previous works : LDDMM [M. I. Miller, L. Younes, and A. Trouvé. Diffeomorphometry and geodesic positioning systems for human anatomy, 2014] Higher-order momentum [S. Sommer M. Nielsen, F. Lauze, and X. Pennec. Higher-order momentum distributions and locally affie lddmm registration. SIAM Journal on Imaging Sciences, 2013] Sparse LDDMM [S. Durrleman, M. Prastawa, G. Gerig, and S. Joshi. Optimal data-driven sparse parameterization of diffeomorphisms for population analysis. In Information Processing in Medical Imaging , pages 123-134. Springer, 2011] Deformation structure evolves with flow No control on deformation structure Previous works Constraints Diffeons [L. Younes. Constrained diffeomorphic shape evolution. Foundations of Computational Mathematics, 2012.] Our model : Deformation modules Purpose : Our model : Deformation modules Purpose : Incorporate constraints in the deformation model Our model : Deformation modules Purpose : Incorporate constraints in the deformation model Merge different constraints in a complex one Sommaire 1 Introduction 2 Deformation modules Definition and first examples Modular large deformations Combining deformation modules 3 Numerical results Deformation modules Definition and first examples A deformation module : Deformation modules Definition and first examples A deformation module : Contains a space of shapes Deformation modules Definition and first examples A deformation module : Contains a space of shapes Can generate vector fields that : Deformation modules Definition and first examples A deformation module : Contains a space of shapes Can generate vector fields that : are of a particular type Deformation modules Definition and first examples A deformation module : Contains a space of shapes Can generate vector fields that : are of a particular type −→ deformation structure Deformation modules Definition and first examples A deformation module : Contains a space of shapes Can generate vector fields that : are of a particular type −→ deformation structure depend on the state of the shape Deformation modules Definition and first examples A deformation module : Contains a space of shapes Can generate vector fields that : are of a particular type −→ deformation structure depend on the state of the shape −→ the deformation structure evolves with the flow Sommaire 1 Introduction 2 Deformation modules Definition and first examples Modular large deformations Combining deformation modules 3 Numerical results Deformation modules Definition and first examples : local translation of scale σ Example of generated vector field Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) Deformation modules Definition and first examples : local translation of scale σ M = (O, H, V, ζ, ξ, c) O is a shape space (S. Arguillère) There exists C > 0 : ∀(o, h) ∈ O × H: |ζ(o, h)|2 V ≤ C c(o, h) Deformation modules Definition and first examples : local scaling of scale σ Deformation modules Definition and first examples : local scaling of scale σ Example of generated vector field Deformation modules Definition and first examples : local scaling of scale σ Example of generated vector field Deformation modules Definition and first examples : local scaling of scale σ Example of generated vector field Deformation modules Definition and first examples : local scaling of scale σ Example of generated vector field z1z2 z3 Deformation modules Definition and first examples : local scaling of scale σ Example of generated vector field z1z2 z3 d1d2 d3 Deformation modules Definition and first examples : local scaling of scale σ Example of generated vector field z1z2 z3 d1d2 d3 Deformation modules Definition and first examples : local scaling of scale σ Deformation modules Definition and first examples : local rotation of scale σ Introduction Definition and first examples : local translation of scale σ and fixed direction Sommaire 1 Introduction 2 Deformation modules Definition and first examples Modular large deformations Combining deformation modules 3 Numerical results Deformation modules Modular large deformations M = (O, H, V, ζ, ξ, c) Deformation modules Modular large deformations Studied trajectories : t → (ot , ht ) ∈ O × H such that ˙ot = ξot (vt ) where vt = ζot (ht ) ∈ ζot (H). Deformation modules Modular large deformations Studied trajectories : t → (ot , ht ) ∈ O × H such that ˙ot = ξot (vt ) where vt = ζot (ht ) ∈ ζot (H). −→ Solutions of ˙ϕv t = vt ◦ ϕv t , ϕv t=0 = Id exist. Deformation modules Modular large deformations Studied trajectories : t → (ot , ht ) ∈ O × H such that ˙ot = ξot (vt ) where vt = ζot (ht ) ∈ ζot (H). −→ Solutions of ˙ϕv t = vt ◦ ϕv t , ϕv t=0 = Id exist. −→ ϕv = modular large deformation. Deformation modules Modular large deformations : an example Sommaire 1 Introduction 2 Deformation modules Definition and first examples Modular large deformations Combining deformation modules 3 Numerical results Deformation modules Combination Deformation modules Combination Deformation modules Combination Features : if ci oi (hi) = |ζi oi (hi)|2 Vi then co(h) = i |ζi oi (hi)|2 Vi = | i ζi oi (hi)|2 V Deformation modules Combination Features : if ci oi (hi) = |ζi oi (hi)|2 Vi then co(h) = i |ζi oi (hi)|2 Vi = | i ζi oi (hi)|2 V Geometrical descriptors are transported by the global vector field Deformation modules Combination Features : if ci oi (hi) = |ζi oi (hi)|2 Vi then co(h) = i |ζi oi (hi)|2 Vi = | i ζi oi (hi)|2 V Geometrical descriptors are transported by the global vector field Coherent mathematical framework : possibility to combine any modules Deformation modules Combination : Example of modular large deformation Sommaire 1 Introduction 2 Deformation modules Definition and first examples Modular large deformations Combining deformation modules 3 Numerical results Deformation modules Matching problem Deformation modules Matching problem Deformation modules Matching problem 1 0 co(h) + g(ϕv t=1 · fsource, ftarget ) v = ζo(h) [N. Charon and A. Trouvé. The varifold representation of non-oriented shapes for diffeomorphic registration, 2013] Deformation modules Matching problem Deformation modules Matching problem Deformation modules Matching problem Deformation modules Matching problem Deformation modules Matching problem Deformation modules Matching problem Conclusion We have presented Conclusion We have presented a coherent mathematical framework Conclusion We have presented a coherent mathematical framework to build modular large deformations. Conclusion We have presented a coherent mathematical framework to build modular large deformations. We showed how easily incorporating constraints in a deformation model Conclusion We have presented a coherent mathematical framework to build modular large deformations. We showed how easily incorporating constraints in a deformation model and merging different constraints in a global one. Conclusion "Is it possible to mechanize human intuitive understanding of biological pictures that typically exhibit a lot of variability but also possess characteristic structure ?" Ulf Grenander Hands : a Pattern Theoric Study of Biological Shapes, 1991 Thank you for your attention !