Fitting Smooth Paths on Riemannian Manifolds - Endometrial Surface

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14323

Résumé

We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using C1 piecewise-Bézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group SO(3) and the special Euclidean group SE(3) for preoperative MRI-based navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.

Fitting Smooth Paths on Riemannian Manifolds - Endometrial Surface

Collection

application/pdf Fitting Smooth Paths on Riemannian Manifolds - Endometrial Surface Antoine Arnould, Pierre-Yves Gousenbourger, Chafik Samir, Pierre-Antoine Absil, Michel Canis

Média

Voir la vidéo

Métriques

167
12
2.95 Mo
 application/pdf
bitcache://40aca6b38901fe3fa7524dd64db6c3798db9c2d5

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">10.23723/11784/14323</identifier><creators><creator><creatorName>Pierre-Antoine Absil</creatorName></creator><creator><creatorName>Antoine Arnould</creatorName></creator><creator><creatorName>Pierre-Yves Gousenbourger</creatorName></creator><creator><creatorName>Chafik Samir</creatorName></creator><creator><creatorName>Michel Canis</creatorName></creator></creators><titles>
            <title>Fitting Smooth Paths on Riemannian Manifolds - Endometrial Surface</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><subjects><subject>Optimization on manifolds</subject><subject>Path fitting on Riemannian manifolds</subject><subject>Bézier functions</subject><subject>MRI-based navigation</subject><subject>Endometrial surface reconstruction</subject></subjects><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">40aca6b38901fe3fa7524dd64db6c3798db9c2d5</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>24719</version>
        <descriptions>
            <description descriptionType="Abstract">
We present a new method to fit smooth paths to a given set of points on Riemannian manifolds using C1 piecewise-Bézier functions. A property of the method is that, when the manifold reduces to a Euclidean space, the control points minimize the mean square acceleration of the path. As an application, we focus on data observations that evolve on certain nonlinear manifolds of importance in medical imaging: the shape manifold for endometrial surface reconstruction; the special orthogonal group SO(3) and the special Euclidean group SE(3) for preoperative MRI-based navigation. Results on real data show that our method succeeds in meeting the clinical goal: combining different modalities to improve the localization of the endometrial lesions.

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

Endometriosis: MRI navigation and surface reconstruction on manifolds GSI2015 A. Arnould, P.-Y. Gousenbourger, C. Samir, P.-A. Absil, M. Canis pierre-yves.gousenbourger@uclouvain.be 30 october 2015 What is endometriosis ? 10 % of women Ovaries • Uterosacral ligaments • Colon • vagina • bladder How to diagnose and cure ? Before surgery location • size • depth MRI TVUS Question 1 : How to merge both techniques ? Question 2 : How to evaluate the size of the cyst ? When endometriosis meets manifolds Answer 1 : MRI navigation as a path on SE(3) R3 o SO(3) S2 o SO 10 15 20 25 30 35 5 10 15 20 20 25 30 35 40 x Bézier path y z 0 0.5 1 1.5 2 2.5 3 0 50 Speed of the Bézier path in the space 2−Norm of the acceleration of the Bézier path in the space 0 0.5 1 1.5 2 2.5 3 0 5 10 Speed of rotation of the Bézier path 2−Norm of the acceleration of rotation of the Bézier path When endometriosis meets manifolds Answer 2 : Endometrial volume reconstruction as path on shape manifold ? ?How to interpolate points on manifolds ? How to interpolate ? Each segment between two consecutive points is a Bézier function. p0 t = 0 p1 t = 1 p2 t = 2 p3 t = 3 Reconstruction : the De Casteljau algorithm b0 b1 b2 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 1 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) β2(b0, b1, b2; 1 2) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) β2(b0, b1, b2; 1 2) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) β2(b0, b1, b2; 1 2) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) β2(b0, b1, b2; 1 2) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) β2(b0, b1, b2; 1 2) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) β2(b0, b1, b2; 1 2) β2(b0, b1, b2; 3 4) Reconstruction : the De Casteljau algorithm b0 b1 b2 | | t 0 11 4 1 2 3 4 β2(b0, b1, b2; 1 4) β2(b0, b1, b2; 1 2) β2(b0, b1, b2; 3 4) Example on the sphere Example on the sphere It’s ugly. Make it smooth ! Smooth interpolation with Bézier (in Rn ) p0 p1 p2 p3 p4 Smooth interpolation with Bézier (in Rn ) p0 p1 p2 p3 p4 Find the optimal position of control points C1 -piecewise Bézier interpolation (in Rn ) pi−1 pi pi+1 C1 -piecewise Bézier interpolation (in Rn ) pi−1 pi pi+1 b+ i = 2pi − b− i Optimal C1 -piecewise Bézier interpolation (in Rn ) Minimization of the mean square acceleration of the path min αi 1 0 ¨β0 2(b− 1 ; t) 2 dt + n−1 i=1 1 0 ¨βi 3(b− i ; t) 2 dt + 1 0 ¨βn 2 (b− n−1; t) 2 dt Optimal C1 -piecewise Bézier interpolation (in Rn ) Minimization of the mean square acceleration of the path min αi 1 0 ¨β0 2(b− 1 ; t) 2 dt + n−1 i=1 1 0 ¨βi 3(b− i ; t) 2 dt + 1 0 ¨βn 2 (b− n−1; t) 2 dt Second order polynomial P(b− i ) Optimal C1 -piecewise Bézier interpolation (in Rn ) Minimization of the mean square acceleration of the path min αi 1 0 ¨β0 2(b− 1 ; t) 2 dt + n−1 i=1 1 0 ¨βi 3(b− i ; t) 2 dt + 1 0 ¨βn 2 (b− n−1; t) 2 dt Second order polynomial P(b− i ) P(b− i ) ! Optimal C1 -piecewise Bézier interpolation (in Rn ) Minimization of the mean square acceleration of the path min αi 1 0 ¨β0 2(b− 1 ; t) 2 dt + n−1 i=1 1 0 ¨βi 3(b− i ; t) 2 dt + 1 0 ¨βn 2 (b− n−1; t) 2 dt Second order polynomial P(b− i ) A B− = C [Logpk (pi)]iP Optimal C1 -piecewise Bézier interpolation (in Rn ) Minimization of the mean square acceleration of the path A B− = C [Logpk (pi)]iP B− = A−1 CP = DP ⇔ b− i = n j=0 Dij 1 pj A result on R2 Optimal C1 -piecewise Bézier interpolation (on M) The control points are given by : b− i = n j=0 Di,jpj Optimal C1 -piecewise Bézier interpolation (on M) The control points are given by : b− i = n j=0 Di,jpj These points are invariant under translation, i.e. : b− i − pref = n j=0 Di,j(pj − pref ) Optimal C1 -piecewise Bézier interpolation (on M) The control points are given by : b− i = n j=0 Di,jpj These points are invariant under translation, i.e. : b− i − pref = n j=0 Di,j(pj − pref ) Transfer to the manifolds setting using the Log as a − b ⇔ Logb(a) Logpref (b− i ) = n j=0 Di,jLogpref (pj) Application 1 : MRI navigation R3 o SO(3) 10 15 20 25 30 35 5 10 15 20 20 25 30 35 40 x Bézier path y z 0 0.5 1 1.5 2 2.5 3 0 50 Speed of the Bézier path in the space 0 0.5 1 1.5 2 2.5 3 0 50 100 2−Norm of the acceleration of the Bézier path in the space 0 0.5 1 1.5 2 0 5 10 Speed of rotation of the Bézi 0 0.5 1 1.5 2 0 5 10 2−Norm of the acceleration of rotation o Application 2 : Endometrial volume reconstruction Conclusions General C1-interpolative method on manifolds... applied in medical imaging. Conclusions General C1-interpolative method on manifolds... applied in medical imaging. It’s light ; It’s fast ; It’s general ; Conclusions General C1-interpolative method on manifolds... applied in medical imaging. It’s light ; It’s fast ; It’s general ; Bézier interpolation can be extended to multidimentional interpolation (surfaces) ; Any questions ? Endometriosis: MRI navigation and surface reconstruction on manifolds GSI2015 A. Arnould, P.-Y. Gousenbourger, C. Samir, P.-A. Absil, M. Canis pierre-yves.gousenbourger@uclouvain.be 30 october 2015