Laurent COHEN February 2017 Fast Marching and Applications 1 01/02/2017 19:16 Laurent D. COHEN, Huawei 2017 1 Directeur de Recherche CNRS CEREMADE, UMR CNRS 7534 Université Paris-9 Dauphine Place du Maréchal de Lattre de Tassigny 75016 Paris, France [email protected]http://www.ceremade.dauphine.fr/~cohen Some joint works with F. Benmansour, Y. Rouchdy, J. Mille, G. Peyré, H. Li , A. Yezzi, Da Chen and J.M. Mirebeau. Huawei, February 3rd, 2017 Laurent D. COHEN Fast Marching and Geodesic Methods. Some Applications
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Laurent COHEN February 2017
Fast Marching and Applications 1
01/02/2017 19:16 Laurent D. COHEN, Huawei 2017 1
Directeur de Recherche CNRSCEREMADE, UMR CNRS 7534 Université Paris-9 DauphinePlace du Maréchal de Lattre de Tassigny 75016 Paris, France
http://www.ceremade.dauphine.fr/~cohenSome joint works with F. Benmansour, Y. Rouchdy, J. Mille, G. Peyré, H. Li , A. Yezzi,
Da Chen and J.M. Mirebeau.Huawei, February 3rd, 2017
Laurent D. COHEN
Fast Marching and Geodesic Methods. Some Applications
Laurent COHEN February 2017
Fast Marching and Applications 2
01/02/2017 19:16 Laurent D. COHEN, Huawei 2017 26
Overview
n Minimal Paths, Fast Marching and Front Propagationn Anisotropic Minimal Paths and Tubular modeln Finding contours as a set of minimal pathsn Application to 2D and 3D tree structuresn Geodesic Density for tree structures
Laurent COHEN February 2017
Fast Marching and Applications 3
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Overview
n Minimal Paths, Fast Marching and Front Propagationn Anisotropic Minimal Paths and Tubular modeln Finding contours as a set of minimal pathsn Application to 2D and 3D tree structuresn Geodesic Density for tree structures
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Fast Marching and Applications 4
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Paths of minimal energy
Looking for a path along which a feature Potential P(x,y) is minimal
example: a vesseldark structureP =gray level
Input : Start point p1=(x1,y1)
End point p2 =(x2,y2)
Image
Output: Minimal Pathp1
p2
∫=L
dssCPCE0
))(()(
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Minimal Paths: Eikonal Equation
STEP 1 : search for the surface of minimal action U of p1 as the minimal energy integrated along a path between start point p1 and any point p in the image
STEP 2: Back-propagation from the end point p2 to the start point p1:
Simple Gradient Descent along
Potential P>0 takes lower values near interesting features : on contours, dark structures, ...
1 pU
∫====
==L
pLCpCpLCpCp dssCPCEpU
0)(;1)0()(;1)0(1 ))(()()( infinf
∫=L
dssCPCE0
))(()(
;1)0(point Start pC =
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Minimal Paths: Eikonal EquationSTEP 1 : minimal action U of p1 as the minimal energy integrated along a path
between start point p1 and any point p in the image
Solution of Eikonal equation:
Example P=1, U Euclidean distance to p1in general, U weighted geodesic distance to p1
∫====
==L
pLCpCpLCpCp dssCPCEpU
0)(;1)0()(;1)0(1 ))(()()( infinf
0)1( and )()( 11 ==∇ pUxPxU pp
;1)0(point Start pC =
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P1
slower
P2 < P1
fasterP=c
Fermat Principle in Geometric Optics : Path followed by light minimizes time
where n>1 is refraction index v=c/n
Snell-Descartes ’law
Minimal paths - 2D simple examples
Examples of shortest paths on univalued or bivalued potential
∫=L
dssCPCE0
))((~)()()( inf
)(;1)0(1 CEpU
pLCpCp
==
=
∫=2
1 ))(( 1 p
pdssCn
cT
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Minimal Paths and Front Propagation
Evolution of t level set of U from p0
n normal vector to a level set of U is in the direction of the
Gradient of U, implies Eikonal Equation :
∫==
=L
pLCpCp dssCPpU
0)(;0)0(0 ))((~)(Action Minimal inf
{ }tpURpt p =∈= )( / )( 02n PropagatioFront L
0)0( and )()( 00 ==∇ pUxPxU pp
),()),((
1),( tntPt
t σσ
σL
L=
∂∂
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FAST MARCHING in 2D: very efficient algorithm O(NlogN) for Eikonal Equation
Level sets of U can be seen as a Front propagation outwards.
Numerical approximation of U(xij) as the solution to the discretized problem with upwind finite difference scheme
This 2nd order equation induces that :action U at {i,j} depends only of the neighbors that have lower actions.
Fast marching introduces order in the selection of the grid points for solving this numerical scheme.
Starting from the initial point p1 with U = 0,the action computed at each point visited can only grow.
Introduced by Sethian / Tsistsiklis
( )( ) 2
,22
1,1,
2,1,1
)(~0),( ),( max
0),( ),( max
jijiji
jiji
xPhxUuxUu
xUuxUu
=−−+
−−
+−
+−
PU ~ =∇ 222 ~ P
yU
xU
=
∂∂
+
∂∂
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Fast Marching Algorithm (Sethian)
n Start: only p0 is trial with U=0.
n Loop: p trial point with minimum Ubecomes alive. neighbors of pbecome trial and are updated.
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J. A. SethianA fast marching level set method for monotonically advancing fronts. P.N.A.S., 93:1591-1595, 1996.
minimal actionpotential Far Trial Alive
InitializationFast Marching Algorithm
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minimal actionpotential Far Trial Alive
46
J. A. SethianA fast marching level set method for monotonically advancing fronts. P.N.A.S., 93:1591-1595, 1996.
Itération #1● Find point xmin (Trial point with smallest value of ).● xmin becomes Alive.● For each of 4 neighbors x of point xmin :
If x is not Alive,Estimate with upwind scheme.
x becomes Trial.
Fast Marching Algorithm
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J. A. SethianA fast marching level set method for monotonically advancing fronts. P.N.A.S., 93:1591-1595, 1996.
minimal actionpotential Far Trial Alive
Itération #2● Find point xmin (Trial point with smallest value of ).● xmin becomes Alive.● For each of 4 neighbors x of point xmin :
If x is not Alive,Estimate with upwind scheme.
x becomes Trial.
Fast Marching Algorithm
Laurent COHEN February 2017
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J. A. SethianA fast marching level set method for monotonically advancing fronts. P.N.A.S., 93:1591-1595, 1996.
minimal actionpotential Far Trial Alive
Itération #k● Find point xmin (Trial point with smallest value of ).● xmin becomes Alive.● For each of 4 neighbors x of point xmin :
01/02/2017 19:16 Laurent D. COHEN, Huawei 2017 50potential
Step #1
Etape #2Descente de gradient sur pour extraire le chemin minimal
Minimal Path between p1 and p2
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L. D. Cohen, R. KimmelGlobal minimum for active contour models : a minimal path approach.International Journal of Computer Vision, 25:57-78, 1997.
potentiel
Minimal Path between p1 and p2
Step #1: U obtained by the FAST MARCHING ALGORITHM
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Step #1: U obtained by the FAST MARCHING ALGORITHM
Etape #2Descente de gradient sur pour extraire le chemin minimal
minimal action
Minimal Path between p1 and p2
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Step #1
Step #2gradient descent on for extraction of minimal path
Minimal Path between p1 and p2
minimal action
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L. D. Cohen, R. KimmelGlobal minimum for active contour models : a minimal path approach.International Journal of Computer Vision, 25:57-78, 1997.
minimal action
Minimal Path between p1 and p2
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L. D. Cohen, R. KimmelGlobal minimum for active contour models : a minimal path approach.International Journal of Computer Vision, 25:57-78, 1997.
minimal path
Is obtained by solving ODE:
simple gradient descent onfrom p2 to p1
ð
minimal action
Minimal Path between p1 and p2
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Step #1
Step #2gradient descent on for extraction of minimal path
Minimal Path between p1 and p2
minimal action
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Step #1
Step #2gradient descent on for extraction of minimal path
Minimal Path between p1 and p2
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Minimal paths for 2D segmentation
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Minimal paths for 2D segmentation
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Simultaneous propagation from both ends
Reference: T. Deschamps and L. D. CohenMinimal paths in 3D images and application to virtual endoscopy.Proceedings ECCV’00, Dublin, Ireland, 2000.
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Minimal Paths
Reference: T. Deschamps and L. D. CohenMinimal paths in 3D images and application to virtual endoscopy.Proceedings ECCV’00, Dublin, Ireland, 2000.
Simultaneous propagation of two fronts until a shock occurs.
meeting point
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Link with Dynamic Programming
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Overview
n Minimal Paths, Fast Marching and Front Propagationn Anisotropic Minimal Paths and Tubular modeln Finding contours as a set of minimal pathsn Application to 2D and 3D tree structuresn Geodesic Density for tree structures
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3D Minimal Path for tubular shapes in 2DCenterline+width
2D in space , 1D for radius of vessel (Li, Yezzi 2007)
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3D Minimal Path for tubular shapes in 2D
2D in space , 1D for radius of vessel
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Typical Retina Image
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Two pairs of user given points
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Extraction by 2D+radius minimal path model
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Overview
n Minimal Paths, Fast Marching and Front Propagationn Anisotropic Minimal Paths and Tubular modeln Finding contours as a set of minimal pathsn Application to 2D and 3D tree structuresn Geodesic Density for tree structures
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Anisotropic Energy
Considers the local orientations of the structures
Describes an infinitesimaldistance along an oriented pathway C, relative to a metric H
Laurent D. COHEN, Huawei 2017
Geodesic Methods for Shape and Surface Processing, Gabriel Peyre and Laurent D. Cohen in Advances in Computational Vision and Medical Image Processing: Methods and Applications, Springer, 2009.
∫=L
dssCsCPCE0
))(' ),(()(
)(' ))(()(' ))(' ),(( sCsCHsCsCsCP T=
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Anisotropic Energy:Eikonal Equation
Laurent D. COHEN, Huawei 2017
Geodesic Methods for Shape and Surface Processing, Gabriel Peyre and Laurent D. Cohen in Advances in Computational Vision and Medical Image Processing: Methods and Applications, Springer, 2009.
∫=L T dssCsCHsCCE0
)(' ))(()(' )(
)()( inf)(;1)0(
1 CEpUpLCpC
p==
=
1)( 11
1)(1 1 =∇∇=∇ −− p
TppHp UHUpU
;1)0(point Start pC =
0)1( and 1 =pU p
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Anisotropic Energy:Gradient descent
Laurent D. COHEN, Huawei 2017
Geodesic Methods for Shape and Surface Processing, Gabriel Peyre and Laurent D. Cohen in Advances in Computational Vision and Medical Image Processing: Methods and Applications, Springer, 2009.
∫=L T dssCsCHsCCE0
)(' ))(()(' )(
)()( inf)(;1)0(
1 CEpUpLCpC
p==
=
))(())(()(' 11 sCUsCHsC p∇−= −
;1)0(point Start pC =
0)1( and 1 =pU p
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Anisotropic Energy:includes Isotropic case
Laurent D. COHEN, Huawei 2017
Geodesic Methods for Shape and Surface Processing, Gabriel Peyre and Laurent D. Cohen in Advances in Computational Vision and Medical Image Processing: Methods and Applications, Springer, 2009.
∫=L T dssCsCHsCCE0
)(' ))(()(' )(
)()( 2 IdpPpH =
))(()(' 1 tCUtC p−∇=
;1)0(point Start pC =
PpU p =∇ )(1
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Anisotropy and Geodesics
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Anisotropy and Geodesics
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Considers the local orientations of the structures0
( ) ( ( ), '( ))L
E C P C s C s ds=∫
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Examples of 3D Minimal Pathsfor tubular shapes in 2D2D in space , 1D for radius of vessel
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Examples of 3D Minimal Pathsfor tubular shapes in 2D2D in space , 1D for radius of vessel
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Examples of 4D Minimal Pathsfor tubular shapes in 3D3D in space , 1D for radius of vessel
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Examples of 4D Minimal Pathsfor tubular shapes in 3D3D in space , 1D for radius of vessel
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Perceptual Grouping using Minimal Paths
Reference: L. D. CohenMultiple Contour Finding and Perceptual Grouping using Minimal Paths.Journal of Mathematical Imaging and Vision, 14:225-236, 2001.
The potential is an incomplete ellipse and 7 points are given(keypoints were found using a Furthest point strategy).
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Perceptual Grouping using a set of Minimal Paths
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Perceptual Grouping using Minimal Paths
Reference: L. D. CohenMultiple Contour Finding and Perceptual Grouping using Minimal Paths.Journal of Mathematical Imaging and Vision, 14:225-236, 2001.
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Perceptual Grouping using Minimal Paths
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Using the orientation with anisotropic geodesics
Perceptual Grouping using Minimal Paths
Anisotropic Geodesics for Perceptual Grouping and Domain Meshing. Sebastien Bougleux and Gabriel Peyr\'e and Laurent D. Cohen. Proc. tenth European Conference on Computer Vision (ECCV'08)}, Marseille, France, October 12-18, 2008.
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Vue Endoscopique d’un arbre vasculaire
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Curvature Penalized Minimal PathMethod with A Finsler Metric
with Da Chen and JM Mirebeau, 2015-2016
n The metric may depend on the orientationn Orientation-lifted metric: the curve length
of Euler elastica can be exactly computed by this metric
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Curvature Penalized Minimal PathMethod with A Finsler Metric
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Curvature Penalized Minimal PathMethod with A Finsler Metric
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Curvature Penalized Minimal PathMethod with A Finsler Metric
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Curvature Penalized Minimal Path Method with A Finsler Metric
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Curvature Penalized Minimal Path Method with A Finsler Metric
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Curvature Penalized Minimal Path Method with A Finsler Metric
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Examples of Remeshing
Original mesh
Uniform Curvatureadapted
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Isotropic vs. Anisotropic Meshing
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Anisotropic Meshing
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Anisotropic Meshing
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Examples of Anisotropic Meshingcontrols density and orientation of triangles
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Geodesic methods for shape recognitionBased on distribution (histogram) of geodesic distances
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Overview
n Minimal Paths, Fast Marching and Front Propagationn Anisotropic Minimal Paths and Tubular modeln Finding contours as a set of minimal pathsn Application to 2D and 3D tree structuresn Geodesic Density for tree structures
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Finding a closed contour by growingminimal paths and adding keypoints
Finding a Closed Boundary by Growing Minimal Paths from a Single Point on 2D or 3D Images. Fethallah Benmansour and Laurent D. Cohen. Journal of Mathematical Imaging and Vision. 2009.
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths and adding keypoints
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The propagationmust be stopped assoon as the domainvisited by the frontshas the sametopology as a ring.
Adding keypoints: Stopping criterion
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths and adding keypoints
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Finding a closed contour by growing minimal paths
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Keypoints and 3D Minimal Pathsfor tubular shapes in 2D(with Li and Yezzi, MICCAI’09)
2D in space , 1D for radius of vessel
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Keypoints and 3D Minimal Pathsfor tubular shapes in 2D2D in space , 1D for radius of vessel
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Keypoints and 3D Minimal Pathsfor tubular shapes in 2D2D in space , 1D for radius of vessel
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Keypoints and 3D Minimal Pathsfor tubular shapes in 2D2D in space , 1D for radius of vessel
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Automatic Keypoint Growing with Mask(with Chen Da)
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Automatic Keypoint Method
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Overview
n Minimal Paths, Fast Marching and Front Propagationn Anisotropic Minimal Paths and Tubular modeln Finding contours as a set of minimal pathsn Application to 2D and 3D tree structuresn Geodesic Density for tree structures
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Geodesic Density
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Geodesic Density
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Geodesic Density
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Geodesic Density
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Geodesic Density: Real example
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Geodesic Density: adaptive voting
Adaptive voting : 1000 end points
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Geodesic Density: adaptive voting
Adaptive voting : 1000 end points
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Conclusion
n Minimally interactive tools for vessels and vascular tree segmentation (tubular branching structures)
n User provides only one initial point and sometimes second end point or stopping parameter
n Fast and efficient propagation algorithmn Models may include orientation and scale of vesselsn Voting approach as a powerful tool to find the structure,
which can be completed with other approach.
Laurent COHEN February 2017
Fast Marching and Applications 98
01/02/2017 19:16 Laurent D. COHEN, Huawei 2017 308