Laurent D. COHEN.

Transcription

1 for biomedical image segmentation Laurent D. COHEN Directeur de Recherche CNRS CEREMADE, UMR CNRS 7534 Université Paris-9 Dauphine Place du Maréchal de Lattre de Tassigny Paris, France Some joint works with Y. Rouchdy, and J. Mille. LJLL, Journée thématique maths, image et biomédecine 18 Décembre /12/ :12 Laurent COHEN, LJLL, Overview Minimal Paths, Fast Marching and Front Propagation Geodesic Density for tree structures Using Geodesic voting with Radius energy Using Geodesic voting as a prior shape Using Geodesic voting as initial deformable tree 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

2 Paths of minimal energy p1 p2 Looking for a path along which a feature Potential P(x,y) is minimal E( C) ( ( 0 example: a vessel dark structure P =gray level Input : Start point p1=(x1,y1) End point p2 =(x2,y2) Image Output: Minimal Path = L P C s)) ds 19/12/ :12 Laurent COHEN, LJLL, Minimal Paths: Eikonal Equation 0 E( C) = L P( C( s)) ds Potential P>0 takes lower values near interesting features : on contours, dark structures,... 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 Start point C (0) = p1; L Up 1 p) = E( C) = P( C( s)) ds ( inf inf C(0) = p1; C( L) = p 0 C(0) = p1; C( L) = p STEP 2: Back-propagation from the end point p2 to the start point p1: Simple Gradient Descent along U p1 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

3 Minimal Paths: Eikonal Equation STEP 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 Start point C (0) = p1; L Up 1 p) = E( C) = P( C( s)) ds ( inf inf 0 Solution C(0) = p1; C of ( L) Eikonal = p equation: C(0) = p1; C( L) = p U p 1 ( x) = P( x) and U p1( p1) = 0 Example P=1, U Euclidean distance to p1 in general, U weighted geodesic distance to p1 19/12/ :12 Laurent COHEN, LJLL, Minimal Paths: Eikonal Equation 0 E( C) = L P( C( s)) ds STEP 2: Back-propagation from the end point p2 to the start point p1: Simple Gradient Descent along U p1 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

4 Minimal Path between p1 and p2 C. Bouzigues image I potential 19/12/ :12 Laurent COHEN, LJLL, Step #1 Minimal Path between p1 and p2 Etape #2 Descente de gradient sur extraire le chemin minimal pour 19/12/ :12 potential Laurent COHEN, LJLL, LJLL, Dec

5 Minimal Path between p1 and p2 Step #1: U obtained by the FAST MARCHING ALGORITHM potentiel L. D. Cohen, R. Kimmel Global minimum for active contour models : a minimal path approach. 19/12/ :12 International Journal Laurent of Computer COHEN, LJLL, Vision, :57-78, Minimal Path between p1 and p2 Step #1: U obtained by the FAST MARCHING ALGORITHM Etape #2 Descente de gradient sur extraire le chemin minimal pour minimal action 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

6 Minimal Path between p1 and p2 Step #1 Step #2 gradient descent on extraction of minimal path for minimal action 19/12/ :12 Laurent COHEN, LJLL, Minimal Path between p1 and p2 minimal action L. D. Cohen, R. Kimmel Global minimum for active contour models : a minimal path approach. 19/12/ :12 International Journal Laurent of Computer COHEN, LJLL, Vision, :57-78, LJLL, Dec

7 Minimal Path between p1 and p2 minimal path Is obtained by solving ODE: simple gradient descent on from p 2 to p 1 minimal action L. D. Cohen, R. Kimmel Global minimum for active contour models : a minimal path approach. 19/12/ :12 International Journal Laurent of Computer COHEN, LJLL, Vision, :57-78, Step #1 Minimal Path between p1 and p2 Step #2 gradient descent on extraction of minimal path for minimal action 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

8 Minimal Path between p1 and p2 Step #1 Step #2 gradient descent on extraction of minimal path for 19/12/ :12 potential Laurent COHEN, LJLL, Overview Minimal Paths, Fast Marching and Front Propagation Geodesic Density for tree structures Using Geodesic voting with Radius energy Using Geodesic voting as a prior shape Using Geodesic voting as initial deformable tree 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

9 Geodesic Density 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density One could give a root point and the set of endpoints for each branch. Goal: user gives only one root point. 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

10 Geodesic Density 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

11 Geodesic Density 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: Biological image 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

12 Geodesic Density: Biological image 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: Shading Zone Problem 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

13 Geodesic Density: Shading Zone Problem Different solutions proposed: - Second step in each shading zone - Multiple source points - Transport Equation - Adaptive Voting: End points adaptively scattered 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: Shading Zone Problem Multiple source points 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

14 Geodesic Density: Transport Equation 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: Transport Equation 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

15 Geodesic Density: Shading Zone Problem Different solutions proposed: - Multiple source points - Second step in each shading zone - End points Scattered in the image 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density Not sensitive to the source point location 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

16 Geodesic Density: Real Biological image 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: Biological image 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

17 Geodesic Density: Biological image 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: Real example 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

18 Geodesic Density: from the image boundary Boundary : 4*256 end points 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: all points From all points : 256*256 end points 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

19 Geodesic Density: adaptive voting Adaptive voting : 1000 end points 19/12/ :12 Laurent COHEN, LJLL, Geodesic Density: adaptive voting Adaptive voting : 1000 end points 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

20 Overview Minimal Paths, Fast Marching and Front Propagation Geodesic Density for tree structures Using Geodesic voting with Radius energy Using Geodesic voting as a prior shape Using Geodesic voting as initial deformable tree 19/12/ :12 Laurent COHEN, LJLL, and centerline (with Y. Rouchdy, ISBI 11) voting using Space + radius distance 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

21 and centerline (with Y. Rouchdy, ISBI 11) voting using Space + radius distance 19/12/ :12 Laurent COHEN, LJLL, and centerline (with Y. Rouchdy, ISBI 11) voting using Space + radius distance 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

22 3D Minimal Path for tubular shapes in 2D 2D in space, 1D for radius of vessel 19/12/ :12 Laurent COHEN, LJLL, and centerline voting using Space + radius distance Different ways of using the voting results. 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

23 and centerline voting using Space + radius distance 19/12/ :12 Laurent COHEN, LJLL, and centerline voting using Space + radius distance 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

24 Overview Minimal Paths, Fast Marching and Front Propagation Geodesic Density for tree structures Using Geodesic voting with Radius energy Using Geodesic voting as a prior shape Using Geodesic voting as initial deformable tree 19/12/ :12 Laurent COHEN, LJLL, and Segmentation with Prior Level set method either edge based or region based 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

25 and Segmentation with Prior Level set initialized with result of voting 19/12/ :12 Laurent COHEN, LJLL, and Segmentation with Prior (with Y. Rouchdy, SSVM 11) Chan Vese Energy including shape Prior given by 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

26 and Segmentation with Prior (with Y. Rouchdy, SSVM 11) Chan Vese Energy including shape Prior given by 19/12/ :12 Laurent COHEN, LJLL, and Segmentation with Prior (with Y. Rouchdy, SSVM 11) Chan Vese Energy including shape Prior given by 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

27 and Segmentation with Prior 3D extension 19/12/ :12 Laurent COHEN, LJLL, and Segmentation 3D extension 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

28 Overview Minimal Paths, Fast Marching and Front Propagation Geodesic Density for tree structures Using Geodesic voting with Radius energy Using Geodesic voting as a prior shape Using Geodesic voting as initial deformable tree 19/12/ :12 Laurent COHEN, LJLL, and Deformable Tree (with Julien Mille, MMBIA 09, ISBI 10) initial image with root point 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

29 and Deformable Tree minimal action map from root point 19/12/ :12 Laurent COHEN, LJLL, and Deformable Tree Initial tree from voting score 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

30 and Deformable Tree Removing unsignificant segments by thresholding the geodesic voting 19/12/ :12 Laurent COHEN, LJLL, and Deformable Tree 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

31 and Deformable Tree Intermediate steps of tree evolution. 19/12/ :12 Laurent COHEN, LJLL, and Deformable Tree final step of tree evolution. 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

32 and Deformable Tree 19/12/ :12 Laurent COHEN, LJLL, and Deformable Tree Energy minimizing Deformable tube 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

33 and Deformable Tree 19/12/ :12 Laurent COHEN, LJLL, Conclusion Minimally interactive tools for vessels and vascular tree segmentation (tubular branching structures) User provides only one initial point Fast and efficient propagation algorithm Voting approach as a powerful tool to find the structure, which can be completed with other approach. 19/12/ :12 Laurent COHEN, LJLL, LJLL, Dec

34 Thank you! Publications on your screen: 19/12/ :12 Laurent COHEN, LJLL, for biomedical image segmentation Laurent D. COHEN Directeur de Recherche CNRS CEREMADE, UMR CNRS 7534 Université Paris-9 Dauphine Place du Maréchal de Lattre de Tassigny Paris, France Some joint works with Y. Rouchdy, and J. Mille. LJLL, Journée thématique maths, image et biomédecine 18 Décembre /12/ :12 Laurent COHEN, LJLL, LJLL, Dec