Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 1 Upgrading the level.

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1 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 1 Upgrading the level set method: point correspondence, topological constraints and deformation priors Jean-Philippe Pons Jean-Philippe.Pons@certis.enpc.fr http://cermics.enpc.fr/~pons C ERTIS École Nationale des Ponts et Chaussées Marne-la-Vallée, France

2 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 2 Acknowledgements Olivier Faugeras, I NRIA Sophia-Antipolis Renaud Keriven, École Nationale des Ponts et Chaussées Mathieu Desbrun, C ALTECH Florent Ségonne, M IT / M GH Gerardo Hermosillo, Siemens Medical Solutions Guillaume Charpiat & Pierre Maurel, École Normale Supérieure Paris

3 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 3 Outline Level sets with point correspondence Level sets with topology control Level sets with deformation priors

4 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 4 Why level sets are cool… No parameterization Automatic handling of topology changes Easy computation of geometric properties Mathematical proofs and numerical stability

5 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 5...and why level sets suck Computationally expensive Narrow band algorithm [Adalsteinsson & Sethian, 95] PDE-based fast local level set method [Peng, Merriman, Osher et al., 99] GPU implementation [Lefohn et al., 04] Fixed uniform resolution Octree-based level sets [Losasso, Fedkiw & Osher, 06] Need a periodic reinitialization Extension velocities [Adalsteinsson & Sethian, 99] Need a mesh extraction step Marching cubes algorithm [Lorensen & Cline, 87]

6 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 6...and why level sets suck (continued) Numerical diffusion Particle level set method [Enright, Fedkiw et al., 02] Limited to codimension 1 Limited to closed surfaces Cannot track a region of interest on the surface Cannot handle interfacial data No point-wise correspondence No control on topology

7 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 7 Outline Level sets with point correspondence Level sets with topology control Level sets with deformation priors

8 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 8 Problem statement Level sets convey a purely geometric description The point-wise correspondence is lost Cannot handle interfacial data Restricts the range of possible applications Workaround: a hybrid Lagrangian-Eulerian method? ?

9 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 9 Back to basics Level set equation: Using a velocity vector field: Transport of an auxiliary quantity: Let be the level set function of an auxiliary surface Region tracking with level sets [Bertalmío, Sapiro & Randall, 99] Open surfaces with level sets [Solem & Heyden, 04] 3D curves with level sets [Burchard, Cheng, Merriman & Osher, 01]

10 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 10 Point correspondence Advecting the point coordinates with the same speed as the level set function Correspondence function pointing to the initial interface System of Eulerian PDEs:

11 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 11 Numerical aspects Reinitialization of the level set function to keep it a signed distance function Run Extension of the correspondence function to keep it constant along the normal Run Projection of the correspondence function to keep it onto the initial interface Take

12 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 12 Results 2D experiments A rotating and shrinking circle Initial interface/data Final interface/data Final correspondence

13 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 13 A shrinking square An expanding square

14 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 14 The merging of two expanding circles A circle in a vortex velocity field

15 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 15 Results 3D experiments A deforming plane A deforming sphere

16 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 16 Cortex unfolding Velocity field? Mean curvature motion Area-preserving tangential velocity field Area-preserving condition Our method Solve the following intrinsic Poisson equation Take Expansion/shrinkage due to tangential motion Expansion/shrinkage due to the association of normal motion and curvature Mean expansion/shrinkage

17 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 17 Example Results Histogram of the Jacobian InitialMean curvature motion Mean curvature motion + area-preservation

18 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 18 Outline Level sets with point correspondence Level sets with topology control Level sets with deformation priors

19 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 19 Level sets with topology control In some applications, automatic topology changes are not desirable Topology-preserving level sets [Han, Xu & Prince, 02] Modified update procedure based on the concept of simple point Topology-consistent marching cubes algorithm Topological dead-ends! Our method: genus-preserving level sets Prevents the formation/closing of handles Allows the objects to split/merge Less sensitive to initial conditions

20 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 20 Application Cortex segmentation from MRI NB: Without topology control, genus = 18

21 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 21 Outline Level sets with point correspondence Level sets with topology control Level sets with deformation priors

22 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 22 Motivation Gradient flows are prone to local minima The gradient = steepest descent direction depends on the choice of an inner product Deformation space = inner product space Gâteaux derivative The gradient is defined by Everybody use We build other inner products to get better descents Related work: Sobolev active contours [Sundaramoorthi, Yezzi & Mennuci, 05]

23 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 23 Our construction A family of inner products symmetric positive definite Motion decomposition: Favoring rigid + scaling motions translation + rotation + scaling + non-rigid

24 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 24 Results Shape warping by minimizing the Hausdorff distance [Charpiat, Faugeras & Keriven, 05] L 2 gradient Gradient with a quasi-rigid prior

25 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 25 Results Shape matching using a quasi-articulated prior

26 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 26 Summary of the contributions Level sets with point correspondence System of Eulerian PDEs Handles normal and tangential velocity fields, large deformations, shocks, rarefactions and topological changes Area-preserving tangential velocity field Genus-preserving level sets In-between traditional level sets and topology-preserving level sets Based on a new concept of digital topology Useful in biomedical image segmentation Gradient flow with deformation priors Generalizes Sobolev active contours Quasi-rigid prior, quasi-articulated prior Improves robustness to local minima

27 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 27 Perspective The level set method has lost much of its simplicity Ongoing work: improving snakes? Computation of geometric quantities Discrete differential geometry, discrete exterior calculus (K. Polthier, P. Schröder, M. Desbrun) Topology changes T-snakes and T-surfaces [McInerney & Terzopoulos, 96] Computational geometry (J.-D. Boissonnat, P. Alliez, L. Kobbelt) Movie preview

28 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 28 Thank you for your attention Questions?

29 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 29 References J.-P. Pons, G. Hermosillo, R. Keriven and O. Faugeras. Maintaining the point correspondence in the level set framework. To appear in Journal of Computational Physics. J.-P. Pons, G. Hermosillo, R. Keriven and O. Faugeras. How to deal with point correspondences and tangential velocities in the level set framework. In Proceedings of ICCV 2003. J.-P. Pons. Methodological and applied contributions to the deformable models framework. PhD thesis, École Nationale des Ponts et Chaussées, 2005. G. Charpiat, P. Maurel, J.-P. Pons, R. Keriven and O. Faugeras. Generalized gradients: priors on minimization flows. To appear in IJCV. G. Charpiat, R. Keriven, J.-P. Pons and O. Faugeras. Designing spatially- coherent minimizing flows for variational problems based on active contours. In Proceedings of ICCV 2005.

30 Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons – MIA06 – September 19 th, 2006 30 The level set method [Osher & Sethian, 88] Interface represented as the zero level set of a higher-dimensional scalar function Link between the motion of the interface and the evolution of the level set function Γ N Eulerian PDE on the cartesian gridLagrangian ODE