Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons...
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Centre d‘Enseignement et de Recherche en Technologies de l‘Information et Systèmes Jean-Philippe Pons – MIA’06 – September 19 th , 2006 1 Upgrading the level set method: point correspondence, topological constraints and deformation priors Jean-Philippe Pons [email protected] http://cermics.enpc.fr/~pons CERTIS École Nationale des Ponts et Chaussées Marne-la-Vallée, France
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Transcript of Centre dEnseignement et de Recherche en Technologies de lInformation et Systèmes Jean-Philippe Pons...
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Jean-Philippe Pons – MIA’06 – September 19th, 2006 1
Upgrading the level set method:point correspondence, topological constraints
and deformation priors
Jean-Philippe [email protected]
http://cermics.enpc.fr/~pons
CERTISÉcole Nationale des Ponts et Chaussées
Marne-la-Vallée, France
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Jean-Philippe Pons – MIA’06 – September 19th, 2006 2
Acknowledgements
Olivier Faugeras, INRIA Sophia-Antipolis
Renaud Keriven, École Nationale des Ponts et Chaussées
Mathieu Desbrun, CALTECH
Florent Ségonne, MIT / MGH
Gerardo Hermosillo, Siemens Medical Solutions
Guillaume Charpiat & Pierre Maurel, École Normale Supérieure Paris
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Jean-Philippe Pons – MIA’06 – September 19th, 2006 3
Outline
Level sets with point correspondence
Level sets with topology control
Level sets with deformation priors
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Why level sets are cool…
No parameterization
Automatic handling of topology changes
Easy computation of geometric properties
Mathematical proofs and numerical stability
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...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]
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Jean-Philippe Pons – MIA’06 – September 19th, 2006 6
...and why level sets suck (continued)
Numerical diffusion Particle level set method [Enright, Fedkiw et al., 02]
Limited to codimension 1Limited to closed surfacesCannot track a region of interest on the surfaceCannot handle interfacial dataNo point-wise correspondence
No control on topology
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Outline
Level sets with point correspondence
Level sets with topology control
Level sets with deformation priors
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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?
?
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Jean-Philippe Pons – MIA’06 – September 19th, 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]
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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:
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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
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Results
2D experiments A rotating and shrinking circle
Initial interface/data Final interface/data Final correspondence
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A shrinking square
An expanding square
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The merging of two expanding circles
A circle in a vortex velocity field
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Jean-Philippe Pons – MIA’06 – September 19th, 2006 15
Results
3D experiments A deforming plane
A deforming sphere
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Jean-Philippe Pons – MIA’06 – September 19th, 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
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ExampleResults
Histogram of the Jacobian
Initial Mean curvature motion Mean curvature motion + area-preservation
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Jean-Philippe Pons – MIA’06 – September 19th, 2006 18
Outline
Level sets with point correspondence
Level sets with topology control
Level sets with deformation priors
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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
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Application
Cortex segmentation from MRI
NB: Without topology control, genus = 18
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Jean-Philippe Pons – MIA’06 – September 19th, 2006 21
Outline
Level sets with point correspondence
Level sets with topology control
Level sets with deformation priors
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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]
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Our construction
A family of inner products symmetric positive definite
Motion decomposition:
Favoring rigid + scaling motions
translation + rotation + scaling + non-rigid
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Results
Shape warping by minimizing the Hausdorff distance [Charpiat, Faugeras & Keriven, 05]
L2 gradient
Gradient with a quasi-rigid prior
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Results
Shape matching using a quasi-articulated prior
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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
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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
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Thank you for your attention
Questions?
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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.
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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
