Upgrading the level set method: point correspondence, topological constraints and deformation priors

1
Upgrading the level set methodpoint
correspondence, topological constraintsand
deformation priors

• Jean-Philippe Pons
• Jean-Philippe.Pons_at_certis.enpc.fr
• http//cermics.enpc.fr/pons
• CERTIS
• École Nationale des Ponts et Chaussées
• Marne-la-Vallée, France

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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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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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...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

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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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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

Final interface/data
Initial 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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Results

• 3D experiments
• A deforming plane
• A deforming sphere

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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
Mean expansion/shrinkage
Expansion/shrinkage due to the association of
normal motion and curvature
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• Example
• Results
• Histogram of the Jacobian

Initial
Mean curvature motion
Mean curvature motion area-preservation
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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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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

N
Eulerian PDE on the cartesian grid
Lagrangian ODE