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Preconditioned Level Set Flows. Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münster. Introduction.

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preconditioned level set flows

Preconditioned Level Set Flows

Martin Burger

Institute for Computational and Applied Mathematics

European Institute for Molecular Imaging (EIMI)

Center for Nonlinear Science (CeNoS)

Westfälische Wilhelms-Universität Münster

introduction
Introduction
  • We often look for geometric objects (with unknown topology) rather than for functions, e.g. in - image segmentation - computer graphics / surface restoration - topology optimization - inverse obstacle scattering- inclusion / cavity / crack detection - …..

Level Set Flows AIP 2007, Vancouver, June 07

introduction1
Introduction
  • Because of the desired flexibility you need, you have to use level set methods (or something similar) Osher & Sethian, JCP 1987, Sethian, Cambridge Univ. Press 1999, Osher & Fedkiw, Springer, 2002

Level Set Flows AIP 2007, Vancouver, June 07

level set methods
Level Set Methods
  • Because of the desired flexibility you need, you have to use level set methods (or something similar) Normal and mean curvature

Level Set Flows AIP 2007, Vancouver, June 07

shape optimization approach
Shape Optimization Approach
  • In a natural way such problems can be formulated as shape optimization problems

where K is a class of admissible shapes (eventually including additional constraints).

  • Mumford-Shah / Chan-Vese functionals in segmentation
  • Least-squares / reciprocity gap in crack / inclusion dectection

Level Set Flows AIP 2007, Vancouver, June 07

level set flows
Level Set Flows
  • Solution by level set flows: find velocities such that resulting evolution of shapes decreases objective functional JSantosa 96, Osher-Santosa 01, Dorn 01/02
  • Geometric flow of the level sets of f can be translated into nonlinear differential equation for f („level set equation“)

Level Set Flows AIP 2007, Vancouver, June 07

shape calculus
Shape Calculus

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  • Formulas for the change of the objective functional are obtained by shape calculus
  • Shape sensitivity is a linear functional of the normal velocity on the boundary

Level Set Flows AIP 2007, Vancouver, June 07

descent directions
Descent Directions
  • Descent directions are obtained by chosing the normal velocity as a representation of the negative shape sensitivity in a Hilbert space
  • Asymptotic expansion for small-time motion with that normal velocity mb 03/04

Level Set Flows AIP 2007, Vancouver, June 07

optimization viewpoint
Optimization Viewpoint
  • Descent method, time stept can be chosen by standard optimization rules (Armijo-Goldstein)
  • Descent method independent ofparametrization, can change topology by splitting
  • Level set method used to perform update
  • Change of scalar product is effective preconditioning !

Level Set Flows AIP 2007, Vancouver, June 07

differential geometry viewpoint
Differential Geometry Viewpoint
  • Interpretation from global differential geometry: - take manifold of shapes with appropriate metric to obtain Riemannian manifold

- tangent spaces of such a manifold can be identified with normal velocities

- Riemannian structure induces scalar product on tangent spaces

  • We usually take the other way: „scalar product on tangent spaces induces Riemannian structure“Michor-Mumford 05

Level Set Flows AIP 2007, Vancouver, June 07

de giorgi viewpoint
De Giorgi Viewpoint

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  • Generalized gradient flow structure in a metric space (concept of minimizing movements)

DeGiorgi 1974, Ambrosio-Gigli-Savare 2005

  • Flow obtained as the limit of variational problems (implicit Euler)
  • We can alternatively chose a metric and expand it to second order !

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics
Shape Metrics

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  • Class of shape metrics obtained by using Hilbert space norms on signed distance functions
  • Expansion yields Sobolev spaces on normal velocities
  • Natural in level set form

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics1
Shape Metrics

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  • Simple Example
  • Shape sensitivity
  • Flow in L2 metric

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics2
Shape Metrics

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  • Resulting Velocity

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics3
Shape Metrics

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  • Possible alternative: H1 metric
  • Resulting Velocity
  • Zero-order flow instead of 2nd order parabolic – very weak step size condition !Sundaramoorthi, Yezzi,Mennucci 06

Level Set Flows AIP 2007, Vancouver, June 07

image segmentation
Image Segmentation
  • Active contour model with L2 and H1 metric Sundaramoorthi

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics4
Shape Metrics
  • Other alternatives: H1/2and H-1/2 metric
  • Sounds complicated, but can be realized as Dirichlet or Neumann traces of Sobolev functions on domain ! Can take harmonic extensions and their H1 scalar product in a surrounding domain
  • Automatic extension, ideal for level set flows

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics5
Shape Metrics
  • H1/2used in topology optimization

Allaire, Jouve et al 06/07

Level Set Flows AIP 2007, Vancouver, June 07

example obstacle problem
Example: Obstacle Problem
  • H-1/2 good choice for source reconstructions, obstacle problems
  • Model Example

Level Set Flows AIP 2007, Vancouver, June 07

shape sensitivities
Shape Sensitivities
  • First shape variationgiven by
  • Second shape variation is coercive at stationary points
  • H-1/2 is the right scalar product !

Level Set Flows AIP 2007, Vancouver, June 07

velocity choice
Velocity choice

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  • Simple calculation shows that velocity for L2 metric is given by
  • By equivalent realization of the H-1/2 metric we obtain alternative choice
  • Immediate extension velocity field

Level Set Flows AIP 2007, Vancouver, June 07

trajectories
Trajectories

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  • Due to existence of a velocity field we can analyze the shape evolution via trajectories
  • Note: from elliptic regularity theory, velocity field is Hölder continuous (existence), but not Lipschitz (no Picard-Lindelöf !!)

Level Set Flows AIP 2007, Vancouver, June 07

trajectories1
Trajectories
  • But the velocity field is almost Lipschitz
  • This is enough for uniqueness (Osgood‘s theorem)
  • No stability estimate for ODE !
  • Thus, topology could change !

Level Set Flows AIP 2007, Vancouver, June 07

complete analysis
Complete Analysis
  • Main result: level set flow is well-defined, independent of level set representation, and converges to a minimizer of the functional
  • Well-definedness by approximation argument and trajectory analysis
  • Independence by existence of uniqueness of trajectories
  • Convergence by energy estimates mb-Matevosyan 07 (?)

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics6
Shape Metrics

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  • Shape deformations nicely controlled by Hausdorff-Metric
  • Not differentiable due to max and sup, hence no reasonable expansion
  • Smooth approximations possibleFaugeras et al 06

Level Set Flows AIP 2007, Vancouver, June 07

shape metrics7
Shape Metrics

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  • For fast methods, use variable metrics – based on second derivatives !
  • Inexact Newton methods: positive definite approximation of second derivativeHintermüller-Ring 04
  • For least-squares problems natural choice: Gauss-Newton (Levenberg-Marquardt, ..)mb 04, Ascher et al 07

Level Set Flows AIP 2007, Vancouver, June 07

levenberg marquardt method
Levenberg-Marquardt Method

Example: cavity detection

Level Set Flows AIP 2007, Vancouver, June 07

levenberg marquardt method1
Levenberg-Marquardt Method
  • No noise
  • Iterations2,4,6,8

Level Set Flows AIP 2007, Vancouver, June 07

levenberg marquardt method2
Levenberg-Marquardt Method

Residual and L1-error

Level Set Flows AIP 2007, Vancouver, June 07

levenberg marquardt method3
Levenberg-Marquardt Method

Residual and L1-error, noisy data

Level Set Flows AIP 2007, Vancouver, June 07

levenberg marquardt method4
Levenberg-Marquardt Method

0.1 % noise

Iterations

5,10,20,25

Level Set Flows AIP 2007, Vancouver, June 07

levenberg marquardt method5
Levenberg-Marquardt Method

1% noise 2% noise

3% noise 4% noise

Level Set Flows AIP 2007, Vancouver, June 07

download and contact
Download and Contact

Papers and talks at

www.math.uni-muenster.de/u/burger

or by email

martin.burger@uni-muenster.de

Based on joint work with:

Norayr Matevosyan, Stan Osher

Thanks for input and suggestions to:

B.Hackl, W.Ring, M.Hintermüller, U.Ascher

Austrian Science Foundation FWF, SFB F 013

Level Set Flows AIP 2007, Vancouver, June 07