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Introduction to Level Set Methods:Part II

Chunming Li

Institute of Imaging Science

Vanderbilt University

URL: www.vuiis.vanderbilt.edu/~licm

E-mail: chunming.li@vanderbilt.edu

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Outline

1. Numerical issues:

• Difference scheme: upwind scheme

• Velocity extension

• Reinitialization

2. Variational level set method:

• Level set evolution without reinitialization.

• Active contour without edges

• Multiphase level set methods

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Numerical Implementation of Numerical Implementation of Level Set EvolutionLevel Set Evolution

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Explicit Euler Scheme

• Consider general evolution equation:

• Update equation at each iteration:

Approximate spatial derivatives by certain difference scheme

Approximate temporal derivatives by forward difference

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Basic Finite Difference Scheme for Spatial Variable

• Backward difference

• Forward difference

• Central difference

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Mean Curvature Motion

• Mean curvature motion

• Update equation:

where

• Mean curvature motion is stable.

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Motion in Normal Direction

• Motion in normal direction:

• Right hand side is approximated by:

where

• Update equation:

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Advection

• Pure advection equation: , with

• Right hand side is approximated by:

where• Update equation:

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Geodesic Active Contour

• Geodesic active contour:

• Update equation:

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General Evolution Equation

• Level set evolution:

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Time Step

• Level set evolution:

• For stable evolution, the time step and spatial step must satisfy the CFL condition:

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Unstable Evolution in Standard Level Set Methods

Evolution of level set function Evolution of zero level setClick here to see the movie Click here to see the movie

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Reinitialization (Redistance)

• “Steady state” means exists, denoted by . Signed distance

function.

• Reinitialization: periodically stop the evolution and repair the degraded level set function as a signed distance function.

• Use upwind scheme to numerically solve the above reinitialization equation.

• Solve to steady state:

Reinitialization equation

• Other reinitialization methods: direct compute SDF or fast marching.

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Velocity Extension

• In some applications, the speed function is only defined on the zero level set (interface)

with

• To extend to the speed function to the entire domain or a narrow band of the zero level set, solve the boundary value problem:

or solve to steady state of the initial value problem:

Note: is the directional derivative of along the normal to the interface

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Summary of Standard Level Set Methods

Reinitialization

Initialization

Evolve level set function

converge?N

Y

stop

Velocity extension

Compute signed distance

Complex upwind scheme and small time step

Solve a PDE

Solve another PDE!

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Variational Level Set FormulationVariational Level Set Formulation

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Drawbacks of Reinitialization

Drawbacks of Reinitialization:

• Still a serious problem: when and how to reinitialize?

(no general answer so far)

• Error in location of the zero level set

• Computationally expensive

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Level Set Evolution without Reinitialization(Li et al, 2005)

• Deviation from a signed distance function:

• Characteristics of signed distance function:

Goal: Find a level set evolution algorithm that can simultaneously move the zero level set while maintaining the signed distance profile throughout the entire evolution.

signed distance function + constant

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Variational Level Set Formulation

Define an energy functional on level set function:

Internal energy:

• Penalize the deviation from a signed distance function

External energy:

• Drive the motion of the zero level set

Gateaux derivative:

Gradient flow (or steepest descent):

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External Energy for Image Segmentation

Weighted length term:

Weighted region term:

Edge indicator function for image I

Define external Energy:

0

0

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Energy Functional and Gradient Flow

Define energy functional:

The gradient flow of the functional is:

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Mechanism of Maintaining Signed Distance

Rewrite the gradient flow of internal energy:

Diffusion rate:

Positive diffusion rate

Decrease gradient (usual diffusion)

Negative diffusion rate

Increase gradient (reverse diffusion)

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Implementation

Use the smoothed Dirac function

Discretization of PDE:

• Forward difference for temporal derivative

• Can use relatively larger time step

• Central difference for spatial derivative

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Experimental Results

Evolution of level set function Evolution of zero level setClick here to see the movieClick here to see the movie

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Flexible and Efficient Initialization

A region-based initialization scheme:

The initial level set function is no longer required to be a signed

distance function in our method

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Experimental ResultsClick here to see the movie

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EvolutionInitial level set functionClick here to see the movie

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Detect Weak Object Boundaries

Click here to see the movie

Microscope image of cells

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Experimental Results:Initialization from Thresholding

Initial level set functionMR image of corpus callosum

Evolution

Click here to see the movie

Image courtesy of Hong Liu,National Institute of Mental Health

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EvolutionInitial level set functionClick here to see the movie

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3D Extension

The proposed level set formulation

and implementation can be easily

extended to 3D.

Click here to see the movie

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Thank you