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

Outline Numerical issues: Difference scheme: upwind scheme 4/24/2017 Outline Numerical issues: Difference scheme: upwind scheme Velocity extension Reinitialization Variational level set method: Level set evolution without reinitialization. Active contour without edges Multiphase level set methods

Numerical Implementation of Level Set Evolution 4/24/2017 Numerical Implementation of Level Set Evolution

Explicit Euler Scheme Consider general evolution equation: 4/24/2017 Explicit Euler Scheme Consider general evolution equation: Approximate spatial derivatives by certain difference scheme Approximate temporal derivatives by forward difference Update equation at each iteration:

Basic Finite Difference Scheme for Spatial Variable 4/24/2017 Basic Finite Difference Scheme for Spatial Variable Backward difference Forward difference Central difference

Mean Curvature Motion Mean curvature motion Update equation: where 4/24/2017 Mean Curvature Motion Mean curvature motion Update equation: where Mean curvature motion is stable.

Motion in Normal Direction 4/24/2017 Motion in Normal Direction Motion in normal direction: Right hand side is approximated by: where Update equation:

Advection Pure advection equation: , with 4/24/2017 Advection Pure advection equation: , with Right hand side is approximated by: where Update equation:

Geodesic Active Contour 4/24/2017 Geodesic Active Contour Geodesic active contour: Update equation:

General Evolution Equation 4/24/2017 General Evolution Equation Level set evolution:

Time Step Level set evolution: 4/24/2017 Time Step Level set evolution: For stable evolution, the time step and spatial step must satisfy the CFL condition:

Unstable Evolution in Standard Level Set Methods 4/24/2017 Unstable Evolution in Standard Level Set Methods Evolution of level set function Evolution of zero level set Click here to see the movie Click here to see the movie

Reinitialization (Redistance) Reinitialization: periodically stop the evolution and repair the degraded level set function as a signed distance function. Solve to steady state: Reinitialization equation “Steady state” means exists, denoted by . Signed distance function. Use upwind scheme to numerically solve the above reinitialization equation. Other reinitialization methods: direct compute SDF or fast marching.

Velocity Extension In some applications, the speed function is only defined on the zero level set (interface) 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: with Note: is the directional derivative of along the normal to the interface

Summary of Standard Level Set Methods 4/24/2017 Summary of Standard Level Set Methods Initialization Compute signed distance Complex upwind scheme and small time step Evolve level set function Solve a PDE Velocity extension Solve another PDE! Reinitialization N converge? Y stop

Variational Level Set Formulation 4/24/2017 Variational Level Set Formulation

Drawbacks of Reinitialization 4/24/2017 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

Level Set Evolution without Reinitialization (Li et al, 2005) 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. Characteristics of signed distance function: signed distance function + constant Deviation from a signed distance function:

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 Gradient flow (or steepest descent): Gateaux derivative:

External Energy for Image Segmentation 4/24/2017 External Energy for Image Segmentation Edge indicator function for image I Define external Energy: Weighted length term: Weighted region term:

Energy Functional and Gradient Flow 4/24/2017 Energy Functional and Gradient Flow Define energy functional: The gradient flow of the functional is:

Mechanism of Maintaining Signed Distance 4/24/2017 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)

Implementation Use the smoothed Dirac function Discretization of PDE: 4/24/2017 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

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

Flexible and Efficient Initialization 4/24/2017 Flexible and Efficient Initialization The initial level set function is no longer required to be a signed distance function in our method A region-based initialization scheme:

Experimental Results Click here to see the movie

Initial level set function Evolution Click here to see the movie

Detect Weak Object Boundaries Microscope image of cells Click here to see the movie

Experimental Results: Initialization from Thresholding 4/24/2017 Experimental Results: Initialization from Thresholding MR image of corpus callosum Initial level set function Image courtesy of Hong Liu, National Institute of Mental Health Evolution Click here to see the movie

Initial level set function Evolution Click here to see the movie

3D Extension The proposed level set formulation Click here to see the movie The proposed level set formulation and implementation can be easily extended to 3D.

Thank you