Lecture 6 : Level Set Method

Introduction

• Developed by– Stanley Osher (UCLA)– J. A. Sethian (UC Berkeley)

• Books– J.A. Sethian: Level Set Methods and Fast

Marching Methods, 1999– S. Osher, R. Fedkiw, Level Set Methods and

Dynamic Implicit Surfaces , 2002

Evolving Curves and Surfaces

Geometry Representation

Explicit Techniques for Evolution

Explicit Techniques - Drawbacks

Implicit Geometries

Discretized Implicit Geometries

Level Set Method: Overview

• Generic numerical method for evolving fronts in an implicit form– Handles topological changes of the evolving interface– Define problem in 1 higher dimension

• Use an implicit representation of the contour C as the zero level set of higher dimensional function - the level set function

Level Set Method: Overview

• Move the level set function, so that it deforms in the way the user expects

• contour = cross section at z=t

Implicit Curve Evolution

Level Set Evolution

• Define a speed function F, that specifies how contour points move in time– Based on application-specific physics such as time,

position, normal, curvature, image gradient magnitude

• Build an initial level set curve

• Adjust over time

• Current contour is defined as

Equation for Level Set Evolution

• Indirectly move C by manipulating

where F is the speed function normal to the curve

Level set equation

Example: an expanding circle

• Level Set representation of a circle

– Setting F=1 causes the circle to expand uniformly

– Observe everywhere– We obtain

• Explicit solution: – meaning the circle has radius r+t at time t

Example: an expanding circle

Motion under curvature

• Complicated shapes?

– Each piece of the curve moves perpendicular to the curve with speed proportional to the curvature

– Since curvature can be either positive or negative , some parts of the curve move outwards while others move inwards

– Example movie file• Setting F = curvature

Level Set Segmentation

• We may think of as signed distance function– Negative inside the curve– Positive outside the curve– Distance function has unit gradient almost

everywhere and smooth

• By choosing a suitable speed function F, we may segment an object in an image

Level Set Segmentation

• Evolving Geometry : (X,t)=0

– Intuitively, move a lot on low intensity gradient area and move little on high intensity gradient area along normal direction

– F : speed function , k : curvature , I : intensity

Segmentation Example

• Arterial tree segmentation

Discretization

• Use upwinded finite difference approximations (first order)

Acceleration Techniques

• Acceleration for fast level set method– Narrow band level set method– Fast marching method

Narrow band level set method

• The efficiency comes from updating the speed function

• We do not need to update the function over the whole image or volume

• Update over a narrow band (2D or 3D)

Fast Marching Method

• Assume the front (level set) propagates always outward or always inward

• Compute T(x,y)=time at which the contour crosses grid point (x,y)

• At any height T, the surface gives the set of points reached at time T

Fast Marching Algorithm

Fast Marching Algorithm

Fast Marching Method

Applications

• Segmentation

• Level Set Surface Editing Operators

• Surface Reconstruction

Segmetation

• 2D

• 3D

Level Set Surface Editing Operators

• SIGGRAPH 2002

Level Set Surface Editing Operators

Surface Reconstruction

• zhao, osher, and fedkiw 2001

A painting interface for interactivesurface deformations