Geodesic Snakes Level-Set Evolution CS7960 Advanced Image - - PowerPoint PPT Presentation

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Geodesic Snakes Level-Set Evolution CS7960 Advanced Image - - PowerPoint PPT Presentation

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Geodesic Snakes Level-Set Evolution CS7960 Advanced Image Processing April 8 th , 2010 Jonathan Bronson Content Motivation Implicit Contour Formulation Hypersurface Embedding Examples Conclusions Motivation Drawbacks of

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Geodesic Snakes

Level-Set Evolution

Jonathan Bronson CS7960 Advanced Image Processing April 8th, 2010

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Content

 Motivation  Implicit Contour Formulation  Hypersurface Embedding  Examples  Conclusions

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Motivation

 Drawbacks of previous Snake formulations:

 Explicit Representation

 Parameterization / Reparameterization issues  Approximating Discrete Derivatives

 Fixed Topology  Extention to 3D very complex (active meshes)

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Motivation

 New Approach:

 Embed contour in higher order surface  Implicit Representation  Insensitive to Topology  Easily extends to 3D

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Mathematical Framework

dC d t =  N : Speed  N : Normal dC d t =  N :Curvature

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Mathematical Framework

 Combining terms simple:

 Still want:

 Ability to slow/stop on edges/lines/etc  Image force term

 Where have we seen this before?

d C d t =  N d C d t =gI   N

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Mathematical Framework

 Anisotropic Diffusion (Perona & Malik)

 Use gradient magnitude for diffusion speed

gI = 1 1∥ ∇  I∥

2

gI =e

−∥ ∇  I∥

2

(Quadratic) (Exponential)

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Mathematical Formulation

 What if we overshoot?  Want to pull toward edges

∂C ∂t =gI   N−∇ gI  ⋅ N  N

Advection Term

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Embedding Contour C(s,t) into Surface u(x,t)

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Embedding

 Embedding function:  Contour:

u x ,t  Cs ,t uC ,t=0

(Zero level-set)

u∈ℜ

3

C∈ℜ

2

u x ,t  Cs ,t

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Embedding

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Embedding Formulation

 How does surface vary over time?

uCt ,t=0 d dt uCt ,t=∂ u ∂ t  ∂C ∂ x ∂ x ∂ t  ∂C ∂ y ∂ y ∂ t ∂ C ∂ z ∂ z ∂ t d dt uCt ,t=∂ u ∂ t ∇ u⋅dC dt

Chain Rule

dC d t =  N= −∇ u

∣∇ u∣

d dt uCt ,t=∂ u ∂ t −∣ ∇ u∣ =0 ∇ u ∇ u

∣

∇ u∣ =∣ ∇ u∣ ∂u ∂ t =∣ ∇ u∣

Hamilton-Jacobi Equation for certain speeds 

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Interpretation

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Summary

 Implicit Solution  Solvable using PDE's (stable)

Parameterization Free

 Seamlessly handles Topological Changes  Extends to 3D in Straightfoward Manner  Common Implementations

 Fast Marching Method  Fast Iterative Method

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Examples

(www.cs.bris.ac.uk)