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Snakes_ Strings_ Balloons and Other Active Contour Models

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					Active Contours (“Snakes”)
                           Goal

• Start with image and initial closed curve
• Evolve curve to lie along “important” features
   –   Edges
   –   Corners
   –   Detected features
   –   User input
                 Applications

• Region selection in Photoshop
• Segmentation of medical images
• Tracking
      Snakes: Active Contour Models

• Introduced by Kass, Witkin, and Terzopoulos
• Framework: energy minimization
  – Bending and stretching curve = more energy
  – Good features = less energy
  – Curve evolves to minimize energy
• Also “Deformable Contours”
              Snakes Energy Equation

• Parametric representation of curve

                      v(s)  x(s), y(s)

• Energy functional consists of three terms


         int v( s )    img v( s )    con v( s )  ds
                      Internal Energy


                  
    int v( s)    ( s) v s ( s)   ( s) v ss ( s)
                                  2                      2
                                                             2
• First term is “membrane” term – minimum
  energy when curve minimizes length
  (“soap bubble”)
• Second term is “thin plate” term – minimum
  energy when curve is smooth
                      Internal Energy


                  
    int v( s)    ( s) v s ( s)   ( s) v ss ( s)
                                  2                      2
                                                             2
• Control  and  to vary between extremes
• Set  to 0 at a point to allow corner
• Set  to 0 everywhere to let curve follow sharp
  creases – “strings”
                   Image Energy

• Variety of terms give different effects
• For example,

              img  w  I ( x, y)  I desired

  minimizes energy at intensity Idesired
               Edge Attraction

• Gradient-based:

              img   w  I ( x, y )
                                         2




• Laplacian-based:
                                         2
               img  w   I ( x, y )
                            2




• In both cases, can smooth with Gaussian
              Constraint Forces

• Spring

                con  k  v  x
                                    2



• Repulsion
                           k
                 con 
                          vx
                                2
                  Evolving Curve

• Computing forces on v that locally minimize
  energy gives differential equation for v
   – Euler-Lagrange formula
               d 2    d    
                  2             0
               ds  v  ds  v  v
                             

• Discretize v: samples (xi, yi)
   – Approximate derivatives with finite differences
• Iterative numerical solver
Corpus Callosum




                  [Davatzikos and Prince]
Corpus Callosum




                  [Davatzikos and Prince]

				
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posted:6/17/2012
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