Image segmentation with PDEs

🗲matt hancock🗲

Image analysis

  • Detection: where is the thing?
  • Recognition: what is the thing?
  • Segmentation: where's the borders of the thing?

Segmentation with PDEs

  • Tell curve/surface how to move → PDE
  • Minimize energy functional → PDE
  • Numerically solve PDE → iterative segmentation method

Example 1

  • Curve velocity is in the outward direction with speed decaying with image edge strength
  • xt = R xs / ||xs|| ×
    f(||D(Gσ∗I)(x)||)

Example 1 (initialization failure)

  • Curve velocity is in the outward direction with speed decaying with image edge strength
  • xt = R xs / ||xs|| ×
    f(||D(Gσ∗I)(x)||)

Example 1 (velocity repairs)

  • Curve velocity is in the outward direction with speed decaying with image edge strength and contracts in bright areas
  • xt = R xs / ||xs|| ×
    f(||D(Gσ∗I)(x)||) ×
    sign(T-I(x))

Example 2

  • Curve velocity is in the outward direction with speed decaying with image edge strength and contracts in bright areas
  • xt = R xs / ||xs|| ×
    f(||D(Gσ∗I)(x)||) ×
    sign(T-I(x))

Implicit curve evolution: level set framework

  • Implicit curve = {x : u(x,t) = 0}
  • Enforce level curve moves in normal direction with speed V(x) yields level set evolution PDE: ut + V ||Du|| = 0
[source]

Example 2

  • Velocity in normal direction decays with image edge strength and contracts in bright areas
  • V(x) =
    f(||D(Gσ∗I)(x)||) ×
    sign(T-I(x))

Some things to read

  • Kass, Michael, Andrew Witkin, and Demetri Terzopoulos. "Snakes: Active contour models." International journal of computer vision (1988)
  • Malladi, Ravi, James A. Sethian, and Baba C. Vemuri. "Shape modeling with front propagation: A level set approach." IEEE transactions on pattern analysis and machine intelligence (1995)
  • Adalsteinsson, David, and James A. Sethian. "A fast level set method for propagating interfaces." Journal of computational physics (1995)