(t)}s>0 minimizing E( s ). Do not confound: t: abscise along the curve. s: artiﬁcial “time” of evolution. Local minimum of: min E( ) Minimization ﬂow: d ds s = E( s )
(t)}s>0 minimizing E( s ). Do not confound: t: abscise along the curve. s: artiﬁcial “time” of evolution. Local minimum of: min E( ) Minimization ﬂow: d ds s = E( s ) Warning: the set of curves is not a vector space. Inner product at : µ, ⇥⇥ = 1 0 µ(t), ⇥(t)⇥|| (t)||dt Riemannian manifold of inﬁnite dimension.
(t)}s>0 minimizing E( s ). Do not confound: t: abscise along the curve. s: artiﬁcial “time” of evolution. Local minimum of: min E( ) Minimization ﬂow: d ds s = E( s ) Warning: the set of curves is not a vector space. Inner product at : µ, ⇥⇥ = 1 0 µ(t), ⇥(t)⇥|| (t)||dt Riemannian manifold of inﬁnite dimension. Numerical implementation: (k+1) = (k) ⇥kE( (k))
normal speed d ds ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) normal ns (t) = s (t) || s (t)|| ⇥s (t) = ns (t), s (t)⇥ 1 || s (t)||2 Normal: Curvature: s ns E( ) only depends on { (t) \ t [0, 1]}.
1 0 W( (t))|| (t)||dt (x, n, ⇥) = W(x)⇥ W(x), n d ds ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) Weighted length: Evolution: 0 s Weight W(x) where W is small. ﬁnite di erences discretization. attraction toward areas
x R2 \ ⇥s (x) = 0 . Level-set curve representation: s (x) = ||x x0 || s s (x) = ||x x0 || s s (x) 0 s (x) 0 Example: circle of radius r Example: square of radius r
x R2 \ ⇥s (x) = 0 . Level-set curve representation: s (x) = ||x x0 || s s (x) = ||x x0 || s s (x) 0 s (x) 0 Union of domains: s = min( 1 s , 2 s ) Intersection of domains: s = max( 1 s , 2 s ) Example: circle of radius r Example: square of radius r s = min( 1 s , 2 s ) s (x) 0
x R2 \ ⇥s (x) = 0 . Level-set curve representation: s (x) = ||x x0 || s s (x) = ||x x0 || s s (x) 0 s (x) 0 Union of domains: s = min( 1 s , 2 s ) Intersection of domains: s = max( 1 s , 2 s ) Popular choice: (signed) distance to a curve ⇥s (x) = ± min t || s (t) x|| Example: circle of radius r Example: square of radius r inﬁnite number of mappings s s . s = min( 1 s , 2 s ) s (x) 0
min t || s (t) x|| Eikonal equation: || s || = 1 with ⇥s ( s (t)) = 0 d ds s = || s ||div W s || s || . Evolution PDE: Comparison with explicit active contours: : 2D instead of 1D equation. + : allows topology change.