Mesh Processing Course : Active Contours

Mesh Processing Course : Active Contours

• 1.
  Active Contours www.numerical-tours.com Gabriel Peyré CEREMADE, Université Paris-Dauphine
• 2.
  Overview • ParametricEdge-based Active Contours • Implicit Edge-based Active Contours • Region-based Active Contours 2
• 3.
  Parametric Active Contours Localminimum: argmin E( ) = L( ) + ⇥R( ) Data Regularization Boundary conditions: fidelity x0 – Open curve: (0) = x0 and (1) = x1 . – Closed curve: (0) = (1). x1 3
• 4.
  Parametric Active Contours Localminimum: argmin E( ) = L( ) + ⇥R( ) Data Regularization Boundary conditions: fidelity x0 – Open curve: (0) = x0 and (1) = x1 . – Closed curve: (0) = (1). Snakes energy: (depends on parameterization) x1 1 1 L( ) = W ( (t))|| (t)||dt, R( ) = || (t)|| + µ|| (t)||dt 0 0 Image f Weight W (x) Curve 3
• 5.
  Geodesic Active Contours Geodesic active contours: (intrinsic) Replace W by W + , 1 E( ) = L( ) = W ( (t))|| (t)||dt 0 (local) minimum of the weighted length L. local geodesic (not minimal path). Weight W (x) Curve 4
• 6.
  Curve Evolution Family ofcurves { s (t)}s>0 minimizing E( s ). s Do not confound: t: abscise along the curve. s+ds s: artificial “time” of evolution. 5
• 7.
  Curve Evolution Family ofcurves { s (t)}s>0 minimizing E( s ). s Do not confound: t: abscise along the curve. s+ds s: artificial “time” of evolution. Local minimum of: min E( ) d Minimization flow: s = E( s ) ds 5
• 8.
  Curve Evolution Family ofcurves { s (t)}s>0 minimizing E( s ). s Do not confound: t: abscise along the curve. s+ds s: artificial “time” of evolution. Local minimum of: min E( ) d Minimization flow: s = E( s ) ds Warning: the set of curves is not a vector space. 1 Inner product at : µ, ⇥⇥ = µ(t), ⇥(t)⇥|| (t)||dt 0 Riemannian manifold of infinite dimension. 5
• 9.
  Curve Evolution Family ofcurves { s (t)}s>0 minimizing E( s ). s Do not confound: t: abscise along the curve. s+ds s: artificial “time” of evolution. Local minimum of: min E( ) d Minimization flow: s = E( s ) ds Warning: the set of curves is not a vector space. 1 Inner product at : µ, ⇥⇥ = µ(t), ⇥(t)⇥|| (t)||dt 0 Riemannian manifold of infinite dimension. Numerical implementation: (k+1) = (k) ⇥k E( (k) ) 5
• 10.
  Intrinsic Curve Evolutions E() only depends on { (t) t [0, 1]}. Intrinsic energy E: evolution along the normal s d ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ns ds speed normal s (t) Normal: ns (t) = || s (t)|| 1 Curvature: ⇥s (t) = ns (t), s (t)⇥ || s (t)||2 6
• 11.
  Mean Curvature Motion 1 No data-fidelity: E( ) = || (t)||dt 0 d Curve-shortening flow. s = E( s ) ds 1 (t) E( + ⇥) = E( ) + , ⇥ (t)⇥dt + O(||⇥||) 0 || (t)|| 1 d (t) ⌅E( ) : t ⇤⇥ || (t)|| dt || (t)|| 0 Mean-curvature motion: s d s (t) = ⇥s (t)ns (t) ds Speed: (x, n, ⇥) = ⇥ 7
• 12.
  Discretization Discretization: = { (i)}N 1 i=0 R2 , with (N ) = (0). ⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)|| k 8
• 13.
  Discretization Discretization: = { (i)}N 1 i=0 R2 , with (N ) = (0). ⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)|| Discrete energy: E( ) = i || (i) (i + 1)|| k 8
• 14.
  Discretization Discretization: = { (i)}N 1 i=0 R2 , with (N ) = (0). ⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)|| Discrete energy: E( ) = i || (i) (i + 1)|| Gradient descent flow: k+1 = k ⇥k E( k ) k E( k ) 8
• 15.
  Discretization Discretization: = { (i)}N 1 i=0 R2 , with (N ) = (0). ⇥µ, ⇥⇤ = i ⇥µ(i), ⇥(i)⇤|| (i) (i + 1)|| Discrete energy: E( ) = i || (i) (i + 1)|| Gradient descent flow: k+1 = k ⇥k E( k ) k 1 Discrete gradient: ⇥E( ) = ⇥ N ⇥( ) ||⇥ || (⇥ )(i) = (i + 1) (i) E( k ) (⇥ )(i) = (i 1) (i) (i) (N )(i) = || (i)|| 8
• 16.
  Geodesic Active ContoursMotion Weighted length: 1 E( ) = L( ) = W ( (t))|| (t)||dt 0 Evolution: d ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) 0 ds (x, n, ⇥) = W (x)⇥ W (x), n attraction toward areas where W is small. s finite di erences discretization. Weight W (x)
• 17.
  Open vs. ClosedCurves 0 s x0 s 0 x1 Weight W (x) Image f (x)
• 18.
  Global Minimum withFast Marching Geodesic distance map: Ux0 (x1 ) = min L( ) (0)=x0 , (1)=x1 Global minimum: Ux0 (x1 ) = L( ) Image f Metric W (x) Distance Ux0 (x) Geodesic curve (t)
• 19.
  Global Minimum withFast Marching Geodesic distance map: Ux0 (x1 ) = min L( ) (0)=x0 , (1)=x1 Global minimum: Ux0 (x1 ) = L( ) Image f Metric W (x) Fast O(N log(N )) algorithm: – Compute Ux0 with Fast Marching. – Solve EDO: Distance Ux0 (x) Geodesic curve (t) d (t) = Ux0 ( (t)) dt (0) = x1
• 20.
  Overview • ParametricEdge-based Active Contours • ImplicitEdge-based Active Contours • Region-based Active Contor 12
• 21.
  Level Sets Level-set curverepresentation: s (x) 0 { s (t) t [0, 1]} = x R ⇥s (x) = 0 . 2 Example: circle of radius r s (x) = ||x x0 || s Example: square of radius r s (x) = ||x x0 || s s (x) 0
• 22.
  Level Sets Level-set curverepresentation: s (x) 0 { s (t) t [0, 1]} = x R ⇥s (x) = 0 . 2 Example: circle of radius r s (x) = ||x x0 || s Example: square of radius r s (x) = ||x x0 || s s (x) 0 Union of domains: s = min( 1 s, s) 2 Intersection of domains: s = max( 1 s, s) 2 s (x) 0 s = min( 1 s, s) 2
• 23.
  Level Sets Level-set curverepresentation: s (x) 0 { s (t) t [0, 1]} = x R ⇥s (x) = 0 . 2 Example: circle of radius r s (x) = ||x x0 || s Example: square of radius r s (x) = ||x x0 || s s (x) 0 Union of domains: s = min( 1 s, s) 2 Intersection of domains: s = max( 1 s, s) 2 s (x) 0 Popular choice: (signed) distance to a curve ⇥s (x) = ± min || s (t) x|| t infinite number of mappings s s. s = min( 1 s, s) 2
• 24.
  Level Sets Evolution Dictionaryparameteric implicit: Position: x= s (t) s (x) Normal: ns (t) = || s (x)|| ⇥s Curvature: s (x) = div (x) || ⇥s || 14
• 25.
  Level Sets Evolution Dictionaryparameteric implicit: Position: x= s (t) s (x) Normal: ns (t) = || s (x)|| ⇥s Curvature: s (x) = div (x) || ⇥s || Evolution PDE: d ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ds d ⇥s (x) ⇥s ⇥s (x) = || ⇥s (x)|| ⇥s (x), , div (x) . ds || ⇥s (x)|| || ⇥s || All level sets evolves together. 14
• 26.
  Proof d Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( ) ds Definition of level-sets: t, ⇥s ( s (t)) = 0 15
• 27.
  Proof d Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( ) ds Definition of level-sets: t, ⇥s ( s (t)) = 0 Deriving with respect to t: ⇤ s ⇤⇥s ⇥s (x), (t) + (x) = 0 for x = s (t) ( ) ⇤s ⇤s 15
• 28.
  Proof d Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( ) ds Definition of level-sets: t, ⇥s ( s (t)) = 0 Deriving with respect to t: ⇤ s ⇤⇥s ⇥s (x), (t) + (x) = 0 for x = s (t) ( ) ⇤s ⇤s ⌅⇤s ( )+( ): (x) = (x, ns (t), ⇥s (t)) ⇤s (x), ns (t) ⌅s 15
• 29.
  Proof d Evolution PDE: ⇥s (t) = (⇥s (t), ns (t), ⇤s (t)) ns (t) ( ) ds Definition of level-sets: t, ⇥s ( s (t)) = 0 Deriving with respect to t: ⇤ s ⇤⇥s ⇥s (x), (t) + (x) = 0 for x = s (t) ( ) ⇤s ⇤s ⌅⇤s ( )+( ): (x) = (x, ns (t), ⇥s (t)) ⇤s (x), ns (t) ⌅s s (x) = || s (x)|| For all x on the curve, d ⇥s (x) ⇥s ⇥s (x) = || ⇥s (x)|| ⇥s (x), , div (x) . ds || ⇥s (x)|| || ⇥s || 15
• 30.
  Implicit Geodesic ActiveContours Evolution PDE: d s s = || s ||div W . ds || s || Comparison with explicit active contours: : 2D instead of 1D equation. + : allows topology change. 16
• 31.
  Implicit Geodesic ActiveContours Evolution PDE: d s s = || s ||div W . ds || s || Comparison with explicit active contours: : 2D instead of 1D equation. + : allows topology change. Re-initialization: ⇥s (x) = ± min || s (t) x|| t Eikonal equation: || s || =1 with ⇥s ( s (t)) = 0 16
• 32.
  Multiple Fluids Dynamics SeeRon Fedkiw homepage. http://physbam.stanford.edu/ fedkiw/ Multiple gaz: Fluid/air interface: 17
• 33.
  Overview • ParametricEdge-based Active Contours • Implicit Edge-based Active Contours • Region-based Active Contours 18
• 34.
  Energy Depending onRegion Optimal segmentation [0, 1]2 = c : min L1 ( ) + L2 ( c ) + R( ) R( ) = | | Data Regularization fidelity Chan-Vese binary model: L1 ( ) = |I(x) c1 |2 dx More general models 19
• 35.
  Energy Depending onRegion Optimal segmentation [0, 1]2 = c : min L1 ( ) + L2 ( c ) + R( ) R( ) = | | Data Regularization fidelity Chan-Vese binary model: L1 ( ) = |I(x) c1 |2 dx More general models Level set implementation: = {x (x) > 0} 2 x H(x) Smoothed Heaviside: H (x) = atan ⇥ x L1 ( ) ⇥ L( ) = H ( (x))||I(x) c1 ||2 dx R( ) R( ) = ||⇥(H )(x)||dx 19
• 36.
  Descent Schemes For agiven c = (c1 , c2 ) R2 : min Ec ( ) = H ( (x))||I(x) c1 ||2 + ⇥ H ( ⇥(x))||I(x) c2 ||2 + ||⇥(H ⇥)(x)||dx 20
• 37.
  Descent Schemes For agiven c = (c1 , c2 ) R2 : min Ec ( ) = H ( (x))||I(x) c1 ||2 + ⇥ H ( ⇥(x))||I(x) c2 ||2 + ||⇥(H ⇥)(x)||dx Descent with respect to : ⇥(k+1) = ⇥(k) k Ec (⇥(k) ) Ec ( ) = H ( (x))G(x) ⇥⇥ G(x) = ||I(x) 2 c1 || ||I(x) 2 c2 || div (x) ||⇥⇥|| 20
• 38.
  Descent Schemes For agiven c = (c1 , c2 ) R2 : min Ec ( ) = H ( (x))||I(x) c1 ||2 + ⇥ H ( ⇥(x))||I(x) c2 ||2 + ||⇥(H ⇥)(x)||dx Descent with respect to : ⇥(k+1) = ⇥(k) k Ec (⇥(k) ) Ec ( ) = H ( (x))G(x) ⇥⇥ G(x) = ||I(x) 2 c1 || ||I(x) 2 c2 || div (x) ||⇥⇥|| Limit 0: ⇥Ec (⇥) { =0} (x)||⇥⇥(x)||G(x) Numerically, use Ec ( ) = || (x)||G(x) 20
• 39.
  Update of c Jointminimization: min Ec ( ) ,c1 ,c2 21
• 40.
  Update of c Jointminimization: min Ec ( ) ,c1 ,c2 Update of : ⇥(k+1) = ⇥(k) k Ec(k) (⇥(k) ) 21
• 41.
  Update of c Jointminimization: min Ec ( ) ,c1 ,c2 Update of : ⇥(k+1) = ⇥(k) k Ec(k) (⇥(k) ) Update of (c1 , c2 ): (k+1) (k+1) (c1 , c2 ) = argmin Ec ( (k) ) c1 ,c2 (k+1) c1 = c( (k) ) I(x)H( (x))dx c( ) = (k+1) c2 = c( (k) ) H( (x))dx 21
• 42.
  Example of Evolution Use de-localized initialization. (0) 22
• 43.
  Conclusion Curve evolution Energy minimization Parametric vs. level set representation. Dictionary to translate – curve properties. – energy gradients. Edge based vs. region based energies. 23