Curvature Motion for Union of Balls

Thomas Lewiner♥♠, Cynthia Ferreira♥,

Marcos Craizer♥ and Ralph Teixeira♣

♥ Department of Mathematics — PUC-Rio♠ Géométrica Project — INRIA Sophia

Antipolis♣ FGV -Rio

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 2/21

Morphological Motions

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 3/21

Expected Properties of Motion No self-intersection No singularities No disconnection Convexification Simplification

Curvature Motion

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 4/21

Curvature Motion

@Q(s;t)@t =K (s;t) ¢N (s;t)

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 5/21

Union of Balls Original model

Modelling and approximation

Curve discretisation(medial axis)

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 6/21

Contributions

Explicit curvature motion for union of balls

Sampling conditions on the union of balls

Derivative approximations for the union of balls

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 7/21

Summary

Medial Axis Curvature Motion

from the Medial Axis

Curvature Motion for Union of Balls

Implementation Issues

Results

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 8/21

Medial Axis

Inner symmetries of a shape Singularities of the distance function

Captures the topology of the shape

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 9/21

Medial Axis of a Union of Balls

Classical Algorithmic Geometry (Amenta et al., CGTA 2001)

Medial axis inside the alpha-shape

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 10/21

Points of the Medial Axis

End Points Bifurcation PointsRegular Points

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 11/21

Balls of the Union

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 12/21

Curvature Motion from Medial Axis:

regular points

8>><

>>:

M t =K (1¡ rv 2)

(1¡ rv 2¡ r rvv )2¡ r 2K 2(1¡ rv 2)N

rt =rK 2(1¡ rv 2)+rv (1¡ rv 2¡ r rvv )(1¡ rv 2¡ r rvv )2¡ r 2K 2(1¡ rv 2)

@Q(s;t)@t =K (s;t) ¢N(s;t)

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 13/21

Regular balls

1st and 2nd derivatives on the medial axis:[Lewiner et al., Sibgrapi 2004]

8>><

>>:

M t =K (1¡ rv 2)

(1¡ rv 2¡ r rvv )2¡ r2K 2(1¡ rv 2)N

rt =rK 2(1¡ rv 2)+rv (1¡ rv 2¡ r rvv )(1¡ rv 2¡ r rvv )2¡ r2K 2(1¡ rv 2)

) T ;N ;K;rv; rvv

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 14/21

Clean end balls

Ellipse of 2 circles

½M t = ¡ K ssrt = ¡ K ss ¡ K

) K ;K ss

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 15/21

Noisy end balls

Ellipse tangent to the circles

½M t = ¡ K ssrt = ¡ K ss ¡ K

îsh

ape

) K ;K ss

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 16/21

Bifurcation balls

Estimate the symmetry set mean evolution of three regular points)

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 17/21

Sampling Conditions

Adjacent balls:

Over-sampling (rarefaction) : add ball

Sub-sampling (numerical) :replace balls by

B (c;r) B (c0; r0)

B ( c+c02

; r+r2)

B ( c+c02

; r+r2)

120 min(r;r0) · kc ¡ c0k · min(r;r0)

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 18/21

Numerical Issues

Inner balls Bifurcation

regular topological change

Avoiding non-existent holes

Numerical validation : fallback to end ball case

$

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 19/21

Comparison with Megawave

[Craizer et al., Math Imaging & Vision, 2004]

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 20/21

Reaction-Diffusion Scale-SpaceQt(s;t) = (®+¯K (s;t)) N (s;t)

T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 21/21

Future works: 3D?

Thank you!