curvature motion for union of balls thomas lewiner ♥♠, cynthia ferreira ♥, marcos craizer ♥...
Post on 19-Dec-2015
213 views
TRANSCRIPT
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!