curvature motion for union of balls thomas lewiner ♥♠, cynthia ferreira ♥, marcos craizer ♥...
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![Page 1: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/1.jpg)
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
![Page 2: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/2.jpg)
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 2/21
Morphological Motions
![Page 3: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/3.jpg)
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
![Page 4: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/4.jpg)
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)
![Page 5: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/5.jpg)
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)
![Page 6: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/6.jpg)
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
![Page 7: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/7.jpg)
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
![Page 8: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/8.jpg)
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
![Page 9: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/9.jpg)
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
![Page 10: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/10.jpg)
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
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T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 11/21
Balls of the Union
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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)
![Page 13: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/13.jpg)
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
![Page 14: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/14.jpg)
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
![Page 15: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/15.jpg)
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
![Page 16: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/16.jpg)
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)
![Page 17: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/17.jpg)
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)
![Page 18: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/18.jpg)
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
$
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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]
![Page 20: Curvature Motion for Union of Balls Thomas Lewiner ♥♠, Cynthia Ferreira ♥, Marcos Craizer ♥ and Ralph Teixeira ♣ ♥ Department of Mathematics — PUC-Rio](https://reader030.vdocuments.site/reader030/viewer/2022032800/56649d3b5503460f94a15df0/html5/thumbnails/20.jpg)
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)
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T. Lewiner, C. Fereira, M. Craizer and R. Teixeira - Curvature Motion for Union of Balls 21/21
Future works: 3D?
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Thank you!