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
Morphological Motions
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
2/21
Expected Properties of Motion

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

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
3/21
Curvature Motion
@Q ( s;t )
@t
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
= K (s; t) ¢N (s; t)
- Curvature Motion for Union of Balls
4/21
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
5/21
Contributions
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

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
6/21
Summary
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
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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
7/21
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
8/21
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
9/21
Points of the Medial Axis
End Points
Bifurcation Points
Regular Points
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
10/21
Balls of the Union
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
11/21
Curvature Motion from Medial Axis:
regular points
@Q ( s;t )
@t
= K (s; t) ¢N (s; t)
8
>
>
< Mt =
>
>
: r =
t
K ( 1¡ r v 2 )
N
( 1¡ r v 2 ¡ r r v v ) 2 ¡ r 2 K 2 ( 1¡ r v 2 )
r K 2 ( 1¡ r v 2 ) + r v ( 1¡ r v 2 ¡ r r v v )
( 1¡ r v 2 ¡ r r v v ) 2 ¡ r 2 K 2 ( 1¡ r v 2 )
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
12/21
8
>
>
< Mt =
Regular balls
>
>
: r =
t
K ( 1¡ r v 2 )
N
( 1¡ r v 2 ¡ r r v v ) 2 ¡ r 2 K 2 ( 1¡ r v 2 )
r K 2 ( 1¡ r v 2 ) + r v ( 1¡ r v 2 ¡ r r v v )
( 1¡ r v 2 ¡ r r v v ) 2 ¡ r 2 K 2 (1¡ r v 2 )
1st and 2nd derivatives on the medial axis:
)
[Lewiner et al., Sibgrapi 2004]
T ; N ; K ; r v ; r vv
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
13/21
Clean end balls
½
M t = ¡ K ss
r t = ¡ K ss ¡ K
Ellipse
2 circles
) of K
;K
ss
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
14/21
Noisy end balls
½
M t = ¡ K ss
r t = ¡ K ss ¡ K
Ellipse tangent to the circles
)
K ; K ss
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
15/21
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
16/21
Sampling
Conditions
1 min(r; r 0) · kc ¡ c0k · min(r; r 0)
20
Adjacent
B (c; rballs:
)

Over-sampling (rarefaction) :
add B
ball( c+ c 0 ; r + r )
2

B (c0; r 0)
2
Sub-sampling (numerical) :
replace
c+ c 0by
B (balls
; r+r )
2
2
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
17/21
Numerical Issues



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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
18/21
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
19/21
Reaction-Diffusion Scale-Space
Qt (s; t) = (® + ¯K (s; t)) N (s; t)
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
20/21
Future works: 3D?
T. Lewiner, C. Fereira, M. Craizer and R. Teixeira
- Curvature Motion for Union of Balls
21/21
Thank you!