SMOOTH SURFACES AND

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SMOOTH SURFACES AND THEIR OUTLINES

• Elements of Differential Geometry
• What are the Inflections of the Contour?
• Koenderinks Theorem

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Informations pratiques

• Présentations http//www.di.ens.fr/ponce/geomvi
  s/lect11.ppt
• http//www.di.ens.fr/ponce/geomvis/lect11.pdf
• Deux cours de plus en Janvier
• Jeudis 5 et 12 Janvier
• Examen le 13 Janvier
• A quand lexamen?

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?1
Chasles absolute conic x12 x22 x32 0,
x4 0.
? T diag(Id,0) d u 2 0.
The absolute quadratic complex
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The join of two points
Note u . v 0
y
x
?
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Perspective projection
r
r
c
c
x

x

x
x
x
x
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Non-linear, 7 images
Non-linear, 20 images
Non-linear, 196 images
Linear, 20 images
2480 points tracked in 196 images
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Canon XL1 digital camcorder, 480720 pixel2
(Ponce McHenry, 2004)
Projective structure from motion Mahamud,
Hebert, Omori Ponce (2001)
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Quantitative comparaison with Pollefeys et al.
(1998, 2002) (synthetic cube with 30cm edges,
corrupted by Gaussian noise)
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Smooth Shapes and their Outlines
Can we say anything about a 3D shape from the
shape of its contour?
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What are the contour stable features??
Reprinted from Computing Exact Aspect Graphs of
Curved Objects Algebraic Surfaces, by S.
Petitjean, J. Ponce, and D.J. Kriegman, the
International Journal of Computer Vision,
9(3)231-255 (1992). ? 1992 Kluwer Academic
Publishers.
folds
cusps
T-junctions
Shadows are like silhouettes..
Reprinted from Solid Shape, by J.J.
Koenderink, MIT Press (1990). ? 1990 by the MIT.
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Differential geometry geometry in the small
The normal to a curve is perpendicular to
the tangent line.
A tangent is the limit of a sequence of secants.
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What can happen to a curve in the vicinity of a
point?
(a) Regular point (b) inflection (c) cusp of
the first kind (d) cusp of the second kind.
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The Gauss Map

• It maps points on a curve onto points on the
  unit circle.

• The direction of traversal of the Gaussian image
  reverts
• at inflections it folds there.

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The curvature
C

• C is the center of curvature
• R CP is the radius of curvature
• ? lim dq/ds 1/R is the curvature.

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Closed curves admit a canonical orientation..
k lt 0
k gt 0
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Twisted curves are more complicated animals..
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A smooth surface, its tangent plane and its
normal.
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Normal sections and normal curvatures
Principal curvatures minimum value k maximum
value k
Gaussian curvature K k k
1
1
2
2
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The local shape of a smooth surface
Elliptic point
Hyperbolic point
K gt 0
K lt 0
Reprinted from On Computing Structural Changes
in Evolving Surfaces and their Appearance, By
S. Pae and J. Ponce, the International Journal of
Computer Vision, 43(2)113-131 (2001). ? 2001
Kluwer Academic Publishers.
Parabolic point
K 0
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The Gauss map
Reprinted from On Computing Structural Changes
in Evolving Surfaces and their Appearance, By
S. Pae and J. Ponce, the International Journal of
Computer Vision, 43(2)113-131 (2001). ? 2001
Kluwer Academic Publishers.
The Gauss map folds at parabolic points.
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Smooth Shapes and their Outlines
Can we say anything about a 3D shape from the
shape of its contour?
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Theorem Koenderink, 1984 the inflections of
the silhouette are the projections of parabolic
points.
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Koenderinks Theorem (1984)
K k k
r
c
Note k gt 0.
r
Corollary K and k have the same sign!
c