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Smooth Surfaces and their Outlines II Elements of Differential Geometry What are the Inflections of the Contour? Koenderink’s Theorem Reading: Smooth Surfaces and their Outlines This file on: feather.ai.uiuc.edu in pub/ponce/lect25.ppt.gz

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Smooth Surfaces and their Outlines II

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Smooth Surfaces and their Outlines II

• Elements of Differential Geometry

• What are the Inflections of the Contour?

• Koenderink’s Theorem

• Reading: Smooth Surfaces and their Outlines

• This file on: feather.ai.uiuc.edu in pub/ponce/lect25.ppt.gz

• Final: Take-home, given on Thursday, due on Monday.

Smooth Shapes and their Outlines

Can we say anything about a 3D shape

from the shape of its contour?

What are the contour stable features??

folds

cusps

T-junctions

Shadows

are like

silhouettes..

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.

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.

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 fold there.

The curvature

C

• C is the center of curvature;

• R = CP is the radius of curvature;

• k = lim dq/ds = 1/R is the curvature.

Closed curves admit a canonical orientation..

k <0

k > 0

Twisted curves are more complicated animals..

A smooth surface, its tangent plane and its normal.

Normal sections and normal curvatures

Principal curvatures:

minimum value k

maximum value k

Gaussian curvature:

K = k k

1

1

2

2

The local shape of a smooth surface

Elliptic point

Hyperbolic point

K > 0

K < 0

K = 0

Parabolic point

The Gauss map

The Gauss map folds at parabolic points.

Smooth Shapes and their Outlines

Can we say anything about a 3D shape

from the shape of its contour?

Theorem [Koenderink, 1984]: the inflections of the silhouette

are the projections of parabolic points.

Koenderink’s Theorem (1984)

K = kk

r

c

Note: k > 0.

r

Corollary: K and k have

the same sign!

c

From Horn’s

Robot Vision