Title: Differential Geometry
1Differential Geometry
2Differential Geometry
- 1. Curvature of curve
- 2. Curvature of surface
- 3. Application of curvature
3Parameterization of Curve
- 1. curve - s arc length
- a(s) ( x(s), y(s) )
- 2. tangent of a curve
- a(s) ( x(s), y(s) )
- 3. curvature of a curve
- a(s) ( x(s), y(s) )
- a(s) -- curvature
4Example (Circle)
- 1. Arc length, s
- 2. coordinates
- 3. tangent
- 4. curvature
5Definition of Curvature
- The normal direction (n) toward the empty side.
Curvature
6Corner Model and Its Signatures
s0
a
b
d
c
a
b
s0
d
c
7Gaussian Filter and Scale Space
8Curvature Scale Space Descriptor
9Curvature of Surfaces
normal curvature
Principal directions and principal curvatures
10Principal Curvatures
plane all directions
sphere all directions
cylinder
ellipsoid
hyperboloid
11Gaussian Curvature and Mean Curvature
12Parabolic points
Parabolic point
elliptic point
hyperbolic point
F.Klein used the parabolic curves for a peculiar
investigation. To test his hypothesis that the
artistic beauty of a face was based on certain
mathematical relation, he has all the parabolic
curves marked out on the Apollo Belvidere. But
the curves did not possess a particularly simpler
form, nor did they follow any general law that
could be discerned.
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15Other Feature Points Edges
- Edges maxima of curvatures
- Zero-crossing of Gaussian/mean curvature
- Ridges and valleys
- Umbilic point principle curvatures are the same
16Lines of Curvature
Principal directions, which gives the maximum and
the minimal normal curvature.
Principal direction
curves along principal directions
PD
PD
PD
17Lines of Curvature
18Curvature Primal Sketches along Lines of Curvature
19Important Formula
- 1. Surface
- 2. surface normal
- 3. the first fundamental form
- 4. the second fundamental form
20Important formula (2)
21Example (sphere)
Z
Y
X
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25Estimate Normal Curvatures of Polygon Mesh
Model
- Normal
- Weighted average of normals of the adjacent faces
- Fit plane
- Curvature
- Fit algebraic surface
- Fit circles
- Curvature Tensor
26Estimate Normal Curvatures of Point Cloud-
Tensor Voting -
- Each point has its estimated normal which is
represented by the eigensystem
27Estimate Normal using Tensor Voting
- Voting
- Estimate normal from positions of receiver and
voter and estimated normal of voter
28Estimate Curvature using Tensor Voting
- Estimate surface
- Point on the unknown surface gets maximum saliency
29Summary
- 1. curvature of curve
- 2. curvature of surface
- Gaussian curvature
- mean curvature
30Surface Description 2(Extended Gaussian Image)
31Topics
1.Gauss map 2.Extended Gaussian
Image 3.Application of EGI
32Gauss map
gauss map
1D
gauss map
2D
Let S?R3 be a surface with an orientation N. The
map N S?R3 takes its values in the unit sphere
The map N S?S2 is called the Gauss map.
33Extended Gaussian Image (EGI)
- Distribution of normals represented in the
spherical coordinates - A weight is expressed the area of the surface
having the given normal
34Characteristics of EGI
- EGI is the necessary and the sufficient condition
for the congruence of two convex polyhedra. - Ratio between the area on the Gaussian sphere and
the area on the object is equal to Gaussian
curvature. - EGI mass on the sphere is the inverse of Gaussian
curvature. - Mass center of EGI is at the origin of the sphere
- An object rotates, then EGI of the object also
rotates. However, both rotations are same.
35Relationship between EGI and Gaussian Curvature
object
Gaussian sphere
small
large
small
(K small)
small
large
large
(K large)
36Gaussian Curvature and EGI Maps
- Since and exist on the tangential plane
at , - we can represent them by a linear combination of
and
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38Implementation of EGI
- Tessellation of the unit sphere
- All cells should have the same area
- have the same shape
- occur in a regular pattern
- geodesic dome based on a regular polyhedron
semi-regular geodesic dome
39Example of EGI
side view
top view
Cylinder
Ellipsoid
40Determination of Attitude using EGI
10
20
0
viewing direction
0
8
5
EGI table
0
8
5
41 The Complex EGI(CEGI)
- Normal distance and area of a 3-D object are
encoded as a complex weight. Pnk associated with
the surface normal nk such that
42 The complex EGI(CEGI)
(note The weight is shown only for normal n1 for
clearly.)
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45Bin Picking System based on EGI
Photometric stereo segmentation Region
selection Photometric stereo EGI
generation EGI matching Grasp planning
Needle map isolated regions target
region precise needle map EGI object attitude
46Calibration
Lookup table for photometric stereo
Hand-eye calibration
47Photometric Stereo Set-up
48Bin-Picking System
49Summary
1. Gauss map 2. Extended Gaussian Image 3.
Characteristics of EGI congruence of two convex
polyhedra EGI mass is the inverse of Gaussian
curvature mass center of EGI is at the origin of
the sphere 4. Implementation of EGI Tessellation
of the unit sphere Recognition using EGI 5.
Complex EGI 6. Bin-picking system based on EGI 7.
Read Horn pp.365-39 pp.423-451