Differential Geometry

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Differential Geometry

• Computer Vision 8

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Differential Geometry

• 1. Curvature of curve
• 2. Curvature of surface
• 3. Application of curvature

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Parameterization 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

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Example (Circle)

• 1. Arc length, s
• 2. coordinates
• 3. tangent
• 4. curvature

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Definition of Curvature

• The normal direction (n) toward the empty side.

Curvature
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Corner Model and Its Signatures
s0
a
b
d
c
a
b
s0
d
c
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Gaussian Filter and Scale Space
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Curvature Scale Space Descriptor
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Curvature of Surfaces
normal curvature
Principal directions and principal curvatures
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Principal Curvatures
plane all directions
sphere all directions
cylinder
ellipsoid
hyperboloid
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Gaussian Curvature and Mean Curvature
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Parabolic 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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Other Feature Points Edges

• Edges maxima of curvatures
• Zero-crossing of Gaussian/mean curvature
• Ridges and valleys
• Umbilic point principle curvatures are the same

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Lines of Curvature
Principal directions, which gives the maximum and
the minimal normal curvature.
Principal direction
curves along principal directions
PD
PD
PD
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Lines of Curvature
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Curvature Primal Sketches along Lines of Curvature
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Important Formula

• 1. Surface
• 2. surface normal
• 3. the first fundamental form
• 4. the second fundamental form

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Important formula (2)
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Example (sphere)
Z
Y
X
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Estimate 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

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Estimate Normal Curvatures of Point Cloud-
Tensor Voting -

• Each point has its estimated normal which is
  represented by the eigensystem

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Estimate Normal using Tensor Voting

• Voting
• Estimate normal from positions of receiver and
  voter and estimated normal of voter

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Estimate Curvature using Tensor Voting

• Estimate surface
• Point on the unknown surface gets maximum saliency

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Summary

• 1. curvature of curve
• 2. curvature of surface
• Gaussian curvature
• mean curvature

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Surface Description 2(Extended Gaussian Image)
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Topics
1.Gauss map 2.Extended Gaussian
Image 3.Application of EGI
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Gauss 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.
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Extended Gaussian Image (EGI)

• Distribution of normals represented in the
  spherical coordinates
• A weight is expressed the area of the surface
  having the given normal

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Characteristics 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.

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Relationship between EGI and Gaussian Curvature
object
Gaussian sphere
small
large
small
(K small)
small
large
large
(K large)
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Gaussian 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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Implementation 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
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Example of EGI
side view
top view
Cylinder
Ellipsoid
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Determination of Attitude using EGI
10
20
0
viewing direction
0
8
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EGI table
0
8
5
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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

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The complex EGI(CEGI)
(note The weight is shown only for normal n1 for
clearly.)
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Bin 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
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Calibration
Lookup table for photometric stereo
Hand-eye calibration
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Photometric Stereo Set-up
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Bin-Picking System
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Summary
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