Equi-affine Differential Signature
•Let us replace the parameter s by the parameter v with the
following relation:
•This is called the equi-affine invariant arc-length.
•Also, we can derive an equi-affine invariant curvature.
•Also, we can derive an equi-affine invariant curvature derivative w.r.t. its
arc-length.
•Example will follow….
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Equi-affine Differential Signature
Example
•Consider a convex curve with the equation y=u(x).
•The simplest equi-affine invariant arc-length will be:
•And the curvature is:
•The curvature derivative will be:
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Equi-affine Differential Signature
Reference
•For more information:
Differential and Numerically Invariant Signature Curves Applied to Object Recognition
Eugenio Calabi, Peter J. Olver, Chehrzad Shakiban, Allen Tannenbaum and Steven Haker
In International Journal of Computer Vision 26(2), 107–135 (1998)
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Equi-affine Parameterization
•Let us replace the parameter s by the parameter v with the
following relation:
•The first derivative w.r.t. s will result in:
•And we will have:
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Equi-affine Parameterization (Cont…)
•Let us get the second derivative according the parameterization v:
•As a result of the above representation, the following relation
holds:
•Prove? Show what that means!
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Numerical Differentiation
Numerical Differentiation
Numerical Differentiation
Numerical Differentiation
Numerical Differentiation
Numerical Differentiation
Numerical Differentiation
Surfaces and 3D Differential Geometry
Lecture #3
Hossam Abdelmunim1 & Aly A. Farag2
1Computer
& Systems Engineering Department, Ain
Shams University, Cairo, Egypt
2Electerical
and Computer Engineering Department,
University of Louisville, Louisville, KY, USA
ECE 643 – Fall 2010
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Parametric Surfaces
We consider a regular surface represented as :
v
It can be represented as a vector as follows:
S (u, v)  [ x(u, v), y(u, v), z (u, v)]T ,
where (u , v)  [0,1]  [0,1]
u
Tangential Plane
•The surface is assumed to have the derivatives w.r.t. the
parameters as follows:
•Where:
Su
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Sv
Tangential Plane and Surface Area
• The vectors Su and Sv forms the tangential plane. So the
surface area which is an intrinsic quantity can be calculated
as follows:
• Note that the differential area vector works in the normal
direction to the surface Su X Sv (Vector Cross Product).
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Geodesic Curvature
• Consider a parametric curve C(s) embedded into the
surface S. This curve will have the following acceleration
vector defined as:
• So, the acceleration vector is represented in terms of the
unit normal vector and the unit tangent which lies in the
same plane with CSS and the surface normal.
• Planar curves have only acceleration in the normal
direction. In this case
. .They are called GEODESICS.
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Principal Curvatures
• Principal curvatures are the minimum and maximum normal
curvatures
given at any point on the surface
measuring the amount of bending of the surface
• The Gaussian and the mean curvatures are defined as
follows:
• And we can use the Gaussian and mean curvatures to
compute the minimum and maximum curvatures.
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Planar Curves and Surface Normal
•Recall again: a curve that results from the intersection of a plane with
a surface S.
•Consider the surface is given by:
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Planar Curves and Surface Normal
(Cont…)
•We can write the equation for the normal vector in terms of the
parameterization u and v as:
•Then substituting for:
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Planar Curves and Surface Normal
(Cont…)
•We have:
•Divide by the magnitude to get the unit normal:
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Planar Curves and Surface Normal
(Cont…)
•Then, we project that 3D vector on the x-y plane or the plane that
includes our curve C. Its unit vector will be:
•This vector is normal to C and working in the outward direction. It
can be written in the following form as well:
Generalize the meaning of the above equation
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Unit Normal and Curvature
•We can show that :
Prove!
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