1. Integral Invariants for Shape
Siddharth Manay1, Byung-Woo Hong2, Anthony Yezzi3 and Stefano Soatto4
1University of California at Los Angeles 2University of Oxford 3Georgia Institute of Technology
Problem: Describing Shape
Example: Local Distance Invariant
Results: Invariants
Design a representation of shape that is invariant to a group (e.g. Euclidian or Affine), local (hence robust to occlusions and deformations), and robust to noise.
Traditional differential invariants are invariant and local, but not robust to noise.
For invariance to rigid transformation in 2D (G=SE(2)).
Where q(p,x) is a local kernel (e.g. a Gaussian kernel with variance s).
We seek a representation based on integral computations rather than derivatives.
Start with closed, planar contours.
Example: Local Area Invariant
Shape
Distance Invariant
Local Area Invariant
Differential Invariant
Results: Shape Matching
Discussion
Since integral invariants are robust, not invariant, to occlusions, large occlusions result in high shape distance.
Unlike smoothing, robust invariants do not destroy information.
Solution: Integral Invariant
For a contour and a group G acting on , is an integral G-invariant if such that
For invariance to rigid transformation in 2D (G=SE(2)).
where h(·, ·) satisfies
Contribution
is the interior of the contour, and Br(p) is a kernel (e.g. Gaussian with variance s or ball with scale r).
In the limit this invariant is a function of curvature.
for
Different choices of kernel h (·, ·) yield different families of invariants.
Defined families of invariants based on integral computations:
Invariant
Local
Robust to noise.
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