Integral Invariants for Shape Siddharth Manay1, Byung-Woo Hong2, Anthony Yezzi3 and Stefano Soatto4 {manay,hong,soatto@cs.ucla.edu ayezzi@ece.gatech.edu} 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. UCLAVISION LABORATORY http://vision.ucla.edu