Discrete Approach to Curve and Surface Evolution Longin Jan Latecki Dept. of Computer and Information Science Temple University Philadelphia Email: latecki@temple.edu

Discrete Curve Evolution P=P0, Discrete Curve Evolution P=P0, ..., Pm Pi+1 is obtained from Pi by deleting the vertices of Pi that have minimal relevance measure K(v, Pi) = K(u,v,w) = |d(u,v)+d(v,w)-d(u,w)| v v w w u u

Discrete Curve Evolution: Preservation of position, no blurring

Discrete Curve Evolution: robustness with respect to noise

Discrete Curve Evolution: extraction of linear segments

Parts of Visual Form (Siddiqi, Tresness, and Kimia 1996) = maximal convex arcs

Discete Cureve Evolution is used in shape similarity retrieval in image databases

Shape similarity measure based on correspondence of visual parts

A video sequence is mapped to a trajectory in a high dimensional space, e.g. by mapping each frame to a feature vector in R37 Discrete curve evolution allows us to determine key frames

Trajectory Simplification 2379 vertices 20 vertices

Mr. Bean

The 10 most relevant frames in Mr. Bean www.videokeyframes.de

Detection of unpredictable events in videos: Mov3.mpg

Alarm threshold = avg rel + 0.1*max. rel

Detection of unpredictable events in videos: seciurity1.mpg

Two most unpredictable frames extracted from Mov3.mpg

Alarm threshold = avg rel + 0.1*max. rel

Two most unpredictable frames extracted from security1.mpg

Discrete Surface Evolution: repeated removal of least relevant vertices

(Lyche and Morken in late 80s): Surface patch f:R2 -> R is represented with radial base splines S given a set of knots T: ||f – G(T)(f)|| = min{||f - g||: g in S} Surface evolution by knot removal Relevance measure of the knot: r(t) = ||f – G(T – {t})(f)||