Discrete Approach Discrete Approach to Curve and Surface Evolutionto Curve and Surface Evolution

Longin Jan Latecki

Dept. of Computer and Information Science

Temple University

Philadelphia

Email: latecki@temple.edu

Discrete Curve EvolutionDiscrete Curve Evolution P=P P=P00, ..., P, ..., Pmm

PPi+1i+1 is obtained from P is obtained from Pii by deleting the by deleting the vertices of Pvertices of Pii that have minimal relevance that have minimal relevance

measure measure K(v, PK(v, Pii) = K(u,v,w) = |d(u,v)+d(v,w)-d(u,w)|) = K(u,v,w) = |d(u,v)+d(v,w)-d(u,w)|

u

v

w u

v

w

Discrete Curve EvolutionDiscrete Curve Evolution: : Preservation of position, no blurringPreservation of position, no blurring

Discrete Curve EvolutionDiscrete Curve Evolution: : robustness with respect to noiserobustness with respect to noise

Discrete Curve EvolutionDiscrete Curve Evolution: : extraction of linear segmentsextraction of linear segments

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

Discete Cureve Discete Cureve Evolution is uEvolution is used sed

in shape in shape similarity retrieval similarity retrieval

in image in image databasesdatabases

Shape similarity measure based on Shape similarity measure based on correspondence of visual partscorrespondence of visual parts

A video sequence is mapA video sequence is mappped to a ed to a trajectory in a high dimensional space,trajectory in a high dimensional space,

e.g. by mapping each frame to a feature e.g. by mapping each frame to a feature vector in Rvector in R3737

Discrete curve evolution allows us to Discrete curve evolution allows us to determine key framesdetermine key frames

Trajectory SimplificationTrajectory Simplification

2379 vertices 20 vertices

Mr. BeanMr. Bean

The 10 most relevant framesThe 10 most relevant frames in Mr. Bean in Mr. Bean

www.videokeyframes.dewww.videokeyframes.de

Detection of unpredictable events in Detection of unpredictable events in videosvideos::

Mov3.mpgMov3.mpg

Alarm threshold = avg rel + 0.1*max. relAlarm threshold = avg rel + 0.1*max. rel

Detection of unpredictable events in Detection of unpredictable events in videosvideos::

seciurity1.mpgseciurity1.mpg

Two most unpredictable frames extracted Two most unpredictable frames extracted fromfrom Mov3.mpg Mov3.mpg

Alarm threshold = avg rel + 0.1*max. relAlarm threshold = avg rel + 0.1*max. rel

Two most unpredictable frames extracted Two most unpredictable frames extracted from security1.mpgfrom security1.mpg

Discrete Surface Evolution: repeated removal of least relevant vertices

(Lyche and Morken in late 80s): (Lyche and Morken in late 80s):

Surface patch f:RSurface patch f:R22 -> R is represented -> R is represented with radial base splines Swith radial base splines S

given a set of knots T:given a set of knots T:

||f – G(T)(f)|| = min{||f - g||: g in S}||f – G(T)(f)|| = min{||f - g||: g in S}

Surface evolution by knot removalSurface evolution by knot removal

Relevance measure of the knot:Relevance measure of the knot:

r(t) = ||f – G(T – {t})(f)||r(t) = ||f – G(T – {t})(f)||