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Discrete Approach to Curve and Surface Evolution

This paper discusses innovative methods for discrete curve and surface evolution, highlighting their robustness against noise and ability to preserve positional accuracy without blurring. It focuses on curve evolution techniques such as extraction of linear segments and maximal convex arcs for shape similarity retrieval in image databases, along with applications in video analysis for key frame detection and unpredictable event identification. The effectiveness of these techniques is evaluated through various case studies, including the analysis of video sequences and surface evolution via knot removal in radial basis spline representations.

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Discrete Approach to Curve and Surface Evolution

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

  2. Discrete Curve Evolution P=P0, ..., PmPi+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

  3. Discrete Curve Evolution: Preservation of position, no blurring

  4. Discrete Curve Evolution: robustness with respect to noise

  5. Discrete Curve Evolution: extraction of linear segments

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

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

  8. Shape similarity measure based on correspondence of visual parts

  9. A video sequence is mapped to a trajectory in a high dimensional space,e.g. by mapping each frame to a feature vector in R37Discrete curve evolution allows us to determine key frames

  10. 2379 vertices 20 vertices Trajectory Simplification

  11. Mr. Bean

  12. The 10 most relevant frames in Mr. Beanwww.videokeyframes.de

  13. Detection of unpredictable events in videos:Mov3.mpg

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

  15. Detection of unpredictable events in videos:seciurity1.mpg

  16. Two most unpredictable frames extracted from Mov3.mpg

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

  18. Two most unpredictable frames extracted from security1.mpg

  19. Discrete Surface Evolution: repeated removal of least relevant vertices

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

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