01/18 Lab meeting

1. 01/18 Lab meeting Fabio Cuzzolin UCLA Vision Lab Department of Computer Science University of California at Los Angeles Los Angeles, January 18 2005

2. PhD student, University of Padova, Department of Computer Science (NAVLAB laboratory) with Ruggero Frezza • Visiting student, ESSRL, Washington University in St. Louis • Visiting student, UCLA, Los Angeles (VisionLab) • Post-doc in Padova, Control and Systems Theory group • Young researcher, Image and Sound Processing Group, Politecnico di Milano • Post-doc, UCLA Vision Lab … past and present

3. Computer vision Discrete mathematics • linear independence on lattices Belief functions and imprecise probabilities • geometric approach • algebraic analysis • combinatorial analysis … the research • object and body tracking • data association • gesture and action recognition research

4. 1 Upper and lower probabilities

5. Past work • Geometric approach to belief functions (ISIPTA’01, SMC-C-05) • Algebra of families of frames (RSS’00, ISIPTA’01, AMAI’03) • Geometry of Dempster’s rule (FSKD’02, SMC-B-04) • Geometry of upper probabilities (ISIPTA’03, SMC-B-05) • Simplicial complexes of fuzzy sets (IPMU’04)

6. The theory of belief functions

7. Uncertainty descriptions • A number of theories have been proposed to extend or replace classical probability: possibilities, fuzzy sets, random sets, monotone capacities, etc. • theory of evidence (A. Dempster, G. Shafer) • belief functions • Dempster’s rule • families of frames

8. Motivations

9. belieffunctions • 3. superadditivity Axioms and superadditivity • probabilities • additivity: if then

10. Example of b.f.

11. belief functions s: 2Θ->[0,1] Belief functions A B1 • ..where m is a mass function on 2Θs.t. B2

12. AiÇBj=A Ai • intersection of focal elements Bj Dempster’s rule • b.f. are combined through Dempster’s rule

13. a1 a3 • s1: • m({a1})=0.7, m({a1 ,a2})=0.3 a2 a4 • s2: • m()=0.1, m({a2 ,a3 ,a4})=0.9 • s1  s2 : • m({a1})=0.19, m({a2})=0.73 • m({a1 ,a2})=0.08 Example of combination

14. Bayes vs Dempster • Belief functions generalize the Bayesian formalism as: • 1- discrete probabilities are a special class of belief functions • 2 - Bayes’ rule is a special case of Dempster’s rule • 3 - a multi-domain representation of the evidence is contemplated

15. algebraic analysis geometric analysis combinatorial analysis probabilistic analysis categorial? My research Theory of evidence

16. Algebra of frames

17. .0 .1 .0 .1 .2 .3 .4 .00 .01 .10 .11 0.00 0.09 0.49 0.90 0.99 0 0.25 0.5 0.75 Family of frames • example: a function yÎ [0,1] is quantized in three different ways • refining • Common refinement 1

18. Lattice structure 1F maximal coarsening QÅW Q W minimal refinement QÄW • order relation: existence of a refining • F is a locally Birkhoff (semimodular with finite length) lattice bounded below

19. Geometric approach to upper and lower probabilisties

20. Belief space • the space of all the belief functions on a given frame • each subset A  A-th coordinate s(A) in an Euclidean space • it has the shape of a simplex

21. Geometry of Dempster’s rule • constant mass loci • foci of conditional subspaces • Dempster’s rule can be studied in the geometric setup too

22. the space of plausibilities isalso a simplex Geometry of upper probs

23. Belief and probabilities • study of the geometric interplay of belief and probability

24. Consistent probabilities • Each belief function is associated with a set of consistent probabilities, forming a simplex in the probabilistic subspace • the vertices of the simplex are the probabilities assigning the mass of each focal element of s to one of its points • the center of mass of P(s) coincides with Smets’ pignistic function

25. possibility measures are a class of belief functions Possibilities in a geometric setup • they have the geometry of a simplicial complex

26. Combinatorial analysis

27. Total belief theorem • generalization of the total probability theorem • a-priori constraint • conditional constraint

28. method: replacing columns through Existence • candidate solution: linear system nn where the columns of A are the focal elements of stot • problem: choosing n columns among m s.t. x has positive components

29. Solution graphs • all the candidate solutions form a graph • Edges = linear transformations

30. New goals... algebraic analysis geometric analysis Theory of evidence combinatorial analysis probabilistic analysis ?

31. Approximations • problem: finding an approximation of s • compositional criterion • the approximation behaves like s when combined through Dempster • probabilistic and fuzzy approximations

32. 1,…, n are indipendent if Indipendence and conflict • s1,…, sn are not always combinable • any s1,…, sn are combinable  are defined on independent frames

33. pseudo Gram-Schmidt • new set of b.f. surely combinable Pseudo Gram-Schmidt • Vector spaces and frames are both semimodular lattices -> admit independence

34. Canonical decomposition • unique decomposition of s into simple b.f. • convex geometry can be used to find it

35. m-1m past and present target association old estimates Am-1 past targets - model associations m-1m Kalman filters  rigid motion constraints Am-1 new estimates Am-1 () Am current targets – model association Am  = Am-1 m-1m Tracking of rigid bodies • data association of points belonging to a rigid body • rigid motion constraints can be written as conditional belief functions  total belief needed

36. Total belief problem and combinatorics • general proof, number of solutions, symmetries of the graph • relation with positive linear systems • homology of solution graphs • matroidal interpretation

37. 2 Computer vision

38. Vision problems • HMM and size functions for gesture recognition (BMVC’97) • object tracking and pose estimation (MTNS’98,SPIE’99, MTNS’00, PAMI’04) • composition of HMMs (ASILOMAR’02) • data association with shape info (CDC’02, CDC’04, PAMI’05) • volumetric action recognition (ICIP’04,MMSP’04)

39. Size functions for gesture recognition

40. Size functions for gesture recognition • Combination of HMMs (for dynamics) and size functions (for pose representation)

41. Size functions • “Topological” representation of contours

42. Measuring functions • Functions defined on the contour of the shape of interest real image family of lines measuring function

43. Feature vectors • a family of measuring functions is chosen • … the szfc are computed, and their means form the feature vector

44. Hidden Markov models • Finite-state model of gestures as sequences of a small number of poses

45. Four-state HMM • Gesture dynamics -> transition matrix A • Object poses -> state-output matrix C

46. EM algorithm • feature matrices: collection of feature vectors along time • two instances of the same gesture A,C EM • learning the model’s parameters through EM

47. Compositional behavior of Hidden Markov models

48. Composition of HMMs • Compositional behavior of HMMS: the model of the action of interest is embedded in the overall model • Example: “fly” gesture in clutter

49. State clustering • Effect of clustering on HMM topology • “Cluttered” model for the two overlapping motions • Reduced model for the “fly” gesture extracted through clustering

50. Kullback-Leibler comparison • We used the K-L distance to measure the similarity between models extracted from clutter and in absence of clutter