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

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Presentation on theme: "01/18 Lab meeting UCLA Vision Lab Department of Computer Science University of California at Los Angeles Fabio Cuzzolin Los Angeles, January 18 2005."— Presentation transcript:

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

2 … past and present 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

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

4 1 Upper and lower probabilities

5 5 Past work Geometric approach to belief functions (ISIPTA01, SMC- C-05) Algebra of families of frames (RSS00, ISIPTA01, AMAI03) Geometry of Dempsters rule (FSKD02, SMC-B-04) Geometry of upper probabilities (ISIPTA03, SMC-B-05) Simplicial complexes of fuzzy sets (IPMU04)

6 The theory of belief functions

7 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 Dempsters rule families of frames

8 8 Motivations

9 9 Axioms and superadditivity probabilities additivity: if then belief functions 3. superadditivity

10 10 Example of b.f.

11 11 belief functions s: 2 Θ ->[0,1] A Belief functions B2B2 B1B1..where m is a mass function on 2 Θ s.t.

12 12 Dempsters rule b.f. are combined through Dempsters rule AiAi BjBj A i B j =A intersection of focal elements

13 13 Example of combination s 1 : m({a 1 })=0.7, m({a 1, a 2 })=0.3 a1a1 a2a2 a3a3 a4a4 s 2 : m( )=0.1, m({a 2, a 3, a 4 })=0.9 s 1 s 2 : m({a 1 })=0.19, m({a 2 })=0.73 m({a 1, a 2 })=0.08

14 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 Dempsters rule 3 - a multi-domain representation of the evidence is contemplated

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

16 Algebra of frames

17 17 Family of frames example: a function y [0,1] is quantized in three different ways refining Common refinement

18 18 Lattice structure minimal refinement 1F1F maximal coarsening F is a locally Birkhoff (semimodular with finite length) lattice bounded below order relation: existence of a refining

19 Geometric approach to upper and lower probabilisties

20 20 it has the shape of a simplex 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

21 21 Geometry of Dempsters rule constant mass loci foci of conditional subspaces Dempsters rule can be studied in the geometric setup too

22 22 the space of plausibilities is also a simplex Geometry of upper probs

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

24 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 25 Possibilities in a geometric setup possibility measures are a class of belief functions they have the geometry of a simplicial complex

26 Combinatorial analysis

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

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

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

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

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

32 32 Indipendence and conflict s 1,…, s n are not always combinable 1,…, n are indipendent if any s 1,…, s n are combinable are defined on independent frames

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

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

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

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

37 2 Computer vision

38 38 Vision problems HMM and size functions for gesture recognition (BMVC97) object tracking and pose estimation (MTNS98,SPIE99, MTNS00, PAMI04) composition of HMMs (ASILOMAR02) data association with shape info (CDC02, CDC04, PAMI05) volumetric action recognition (ICIP04,MMSP04)

39 Size functions for gesture recognition

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

41 41 Size functions Topological representation of contours

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

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

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

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

46 46 EM algorithm feature matrices: collection of feature vectors along time EM A,C learning the models parameters through EM ztwo instances of the same gesture

47 Compositional behavior of Hidden Markov models

48 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 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 50 Kullback-Leibler comparison We used the K-L distance to measure the similarity between models extracted from clutter and in absence of clutter

51 Model-free object pose estimation

52 52 Model-free pose estimation Pose estimation: inferring the configuration of a moving object from one or more image sequences Most approaches in the literature are model-based: they assume some knowledge about the nature of the body (articulated, deformable, etc) and some sort of model T=0 t=T

53 53 Model-free scenario Scenario: the configuration of an uknown object is desired, given no a-priori information about the nature itself of the body The only info available is carried by the images We need to build a map from image measurements to body poses This can be done by using a learning technique, based on training data

54 54 Evidential model The evidential model is built during the training stage, when the feature-pose maps are learned approximate feature spaces training set of sample poses feature-pose maps (refinings)

55 Feature extraction From the blurred image the region with color similar to the region of interest is selected, and the bounding box is detected

56 56 Estimates from the combined model Ground truth versus estimates for two components of the pose

57 JPDA with shape information for data association

58 58 JPDA with shape info robustness: clutter does not meet shape constraints occlusions: occluded targets can be estimated JPDA model: independent targets Shape model: rigid links Dempsters fusion

59 59 Triangle simulation the clutter affects only the standard JPDA estimates

60 60 Body tracking Application: tracking of feature points on a moving human body

61 Volumetric action recognition

62 62 Volumetric action recognition problem: recognizing the action performed by a person viewed by a number of cameras step 1: modeling the dynamics of the motion step 2: extracting image features 2D approaches: features are extracted from single views -> viewpoint dependence volumetric approach: features are extracted from a volumetric reconstruction of the moving body

63 63 Multiple sequences synchronized views from different cameras, chromakeying

64 64 Volumetric intersection silhouette extraction of the moving object from all views 3D object shape reconstruction through intersection of occlusion cones more views -> more details

65 65 3D feature extraction locations of torso, arms, and legs of the moving person k-means clustering to separate bodyparts


67 67 Modeling and recognition model of the walking action … classification: each new feature matrix is fed to all the learnt models, generating a set of likelihoods HMM 1 HMM 2 HMM n

68 3 Combinatorics

69 69 Independence on lattices three distinct independence relations modularity equivalent formulations LI3 LI2=LI3 LI1 semimodular latticemodular lattice LI2 LI1

70 70 Scheme of the proof

71 4 Conclusions


73 73 …concluding the ToE comes from a strong critics to the Bayesian framework useful for sensor fusion problems under incomplete information real problem solutions stimulate the extension of the formalism complex objects mathematically rich Young theory need completion

74 74 In the near future.. search for a metric on the space of dynamical systems – stochastic models sistematic description of the geometric approach to non-additive measures understand the intricate relations between probability and combinatorics

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