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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

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… 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

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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

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1 Upper and lower probabilities

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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)

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The theory of belief functions

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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

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8 Motivations

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

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10 Example of b.f.

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

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

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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

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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

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

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Algebra of frames

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

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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

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Geometric approach to upper and lower probabilisties

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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

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

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

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

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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

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

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Combinatorial analysis

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

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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

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

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

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

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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

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

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

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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

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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

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2 Computer vision

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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)

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Size functions for gesture recognition

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

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41 Size functions Topological representation of contours

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

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

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

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

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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

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Compositional behavior of Hidden Markov models

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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

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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

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

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Model-free object pose estimation

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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

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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

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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)

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

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

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JPDA with shape information for data association

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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

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59 Triangle simulation the clutter affects only the standard JPDA estimates

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

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Volumetric action recognition

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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

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63 Multiple sequences synchronized views from different cameras, chromakeying

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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

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

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66 Feature matrices two instances of the action walking TORSO COORDINATES ABDOMEN COORDINATES RIGHT LEG COORDINATES LEFT LEG COORDINATES X Y Z

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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

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3 Combinatorics

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

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70 Scheme of the proof

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4 Conclusions

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72 from real to abstract OBJECT TRACKINGDATA ASSOCIATION MEASUREMENT CONFLICT POINTWISE ESTIMATE CONDITIONAL CONSTRAINTS ALGEBRAIC ANALYSIS GEOMETRIC APPROACH TOTAL BELIEF the solution of real problems stimulates new theoretical issues

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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

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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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