1. Efficient Discriminative Learning of Parts-based Models M. Pawan Kumar Andrew Zisserman Philip Torr http://www.robots.ox.ac.uk/~vgg http://cms.brookes.ac.uk/research/visiongroup Aim: To efficiently learn parts-based models which discriminate between positive and negative poses of the object category Results - Sign Language Efficient Reformulation Results - Buffy Parts-based Model G = (V, E)Restricted to Tree The Learning Problem Q(f) = ∑ Q a (f(a)) + ∑ Q ab (f(a), f(b)) f : V Pose of V (h values) Q a (f(a)) : Unary potential for f(a) Computed using features Q ab (f(a), f(b)): Pairwise potential for validity of (f(a),f(b)) Restricted to Potts Q a (f(a)) : w a T  (f(a))Q a (f(a),f(b)) : w ab T  (f(a),f(b)) Q(f) : w T  (f) min ||w|| + C∑  i w T  (f + i ) +  ≥ 1 -  + i w T  (f - ij ) +  ≤ -1 +  - i Maximize margin, minimize hinge loss High energy for all positive examples Low energy for all negative examples Related Work Local Iterative Support Vector Machine (ISVM-1) Start with a small subset of negative examples (1 per image) Solve for w and b Replace negative examples with current MAP estimates Converges to local optimum Start with a small subset of negative examples (1 per image) Solve for w and b Add current MAP estimates to set of negative examples Converges to global optimum Global Iterative Support Vector Machine (ISVM-2) Drawback: Requires obtaining MAP estimate of each image at each iteration (computationally expensive) Our: 86.4% Buehler et al.,2008: 87.7% Our: 39.2% Ferrari et al.,2008: 41.0% 100 training images, 95 test images ISVM-1 ISVM-2 Our ISVM-1 ISVM-2 Our 196 training images, 204 test images ISVMs run for twice as long For all j (exponential in |V|) = 1, if (f(a),f(b))  L ab, = 0, otherwise. b a w T  (f - ij ) +  ≤ -1 +  - i, for all j M i ba (k) ≥ w b  b (l), for all l M i ba (k) ≥ w b  b (l) + w ab, for all (k,l)  L ab w a  a (k) + ∑ b M i ba (k) +  ≤ -1 +  - i Exponential in |V| Linear in |V| Linear in h Linear in |L ab | b a b a max  ab T 1 -  ab T K ab  ab s.t.  ab T y = 0,  ab ≥ 0 0 ≤ ∑  i ab (k) + ∑  i ab (k,l) ∑  i ab (k) + ∑  i ab (k,l) ≤ C Problem (1) ∑ k  i ab (k) = ∑ l  i ba (l) Constraint (3)Results in a large minimal problem Dual Decomposition Master Problem(1)Problem (2) minimal problem size = 2 Update Lagrange multiplier of (3) SVM-like problems Modified SVM Light min ∑ i g i (x), subject to x  P min ∑ g i (x i ), s.t. x i  P, x i = x max min ∑ g i (x i ) + i (x i - x), s.t. x i  P KKT Condition: ∑ i = 0 Solve min ∑ g i (x i ) + i x i i = i +  x i * Project Problem (1) learns the unary weight vector w a and pairwise weight w ab Problem (2) learns the unary weight vector w b and pairwise weight w ab M i ba (k) analogous to messages in Belief Propagation (BP) Efficient BP using distance transform: Felzenszwalb and Huttenlocher, 2004 Solving the Dual Implementation Details FeaturesShape: HOG Appearance: (x,x 2 ), x = fraction of skin pixels DataPositive examples: Provided by user Negative examples: All other poses OcclusionEach putative pose can be occluded (twice the number of labels)  (f(a),f(b)) max  ba T 1 -  ba T K ba  ba s.t.  ba T y = 0,  ba ≥ 0 0 ≤ ∑  i ba (k) + ∑  i ba (k,l) ∑  i ba (k) + ∑  i ba (k,l) ≤ C Problem (2)