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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 LanguageEfficient Reformulation
Results - Buffy
Parts-based Model G = (V, E) Restricted to Tree
The Learning Problem
Q(f) = ∑ Qa(f(a)) + ∑ Qab(f(a), f(b))
f : V Pose of V (h values)
Qa(f(a)) : Unary potential for f(a) Computed using featuresQab(f(a), f(b)): Pairwise potential
for validity of (f(a),f(b)) Restricted to Potts
Qa(f(a)) : waT(f(a)) Qa(f(a),f(b)) : wab
T(f(a),f(b)) Q(f) : wT(f)
min ||w|| + C∑ i
wT(f+i) + ≥ 1 - +
i
wT(f-ij) + ≤ -1 + -
i
Maximize margin, minimize hinge lossHigh energy for all positive examplesLow energy for all negative examples
Related WorkLocal 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)) Lab, = 0, otherwise.
b
a
wT(f-ij) + ≤ -1 + -
i, for all j
Miba(k) ≥ wbb(l), for all l
Miba(k) ≥ wbb(l) + wab,
for all (k,l) Lab
waa(k) + ∑b Miba(k) + ≤ -1 + -
i
Exponential in |V|
Linear in |V|
Linear in h
Linear in |Lab|
b
a
b
a
max abT1 - ab
TKabab
s.t. abTy = 0, ab ≥ 0
0 ≤ ∑ iab(k) + ∑ i
ab(k,l)
∑ iab(k) + ∑ i
ab(k,l) ≤ C
Problem (1)
∑k iab(k) = ∑ li
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 SVMLight
min ∑ i gi(x), subject to x P
min ∑ gi(xi), s.t. xi P, xi = x max min ∑ gi(xi) + i(xi - x), s.t. xi P
KKT Condition: ∑ i = 0 Solve min ∑ gi(xi) + ixi i = i +xi* Project
Problem (1) learns the unary weight vector wa and pairwise weight wab
Problem (2) learns the unary weight vector wb and pairwise weight wab
Miba(k) analogous to messages in Belief Propagation (BP)
Efficient BP using distance transform: Felzenszwalb and Huttenlocher, 2004
Solving the Dual
Implementation DetailsFeatures Shape: HOG Appearance: (x,x2), x = fraction of skin pixelsData Positive examples: Provided by user Negative examples: All other posesOcclusion Each putative pose can be occluded (twice the number of labels)
(f(a),f(b))
max baT1 - ba
TKbaba
s.t. baTy = 0, ba ≥ 0
0 ≤ ∑ iba(k) + ∑ i
ba(k,l)
∑ iba(k) + ∑ i
ba(k,l) ≤ C
Problem (2)