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Using Image Priors in Maximum Margin
ClassifiersTali Brayer
Margarita Osadchy
Daniel Keren
Object DetectionProblem:
Locate instances of object category in a given image. Asymmetric classification
problem!
Background Object (Category)
Very large Relatively small
Complex (thousands of categories)
Simple (single category)
Large prior to appear in an image
Small prior
Easy to collect (not easy to learn from examples)
Hard to collect
All images
Intuition
Denote H to be the acceptance region of a classifier. We propose to minimize
Pr(All images) ( Pr(bkg)) in H except for the object samples.
Background
Object class
All images Background
We have a prior on the distribution of all natural images
Other work:
Combine small labeled training set with large unlabeled set – semi-supervised learning: EM with generative mixture models, Fisher kernel,self-training, co-training, transductive SVM, and graph-based methods…
All good for the symmetric case, but
We have more information: marginal background
Image smoothness measureLower probability
Lower probability
Distribution of Natural Images – “Boltzmann-like”
dxdyIII yx22exp)Pr(
lklkxlk
,
2,
22exp)xPr(
In frequency domain:
Linear SVM
Maximal margin
Enough training data
Class 1
Class 2
Not Enough training data
Linear SVM
Class 1
Class 2
0 xwx b
MM classifier with Prior
0xwx T b
margin wide3)
H samples positive )2
Hin images) natural(min)1
P
Class 1
Class 2
Minimize the probability of natural images over H
After some manipulations it reduces to
n
H
n
iii
bwdxdxxd ...expmin 1
1
2
,
n
i i
ib
dw
b
1
2,werfc
2min
Q
Random w with unit norm and random b from [-0.5, 0.5]
% o
f im
ages
tha
t w
x+b>
0
Relation between the number of natural random images in the positive half-space and the integral
Training Algorithm
erfc2
..1 ,0
..11 wx..
wmin
1
2
,
1
2
,
n
i i
i
i
ii
Mii
bw
d
w
-b
Mi
Mibts
C
Probability constraint:
(δ→0)
Convex Constraint
ii
n
i i
i
n
i i
i
dDDb
d
wb
d
w
b
1 0, w
0 ,2
erfc
21
1
2
1
2
1
convex
n
i i
i
dw
b
1
2erfc
2
Results Tested categories: cars (side view), faces. Training: 5/10/20/60/(all available data) object’s
images. All available background images. Test:
Face set: 472 faces, 23,573 bkg. Images Cars test: 299 cars, 10,111 bkg. images
Ran 50 trials for each set with different random choices of training data.
Weighted SVM was used to deal with the asymmetry in class sizes.
UIUC
CBCL
Average recognition rate(%): Faces
5 10 60 all
Weighted Linear SVM
70 72.5 75.2 77
Weighted Kernel SVM
69.7 72.6 79.6 83
MM_prior
Linear72.7 75 78 80.3
MM_prior Kernel
71.7 75.2 79.1 -
Average recognition rate(%): Cars
5 10 60 all
Weighted Linear SVM
89.24 91.8 92.8 93.7
Weighted Kernel SVM
90 92.9 95.4 96
MM_prior Linear
91.3 93 94.3 95.3
MM_prior Kernel
89.4 93.2 95.8 -
Future Work Video. Explore additional and more robust features. Refining the priors (using background
examples). Kernelization.