classification using intersection kernel svms is efficient joint work with subhransu maji and alex...
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Classification using intersection kernel SVMs is efficient
Joint work with Subhransu Maji and Alex Berg (CVPR’08)
Jitendra Malik UC Berkeley
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Boosted decision trees + Very fast evaluation - Slow training (esp. multi-class)Linear SVM + Fast evaluation + Fast training - Low accuracy unless very good featuresNon-linear kernelized SVM + Better accuracy than linear . Medium training - Slow evaluation
This work
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Support Vector Machines
Linear Separators (aka. Perceptrons)
B2
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Support Vector Machines
Other possible solutions
B2
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Support Vector Machines
Which one is better? B1 or B2? How do you define better?
B1
B2
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Support Vector Machines
Find hyperplane maximizes the margin => B1 is better than B2
B1
B2
b11
b12
b21
b22
margin
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Kernel Support Vector Machines
Kernel :•Inner Product in Hilbert Space
•Can Learn Non Linear Boundaries
2
2( , ) exp( )
2
x zK x z
σ−
= −
( , ) ( ) ( )TK x z x z=Φ Φ
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Feature Representation
Discriminative Classifier
(+ examples) (- examples)
Training Stage
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Our Multiscale HOG-like feature
Concatenate orientation histograms for each orange region.Differences from HOG: -- Hierarchy of regions -- Only performing L1 normalization once (at 16x16)
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Comparison to HOG (Dalal & Triggs)
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Smaller Dimensional (1360 vs. 3780) Simple Implementation (Convolutions) Faster to compute
+ No non-local Normalization
+ No gaussian weighting
+ No color normalization
Comparison to HOG (Dalal & Triggs)
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What is the Intersection Kernel?
Histogram Intersection kernel between histograms a, b
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What is the Intersection Kernel?
Histogram Intersection kernel between histograms a, b
K small -> a, b are differentK large -> a, b are similar
Intro. by Swain and Ballard 1991 to compare color histograms.Odone et al 2005 proved positive definiteness.Can be used directly as a kernel for an SVM.Compare to
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linear SVM, Kernelized SVM, IKSVM
Decision function is where:
Linear:
Non-linearUsingKernel
HistogramIntersectionKernel
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Kernelized SVMs slow to evaluate
Arbitrary Kernel
HistogramIntersectionKernel
Feature corresponding to a support vector l
Feature vector to evaluate
Kernel EvaluationSum over all support vectors
SVM with Kernel Cost: # Support Vectors x Cost of kernel comp.IKSVM Cost: # Support Vectors x # feature dimensions
Decision function is where:
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The Trick
Decision function is where:
Just sort the support vectorvalues in each coordinate, andpre-compute
To evaluate, find position ofin the sorted support vectorvalues (cost: log #sv)look up values, multiply & add
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The Trick
Decision function is where:
Just sort the support vectorvalues in each coordinate, andpre-compute
To evaluate, find position ofin the sorted support vectorvalues (cost: log #sv)look up values, multiply & add
#support vectors x #dimensions
log( #support vectors ) x #dimensions
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The Trick 2
For IK hi is piecewise linear, and quite smooth, blue plot. We can approximate with fewer uniformly spaced segments, red plot. Saves
time & space!
Decision function is where:
#support vectors x #dimensionslog( #support vectors ) x #dimensions
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The Trick 2
Decision function is where:
#support vectors x #dimensionslog( #support vectors ) x #dimensions
constant x #dimensions
For IK hi is piecewise linear, and quite smooth, blue plot. We can approximate with fewer uniformly spaced segments, red plot. Saves
time & space!
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Timing Results
Time to evaluate 10,000 feature vectors
IKSVM with our multi-scale version of HOG featuresbeats Dalal & Triggs. Alsofor Daimler Chrysler data. Current Best on these datasets.
Linear SVM with our multi-scale Version of HOG featureshas worse classification perf.than Dalal & Triggs.
reduced memory!
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Distribution of support vector values and hi
Distribution of
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Best Performance on Pedestrian Detection,Improve on Linear for Many Tasks
INRIA PedestriansDaimler Chrysler Pedestrians
Caltech 101 with “simple features” Linear SVM 40% correct IKSVM 52% correct
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Classification Errors
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Results – ETHZ DatasetDataset: Ferrari et al., ECCV 2006 255 images, over 5 classes training = half of positive images for a class + same number from the other classes (1/4 from each) testing = all other images large scale changes; extensive clutter
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Method Applelogo Bottle Giraffe Mug Swan Avg
PAS* 65.0 89.3 72.3 80.6 64.7 76.7
Our 86.1 81.0 62.1 78.0 100 81.4
Beats many current techniques without any changes to our features/classification framework.
Recall at 0.3 False Positive per Image Shape is an important cue (use Pb instead of OE)
Results – ETHZ Dataset
*Ferarri et.al, IEEE PAMI - 08
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Other kernels allow similar trick
Decision function is where:
IKSVM SVM
hi not piece-wise linear,but we can still use anapproximation for fastevaluation.
hi are piece-wise linear,uniformly spacedpiece-wise linear approx.is fast.
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Conclusions Exactly evaluate IKSVM in O(n log m) as opposed to O(nm)
Makes SV cascade or other ordering schemes irrelevant for intersection kernel
Verified that IKSVM offers classification performance advantages over linear
Approximate decision functions that decompose to a sum of functions for each coordinate (including Chi squared)
Directly learn such classification functions (no SVM machinery) Generalized linear svm beats linear SVM in some applications
often as good as more expensive RBF kernels Showed that relatively simple features with IKSVM beats Dalal
& Triggs (linear SVM), leading to the state of the art in pedestrian detection.
Applies to best Caltech 256, Pascal VOC 2007 methods.
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Classification Using Intersection Kernel Support Vector Machines is efficient.Subhransu Maji and Alexander C. Berg and Jitendra Malik.Proceedings of CVPR 2008, Anchorage, Alaska, June 2008.
Software and more results available at
http://www.cs.berkeley.edu/~smaji/projects/fiksvm/