asymmetric boosting for face detection

Asymmetric Boosting for Face Detection

Minh-Tri PhamPh.D. Candidate and Research AssociateNanyang Technological University, Singapore

presented by

Overview

• Online Asymmetric Boosting

• Fast Training and Selection of Haar-like Features using Statistics

• Detection with Multi-exit Asymmetric Boosting

Online Asymmetric Boosting

CVPR’07 oral paper:Minh-Tri Pham and Tat-Jen Cham. Online Learning Asymmetric Boosted Classifiers for Object Detection. In Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), Minneapolis, MN, 2007.

objectregion

Motivation• Usual goal of object detector:

– Focused on accuracy• General detectors are designed to

deal with different input spaces• Only one input space is used per

application

input space 1

input space 2

input space 3

global input spaceOffline learned

Online learned?

non-objectregion

Supervisor-Student paradigmSupervisor:

Student:

Fast but limited Student Detector

Slow but general

Supervisor Detector

Input Output

• Supervisor = existing object detector

• Student = online-learned object detector

• Less complex model

Faster detection speed

Problem overview• Common appearance-based approach:

– Classify a patch using a cascade or tree of boosted classifiers (Viola-Jones and variants):

– F1, F2, …, FN: boosted classifiers

• Main challenges for online learning a boosted classifier:– Asymmetric: P(non-object) >> P(object)– Online data

F1 F2 FN….passpasspass pass

reject reject reject

object

non-object

Review of current methods• Online learning for boosting:

– Online Boosting of Oza (2005)

• Replace offline weak learners with online weak learners

• Propagate weights similarly to AdaBoost

• Only works well when P(non-object) ≈ P(object)

• P(non-object) >> P(object):– Viola and Jones (2002)– Ma and Ding (2003)– Hou et. al. (2006)

• Reweigh positives higher and negatives lower

• Offline learning only

Asymmetric Online Boosting• Incorporate asymmetric reweighing scheme into Online Boosting

Skewness balancing:• New reweighting scheme giving

better accuracy

Polarity balancing:• Faster learning convergence rate

• Skewness:– Measure the degree of asymmetry of the class probability distribution– Defined as:

• = logP(negative) – logP(positive)• Viola-Jones’ reweighing scheme:

– Reweigh positives the same amount more than negatives on every weak learner

– km = reweighing amount on the m-th weak learner– k = total reweighing amount

MM kkkk ...21

Skewness balancing

weak learners

skewness

Initial skewness:1 > 0After reweighing:

1’ = 1 - log k1After training weak learner 1:

2 ≈ 0After reweighing:2’ = 2 - log k2

After training weak learner 2:




: negative example: positive example

• Our approach:– Reweigh positives more than negatives differently, to have equal skewness

presented to every weak learner

– m = skewness after training the (m-1)-th weak learner– km = reweighing amount on the m-th weak learner– k = total reweighing amount

Skewness balancing

weak learners

skewness

Initial skewness:1 > 0

After reweighing:1’ = 1 - log k1








m

m

iim mM

mMkkmM

k 1

loglog1

1exp1

1

Skewness balancing• Effective for initial boosted

classifiers in the cascade– Better accuracy faster detection

speed

• Effectiveness degraded as boosted classifiers get more complicated

ROC curve for 4-feature boosted classifier

ROC curve for 200-feature boosted classifier

Polarity balancing• After training a weak learner with AdaBoost:– classified weights = mis-classified weights– positive weights = negative weights (if weak learner is optimal)

• To maintain online AdaBoost’s properties:

– Online Boosting ensures asymptotically:

• classified weights = mis-classified weights

– Our method ensures asymptotically:

• classified weights = mis-classified weights

• positive weights = negative weights

Faster convergence rate

TP TN

FN FP

positive negative

Correctly classified

Wrongly classified

Weight distribution after training a weak learner

• Learning time:– About 5-30% faster with Polarity balancing

Polarity balancing

Online Learning a 20-feature boosted classifier

Overall performance• ROC curves:

– Similar results

Online Learning a Face Detector• Video clip:

– Length: 20 minutes– Resolution: 352x288– 25fps

• Learn online from the first 10 minutes – using OpenCV’s face detector as supervisor

• Test with the remaining 10 minutes

OpenCV’s face detectorDetection speed: 15fps

Our online-learned face detectorDetection speed: 30fps

Online Learning a Face Detector• Distribution of weak learners over the cascade:

1 4 7 10 13 16 19 22 250

40

80

120

160

200

Viola-Jones'04

Online-learned face detector

Number of boosted classifiers

Num

ber o

f wea

k cl

assi

fiers

Concluding remarks• Skewness balancing:

– Effective for early boosted classifiers• Better accuracy faster detection speed

• Polarity balancing: – Reduction in learning time about 5-30% empirically

• Online learning an object detector from an offline counterpart:– Worst case:

• detection accuracy and speed similar– Average case:

• detection speed can be faster (twice as much)

Fast Training and Selection of Haar-like Features using Statistics

ICCV’07 oral paper:Minh-Tri Pham and Tat-Jen Cham. Fast Training and Selection of Haar Features using Statistics in Boosting-based Face Detection. In Proc. International Conference on on Computer Vision (ICCV), Rio de Janeiro, Brazil, 2007.

• Won Travel Grant Award• Won Second Prize, Best Student Paper in Year 2007 Award, Pattern Recognition and Machine

Intelligence Association (PREMIA), Singapore

Motivation

• Face detectors today– Real-time detection

speed

…but…

– Weeks of training time

18

Factor

Description Common value

N number of examples 10,000

M number of weak classifiers in total

4,000 - 6,000

T number of Haar-like features

40,000

Why is Training so Slow?

• Time complexity: O(MNT log N)– 15ms to train a feature classifier– 10 minutes to train a weak classifier– 27 days to train a face detector

A view of a face detector training algorithm

for weak classifier m from 1 to M:…update weights – O(N)for feature t from 1 to T:

compute N feature values – O(N)sort N feature values – O(N log N)train feature classifier – O(N)

select best feature classifier – O(T)…

19

Why Should the Training Time be Improved?• Tradeoff between time and generalization

– E.g. training 100 times slower if we increase both N and T by 10 times

• Trial and error to find key parameters for training– Much longer training time needed

• Online-learning face detectors have the same problem

20

Existing Approaches to Reduce the Training Time• Sub-sample Haar-like feature set

– Simple but loses generalization

• Use histograms and real-valued boosting (B. Wu et. al. ‘04)– Pro: Reduce from O(MNT log N) to O(MNT)– Con: Raise overfitting concerns:

• Real AdaBoost not known to be overfitting resistant• Weak classifier may overfit if too many histogram bins are used

• Pre-compute feature values’ sorting orders (J. Wu et. al. ‘07)– Pro: Reduce from O(MNT log N) to O(MNT)– Con: Require huge memory storage

• For N = 10,000 and T = 40,000, a total of 800MB is needed.

21

A view of a face detector training algorithm

for weak classifier m from 1 to M:…update weights – O(N)for feature t from 1 to T:

compute N feature values – O(N)sort N feature values – O(N log N)train feature classifier – O(N)


Factor




4,000 - 6,000


40,000

Why is Training so Slow?

• Time complexity: O(MNT log N)– 15ms to train a feature classifier– 10min to train a weak classifier– 27 days to train a face detector

• Bottleneck:– At least O(NT) to train a weak

classifier

• Can we avoid O(NT)?

22

Our Proposal

• Fast StatBoost: To train feature classifiers using statistics rather than using input data– Con:

• Less accurate… but not critical for a feature classifier

– Pro: • Much faster training time:

Constant time instead of linear time

23

Fast StatBoost

24

• Training feature classifiers using statistics:– Assumption: feature value v(t) is normally

distributed given face class c is known – Closed-form solution for optimal threshold

• Fast linear projections of the statistics of a window’s integral image into 1D statistics of a feature value

Non-faceFace

Optimalthreshold

Featurevalue

)()( tTt gmJ )()(2)( tTtt gg J

constant time to train a feature classifier

: Haar-like feature, a sparse vector with less than 20 non-zero elements

: mean vector and covariance matrix ofJJm , J

)(tg

: random vector representing a window’s integral imageJ : mean and variance of feature value v(t)2)()( , tt

Fast StatBoost• Integral image’s statistics are obtained directly from the weighted input data

– Input: N training integral images and their current weights w(m):

– We compute:• Sample total weight:

• Sample mean vector:

• Sample covariance matrix:

NNmN

mm ccc ,,,...,,,,,, )(22

)(2

)(1 JwJwJw 11

ccn

nmncc

n

wz:

)(1ˆˆ Jm

25

ccn

mnc

n

wz:

)(ˆ

Tcc

ccn

Tnn

mncc

n

wz mmJJ ˆˆˆˆ:

)(1

Factor




4,000 - 6,000


40,000

d number of pixels of a window

300-500

Fast StatBoost• To train a weak classifier:

– Extract the class-conditional integral image statistics

• Time complexity: O(Nd2)• Factor d2 negligible because fast algorithms

exist, hence in practice: O(N)

– Train T feature classifiers by projecting the statistics into 1D:

• Time complexity: O(T)

– Select the best feature classifier• Time complexity: O(T)

• Time complexity: O(N+T)

A view of our face detector training algorithm

for weak classifier m from 1 to M:…update weights – O(N)Extract statistics of integral image – O(Nd2)for feature t from 1 to T:

project statistics into 1D – O(1)train feature classifier – O(1)


26

Experimental Results• Setup

– Intel Pentium IV 2.8GHz– 19 types 295,920 Haar-like

features

• Time for extracting the statistics:– Main factor: covariance matrices

• GotoBLAS: 0.49 seconds per matrix

• Time for training T features:– 2.1 seconds

(1) (2)

(17)

(7)

(3) (4) (5) (6)

(14)(15)

(16)

(8) (9)(10) (11) (12) (13)

(18) (19)

Edge features: Corner features:

Diagonal line features:

Line features: Center-surround features:

Nineteen feature types used in our experiments

Total training time: 3.1 seconds per weak classifier with 300K features• Existing methods: 1-10 minutes with 40K features or fewer

27

Experimental Results• Comparison with Fast AdaBoost (J. Wu et. al. ‘07), the fastest known

implementation of Viola-Jones’ framework:

28

0 50000 100000 150000 200000 250000 30000002468

1012

training time of a weak classifier

Fast AdaBoostFast StatBoost

number of features (T)

seco

nds

(s)

Experimental Results• Performance of a cascade:

ROC curves of the final cascades for face detection

Method Total training time

Memory requirement

Fast AdaBoost (T=40K)

13h 20m 800 MB

Fast StatBoost (T=40K)

02h 13m 30 MB

Fast StatBoost (T=300K)

03h 02m 30 MB

29

Conclusions

• Fast StatBoost: use of statistics instead of input data to train feature classifiers

• Time:– Reduction of the face detector training time from up to a month to 3 hours– Significant gain in both N and T with little increase in training time

• Due to O(N+T) per weak classifier

• Accuracy:– Even better accuracy for face detector

• Due to much more members of Haar-like features explored

30

Detection with Multi-exit Asymmetric Boosting

CVPR’08 poster paper:Minh-Tri Pham and Viet-Dung D. Hoang and Tat-Jen Cham. Detection with Multi-exit Asymmetric Boosting. In Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage, Alaska, 2008.

• Won Travel Grant Award

Problem overview• Common appearance-based approach:

– F1, F2, …, FN : boosted classifiers

– f1,1, f1,2, …, f1,K : weak classifiers– : threshold

F1 F2 FN….passpasspass pass


object

non-object

pass

reject

F1

…. +++ yes

no

f1,1 f1,2 f1,K > ?

Objective

• Find f1,1, f1,2, …, f1,K, and such that:– – – K is minimized proportional to F1’s evaluation time

pass

reject

F1

…. +++ yes

no

f1,1 f1,2 f1,K > ?

01

01

)()(

FFRRFFAR

K

ii xfsignxF

1,11 )()(

Existing trends (1)

Idea• For k from 1 until convergence:

– Let

– Learn new weak classifier f1,k(x):

– Let

– Adjust to see if we can achieve FAR(F1) <= 0 and FRR(F1) <= 0:• Break loop if such exists

Issues• Weak classifiers are sub-

optimal w.r.t. training goal.• Too many weak classifiers

are required in practice.

k

ii xfsignxF

1,11 )()(

)()(minargˆ11,1

,1

FFRRFFARfkf

k

k

ii xfsignxF

1,11 )()(

Existing trends (2)

Idea• For k from 1 until convergence:

– Let

– Learn new weak classifier f1,k(x):

– Break loop if FAR(F1) <= 0 and FRR(F1) <= 0Pros• Reduce FRR at the

cost of increasing FAR – acceptable for cascades

• Fewer weak classifiers

k

ii xfsignxF

1,11 )()(

)()(minargˆ11,1

,1

FFRRFFARfkf

k

Cons• How to choose ?• Much longer training

time

Solution to con• Trial and error:

• choose such that K is minimized.

Our solution

36

Why?

Learn every weak classifier using the same asymmetric goal:

where

)(,1 xf k

,)()(minargˆ11,1

,1

FFRRFFARfkf

k

.0

0

Because…• Consider two desired bounds (or targets) for learning a boosted classifier

– Exact bound: and– Conservative bound:

• (2) is more conservative than (1) because (2) => (1).

0)( MFFAR 0)( MFFRR

00

0 )()(

MM FFRRFFAR

:)(xFM

(2)(1)

0 1

1

0

= 1

H1

H2

H200H201

H3

H4

0

Q1Q2

Q200

Q201

Q3Q4

FAR

FRR

exact bound

conservativebound

FRR0 1

1

= 0/0

FAR

H1

H2

H3

H39

H40

0

0

H41

Q1

Q2

Q3Q39

Q41

Q40

exact bound

conservativebound

At for every new weak classifier learned, the ROC operating

point moves the fastest toward the conservative bound

,0

0

Multi-exit BoostingA method to train a single boosted classifier with multiple exit nodes:

: a weak classifier : a weak classifier followed by a decision to continue or reject – an exit node

f1 f2 f3 f4 f5 f6 f7 f8 object

non-obj

pass pass passreject reject reject

fi fi

+ + + + + + +

.0

0

• Features:• Weak classifiers are trained with the same goal:• Every pass/reject decision is guaranteed with and• The classifier is a cascade.• Score is propagated from one node to another.

• Main advantages:• Weak classifiers are learned (approximately) optimally.• No training of multiple boosted classifiers.• Much fewer weak classifiers are needed than traditional cascades.

0FAR .0FRR

F2F1 F3

ResultsGoal () vs. Number of weak classifiers (K)

• Toy problem: To learn a (single-exit) boosted classifier F for classifying face/non-face patches such that FAR(F) < 0.8 and FRR(F) < 0.01– Best goal:

– Ours chooses:

• Similar results were obtained for tests on other desired error rates.

.8001.08.0

].100,10[

Ours vs. Others (in Face Detection)

• Use Fast StatBoost as base method for fast-training a weak classifier.

Method No of weak

classifiers

No of exit

nodes

Total training

time

Viola Jones [3] 4,297 32 6h20m

Viola Jones [4] 3,502 29 4h30m

Boosting chain [7] 959 22 2h10m

Nested cascade [5] 894 20 2h

Soft cascade [1] 4,871 4,871 6h40m

Dynamic cascade [6] 1,172 1,172 2h50m

Multi-exit Asymmetric Boosting

575 24 1h20m

Ours vs. Others (in Face Detection)• MIT+CMU Frontal Face Test set:

Conclusion

• Multi-exit Asymmetric Boosting trains every weak classifier approximately optimally.

– Better accuracy

– Much fewer weak classifiers

– Significantly reduces training time• No more trial-and-error for training a boosted classifier

42

Margin-based Bounds on an Asymmetric Error of a Classifier

CVPR’08 poster paper:Minh-Tri Pham and Viet-Dung D. Hoang and Tat-Jen Cham. Detection with Multi-exit Asymmetric Boosting. In Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), Anchorage, Alaska, 2008.

• Won Travel Grant Award

Motivation

• A number of cost-sensitive learning methods have been proposed to deal with binary classification with an imbalanced dataset:– Cost-sensitive decision trees (Knoll et. al. ‘94)– Cost-sensitive neural networks (Kuka and Kononenko ‘98)– Imbalanced SVM (Veropoulos et. al. ‘99)– Asymmetric Boosting (Karakoulas and Taylor ‘99, Viola and Jones ‘02)

• Their objective function has the same form of an asymmetric error:

– where is the prediction input xare given

• Bounds on the generalization error of a classifier exist, but bounds on an asymmetric error have not been proposed yet.

1|0)(1|0)(minarg 21

yxfPyxfPfFf

21,))((

xfsign

FRR FAR

Why bounding an Asymmetric Error?

• Generalization error is a special case of an asymmetric error.– Consider , we get:

• It helps to explain how the classifier performs on unknown data in problems with imbalanced prior probabilities.

1|0)(1|0)(minarg 21

yxfPyxfPfFf

)1(),1( 21 yPyP

yxfsignPxyfPfFfFf

))((minarg0)(minarg

This work’s contribution...• To give bounds on an asymmetric error of a binary classifier:

Summary

• Online Asymmetric Boosting– Integrates Asymmetric Boosting with Online Learning

• Fast Training and Selection of Haar-like Features using Statistics– Dramatically reduce training time from weeks to a few hours

• Multi-exit Asymmetric Boosting– Approximately minimizes the number of weak classifiers

Thank You!

Backup Slides

Online Asymmetric Boosting

CVPR’07 oral paper:Minh-Tri Pham and Tat-Jen Cham. Online Learning Asymmetric Boosted Classifiers for Object Detection. In Proc. IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), Minneapolis, MN, 2007.

AdaBoost (Freund-Schapire’96)

Offline Weak

Learner1

Offline Weak

Learner2

Wrongly classified

Wrongly classified




Stage 1 Stage 2

Asymmetric Boost (Viola-Jones’02)

Offline Weak

Learner1

Offline Weak

Learner2


Stage 1 Stage 2

• To address P(non-object) >> P(object):• Weight positives k times more than negatives

Online Boosting (Oza-Rusell’01)

Online Weak

Learner1

Online Weak

Learner2



• To learn data online:• If wrongly classified: increase weight; otherwise : decrease weight

Wrongly classified

Asymmetric Online Boosting

OnlineWeak

Learner2

OnlineWeak

Learner1


• Incorporate the asymmetric reweighting scheme into Online Boosting• Increase positive weights, decrease negative weights

Wrongly classified


• Learning time:– About 5-30% faster with Polarity balancing

Polarity balancing

Online Learning a 20-feature boosted classifier

Experimental Results

ROC curves of boosting with 15 weak classifiers ROC curves of boosting with 200 weak classifiers

56

• Performance of a single boosted classifier:

Experiments: UCI-KDD• Symmetric dataset:

– P(positive) ≈ P(negative)• Ours vs. Online Boosting

• Accuracy performance:– Ours better on under-

complete datasets• Promoters• German Credit• Synthetic 20-dim

– Similar on over-complete datasets

• Learning time:– About 5-40% reduction in

time to achieve the same level of error

Online OurDataset AdaBoost Boosting methodPromoters 0.8455 0.7136 0.7429Breast Cancer 0.9445 0.9573 0.9552German Credit 0.735 0.6879 0.7013Chess 0.9517 0.9476 0.9501Mushroom 0.9966 0.9987 0.9978Cencus Income 0.9365 0.9398 0.9372Synthetic 5-dim 0.9382 0.9049 0.9251Synthetic 20-dim 0.8923 0.7972 0.8404

Learning curve for Mushroom dataset

Accuracy of different datasets

Experiments: Face Detector• Asymmetric dataset:

– P(face) << P(non-face)

• Training set:– BioID, ARFace

• Test set:– MIT+CMU

• Patch size: 24x24• Haar feature pool: 5000• Cascade: 25 boosted

classifiers

Relation with Other MethodsMethod No of face

examplesNo of

featuresTime to train a weak classifier

No of weak classifiers

Total training

time

CPU speed

Viola-Jones ‘01 9,500 40,000 > 10 min 4,297 weeks 400 MHz

Li et. al. ‘02 6,000 - - 2,546 weeks 700 MHz

B. Wu el. al. ‘04 20,000 - - 756 1-2 weeks 1.4GHz

Huang el. al. ‘07 30,000 - < 1 min - 2 days 3.0 GHz

J. Wu el. al. ’07 5,000 40,000 12.4s 3,870 13h 20m 2.8 GHz

Fast StatBoost 5,000 295,920 3.1s 3,502 3h02m 2.8 GHz

59

Fast Training and Selection of Haar-like Features using Statistics

ICCV’07 oral paper:Minh-Tri Pham and Tat-Jen Cham. Fast Training and Selection of Haar Features using Statistics in Boosting-based Face Detection. In Proc. International Conference on on Computer Vision (ICCV), Rio de Janeiro, Brazil, 2007.

How good are Gaussian assumptions?

61

Weak classifier 1’s feature value distributionin an ensemble of 200 weak classifiers


62



63



64


Fast StatBoost• Scalar estimators are reliable, despite being computed from high-

dimensional estimators– Scalar mean estimator :

– Scalar variance estimator :

65

)()()( ˆˆ tc

tTc

tc gm

N

VarVartctT

ctc

2)()()( ˆˆ gm

2)()()(2)( ˆˆ tc

tc

Ttc

tc gg J

1

2ˆˆ4)(

)()(2)(

N

VarVartct

cTt

ctc

gg J

)(ˆ tc

2)(ˆ tc

(Ahn-Fessler ‘03)

Relation with Other Methods• Real-valued weak classifier

– Our feature classifier can produce real values:• Instead of ±1, return:

• Online learning:– Gains little benefit from this technique due to the same time complexity:

• O(T) per update of one example

• Application to features beyond Haar-like:– Only requirement: a feature value is a linear projection of the input.

• Joint Haar-like features (Li et. al. ‘02)• Extended Haar-like features (Lienhart et. al. ‘03)• Sparse Granular (Huang et. al. ’07)

1|

1|log5.0 )(

)(

cvPcvP

t

t

66

Face Detectors today• Typical approach for

object detection (e.g. Viola-Jones face detector)– Scan image with probe

window (x,y,s) at different positions and scales

– Binary classify each window patch into• face, or• non-face

input space

non-face

face0

1

67

Face Detectors today• Cascade (or tree) of boosted classifiers with increasing

complexities:

• Boosted classifier:

– Parameters:• fq,m(x) = weak classifier, selected from a set of feature classifiers• aq,m = voting coefficient

– Discrete AdaBoost: • strongly resistant to overfitting (Rudin-Schapire-Daubechies ’03-’07)

F1 F2 FQ….passpasspass pass


face

non-face

rejectpass

xfasignxFqM

mmqmqq :1

:1)()(

1,,

68

Face Detectors today• Haar-like feature classifier:

o

Feature value v(t) = Haar-like feature H(t) o Image window I

– v(t) can be computed extremely fast using image integrals

– {H(t)} is generated by• scaling and translation• of a few feature types

– Classify window by thresholding v(t) : fq,m(x) = pt sign(v(t) > t)– t : threshold value– : parity bit

y x

xytxy

tv ,)(,

)( IH

Four feature types used by Viola-Jones

1,1 tp

69

Our Approach• To train feature classifiers using statistics:

– Assumption: feature value v(t) is Gaussian-distributed given face class c is known – Solve:

– Non-convex, but a close-form solution exists (Duda el. al. ’02)– How to estimate the class-conditional statistics ?

)(1

)(1

)(1

)(1 ,,, tttt

)(tv

pcvPpcvPp tt

ptt

)()(

,minarg,

)()()( ,| tp

tp

tvgpcP

)()()( ,| tp

tp

tvgpcP

70

Our Approach• Linear relationships among the variables:

– Denote by a vector consisting of all the elements of matrix A arranged in a pre-defined order.

v(t) is a linear projection of image window I with direction of projection H(t) :

Image integral J is a linear transform of I:• By definition (Viola-Jones ‘01):

• Hence, , where B is a non-singular transformation matrix.

v(t) is a linear projection of image integral J, too:

• where is a sparse vector with very few non-zero elements (Viola-Jones ‘01)

IHIH

Tt

y xxy

txy

tv )(,

)(,

)(

y

y

x

xxyxy

1' 1'',', IJ

A

IBJ

JgJBHIH

TtTtTttv )(1)()()(

)(1)( tTt HBg

71

Our Approach• Applying the sparsity of g(t) to statistics:

– Given , computing is extremely fast due to the sparsity of g(t).

– Computing the statistics of v(t) :

• where and are the mean vector and the covariance matrix of image integral If ’s statistics are known, computing and is extremely fast as well

Same for class-conditional statistics• How to estimate the class-conditional statistics of ?

Jg

Tttv )()(

JmgJg

TtTttt v )()()()(

)()()()(2)(2)(2)( tTttTTTtttt vv gggJJJJg J

Jm J J

J

J )(t )(t

)(1

)(1

)(1

)(1 ,,, tttt

J

72

Our Approach• Estimation of the class-conditional image integral statistics

– Information given at the m-th weak classifier:• N input windows and their corresponding classes, • weighted by vector w(m):

• Let for all n from 1 to N.

– Direct estimation from the current input data:

NNmN

mm ccc ,,,...,,,,,, )(22

)(2

)(1 IwIwIw 11

nn IBJ

Tcc

ccn

Tnn

mncc

ccnn

mncc

ccn

mnc

n

n

n

wz

wz

wz

mmJJ

Jm

ˆˆˆˆ

ˆˆ

ˆ

:

)(1

:

)(1

:

)(

73

asymmetric boosting for face detection

Documents