Research Article A Modified Adaboost Algorithm to Reduce ...downloads.hindawi.com/journals/mpe/2016/5289413.pdf · 3. Probabilistic Weighting Adjusted Adaboost e main objective of

Research ArticleA Modified Adaboost Algorithm to Reduce False Positives inFace Detection

Cesar Niyomugabo Hyo-rim Choi and Tae Yong Kim

GSAIM Chung-Ang University Seoul Republic of Korea

Correspondence should be addressed to Tae Yong Kim kimtycauackr

Received 3 June 2016 Accepted 7 September 2016

Academic Editor Jinyang Liang

Copyright copy 2016 Cesar Niyomugabo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present amodified Adaboost algorithm in face detection which aims at an accurate algorithm to reduce false-positive detectionrates We built a new Adaboost weighting system that considers the total error of weak classifiers and classification probability Theprobability was determined by computing both positive and negative classification errors for eachweak classifierThenewweightingsystem gives higher weights to weak classifiers with the best positive classifications which reduces false positives during detectionExperimental results reveal that the original Adaboost and the proposedmethod have comparable face detection rate performancesand the false-positive results were reduced almost four times using the proposed method

1 Introduction

Face detection is a computer technology that determines thelocation size and posture of the face in a given image or videosequence [1] Face detection is an active research area in thecomputer vision community locating a human face in animage plays a key role in applications like face recognitionvideo surveillance human computer interfaces databasemanagement and querying image databases [2 3] The mostsuccessful face detectionmethod was developed by Viola andJones based on theAdaboost learning algorithm [4] Althoughtheir method was successful in face detection it faces falsealarm challenges which may increase in the presence of acomplex background False positives in an application canbe a source of errors and need additional postprocessing toremove them As our contribution to reduce the numberof false positives we propose a probabilistic approach tomodify the weighting system of the Adaboost algorithmwhich includes the expansion of key ideas and supportingexperimental results over the preliminary version [5]

2 Face Detection and Adaboost

A number of promising face detection algorithms have beenpublished Among these the Adaboost method stands out

because it is often referred to by other face detection studiesIn this sectionwe present the outline andmain points of someof face detection algorithms

21 Face Detection Madhuranath developed the ldquomodifiedAdaboost for face detectionrdquo In their method multiple strongclassifiers based on different Haar-like types trained onthe same set of input images are combined into a singlemodified-strong classifier [6] Viola and Jones [4] presentedthe fundamentals of their face detection algorithm Thisalgorithm only detects frontal upright faces however amodified algorithmwas presented in 2003 that detects profileand rotated views [7] In ldquoFace Detection Using a NeuralNetworkrdquo [8] the authors computed an image pyramid todetect faces at multiple scales A fixed size subwindow wassubjected to each image in the pyramid The content ofthe subwindow is corrected for nonuniform lightening andsubjected to histogram equalization The processed contentis passed to parallel neural networks that carry out the actualface detection The outputs are combined using logical ANDto reduce the amount false detection rate In its first form thisalgorithm also only detects frontal upright faces

Schneiderman and Kanade [9] calculated an image pyra-mid and fixed size subwindow scans through each layer ofthis pyramid The content of the subwindow was wavelet

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5289413 6 pageshttpdxdoiorg10115520165289413

2 Mathematical Problems in Engineering

analyzed and histograms were prepared for the differentwavelet coefficients These coefficients were fed to differenttrained parallel detectors sensitive to various orientations ofthe object The orientation of the object is determined bythe detector that yields the highest output In contrast to thebasic ViolandashJones algorithm this algorithm detects profileviews AL-Allaf reviewed face detection studies based ondifferent ANN approaches [10] whereas C S Patil and AJ Patil combined skin color information and Support VectorMachine to detect faces [11]

One of the fundamental problems with real-time objectdetection is that the size and position of a given object withinan image are unknown An image pyramid was computed toovercome this obstacle and a detector scans each image inthe pyramid However this process is rather time consumingthus Viola and Jones presented a new approach based onAdaboost algorithm to solve the problem However one ofthe disadvantages was a high false-positive rate

22 Adaboost Learning Algorithm First introduced by Fre-und and Schapire [12] the Adaboost algorithm is shortfor Adaptive Boosting This algorithm strengthens overallperformancewhenusedwith otherweak learning algorithmsA weak learning algorithm consists of a learning algorithmthat classifies the input data better than random

The Adaboost algorithm is adaptive in that misclassifieddata from previous classifiers are boosted during trainingby assigning them higher weight than that of the correctlyclassified data The training database is input data set andassociated classification labels Adaboost repeatedly calls aweak learning algorithm over the training data setMost opti-mal parameters of weak learning algorithms are computed ateach stage which minimizes the classification error A weaklearning classifier with optimal parameters at a given trainingstage is called a best weak classifier [4] The input dataset is initially weighted equally however the weak learningalgorithm puts emphasis on the misclassified data more thanthe correctly classified data during the training process Thisis accomplished by raising the weights of the misclassifieddata during each stage with respect to the correctly classifieddata The main steps for the Adaboost algorithm to classifydata efficiently are presented in the following

Pseudocode for the Adaboost Learning Algorithm(1) Given data and corresponding labels (119909

1 1199101)

(119909119899 119910119899) where 119909 is the input data and 119910 = 1 0 are

the labels for positive and negative examples with thesize of the input data being 119899

(2) Initialize weights 119908 of the input data equally 1199081119894=

12119899 where 119894 = 1 119899 and 119894 is the 119894th index of thedata

(3) For 119905 = 1 119879 where 119879 is the number of stages oftraining

(31) normalize the weights 119908 of the 119905th stage

119908119905119894=

119908119905119894

sum119895=119899

119895=1119908119905119895

(1)

(32) select the best weak classifier ℎ119905() in terms of

parameters of the 119905th stage 119903119905which minimizes

the classification error between the weak classi-fier output ℎ

119905(119909119894 119903119905) over the 119894th index of the data

and the corresponding label 119910119894 over all indices

119868 of the data 119868 = 1 119899

120576119905=

119894=119899

sum119894=1

119908119905119894

1003816100381610038161003816ℎ119905 (119909119894 119903119905) minus 1199101198941003816100381610038161003816 (2)

(33) compute the 119905th stage exponent 120573119905

120573119905=120576119905

1 minus 120576119905

(3)

a weak classifier ℎ() which classifies the inputdata better than random will result in 120576

119905which

is less than 05 thus 120573119905will be less than 10

(34) classify the 119894th index of the data119909119894with thisweak

classifier ℎ119905() compare with the actual label 119910

119894

and store the error in classification 119890119894over all

indices 119868 = 1 119899 of the data

119890119894=

0 if ℎ119905(119909119894 119903119905) = 119910119894

1 if ℎ119905(119909119894 119903119905) = 119910119894

(4)

(35) update the weights 119908 of the input data duringthis 119905th stage using the exponents 120573

119905computed

in step (33) Since theweights are being normal-ized in step (31) the weights of the incorrectlyclassified data are boosted this is the basic ideabehind Adaboost

119908119905+1119894= 1199081199051198941205731minus119890119894

119905 (5)

(4) The final strong classifier 119862(119909) is the weightedmajority of the individual weak classifiers chosen instep (32) of each stage 119905

119862 (119909) =

1 if119879

sum119905=1

120572119905ℎ119905(119909 119903119905) ge1

2

119879

sum119905=1

120572119905

0 otherwise(6)

where 119862(119909) is the strong classifier 120572119905= log(1120573

119905) is

the weight ℎ119905(119909 119903119905) is the weak classifier of the 119905th

stage 119909 is the input data 119903119905are the parameters and

ℎ119905() is the Haar-like feature type

Freund and Schapire [12] showed that the training erroron the final hypothesis is upper bounded and if the individualweak hypothesis classifies the input data better than randomthe training error decreased exponentially with an increasein the number of stages The generalization property ofAdaboost is a gradient-descent method in the space of weakclassifiers as shown by Schapire and Singer [13]

Mathematical Problems in Engineering 3

23 Face Classification Using Adaboost Faces and nonfaceswith their corresponding labels become the input data setfor the Adaboost algorithm Each Haar-like feature ℎ() isevaluated as the sum of the pixels in the image correspondingto the white portion subtracted from the sum of the pixelsin the image corresponding to the black portion A weakclassifier has been correctly classified if the value of theHaar-like feature at a particular location (119909 119910) on the image isgreater than the threshold 120579 and the polarity 119901 determinesthe sign of this inequality in

119901 lowast ℎ (119909 119910) gt 119901 lowast 120579 (7)

Classification of the weak classifier depends on the evalu-ation of the Haar-like feature ℎ() on the image 119867 at thelocation (119909 119910) the threshold 120579 and polarity 119901 Image 119867location (119909 119910) polarity 119901 and threshold 120579 are the weakclassifier parameters and the output of the weak classifier isrepresented as ℎ(119867 119909 119910 119901 120579) Thus the data in the Adaboostlearning algorithm is image119867 and the parameters become thelocation polarity and threshold (119909 119910 119901 and 120579) During thetraining stages of a single weak classifier theHaar-like featureℎ() is evaluated at each location (119909 119910) across all 119899 images119867119894 119894 = 1 119899 The weighted sum of the error between the

correct and the actual classification |ℎ119905(119867119894 119909 119910 119901 120579) minus 119910

119894| for

all location (119909 119910) is computedThe location (119909119905 119910119905) threshold

120579119905 and polarity 119901

119905of the Haar-like feature ℎ

119905() of the 119905th

stage which minimizes the weighted error are chosen as theweak classifier parameters ℎ

119905(119867119894 119909119905 119910119905 119901119905 120579119905)Thisminimum

weighted error for the 119905th stage represented as 120576119905 is shown in

120576 = min119909119910119901120579

119894=119899

sum119894=1

119908119894

1003816100381610038161003816ℎ119905 (119867119894 119909119894 119910119894 119901119894 120579119894) minus 1199101198941003816100381610038161003816 (8)

The exponent 120573119905used for updating the weights is computed

based on this weighted error as shown in

120573119905=120576119905

1 minus 120576119905

(9)

Typically 120573119905will be less than unity The classification of the

image119867119894based on the optimal weak classifier ℎ

119905(119867119894 119909119905 119910119905 120579119905)

is performed as shown in

119890119894=

0 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

1 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

(10)

The classification error 119890119894is used to update the weights of the

(119905 + 1)th stage 119882119905+1119894

based on the weights of the 119905th stageand 119908

119905119894according to the Adaboost method of updating the

weights as shown in

119908119905+1119894= 1199081199051198941205731minus119890119894

119905 (11)

As 120573119905is less than unity the correctly classified images are

weighted lower than the misclassified images The process ofnormalizing the weights in the next training stage results inlower weights of the misclassified images than those of thepresent stage When all training stages 119879 are complete the

final strong classifier is the weighted majority of the optimalweak classifiers ℎ

119905(119867119894 119909119905 119910119905 119901119905 120579119905) of each stage as shown in

119867(119909) =

1 if119879

sum119905=1

120572119905ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) ge1

2

119879

sum119905=1

119886119905

0 otherwise(12)


119905) is the

weight ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) is the weak classifier of 119905th stage

120579119905is the threshold and ℎ

119905() is the Haar-like feature type

3 Probabilistic Weighting Adjusted Adaboost

The main objective of our proposed method is to reduce thenumber of false-positive results We want to reduce the num-ber of regions in the image that are classified falsely as facesTherefore our contribution changes the weighting systemof the original Adaboost algorithm based on a probabilisticapproach [14]

31 Best Weak Classifier Selection As in the ViolandashJonesmethod we used training data made of positive (croppedface) and negative (random images without faces) images andused Haar-like features to build the full dictionary of weakclassifiers Note that weight was normalized in step (31) ofthe pseudocode for the Adaboost learning algorithm whichmakes the total weighted error sum to 1 As in the ViolandashJones method a weak classifier is selected as the ldquobest weakclassifierrdquo once its total weighted error 120576

119905is less than 05 [4]

In the proposed procedure a weak classifier was classified asthe best one when the weighted positive error was less than005 to keep the positive detection rate at about 95

For a given pattern 119909119894each best weak classifier ℎ

119895provides

ℎ119895(119909119894) isin 1 0 and the final decision of the committee 119867 of

selected best weak classifiers is 119867(119909119894) which can be written

as the weighted sum of the decision of the best weak classifieras follows

119867(119909119894) = 1205721ℎ1(1199091) + 1205722ℎ2(1199092) + sdot sdot sdot + 120572

119898ℎ119898(119909119894) (13)

where ℎ1 ℎ2 ℎ

119898denote 119898 best weak classifiers selected

from the pool The constants 1205721 1205722 120572

119898are weights

assigned to each classifier decision in the committee Recallthat every ℎ

119895just answers ldquoyesrdquo (1) or ldquonordquo (0) to a classi-

fication problem and the result is a linear combination ofclassifiers followed by a nonlinear decision (sign function)

In the ViolandashJones method the weight given to each bestclassifier only depends on the total error that each classifiercommitted in a training set as shown in (9) In the originalAdaboost method if two best weak classifiers have the sameerror their opinions are given the sameweight nomatter howdifferent their probabilities of classifying positive or negativeimages may be We introduce a new weighting system theweight given to the opinion (ldquoyesrdquo or ldquonordquo) of the best weakclassifier considers the ability of the best weak classifier toclassify positive images on one side as well as negative imagesTherefore this information will be very useful to build asystem that reduces the false-positive rate by giving moreweight to the best weak classifier with a high probability ofclassifying the positive images correctly


32 BestWeak Classifier Classification Probability andWeight-ing A weak classifier from the pool is voted to be the bestweak classifier once once it classifies the input data betterthan random [4] Now let us consider that the output froma given best weak classifier is made for a given number ofwellclassified images false-positive (negative images classified aspositive) and false-negative images (positive images classifiedas negative) The ldquofalse-positive error probabilityrdquo and theldquofalse-negative error probabilityrdquo can be computed as follows[14]

119875FP =119864FP119864Total

119875FN =119864FN119864Total

(14)

where 119864Total is total error 119864FP is the false-positive error119864FN is the false-negative error 119875FP is the false-positive errorprobability and 119875FN is the false-negative error probability

After calculating the probabilities we used them tobuild a new weighting system that considers both errorand probabilities for weighting each classifierrsquos opinion Infact two classifiers with the same error but with differentclassification probabilities will have different weights becausethe probability is considered while assigning weights to theclassifiers The weight in the original method (12) can berewritten as

120572 = log(1 minus 120576119905

120576119905

) (15)

where 120576119905is the total error of a considered classifier at stage

119905 The new alpha called ldquoprobabilistic alphardquo is computed asfollows considering the previously calculated probabilities

1205721015840= [log(

1 minus 120576119905

120576119905

)] times (1 minus 119875FN119875FP

) (16)

The proposed alpha is inversely proportional to the false-positive error probability thus when a given classifier has ahigh false-positive error rate its weight is lowered otherwisethe inverse is true Once a best weak classifier produces high119875FP it is given a relatively small weight which produces astrong classifier that reduces the number of false positivesNote also that when a classifier produces an increasing 119875FNthe value of the numerator of the term after themultiplicationsign in (16) decreases and this also lowers the weight of theproposed alpha The probabilistic alpha on updating weightallows greater update frommisclassified images compared tothe original method This is because the proposed alpha willalways be greater than the original alpha according to (16)Hence misclassified images are relatively highly weightedusing the proposed method therefore accuracy is higher forthe best weak classifiers The following shows the pseudocodefor the proposed algorithm

Pseudocode for the Proposed Algorithm

(1) Images (1199091 1199101) (119909

119899 119910119899) where 119910 = 1 0 for face

and nonface images are given

(2) Initialize weights 1199081119894= 1119898 1119897 where 119898 and 119897

are the numbers of positive and negative imagesrespectively

(3) For 119905 = 1 119879

(31) normalize the weights w of the 119905th stage

119908119905119894=

119908119905119894

sum119895=119899

119895=1119908119905119895

(17)

(32) select the best weak classifier ℎ119905()with respect to

the weighted error

119864Total = 120576119905 = min119894=119899

sum119894=1

119908119905119894119888119894 (18)

where

119888119894=

0 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

1 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

(19)

(33) compute false-positive error over all indices 119894 offace images 119894 = 1 119875

119864FP =119894=119875

sum119894=1

119908119905119894119888119894 (20)

(34) compute false-negative error over all indices 119894 ofnonface images 119894 = 1 119873

119864FN =119894=119873

sum119894=1

119908119905119894119888119894 (21)

(35) compute weak classifier classification probabili-ties

119875FP =119864FP119864Total

119875FN =119864FN119864Total

(22)

(36) compute alpha 1205721015840119905

1205721015840

119905= [log(

1 minus 120576119905

120576119905


) (23)

(37) update the weights

119908(119898+1)

119894= 119908119898

119894119890(1205721015840

119905119888119894) (24)

(4) The final strong classifier119867(119909) is

119867(119909)

=

1 if119879

sum119905=1

1205721015840119905ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) ge1

2

119879

sum119905=1

1205721015840

119905

0 otherwise

(25)

where 1205721015840119905is the probabilistic weight and ℎ

119905() is the

weak classifier of the 119905th stage


Table 1 Comparison of the ViolandashJones and the proposed algorithm the number of false-positive results is almost four times lower usingthe proposed algorithm

True positive(detecting faces) False positive False negative

(missing faces)Precision

TP(TP + FP)ViolandashJones algorithm 220 495 20 308Proposed algorithm 216 115 24 653

(a) (b)

Figure 1 Detected faces by the ViolandashJones method (a) and the proposed method (b)

4 Experimental ClassificationResults and Analysis

We trained and tested data sets such as the CMUMITface data set for training and Georgia Tech face databasemade of 50 distinct persons (15 images per subject) fortestingTheGeorgia Tech imageswere taken at different timeswith different lighting facial expressions (openclosed eyessmilingnot smiling) and facial details (glassesno glasses)All images were taken against a complex background withthe subjects in an upright and frontal position (with tolerancefor some side movement) We evaluate the performance ofthe algorithms using the precision The precision for a classis the number of true positives divided by the sum of truepositives and false positives which are items incorrectlylabeled as belonging to the classHigh precisionmeans that analgorithm returned substantially more relevant results thanirrelevant

Experimental results obtained from the ViolandashJones andthe proposed probabilistic method are described in Table 1The number of false-positive results is reduced about fourtimes using the proposed method and the precision of theproposed algorithm is twice as high as that of ViolandashJonesalgorithm without much deteriorating the accuracy Theprocessing speed is around 30 fps for both methods

The faces detected by the ViolandashJones method and theproposed method are shown in Figure 1 Figure 1(a) showsa false-positive result and Figure 1(b) shows the same facedetected without the false-positive by the proposed proba-bilistic weighting

5 Conclusion

The challenge in face detection using Adaboost is the numberof false-positive results that accompany actual faces detectedon a complex background In this study we presenteda probabilistic weighting adjusted Adaboost algorithm fordetecting faces in a complex background that reduced false-positive errors We want to reduce the number of regions inan image that are classified as faces but that are not facesTherefore we propose a modified version of the Adaboostalgorithm that classified weak classifier probabilities forweighting the decisions of each best weak classifier Classifierswith the same total error have the same weight in the originalAdaboost algorithm However classifiers with the sameerror in our proposed probabilistic approach have differentprobabilities and different weights In this new weightingsystem the classifierrsquos weight is inversely proportional to thefalse-positive error probability thus when a given classifierhas a high probability of false-positive errors its weightis decreased Experimental results reveal that the proposedalgorithm reduces the number of false positives almost fourtimes compared to that of the original Adaboost

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research was supported by the Basic Science ResearchProgram through the National Research Foundation of


Korea (NRF) funded by the Ministry of Education Scienceand Technology (NRF-2015R1D1A1A01058394) and by theChung-Ang University Research Grants in 2016

References

[1] C Zhipeng H Junda and Z Wenbin ldquoFace detection systembased on skin colorrdquo in Proceedings of the International Confer-ence on Networking and Digital Society May 2010

[2] S K Singh D S Chauhan M Vasta and R Singh ldquoA robustcolor based face detection algorithmrdquo Tamkang Journal ofScience and Engineering vol 6 pp 227ndash234 2003

[3] R Rai D Katol and N Rai ldquoSurvey paper on vehicle theftdetection through face recognition systemrdquo International Jour-nal of Emerging Trends amp Technology in Computer Science vol3 no 1 pp 256ndash258 2014

[4] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[5] C Niyomugabo and T Kim ldquoA probabilistic approach toadjust Adaboost weightsrdquo in Proceedings of the SAI ComputingConference (SAI rsquo16) pp 1357ndash1360 London UK July 2016

[6] HMadhuranath T R Babu and S V Subrahmanya ldquoModifiedadaboost method for efficient face detectionrdquo in Proceedings ofthe 12th International Conference on Hybrid Intelligent Systems(HIS rsquo12) IEEE Pune India 2012

[7] P Viola and M J Jones ldquoFast multi-view face detectionrdquoMitsubish Electronic Research Laboratories TR2003-096 2003

[8] H A Rowley S Baluja and T Kanade ldquoNeural network-based face detectionrdquo IEEETransactions on PatternAnalysis andMachine Intelligence vol 20 no 1 pp 23ndash38 1998

[9] H Schneiderman and T Kanade ldquoA statistical method for 3Dobject detection applied to faces and carsrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo00) Bangalore India 2000

[10] O N AL-Allaf ldquoReview of face detection systems based arti-ficial neural networks algorithmsrdquo The International journal ofMultimedia amp Its Applications vol 6 no 1 pp 1ndash16 2014

[11] C S Patil and A J Patil ldquoA review paper on facial detectiontechnique using pixel and color segmentationrdquo InternationalJournal of Computer Applications vol 62 no 1 pp 21ndash24 2013

[12] Y Freund and R E Schapire ldquoA decision-theoretic generaliza-tion of on-line learning and an application to boostingrdquo Journalof Computer and System Sciences vol 55 no 1 pp 119ndash139 1997

[13] R E Schapire and Y Singer ldquoImproved boosting algorithmusing confidence-rated predictionrdquo in Proceeding of the 11thAnnual Conference on Computation LearningTheory pp 80ndash91Madison Wis USA 1998

[14] R V Hogg and E A Tanis Probability and Statistical InferencePrentice Hall 2006

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analyzed and histograms were prepared for the differentwavelet coefficients These coefficients were fed to differenttrained parallel detectors sensitive to various orientations ofthe object The orientation of the object is determined bythe detector that yields the highest output In contrast to thebasic ViolandashJones algorithm this algorithm detects profileviews AL-Allaf reviewed face detection studies based ondifferent ANN approaches [10] whereas C S Patil and AJ Patil combined skin color information and Support VectorMachine to detect faces [11]

One of the fundamental problems with real-time objectdetection is that the size and position of a given object withinan image are unknown An image pyramid was computed toovercome this obstacle and a detector scans each image inthe pyramid However this process is rather time consumingthus Viola and Jones presented a new approach based onAdaboost algorithm to solve the problem However one ofthe disadvantages was a high false-positive rate

22 Adaboost Learning Algorithm First introduced by Fre-und and Schapire [12] the Adaboost algorithm is shortfor Adaptive Boosting This algorithm strengthens overallperformancewhenusedwith otherweak learning algorithmsA weak learning algorithm consists of a learning algorithmthat classifies the input data better than random

The Adaboost algorithm is adaptive in that misclassifieddata from previous classifiers are boosted during trainingby assigning them higher weight than that of the correctlyclassified data The training database is input data set andassociated classification labels Adaboost repeatedly calls aweak learning algorithm over the training data setMost opti-mal parameters of weak learning algorithms are computed ateach stage which minimizes the classification error A weaklearning classifier with optimal parameters at a given trainingstage is called a best weak classifier [4] The input dataset is initially weighted equally however the weak learningalgorithm puts emphasis on the misclassified data more thanthe correctly classified data during the training process Thisis accomplished by raising the weights of the misclassifieddata during each stage with respect to the correctly classifieddata The main steps for the Adaboost algorithm to classifydata efficiently are presented in the following

Pseudocode for the Adaboost Learning Algorithm(1) Given data and corresponding labels (119909

1 1199101)

(119909119899 119910119899) where 119909 is the input data and 119910 = 1 0 are

the labels for positive and negative examples with thesize of the input data being 119899

(2) Initialize weights 119908 of the input data equally 1199081119894=

12119899 where 119894 = 1 119899 and 119894 is the 119894th index of thedata

(3) For 119905 = 1 119879 where 119879 is the number of stages oftraining

(31) normalize the weights 119908 of the 119905th stage

119908119905119894=

119908119905119894

sum119895=119899

119895=1119908119905119895

(1)

(32) select the best weak classifier ℎ119905() in terms of

parameters of the 119905th stage 119903119905which minimizes

the classification error between the weak classi-fier output ℎ

119905(119909119894 119903119905) over the 119894th index of the data

and the corresponding label 119910119894 over all indices

119868 of the data 119868 = 1 119899

120576119905=

119894=119899

sum119894=1

119908119905119894

1003816100381610038161003816ℎ119905 (119909119894 119903119905) minus 1199101198941003816100381610038161003816 (2)

(33) compute the 119905th stage exponent 120573119905

120573119905=120576119905

1 minus 120576119905

(3)

a weak classifier ℎ() which classifies the inputdata better than random will result in 120576

119905which

is less than 05 thus 120573119905will be less than 10

(34) classify the 119894th index of the data119909119894with thisweak

classifier ℎ119905() compare with the actual label 119910

119894

and store the error in classification 119890119894over all

indices 119868 = 1 119899 of the data

119890119894=

0 if ℎ119905(119909119894 119903119905) = 119910119894

1 if ℎ119905(119909119894 119903119905) = 119910119894

(4)

(35) update the weights 119908 of the input data duringthis 119905th stage using the exponents 120573

119905computed

in step (33) Since theweights are being normal-ized in step (31) the weights of the incorrectlyclassified data are boosted this is the basic ideabehind Adaboost

119908119905+1119894= 1199081199051198941205731minus119890119894

119905 (5)

(4) The final strong classifier 119862(119909) is the weightedmajority of the individual weak classifiers chosen instep (32) of each stage 119905

119862 (119909) =

1 if119879

sum119905=1

120572119905ℎ119905(119909 119903119905) ge1

2

119879

sum119905=1

120572119905

0 otherwise(6)


119905) is

the weight ℎ119905(119909 119903119905) is the weak classifier of the 119905th

stage 119909 is the input data 119903119905are the parameters and

ℎ119905() is the Haar-like feature type

Freund and Schapire [12] showed that the training erroron the final hypothesis is upper bounded and if the individualweak hypothesis classifies the input data better than randomthe training error decreased exponentially with an increasein the number of stages The generalization property ofAdaboost is a gradient-descent method in the space of weakclassifiers as shown by Schapire and Singer [13]






119894| for


120579119905 and polarity 119901


119905() of the 119905th


119905(119867119894 119909119905 119910119905 119901119905 120579119905)Thisminimum


120576 = min119909119910119901120579

119894=119899

sum119894=1

119908119894

1003816100381610038161003816ℎ119905 (119867119894 119909119894 119910119894 119901119894 120579119894) minus 1199101198941003816100381610038161003816 (8)



120573119905=120576119905

1 minus 120576119905

(9)



119905(119867119894 119909119905 119910119905 120579119905)


119890119894=

0 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

1 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

(10)


(119905 + 1)th stage 119882119905+1119894



weights as shown in

119908119905+1119894= 1199081199051198941205731minus119890119894

119905 (11)





119867(119909) =

1 if119879

sum119905=1

120572119905ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) ge1

2

119879

sum119905=1

119886119905

0 otherwise(12)


119905) is the










119895provides




119867(119909119894) = 1205721ℎ1(1199091) + 1205722ℎ2(1199092) + sdot sdot sdot + 120572

119898ℎ119898(119909119894) (13)

where ℎ1 ℎ2 ℎ



119898are weights







119875FP =119864FP119864Total

119875FN =119864FN119864Total

(14)



120572 = log(1 minus 120576119905

120576119905

) (15)



1205721015840= [log(

1 minus 120576119905

120576119905


) (16)



(1) Images (1199091 1199101) (119909

119899 119910119899) where 119910 = 1 0 for face




(3) For 119905 = 1 119879


119908119905119894=

119908119905119894

sum119895=119899

119895=1119908119905119895

(17)


the weighted error

119864Total = 120576119905 = min119894=119899

sum119894=1

119908119905119894119888119894 (18)

where

119888119894=

0 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

1 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

(19)


119864FP =119894=119875

sum119894=1

119908119905119894119888119894 (20)


119864FN =119894=119873

sum119894=1

119908119905119894119888119894 (21)


119875FP =119864FP119864Total

119875FN =119864FN119864Total

(22)

(36) compute alpha 1205721015840119905

1205721015840

119905= [log(

1 minus 120576119905

120576119905


) (23)


119908(119898+1)

119894= 119908119898

119894119890(1205721015840

119905119888119894) (24)


119867(119909)

=

1 if119879

sum119905=1

1205721015840119905ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) ge1

2

119879

sum119905=1

1205721015840

119905

0 otherwise

(25)


119905() is the







(a) (b)






5 Conclusion


Competing Interests


Acknowledgments




References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894| for


120579119905 and polarity 119901


119905() of the 119905th


119905(119867119894 119909119905 119910119905 119901119905 120579119905)Thisminimum


120576 = min119909119910119901120579

119894=119899

sum119894=1

119908119894

1003816100381610038161003816ℎ119905 (119867119894 119909119894 119910119894 119901119894 120579119894) minus 1199101198941003816100381610038161003816 (8)



120573119905=120576119905

1 minus 120576119905

(9)



119905(119867119894 119909119905 119910119905 120579119905)


119890119894=

0 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

1 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

(10)


(119905 + 1)th stage 119882119905+1119894



weights as shown in

119908119905+1119894= 1199081199051198941205731minus119890119894

119905 (11)





119867(119909) =

1 if119879

sum119905=1

120572119905ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) ge1

2

119879

sum119905=1

119886119905

0 otherwise(12)


119905) is the










119895provides




119867(119909119894) = 1205721ℎ1(1199091) + 1205722ℎ2(1199092) + sdot sdot sdot + 120572

119898ℎ119898(119909119894) (13)

where ℎ1 ℎ2 ℎ



119898are weights







119875FP =119864FP119864Total

119875FN =119864FN119864Total

(14)



120572 = log(1 minus 120576119905

120576119905

) (15)



1205721015840= [log(

1 minus 120576119905

120576119905


) (16)



(1) Images (1199091 1199101) (119909

119899 119910119899) where 119910 = 1 0 for face




(3) For 119905 = 1 119879


119908119905119894=

119908119905119894

sum119895=119899

119895=1119908119905119895

(17)


the weighted error

119864Total = 120576119905 = min119894=119899

sum119894=1

119908119905119894119888119894 (18)

where

119888119894=

0 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

1 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

(19)


119864FP =119894=119875

sum119894=1

119908119905119894119888119894 (20)


119864FN =119894=119873

sum119894=1

119908119905119894119888119894 (21)


119875FP =119864FP119864Total

119875FN =119864FN119864Total

(22)

(36) compute alpha 1205721015840119905

1205721015840

119905= [log(

1 minus 120576119905

120576119905


) (23)


119908(119898+1)

119894= 119908119898

119894119890(1205721015840

119905119888119894) (24)


119867(119909)

=

1 if119879

sum119905=1

1205721015840119905ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) ge1

2

119879

sum119905=1

1205721015840

119905

0 otherwise

(25)


119905() is the







(a) (b)






5 Conclusion


Competing Interests


Acknowledgments




References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875FP =119864FP119864Total

119875FN =119864FN119864Total

(14)



120572 = log(1 minus 120576119905

120576119905

) (15)



1205721015840= [log(

1 minus 120576119905

120576119905


) (16)



(1) Images (1199091 1199101) (119909

119899 119910119899) where 119910 = 1 0 for face




(3) For 119905 = 1 119879


119908119905119894=

119908119905119894

sum119895=119899

119895=1119908119905119895

(17)


the weighted error

119864Total = 120576119905 = min119894=119899

sum119894=1

119908119905119894119888119894 (18)

where

119888119894=

0 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

1 if ℎ119905(119867119894 119909119894 119910119894 119901119905 120579119894) = 119910119894

(19)


119864FP =119894=119875

sum119894=1

119908119905119894119888119894 (20)


119864FN =119894=119873

sum119894=1

119908119905119894119888119894 (21)


119875FP =119864FP119864Total

119875FN =119864FN119864Total

(22)

(36) compute alpha 1205721015840119905

1205721015840

119905= [log(

1 minus 120576119905

120576119905


) (23)


119908(119898+1)

119894= 119908119898

119894119890(1205721015840

119905119888119894) (24)


119867(119909)

=

1 if119879

sum119905=1

1205721015840119905ℎ119905(119867119905 119909119905 119910119905 119901119905 120579119905) ge1

2

119879

sum119905=1

1205721015840

119905

0 otherwise

(25)


119905() is the







(a) (b)






5 Conclusion


Competing Interests


Acknowledgments




References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(a) (b)






5 Conclusion


Competing Interests


Acknowledgments




References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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