Research Article Robust Eye Localization by...

Research ArticleRobust Eye Localization by Combining Classification andRegression Methods

Pak Il Nam1 Ri Song Jin1 and Peter Peer2

1 Institute of Mathematics State Academy of Sciences Democratic Peoplersquos Republic of Korea2 Faculty of Computer and Information Science University of Ljubljana Slovenia

Correspondence should be addressed to Pak Il Nam pakilnamyahoocom

Received 24 November 2013 Accepted 4 March 2014 Published 30 March 2014

Academic Editors Y Dimakopoulos and Z Huang

Copyright copy 2014 Pak Il Nam et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Eye localization is an important part in face recognition system because its precision closely affects the performance of the systemIn this paper we analyze the limitations of classification and regression methods and propose a robust and accurate eye localizationmethod combining these twomethodsThe classification method in eye localization is robust but its precision is not so high whilethe regression method is sensitive to the initial position but in case the initial position is near to the eye position it can convergeto the eye position accurately Experiments on BioID and LFW databases show that the proposed method gives very good resultson both low and high quality images

1 Introduction

Because face images should be normalized based on thecoordinates of eyes in most face recognition systems eyelocalization is an important part in face recognition systemsIts precision closely affects the performance of face recogni-tion [1 2] Eye localization methods considering geometricproperties of eyes such as edges shape and probabilisticcharacteristics are high in precision in normal conditionsbut they are sensitive to illumination pose expression andglasses [3ndash6]

State-of-the-art methods in eye localization are basedon boosting classification regression boosting and cascadeboosting and SVM and other variants [1 2 7ndash11] In partic-ular the method in [1] is very effective guaranteeing highprecision even in unconstrained environment It integratesthe following three characteristics

(i) probabilistic cascade(ii) two-level localization framework(iii) extended local binary pattern (ELBP)

In eye localization the boundary between the positiveand negative samples is ambiguous especially in low qualityimages Thus positive samples with low quality are easily

rejected by the thresholds in the cascade and fail to contributeto the final result In [1] the authors introduced a qualityadaptive cascade that works in a probabilistic framework (Pcascade) In the P cascade framework all image patches havea chance to contribute to the final result and their contri-butions are determined by their corresponding probabilityIn this way P cascade can adapt to face images of arbitraryquality Furthermore they constructed two-level localizationframeworkwith a coarse-to-fine localization for the system tobe robust and accurate Figure 1 shows the size and geometryof the eye training samples for two-level stacked classifiers

In order to enhance the classification ability they intro-duced ELBP as improvement to LBP and multiblock LBP(MB-LBP) to improve the systemrsquos precision Figure 2 showsthe ELBP used in [1]

In this paper we propose an eye localization method withtwo-level localization framework which is both robust andaccurate even in unconstrained environment In the coarselevel we use the classificationmethod similar to [1] primarilyupgrading it with pyramid structure and postprocessingMoreover in the fine level we apply shape regression methodsimilar to [12] where we improve the robustness primarily byusing the coarse level information normalization based ontwo-eye centers and shape initialization

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2014 Article ID 804291 7 pageshttpdxdoiorg1011552014804291

2 ISRN Applied Mathematics

Coarse level Fine level

Figure 1 The size and geometry of training samples for two-level stacked classifiers

Original LBP

(a)

Changed aspect ratio

(b)

Rotated example

(c)

Figure 2 LBP and two ELBP operators

This paper is organized as follows Section 2 introducesproposed method through a definition of an eye centeranalysis of the limitations of classification and regressionmethod and discussion of the two-level approach to eyelocalization namely the coarse level using classifier and finelevel using regression Section 3 conducts experiments onhigh and low quality databases to illustrate the superioritiesof the proposed method Section 4 concludes the paper

2 Proposed Eye Localization Method

21 Definition of the Eye Center In general eye center isconsidered as the center of the pupil But according to theeye gaze direction the center of pupil is offset to some extentFurthermore for closed eyes the center of pupil is not seen

Figure 3 Definition of the eye center

In order to estimate the eye center more accurately weintroduce a definition of the eye center In Figure 3 119860 =

(119860119909 119860119910) and 119861 = (119861

119909 119861119910) are left and right end points of the

eye and 119863 = (119863119909 119863119910) and 119864 = (119864

119909 119864119910) are upper and lower

ISRN Applied Mathematics 3

Figure 4 Example of the eye localization failure by regressionmethod

points of the pupil respectivelyThen eye center119862 = (119862119909 119862119910)

is defined as

119862119909=(119860119909+ 119861119909)

2

119862119910=

(119863119910+ 119864119910)

2

(1)

This definition approximately coincides with the centerof pupil in most cases and is able to give a robust eye centerposition even in case when pupil is offset much from theeye center The definition can also give the state of two eyesnamely closeness and openness If the center of pupil isneeded one can estimate it using the points D and E easily

The problem however is finding the method that canestimate the points A B D and E We are going to esti-mate these points by the combination of classification andregression method In general the classification method isrobust but not so accurate The classification method whichdetermines whether input data are positive or negative isunable to express correctly how far the data are from the eyecenter If the data which are very near to the eye center areused as negative examples the training error increases andlow quality eye examples may be misrecognized as noneyeexamples Unlike this regression methods train the distancefrom current position to the eye center and consequently havebetter possibility for precision improvement A drawback ofregression method is its sensitivity to the initial positionnamely in case that initial position is far from the eye centerand there exist some patterns similar to the eye near to theinitial position and it may not converge to the eye center(Figure 4) In Figure 4 green triangles represent an initialshape and red squares represent the final shape obtained byregression method [12] But if an initial shape is near enoughto two-eye centers the result is better

In order to solve this sensitivity problem we propose arobust and accurate eye localization method by the com-bination of classification and regression methods We firstestimate initial positions of two eyes by the coarse localizationand then find around initial positions more accurate eyecenters by the regression method

Figure 5 Extension of face detection region and pyramid construc-tion

22 Coarse Eye Localization Because the main problem incoarse eye localization is robustness rather than precisionunder many factors such as eyebrow hair and glasses weperform the coarse eye localization by primarily payingattention to robustness

221 Feature Selection and Classifier Construction We usethe same size and geometry of the eye training samples as inthe coarse level of [1] (Figure 1) Furthermore we use ELBP asthe eye detection feature (Figure 2) Compared to LBP ELBPhas two radiuses and an angle which makes the shape ofELBP a rotated ellipse Given a 20 times 20 image the dimensionof ELBP feature set is very high when using 4 orientationsConsidering the characteristic of ELBP we use gentle boostalgorithm to select efficient features and construct classifierWe also use P cascade with rejection stage and probability

222 Eye Detection by Image Pyramid and Postprocessing Ingeneral eye detection is performed in the face region afterface is detected In literature on eye detection authors usuallydo not construct image pyramid because the ratio betweenthe size of face region and the size of corresponding eye regionusually falls in certain range But this is not always satisfiedand for some faces these ratios may exceed the expectedrange considerably Moreover if face region does not containtwo eyes eye detection fails As shown in Figure 1 trainingsamples we use are extracted under the condition that thedistance between two eyes is constant Thus image pyramidstructure is certainly needed for eye detection that is robustto scale variation of face detection regions

In order to perform such robust eye localization we firstextend the face detection region in all directions so that itcontains two eyes completely and normalize face region to 3image pyramids 60 times 60 55 times 55 and 50 times 50 and then eyedetector scans each region Figure 5 shows the extension offace detection region and pyramid construction

Then we calculate the maximum rejection stage corre-sponding to P cascade from 3 image pyramids and find eyecandidate positions with the maximum value As shown inFigure 6 most eye candidate positions (each white pixel) areusually around eye center but some candidates might bearound either the frame of glasses eyebrow or similar

Considering that most candidates are around eye centerand their classifier probabilities are usually high we firstperform hierarchical clustering [13] on themThen we deter-mine the cluster having the largest number of candidates as


(a) (b) (c)

Figure 6 Detected eye candidate positions

the best cluster If two clusters have the same number ofcandidates we select the best cluster by comparing theirmeanclassifier probabilities Furthermore we select 119898 candidateswith the highest classifier probability in the best clusterand determine the best eye candidate position by weightedarithmetic average

119890lowast

=

sum119890119894isin119890rank 119898

119901119894119890119894

sum119890119894isin119890rank 119898

119901119894

(2)

where 119890119894is a candidate in face image 119901

119894is the classifier

probability of 119890119894 and 119890rank 119898 is the first119898 candidates with the

highest classifier probabilities in the best cluster (119898 = 5 in ourcase) With this method even in case that some noneyes arerecognized as eyes with high probabilities we can accuratelyestimate eye centers

23 Fine Eye Localization by Explicit Shape Regression Aftereye candidate position is determined fine eye localization forhigher precision is performed As described in Section 21 weconvert eye localization problem into the estimation of leftand right end points of an eye and upper and lower pointsof a pupil Moreover regarding robustness we are convincedthat considering two eyes at the same time is better thanconsidering each eye individually

On the basis of these considerations we apply the explicitshape regression used in face landmark detection to refine eyelocalization [12]

231 Explicit Shape Regression Given a facial image 119868 andan initial face shape 1198780 each regressor computes a shapeincrement 120575119878 from image features and then updates the faceshape in a cascaded manner

119878119905

= 119878119905minus1

+ 119877119905

(119868 119878119905minus1

) 119905 = 1 sdot sdot sdot 119879 (3)

where the 119905-th weak regressor 119877119905 updates the previous shape119878119905minus1 to the new shape 119878119905

The regressor 119877119905 depends on both image 119868 and previousestimated shape 119878119905minus1 Given 119873 training examples (119868

119894 119878119894)119873

1

the regressor is obtained by explicitly minimizing the sum ofalignment errors

119877119905

= argmin119877

119873

sum

119894=1

10038171003817100381710038171003817119878119894minus (119878119894

119905minus1

+ 119877 (119868119894 119878119894

119905minus1

))10038171003817100381710038171003817 (4)

where 119878119894

119905minus1 is the estimated shape in previous stageThe regressors (1198771 119877119905 119877119879) are sequentially learnt

until the training error no longer decreases In [12] eachweakregressor119877119905 is learnt by a second level boosted regression thatis 119877119905 = (1199031 119903119896 119903119870)

Shape indexed image features used in [12] are indexedonly relative to 119878

119905minus1 and no longer change when those 119903rsquosare obtained Very simple and efficient Fern is used as eachprimitive regressor 119903119894 A Fern is a composition of 119865 featuresand thresholds that divide the feature space and all trainingsamples into 2119865 bins Each bin 119887 is associatedwith a regressionoutput 120575119878

119887that minimizes the alignment error of training

samples Ω119887falling into the bin

120575119878119887=

1

1 + 1205731003816100381610038161003816Ω119887

1003816100381610038161003816

sum119894isinΩ119887

(119878119894minus 119878119894)

1003816100381610038161003816Ω1198871003816100381610038161003816

(5)

where 120573 is a free shrinkage parameter When the bin hassufficient training samples 120573 has little effect otherwise itadaptively reduces the estimation Efficient feature selectionand Fern construction are performed by using fast correlationcomputation Methods that generate feature set select effi-cient features and construct Ferns are the same as in [12] thusin continuation we discuss our contributions consideringcharacteristics of fine eye localization

232 Normalization Based on Two-Eye Centers In orderto apply explicit shape regression to eye localization werandomly generate 119875 pixels and calculate 1198752 pixel differencefeatures Each pixel is indexed relative to the currently esti-mated shape 119878119905minus1 rather than the original image coordinatesA similarity transform to normalize the current shape to amean shape is needed Given the current shape 119878

119894

119905minus1 two-eyecenters 119871 = (119871

119909 119871119910) and 119877 = (119877

119909 119877119910) are easily calculated

by (1) Then rotation and scale transformation based ontwo-eye centers can normalize the current shape to a meanshapeDenoting the distance between two eyes inmean shapeby dist rotation angle 120579 and scale constant 120572 are determinedas follows

120579 = minus arctan(119877119910minus 119871119910)

(119877119909minus 119871119909)

120572 =dist

119871 minus 119877

(6)


Input Face detection

Pyramid +

Pyramid +

PostprocessingNormalization Fine localization

coarse left eye localization

coarse right eye localization

by two-eye candidates

by shape regression

image

Figure 7 Eye localization flowchart

Then the transformation that normalizes the currentshape to a mean shape is

119879 (119909 119910 119888 120572 120579) = 120572119877120579(119909 minus 119888

119909

119910 minus 119888119910

) (7)

where 119877120579(119909

119910 ) = (cos 120579 minus sin 120579sin 120579 cos 120579 ) (

119909

119910 ) and 119888 = (119888119909 119888119910) is a rotationcenter

119879minus1 the inverse of 119879 is then

119879minus1

(119906 V 119888 120572 120579) = (119888119909

119888119910

) +1

120572119877minus120579(119906

V) (8)

233 Shape Initialization Asdescribed in Section 21 regres-sion method is somewhat sensitive to initial point In otherwords farther away fromobject position the initial points arethe smaller the convergence tendency to the points is If we setinitial shape using only face detection information becauseof instability of face detection information in some cases itsshape (especially scale) may be quite different from the realshape

To solve this problem we use coarse eye localizationinformation described in Section 22 But because coarse eyelocalization information is also not completely accurate weobserve the following According to the experimental resultin [1] even for LFW and VS (Video Surveillance) databaseswhich have low quality images the success rate at normalizederror err lt 025 (9) ismore than 99 percent (this result is fromthe cumulative curves for eye localization in [1]) Surely forimages that are not low quality the result is nearly 100 percentConsequently in order to model coarse eye localizationinformation statistically we add zero-mean Gaussian noisewith constant variance to real eye centers (ground truth)so that their (noisy two eyes) normalized errors are smallerthan 025 Based on noisy two-eye positions and (7) (8) wenormalize the face regions to constant size and then thesenormalized images and corresponding real eye positions areused to explicit shape regression training

After training by using the trained regression functionand the coarse positions of two eyes we find left and rightend points of an eye and upper and lower points of a pupiland determine two-eye centers using (1) Because of usingthe coarse positions of two eyes the scales of the eyes aredetermined very robust and their initial positions are alsonear to the real positions Consequently we overcome thelimitation of the regression method

Figure 7 shows the flowchart of proposed eye localizationsystem After performing face detection the left and right eye

detectors scan tree level pyramid images get eye candidatepoints and determine the best candidate points for two eyesby using the proposed postprocessingThenwe normalize theinput image based on the candidates of two eyes find the leftand right end points of an eye and upper and lower points ofa pupil by regression method and output the best positionsof two-eye centers

3 Experimental Results

To train an eye detector we construct the training set whichcontains 12000 face images from various databases includingColorFERETMUCT PICS andCVLFaceDBThese trainingimages are considered as high quality images The test setis divided into two categories high quality and low qualityBioID database is used for high quality evaluation and LFWdatabase is used for low quality evaluation BioID databaseconsists of 1521 images while LFWdatabase consists of 13233images

The normalized error is used to evaluate the errorbetween the localized eye positions and the ground truth

err =max (10038171003817100381710038171003817119897119892 minus 119897

1003817100381710038171003817100381710038171003817100381710038171003817119903119892minus 11990310038171003817100381710038171003817)

10038171003817100381710038171003817119897119892minus 119903119892

10038171003817100381710038171003817

(9)

where 119897119892and 119903119892are the ground truth positions of the left and

right eyes and 119897 and 119903 are the eyes positions localized by analgorithm respectively Considering that similarity drops inface recognition if normalized eye localization error is morethan 5 we use 5 and 10 normalized error to evaluate eyelocalization performance

31 Coarse Eye Localization Because left eye and right eye arealmost symmetric wemay construct eye detector only for lefteye and therefore we can use 24000 eye training samples forcoarse eye localization training

Coarse eye localization training is similar as in [1] and itdiffers in our additional steps using pyramid structure andpostprocessing To show the effectiveness of the proposedpyramid structure and postprocessing we trained coarselevel method described in [1] and compared the performancewith the proposed method Tables 1 and 2 show the perfor-mance of both methods on BioID and LFW databases Asshown the proposedmethod effectively improves the perfor-mance


Table 1 Comparison of coarse eye localization methods on BioID

err lt 005 err lt 01Coarse level from [1] 614 9614Proposed method 669 9666

Table 2 Comparison of coarse eye localization methods on LFW


Table 3 Comparison of eye localization methods on BioID

err lt 005 err lt 01[1] 705 979Proposed method 926 9926

Table 4 Comparison of eye localization methods on LFW


32 Fine Eye Localization Training images used in shaperegression training are the same as in coarse eye localizationtraining with addition of Gaussian noise to real eye positionsof each image The images are normalized as described inSection 23 Each training sample consists of a training imagean initial shape and a ground truth shape To achievebetter generalization ability we augment the training databy randomly sampling multiple (20 in our implementation)shapes from other annotated images as initial shapes of eachtraining image This is very effective in terms of robustnessagainst large posevariation and rough initial shapes duringthe testing We run the regressor several times (5 timesin our implementation) and take the median result as thefinal estimation We determined the parameters by cross-validation and set them to 119865 = 5 120573 = 1000 119879 = 9 119870 = 500119875 = 500 and dist = 30 as they represent good tradeoffbetween computational cost and precision

Because the reported results of method [1] are the highestin literature we compare our results with them Tables 3 and4 show the results our method gets better results for bothhigh and low quality images Particularly for LFW databaseour method gives much better results and they show thatour method works very well even for low quality images inunconstrained environment

Furthermore the numbers prove that results are sub-stantially improved by the combination of classification andregression methods

4 Conclusion

In this paper we proposed an eye localization method of highprecision by the combination of classification and regression

methods The proposed method is based on facts that classi-fication method is robust but less accurate while regressionmethod is less robust but very accurate because it hasmore information about object position Proposed methodis both robust and accurate and gives the highest precisionin terms of normalized error We believe that the conceptof the proposed method can be widely used not only in eyelocalization problem but in many other object localizationproblems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D Yi Z Lei and S Z Li ldquoA robust eye localization methodfor low quality face imagesrdquo in Proceedings of the InternationalJoint Conference on Biometrics (IJCB rsquo11) pp 1ndash6 WashingtonDC USA October 2011

[2] P Wang M B Green J Qiang and J Wayman ldquoAutomaticeye detection and its validationrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognitionmdashWorkshops (CVPR rsquo05) vol 3 pp 164ndash170 SanDiego Calif USA June 2005

[3] Z-H Zhou and X Geng ldquoProjection functions for eye detec-tionrdquo Pattern Recognition vol 37 no 5 pp 1049ndash1056 2004

[4] Q Chen K Kotani F Lee and T Ohmi ldquoA robust eye detectionapproach based on edge related informationrdquo InternationalJournal of Computer Science and Network Security vol 9 no9 pp 22ndash27 2009

[5] S Asteriadis N Nikolaidis A Hajdu and I Pitas ldquoA novel eye-detection algorithmutilizing edge-related geometrical informa-tionrdquo in Proceedings of the European Signal Processing Confer-ence Florence Italy September 2006

[6] Q Chen K Kotani F Lee and T Ohmi ldquoAn accurate eyedetection method using elliptical separability filter and com-bined featuresrdquo International Journal of Computer Science andNetwork Security vol 9 no 8 pp 65ndash72 2009

[7] Y Ma X Ding Z Wang and N Wang ldquoRobust preciseeye location under probabilistic frameworkrdquo in Proceedings ofthe 6th IEEE International Conference on Automatic Face andGesture Recognition (FGR rsquo04) pp 339ndash344 Seoul Korea May2004

[8] Z Niu S Shan S Yan X Chen and W Gao ldquo2D cascadedAdaBoost for eye localizationrdquo in Proceedings of the 18thInternational Conference on Pattern Recognition (ICPR rsquo06) vol2 pp 1216ndash1219 Hong Kong August 2006

[9] X Tan F Song Z-H Zhou and S Chen ldquoEnhanced picto-rial structures for precise eye localization under uncontrolledconditionsrdquo in Proceedings of the IEEE Conference on ComputerVision and Pattern Recognition (CVPR rsquo09) pp 1621ndash1628Miami Fla USA June 2009

[10] X Tang Z Ou T Su H Sun and P Zhao ldquoRobust preciseeye location by adaboost and svm techniquesrdquo in Advances inNeural Networks vol 3497 of LectureNotes in Computer Sciencepp 93ndash98 Springer Berlin Germany 2005

[11] M Everingham and A Zisserman ldquoRegression and classifica-tion approaches to eye localization in face imagesrdquo in Proceed-ings of the 7th International Conference on Automatic Face and


Gesture Recognition (FGR rsquo06) pp 441ndash446 Southampton UkApril 2006

[12] X Cao Y Wei F Wen and J Sun ldquoFace alignment by explicitshape regressionrdquo in Proceedings of the IEEE Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2887ndash2894 Providence RI USA June 2012

[13] T Hastie The Elements of Statistical Learning Springer BerlinGermany 2008

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Coarse level Fine level

Figure 1 The size and geometry of training samples for two-level stacked classifiers

Original LBP

(a)

Changed aspect ratio

(b)

Rotated example

(c)

Figure 2 LBP and two ELBP operators

This paper is organized as follows Section 2 introducesproposed method through a definition of an eye centeranalysis of the limitations of classification and regressionmethod and discussion of the two-level approach to eyelocalization namely the coarse level using classifier and finelevel using regression Section 3 conducts experiments onhigh and low quality databases to illustrate the superioritiesof the proposed method Section 4 concludes the paper

2 Proposed Eye Localization Method

21 Definition of the Eye Center In general eye center isconsidered as the center of the pupil But according to theeye gaze direction the center of pupil is offset to some extentFurthermore for closed eyes the center of pupil is not seen

Figure 3 Definition of the eye center

In order to estimate the eye center more accurately weintroduce a definition of the eye center In Figure 3 119860 =

(119860119909 119860119910) and 119861 = (119861

119909 119861119910) are left and right end points of the

eye and 119863 = (119863119909 119863119910) and 119864 = (119864

119909 119864119910) are upper and lower




is defined as

119862119909=(119860119909+ 119861119909)

2

119862119910=

(119863119910+ 119864119910)

2

(1)












(a) (b) (c)



119890lowast

=

sum119890119894isin119890rank 119898

119901119894119890119894

sum119890119894isin119890rank 119898

119901119894

(2)








119878119905

= 119878119905minus1

+ 119877119905

(119868 119878119905minus1

) 119905 = 1 sdot sdot sdot 119879 (3)



119894 119878119894)119873

1


119877119905

= argmin119877

119873

sum

119894=1

10038171003817100381710038171003817119878119894minus (119878119894

119905minus1

+ 119877 (119868119894 119878119894

119905minus1

))10038171003817100381710038171003817 (4)

where 119878119894







120575119878119887=

1

1 + 1205731003816100381610038161003816Ω119887

1003816100381610038161003816


(119878119894minus 119878119894)

1003816100381610038161003816Ω1198871003816100381610038161003816

(5)



119894


119909 119871119910) and 119877 = (119877




(119877119909minus 119871119909)

120572 =dist

119871 minus 119877

(6)



Pyramid +

Pyramid +





by shape regression

image



119879 (119909 119910 119888 120572 120579) = 120572119877120579(119909 minus 119888

119909

119910 minus 119888119910

) (7)

where 119877120579(119909


119909



119879minus1

(119906 V 119888 120572 120579) = (119888119909

119888119910

) +1

120572119877minus120579(119906

V) (8)









err =max (10038171003817100381710038171003817119897119892 minus 119897

1003817100381710038171003817100381710038171003817100381710038171003817119903119892minus 11990310038171003817100381710038171003817)

10038171003817100381710038171003817119897119892minus 119903119892

10038171003817100381710038171003817

(9)

















4 Conclusion





References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












is defined as

119862119909=(119860119909+ 119861119909)

2

119862119910=

(119863119910+ 119864119910)

2

(1)












(a) (b) (c)



119890lowast

=

sum119890119894isin119890rank 119898

119901119894119890119894

sum119890119894isin119890rank 119898

119901119894

(2)








119878119905

= 119878119905minus1

+ 119877119905

(119868 119878119905minus1

) 119905 = 1 sdot sdot sdot 119879 (3)



119894 119878119894)119873

1


119877119905

= argmin119877

119873

sum

119894=1

10038171003817100381710038171003817119878119894minus (119878119894

119905minus1

+ 119877 (119868119894 119878119894

119905minus1

))10038171003817100381710038171003817 (4)

where 119878119894







120575119878119887=

1

1 + 1205731003816100381610038161003816Ω119887

1003816100381610038161003816


(119878119894minus 119878119894)

1003816100381610038161003816Ω1198871003816100381610038161003816

(5)



119894


119909 119871119910) and 119877 = (119877




(119877119909minus 119871119909)

120572 =dist

119871 minus 119877

(6)



Pyramid +

Pyramid +





by shape regression

image



119879 (119909 119910 119888 120572 120579) = 120572119877120579(119909 minus 119888

119909

119910 minus 119888119910

) (7)

where 119877120579(119909


119909



119879minus1

(119906 V 119888 120572 120579) = (119888119909

119888119910

) +1

120572119877minus120579(119906

V) (8)









err =max (10038171003817100381710038171003817119897119892 minus 119897

1003817100381710038171003817100381710038171003817100381710038171003817119903119892minus 11990310038171003817100381710038171003817)

10038171003817100381710038171003817119897119892minus 119903119892

10038171003817100381710038171003817

(9)

















4 Conclusion





References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)



119890lowast

=

sum119890119894isin119890rank 119898

119901119894119890119894

sum119890119894isin119890rank 119898

119901119894

(2)








119878119905

= 119878119905minus1

+ 119877119905

(119868 119878119905minus1

) 119905 = 1 sdot sdot sdot 119879 (3)



119894 119878119894)119873

1


119877119905

= argmin119877

119873

sum

119894=1

10038171003817100381710038171003817119878119894minus (119878119894

119905minus1

+ 119877 (119868119894 119878119894

119905minus1

))10038171003817100381710038171003817 (4)

where 119878119894







120575119878119887=

1

1 + 1205731003816100381610038161003816Ω119887

1003816100381610038161003816


(119878119894minus 119878119894)

1003816100381610038161003816Ω1198871003816100381610038161003816

(5)



119894


119909 119871119910) and 119877 = (119877




(119877119909minus 119871119909)

120572 =dist

119871 minus 119877

(6)



Pyramid +

Pyramid +





by shape regression

image



119879 (119909 119910 119888 120572 120579) = 120572119877120579(119909 minus 119888

119909

119910 minus 119888119910

) (7)

where 119877120579(119909


119909



119879minus1

(119906 V 119888 120572 120579) = (119888119909

119888119910

) +1

120572119877minus120579(119906

V) (8)









err =max (10038171003817100381710038171003817119897119892 minus 119897

1003817100381710038171003817100381710038171003817100381710038171003817119903119892minus 11990310038171003817100381710038171003817)

10038171003817100381710038171003817119897119892minus 119903119892

10038171003817100381710038171003817

(9)

















4 Conclusion





References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Pyramid +

Pyramid +





by shape regression

image



119879 (119909 119910 119888 120572 120579) = 120572119877120579(119909 minus 119888

119909

119910 minus 119888119910

) (7)

where 119877120579(119909


119909



119879minus1

(119906 V 119888 120572 120579) = (119888119909

119888119910

) +1

120572119877minus120579(119906

V) (8)









err =max (10038171003817100381710038171003817119897119892 minus 119897

1003817100381710038171003817100381710038171003817100381710038171003817119903119892minus 11990310038171003817100381710038171003817)

10038171003817100381710038171003817119897119892minus 119903119892

10038171003817100381710038171003817

(9)

















4 Conclusion





References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















4 Conclusion





References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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