Bilinear Discriminant Analysis for Face Recognition

Muriel Visani1, Christophe Garcia1 and Jean-Michel Jolion2

1 France Telecom Division R&D, TECH/IRIS 2 Laboratoire LIRIS, INSA Lyon

4, rue du Clos Courtel 20, Avenue Albert Einstein

35512 Cesson-Sevigne, France Villeurbanne, 69621 cedex, France

muriel.visani,christophe.garcia@rd.francetelecom.com Jean-Michel.Jolion@insa-lyon.fr

Abstract

In this paper, a new statistical projection method called Bilinear Discriminant Analysis (BDA) is presented. Theproposed method efficiently combines two complementary versions of Two-Dimensional-Oriented Linear Discrim-inant Analysis (2DoLDA), namely Column-Oriented Linear Discriminant Analysis (CoLDA) and Row-OrientedLinear Discriminant Analysis (RoLDA), through an iterative algorithm using a generalized bilinear projection-based Fisher criterion. A series of experiments was performed on various international face image databases inorder to evaluate and compare the effectiveness of BDA to RoLDA and CoLDA. The experimental results indicatethat BDA is more efficient than RoLDA, CoLDA and 2DPCA for the task of face recognition, while leading to asignificant dimensionality reduction.

1 Introduction

In the eigenfaces [1] (resp. fisherfaces [2]) method, the 2D face images of size h×w are first transformed into 1Dimage vectors of size h · w, and then a Principal Component Analysis (PCA) (resp. Linear Discriminant Analysis(LDA)) is applied to the high-dimensional image vector space, where statistical analysis is costly and may beunstable. To overcome these drawbacks, Yang et al. [3] proposed the Two Dimensional PCA (2D PCA) methodthat aims at performing PCA using directly the face image matrices. It has been shown that 2D PCA is moreefficient [3] and robust [4] than the eigenfaces when dealing with face segmentation inaccuracies, low-qualityimages and partial occlusions.In [5], we proposed the Two-Dimensional-Oriented Linear Discriminant Analysis (2DoLDA) approach, that con-sists in applying LDA on image matrices. We have shown on various face image databases that 2DoLDA is moreefficient than both 2D PCA and the Fisherfaces method for the task of face recognition, and that it is more robustto variations in lighting conditions, facial expressions and head poses.In this paper, we propose a novel supervised projection method called Bilinear Discriminant Analysis (BDA)that achieves better recognition results than 2DoLDA while substantially reducing the computational cost of therecognition step, using a generalized Fisher criterion relying on bilinear projections.The remainder of the paper is organized as follows. In section 2, we remind the theory and algorithm of 2DoLDA.In section 3, we describe in details the principle and algorithm of the proposed BDA method, pointing out itsadvantages over previous methods. In section 4, a series of three experiments, on different international data sets,is presented to demonstrate the effectiveness and robustness of BDA and compare its performances with respect toRoLDA, CoLDA and 2DPCA. Finally, conclusions are drawn in section 5.

2 Two-Dimensional Oriented Linear Discriminant Analysis (2Do LDA)

In this section, we remind the 2DoLDA theory and algorithm. For more details refer to [5]. 2DoLDA may beimplemented in two different ways: Row-oriented LDA (RoLDA) and Column-oriented LDA (CoLDA). Let usfirst present RoLDA.The model is constructed from a training set containing n face images stored as h × w matrices Xi labelled bytheir corresponding identity Ωc. The set of images corresponding to one person is called a class. The training setcontains C classes. Let us consider a projection matrix P , of size w× k. The projection XP

i of Xi with P is givenby: XP

i = Xi ·P (1)Using RoLDA, the signature of Xi is XP

i , of size h× k. Our aim is to determine the matrix P jointly maximizingseparation between signatures from different classes and minimizing separation between signatures from the sameclass. Under the assumptions that the rows of the images are multinormal and that rows from different classes havethe same within-class covariance, P maximizes the generalized Fisher criterion [5]: J(P ) = |P T SbP |

|P T SwP | (2)Sw and Sb being respectively the generalized within-class and between-class covariance matrices of the training

set:Sw =

C∑c=1

∑

Xi∈Ωc

(Xi − Xc)T (Xi − Xc) and Sb =C∑

c=1

nc(Xc − X)T (Xc − X) (3)

and Xc and X are respectively the mean of the nc samples of class Ωc and the mean of the whole training set. If Sw

is non-singular (which is generally verified as w << n), the k columns of P , named Row-Oriented DiscriminantComponents (RoDCs) and maximizing criterion (2) are the eigenvectors of S−1

w Sb with largest eigenvalues. Anumerically stable way to compute them is given in [6].Analogeously, for Column-Oriented Linear Discriminant Analysis (CoLDA), the considered projection is:

XQi = QT ·Xi, (4)

where Q is a projection matrix of size h × k. Under the assumptions of multinormality and homoscedasticity ofthe columns, we can consider the following generalized Fisher criterion:

J(Q) =|QT ΣbQ||QT ΣwQ| (5)

where Σw and Σb are respectively the within-class and between-class covariance matrices of the (XTi )i∈1...n:

Σw =C∑

c=1

∑

Xi∈Ωc

(Xi − Xc)(Xi − Xc)T and Σb =C∑

c=1

nc(Xc − X)(Xc − X)T . (6)

If Σw is non-singular, the k columns of Q, named Column-Oriented Dicriminant Components and maximizingcriterion (5) are the k eigenvectors of Σ−1

w Σb with largest eigenvalues.There are at most C−1 RoDCs and CoDCs corresponding to non-zero eigenvalues; their number k can be selectedusing the Wilks Lambda criteria, which is also known as the stepwise discriminant analysis [7]. This analysisshows that the number k of 2Do-DCs required by both methods is comparable, generally inferior to 15, even if thenumber of classes is large, as shown in Figure 2(a), reporting an experiment performed on 107 classes.Recognition is performed in the projection space defined by RoLDA (resp. CoLDA): two faces images Xa and Xb

are compared using the Euclidean distance between their projections XPa and XP

b (resp. XQa and XQ

b ).

3 Bilinear Discriminant Analysis (BDA)

3.1 Why Combining CoLDA and RoLDA?

We conducted four experiments highlighting the complementarity of RoLDA and CoLDA. In the following, all theface images are centered and cropped to a size of h× w = 75× 65 pixels.The first two experiments are performed on subsets of the Asian Face Image Database PF01 [8] containing 107persons; they illustrate the fact that depending on the training and test data, RoLDA can significantly outperformCoLDA or the contrary. In the first experiment, the training set (see Figure 1(a)) contains 5 near-frontal views perperson (535 images). The test set (see Figure 1(b)) contains 4 views per person, with stronger non-frontal posesthan in the training set.

(a) (b) (c) (d)Figure 1: Subsets of the IML database used for experiments 1-2. (a): Extract of the training set used for the firstexperiment, (b): extract of the test set used for the first experiment, (c): Extract of the training set used for thesecond experiment, (d): extract of the test set used for the second experiment.

Figure 2(a) shows that both CoLDA and RoLDA provide more than 92% recognition rate, and outperform 2DPCA. However, RoLDA is more efficient than CoLDA, with 4,5% improvement of the recognition rate betweenthe respective maxima. In the second experiment, the training set (see Figure 1(c)) containing 3 views per personand the test set (see Figure 1(d)) containing 2 views per person are randomly taken from the subset of the IMLdatabase containing facial expressions: neutral, happy, surprised, irritated and closed eyes. Figure 2(b) showsthat both RoLDA and CoLDA outperform 2D PCA, and that CoLDA is more efficient than RoLDA (with 5,6%improvement of the recognition rate between the maxima) on that particular and complex -the recognition rates areinferior to 60%- subset of the IML database.The third and fourth experiments provide further comparison of the performances of CoLDA and RoLDA; theyare performed on the Yale Face Database [2] (see Figure 3(a-b)) containing 11 views of each of 15 persons.

0,7

0,75

0,8

0,85

0,9

0,95

1

0 5 10 15 20

Number of projection vectors

Rec

ogni

tion

Rat

e

RoLDA

CoLDA

2D PCA

0,4

0,45

0,5

0,55

0,6

0,65

0 5 10 15 20

Number of projection vectors

Rec

og

nit

ion

Rat

e

RoLDA

CoLDA

2D PCA

(a) (b)Figure 2: Compared recognition rates of RoLDA, CoLDA and 2D PCA on a subset of the IML database showing(a) head pose changes (b) facial expression changes, when varying the number k of projection vectors.

These views present occlusions (”glasses”, ”noglasses”) and dissimilarities in the lighting conditions and facialexpressions. In the third experiment, the Yale face database is randomly partitioned into a training set containingfour views per person, and a test set containing six views per person. To ensure homoscedasticity, the viewsof each set are consistent among the classes, e.g. all the ”wink” views are included in the test set, and all the”neutral” in the training set. This operation is repeated several times. From each partition we compute a confusionmatrix. The confusion matrices from five different random partitions, with k = C−1 = 14, are shown in Table 1.In each confusion matrix, the top left cell contains the number of faces correctly classified by both CoLDA andRoLDA. The top right element is the number of faces correctly classified by RoLDA, but misclassified by CoLDA.The bottom left element is the number of faces correctly classified by CoLDA, but misclassified by RoLDA. Thebottom right cell contains the number of faces misclassified by both CoLDA and RoLDA.1611 1053 35 1171 64 872 1614 555 187 263(a) (b) (c) (d) (e)

Table 1: Confusion matrices on random partitions of the Yale Face Database.

Table 1(a) shows that, on the first random partition of the Yale database, the performances of RoLDA and CoLDAare comparable (the recognition rates are respectively 53+10

53+10+11+16 = 70% and 71,1%). However, classificationresults are very different: 21 samples (23,3% of the test set) are correctly classified by one method, but not by theother, and 82,2% of the test faces are recognized by at least one of the two methods. Table 1(b-c) illustrates thefact that RoLDA generally outperforms CoLDA. Table 1(d-e) shows that in some configurations where the rate ofmisclassification by both methods is high (respectively 16

90 = 17, 8% and 20% for partitions (d) and (e)), CoLDAoutperforms RoLDA.The fourth experiment provides further qualitative analysis. The training set (see Figure 3(a)) contains four viewsfor all the 15 subjects, with variations in the lighting conditions and facial expressions. Then, seven test sets arebuilt (see Figure 3(b)), corresponding to the remaining views. Figure 3(c) compares the performances of RoLDAand CoLDA on these test sets. Figure 3(c) shows that, for a fixed training set containing few variations, RoLDA isgenerally more efficient than CoLDA, but in some cases CoLDA drastically outperforms RoLDA, especially whenthe test set contains dissimetries of the image following the vertical axis (”leftlight” and ”rightlight”). CoLDA canalso slightly outperform RoLDA when the test set shows strong facial expression changes, e.g. ”surprised”.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

w ink surprised leftlight rightlight sad sleepy occlusion

Reco

gn

itio

n R

ate RoLDA

CoLDANormal Centerlight Happy Glasses / Noglasses

(a)

Wink Surprised Leftlight Rightlight Sad Sleepy Occlusion

(b) (c)Figure 3: (a): Extract of the training set used for the fourth experiment (b): extract of the 7 test sets of the fourthexperiment. A subject not wearing eyeglasses in the training set wears eyeglasses in the ”occlusion” set, andvice-versa. (c): Compared recognition rates of CoLDA and RoLDA on partitions of the Yale Database.

Choosing between CoLDA and RoLDA therefore requires a preliminary qualitative analysis of the training andtest sets, which is a difficult task. As both RoLDA and CoLDA are very performant but give different recognitionresults, efficiently combining them can lead to a highly performant method.In 2DoLDA, considering image matrices instead of vectors (as in the Fisherfaces method) when performing LDAleads to a reduced computational cost when building the model, and to a reduced storage cost [5]. But the size ofthe matrices XP

a and XPb (resp. XQ

a and XQb ) is h × k for RoLDA (resp. k × w for CoLDA) and may be large.

As exposed in the following section, using BDA for recognition leads to a drastic reduction in the signatures size,and therefore reduces the computational cost during the recognition step, which is often online.

3.2 Description of the method

Let us consider two projection matrices Q ∈ Rh×k and P ∈ Rw×k. The projected sample XQ,Pi of Xi on (Q,P )

is given by: XQ,Pi = QT XiP (7)

Let us define the signature of Xi by its bilinear projection XQ∗,P∗i (of size k×k) onto (Q∗, P ∗). We are searching

for the optimal pair of matrices (Q∗, P ∗) maximizing separation between signatures from different classes whileminimizing separation between signatures from the same class:

(Q∗, P ∗) = Argmax(Q,P )∈Rh×k×Rw×k

|SQ,Pb ||SQ,P

w | = Argmax(Q,P )∈Rh×k×Rw×k

|∑Cc=1 nc(X

Q,Pc −XQ,P )T (XQ,P

c −XQ,P )||∑C

c=1

∑i∈Ωc

(XQ,Pi −XQ,P

c )T (XQ,Pi −XQ,P

c )|(8)

SQ,Pw and SQ,P

b being the within-class and between-class covariance matrices of the set (XQ,Pi )i∈1,...,n.

This objective function is biquadratic and has no analytical solution. We therefore propose an iterative procedurethat we call Bilinear Discriminant Analysis.Let us expand the expression (8):

(Q∗, P ∗) = Argmax(Q,P )∈Rh×k×Rw×k

[|ΣC

c=1nc(PT (Xc−X)T QQT (Xc−X)P )|

|ΣC

c=1Σi∈Ωc (P T (Xi−Xc)T QQT (Xi−Xc)P )|

](9)

For any fixed Q ∈ Rh×k, using equation (9), the objective function (8) can be rewritten:

P ∗ = ArgmaxP∈Rw×k

[|P T

hΣC

c=1nc(XQc −XQ)T (XQ

c −XQ)iP |

|P ThΣC

c=1Σi∈Ωc (XQi −XQ

c )T (XQi −XQ

c )iP |

]= Argmax

P∈Rw×k

|P T SQb P |

|P T SQw P | (10)

SQw and SQ

b being respectively the generalized within-class covariance matrix and the generalized between-classcovariance matrix of the set (XQ

i )i∈1...n (from equation (4)). Therefore the columns of the matrix P ∗ are thek eigenvectors of SQ

w−1

SQb with largest eigenvalues, obtained by applying RoLDA on the set of the projected

samples XQi . Let us denote A = PT (Xc −X)T Q, matrix of size k × k. Given that, for every square matrix A,

|AT A| = |AAT |, the objective function (8) can be rewritten:

(Q∗, P ∗) = Argmax(Q,P )∈Rh×k×Rw×k

[|ΣC

c=1nc(QT (Xc−X)PP T (Xc−X)T Q)|

[ΣC

c=1Σi∈Ωc (QT (Xi−Xc)PP T (Xi−Xc)T Q)|

](11)

For a fixed P ∈ Rw×k, using equation (11) the objective function (8) can be rewritten Q∗ = ArgmaxQ∈Rh×k

|QT ΣPb Q|

|QT ΣPwQ| ,

ΣPw and ΣP

b being the generalized within-class and between-class covariance matrices of the set ((XPi )T )i∈1...n.

Therefore, the columns of Q∗ are the k eigenvectors of (ΣPw)−1ΣP

b with largest eigenvalues, obtained by applyingCoLDA on the set of the projected samples (XP

i )i∈1...n.Recognition is performed in the projection space defined by BDA: two face image Xa and Xb are compared usingthe Euclidean distance between their bilinear projections X

(Q,P )a and X

(Q,P )b .

We can note that the computational cost of one comparison is o(k2) for BDA, versus o(h · k) for RoLDA and 2DPCA, and o(w · k) for CoLDA; therefore BDA drastically reduces the computational cost of the recognition step.

3.3 Algorithm of the BDA approach

Let us initialize Q0 = Ih, Ih being the identity matrix of Rh×h, and k0=C−1. The proposed algorithm for BDA is:1. For i ∈ 1, . . . , n, compute XQt

i = (Qt)T Xi.2. Apply RoLDA on (XQt

i )i∈1,...,n: compute SQtw , SQt

b and, from (SQtw )−1 ·SQt

b , compute Pt, of size w×kt;3. For i ∈ 1, . . . , n, compute XPt

i = XiPt.4. Apply CoLDA on (XPt

i )i∈1,...,n: compute ΣPtw , ΣPt

b , and, from (ΣPtw )−1 ·ΣPt

b , compute Qt, of size h×kt;

5. Compute χ2 = −(n− h+C2 − 1)ln(

∏C−1j=kt+1

11+λj

).

6. if χ2 < p−value[χ2 ((h−kt)(C−kt−1))

], then t← t+1, kt ← kt−1−1, and return to step.1.

7. else kt ← kt−1, Q← Qt−1 and P ← Pt−1

The stopping criterion (steps 6-7) derives from the Wilks Lambda criterion,testing the discriminatory power of theC − 1 − kt eigenvectors of (ΣPt

w )−1 · ΣPt

b removed at step 4, when keeping only the kt first columns of Qt. Weconsider the following test: H0: at least one of the eigenvectors kt+1, . . . C−1 is discriminative, and H1: non H0.Under H0, it can be easily shown that −(n− h+C

2 − 1)ln(∏C−1

j=kt+11

1+λj) corresponds to a χ2 distribution, with

(h− kt)(C − kt− 1) degrees of freedom. The p-value can be chosen at confidence level of 5%. If χ2 < p−value,the C − kt − 1 last eigenvectors (with smallest eigenvalue) can be removed and the stepwise analysis goes on. Ifχ2 > p−value, the eigenvector kt + 1 = kt−1 is discriminative and should be kept.

4 Experimental Results

Three experiments are performed on the Asian Face Image Database PF01 [8], the FERET [9]1 face database, andthe ORL Database [10], to assess the effectiveness of BDA and compare it with RoLDA, CoLDA and 2D-PCA.The first experiment is performed on the subset of the IML database containing different facial expressions (seeFigure 1(c-d)). Figure 4(d) shows that BDA outperforms RoLDA, CoLDA and 2D PCA with at most 10% recog-nition rate improvement when k = 14.The second experiment, performed on FERET, aims at testing the generalization power of BDA, and comparingit to RoLDA, CoLDA and 2D PCA. Indeed, LDA-based methods are known to be more efficient when comparingfaces of known people, but provide worse generalization results than unsupervised methods. The training set (seeFigure 4(a)) contains 818 images of 152 persons with at least 4 views / person, taken on different days, underdifferent lighting conditions. Two test sets (see Figure 4(b-c)) are compared, each one containing 200 persons withone view per person, taken from FERET but not present in the training set. From one test set to the other, thefacial expressions vary. From Figure 4(e), we can conclude that when the training set contains many classes andimportant variations inside the classes, BDA provides better generalization than the other methods.

IML - facial expressions

0,4

0,45

0,5

0,55

0,6

0,65

0,7

7 9 11 13 15Number of projection vectors

Reco

gn

itio

n R

ate

BDA

RoLDA

CoLDA

2D PCA

FERET - Comparison of the two test sets

0,7

0,75

0,8

0,85

0,9

8 10 12 14Number of projection vectors

Reco

gn

itio

n R

ate

BDA

RoLDA

CoLDA

2D PCA

(a) (b) (c)

(d) (e)

Figure 4: (a): Extract of the training set used for the second experiment. (b-c): Extracts of the two test setscompared in the second experiment. (d): Compared recognition rates of BDA, RoLDA, CoLDA and 2D PCA, onthe subset of IML with expression variations. (e): Compared classification rates of BDA, RoLDA, CoLDA and 2DPCA on the subset of the FERET database, when varying the number k of projection vectors.

For the third experiment, the ORL database has been randomly partitioned into a training set containing 5 views,and a test set containing the 5 remaining views, for each of the 40 persons. This operation has been repeated 7times and BDA, RoLDA and CoLDA have been applied. Figure 5 shows that BDA provides better recognitionrates than RoLDA and CoLDA on all the random partitions, whenever RoLDA outperforms CoLDA (partitions(a-b) and (d-g)) or CoLDA outperforms RoLDA (partition (c)). The results are computed from the optimal numberof projection vectors, which is k = 14 for the three methods.For further analysis, the contingency table summed over partitions (a-g) is given in Table 2. The total number oftested faces is 7·200 = 1400. The logical symbol ”e” stands for ”not”, i.e. the element corresponding to the secondrow and first column of the table is the number of faces recognized by RoLDA, but misclassified by CoLDA.

1Portions of the research in this paper use the FERET database of facial images collected under the FERET program.

0,85

0,87

0,89

0,91

0,93

0,95

0,97

0,99

a b c d e f g

Rec

ogn

itio

n R

ate BDA

RoLDA

CoLDA

13471400Total

1748RoLDA ∩ CoLDA

1422RoLDA ∩ CoLDA

2433RoLDA ∩ CoLDA

12921297RoLDA ∩ CoLDA

∩ BDA

Figure 5: Compared recognition rates of BDA, Table 2: Contingency table summed overRoLDA and CoLDA on 7 random partitions partitions (a-g) of the ORL database.

of the ORL database.

The second row and second column element is the number of samples correctly classified by RoLDA and BDA,but misclassified by CoLDA. From table 2 we can see that BDA correctly classifies 1292

1297 = 99.6% of the samplesthat were correctly classified by both RoLDA and CoLDA. Moreover, it recognizes the major part of the samplesthat were correctly classified by only one of the two methods (72.7% for RoLDA and 63.6% for CoLDA). It alsocorrectly classifies 35,4% of the samples that were misclassified by both methods, which shows the efficiency ofthe BDA iterative algorithm. It should be noted that the size of the signature for one sample is 75 · 14 = 1050 forRoLDA, 65 · 14 = 910 for CoLDA, and only 142 = 196 for BDA.

5 Conclusion

In this paper, we have proposed a new class-based projection method, called Bilinear Discriminant Analysis, thatcan be successfully applied to face recognition. This technique, by maximizing a generalized Fisher criterion lyingon a bilinear projection and computed directly from face image matrices, constructs a discriminant projectionmatrix.It has already been shown [3,4] that 2D PCA outperforms the traditional eigenfaces method. We have alreadyhighlighted [5] that 2DoLDA outperforms 2D PCA and the fisherfaces method. In this paper, we have shown onvarious international databases that BDA is more efficient than 2DoLDA.Moreover, BDA, by providing reduced size image signatures, allows an important computational gain compared to2DoLDA and 2D PCA during the recognition step.

Aknowledgement

This research was supported by the European Commission under contract FP6-001765 aceMedia.

6 References

[1] M.A. Turk and A.D. Pentland, ”Eigenfaces for Recognition”. Journal of Cognitive Neuroscience, 3(1), pages 71-86, 1991.[2] P.N. Belhumeur, J.P. Hespanha and D.J. Kriegman, ”Eigenfaces vs Fisherfaces : Recognition Using Class Specific Linear

Projection”. In IEEE Transactions on Pattern Analysis and Machine Intelligence, Special Issue on Face Recognition, 17(7),pages 711-720, 1997.

[3] J. Yang, D. Zhang and A.F. Frangi, ”Two-Dimensional PCA: A New Approach to Appearance-Based Face Representationand Recognition”. In IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(1), pages 131-137, Jan. 2004.

[4] M. Visani, C. Garcia and C. Laurent, ”Comparing Robustness of Two-Dimensional PCA and Eigenfaces for Face Recogni-tion”. In Proceedings of the 1st International Conference on Image Analysis and Recognition (ICIAR 04), Springer LectureNotes in Computer Science (LNCS 3211), A. Campilho, M. Kamel (eds), pages 717-724 Sept. 2004.

[5] M. Visani, C. Garcia and J.M. Jolion, ”Two-Dimensional-Oriented Linear Discriminant Analysis for Face Recognition”. InProceedings of the International Conference on Computer Vision and Graphics (ICCVG 2004), to appear in ComputationalImaging and Vision series, Warsaw, Poland, Sept. 2004.

[6] D.L. Swets and J. Weng, ”Using Discriminant Eigenfeatures for Image Retrieval”. In IEEE Transactions on Pattern Analysisand Machine Intelligence, 18(8), pages 831-837, 1996.

[7] R.L. Jenrich, ”Stepwise Discriminant Analysis”, Standard Statistical Methods for Digital Computers, pages 76-95 Rals-ton, A and Wilf, H.S. (eds), Wiley, New York, USA, 1977.

[8] B.W. Hwang, M.C. Roh and S.W. Lee, ”Performance Evaluation of Face Recognition Algorithms on Asian Face Database”.In IEEE International Conference on Automatic Face and Gesture Recognition, pages 278-283, Seoul, Korea, May 2004.

[9] P.J. Phillips, H. Wechsler, J. Huang and P. Rauss, ”The FERET Database and Evaluation Procedure for Face Recognition

Algorithms”. In Image and Vision Computing, 16(5), pages 295-306, 1998.[10] F. Samaria and A. Harter, ”Parameterisation of a Stochastic Model for Human Face Identification”. In Proceedings of 2nd

IEEE Workshop on Applications of Computer Vision, Sarasota FL, Dec. 1994.