On the Bin Number Choice of Joint Histogram Estimation Applied

to Mutual Information based Face Recognition

Abdenour Hacine-Gharbi

and Philippe Ravier

LMSE Laboratory, Bordj Bou Arreridj University, Elanasser-Bordj Bou Arréridj 34030, Algeria

PRISME laboratory, University of Orléans, BP 6744 Orléans Cedex 2, France

Keywords: Face Recognition, Local Methods, Holistic Methods, Template Matching, Similarity, Mutual Information,

Histogram Approach, Bin Number Selection.

Abstract: In this paper, we investigate the binning problem of joint histogram estimation applied to mutual

information based face recognition application. Classical approaches for histograms estimation tend to

empirically fix the bin numbers. We evaluate in this work some state of the art rules for automatically

choosing the bin numbers. The face recognition problem has been studied in the case of local and holistic

methods. The choice’s performance has been evaluated using AT&T database with single sample in the

training set. The results show that better accuracy recognition rates can be achieved with data driven bin

number choices rather than fixed bin numbers. In the local method, the results show a higher robustness of

the automatic vs fixed bin number choice when the regions become smaller.

1 INTRODUCTION

Mutual Information (MI) has been generally used as

a measure of statistical dependency between two

random variables (Cover and Thomas, 2006) and has

therefore become as a popular similarity measure in

different signal and image processing applications

(Drugman, Gurban and Thiran, October 2007)

(Pluim, Maintz and Viergever, 2003). More

particularly, it represents the amount of the shared

information between random variables.

However the MI computation from data requires the

estimation of joint and marginal probability density

functions (pdfs) (Cover and Thomas, 2006), which

are not known practically and have to be estimated

with finite number of samples (Moddemeijer, 1989)

(Jain, Duin and Mao, 2000). This estimation can be

performed by different approaches such as

histogram (Moddemeijer, 1989)(Hacine-Gharbi et

al., 2012), Parzen Window (Kwak and Choi, 2002),

Gaussian Mixture Models GMM (Ait Kerroum,

Hammouch and Aboutajdine, 2010).

The histogram based approach for the mutual

information has been extensively used for its

undeniable advantages in terms of simplicity and

computational complexity (Legg et al., July 2007).

In the last decades, this approach has been widely

applied in the biomedical (Legg et al., 2013) (Pluim,

Maintz and Viergever, 2003) and biometric

(Nabatchian, Abdel-Raheem and Ahmadi, 2011)

fields as a measure of intensity similarity between

images for the image comparison task. In this

context, the evaluation of intensity similarity

between images is achieved by the joint intensity

histogram based mutual information estimation.

However, the bin number or equivalently bin width

is a crucial parameter that must be carefully selected

for the histogram construction in order to avoid a

large bias and high mean squared error (MSE)

estimation of mutual information (Hacine-Gharbi et

al., 2013) (Panzeri et al., 2007). In (Legg et al., July

2007) the authors have used the Sturges’ rule in a

computer vision problem for selecting the

appropriate bin number for histograms estimation.

Using the Sturges’ rule offers an improvement of the

accuracy and efficiency of the registration process

when applied on Fundus eye images. In (Legg et al.,

2013), the same authors state that there is no

definitive answer on the question of how many bins

should be used when constructing a histogram.

In the field of facial biometric recognition, a

mutual information based method has been proposed

in (Makaremi and Ahamdi, 2009). This method

locally compares the images by performing the

analysis on local regions, separately. For each

777

Hacine-Gharbi A. and Ravier P..

On the Bin Number Choice of Joint Histogram Estimation Applied to Mutual Information based Face Recognition.

DOI: 10.5220/0004925607770782

In Proceedings of the 3rd International Conference on Pattern Recognition Applications and Methods (ICPRAM-2014), pages 777-782

ISBN: 978-989-758-018-5

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

region, the mutual information is computed as a

measure of similarity between two images at the

same location. In the next, the averaged of the

mutual information estimates is considered. The

results show that the accuracy rate is dependent of

the bin number when chosen as a rule of thumb.

So, in the face recognition literature relating to

Mutual Information, most of papers use a fixed

number of bins. No rule for selecting the optimal

number of bins has been proposed. Hence, in this

paper, firstly we investigate the problem of choosing

the bin number for face recognition application

using different rules. Secondly, we investigate the

advantage of a global MI estimation instead of a

mean local MI estimation for the application

proposed in (Makaremi and Ahamdi, 2009). This is

motivated by the fact that previous studies already

shown the interest of comparing local and global

methods in the field of multimodal image

registration (Pluim, Maintz and Viergever,

2003)(Hermosillo and Faugeras, 2001).

2 MUTUAL INFORMATION

BETWEEN IMAGES

The Mutual Information (MI) quantifies the

information shared between two random variables.

Therefore, it is used generally to measure a

statistical dependency between variables (Cover and

Thomas, 2006). High mutual information means that

the variables are very dependent.

For two discrete random variables X and Y, with

joint Probability Distribution Function (PDF)





x,y



and marginal distributions p







and







respectively, the mutual information I(X;Y) of

X and Y or simply I

is defined as:



X;Y





∑∑



x,y



log







,













.









bit



(1)

In this paper, we exploit the histogram of

intensities of images to estimate the PDFs of facial

images. However, bin partitioning in histogram can

either be adaptive or uniform (Darbellay and Vajda,

1999) (Hacine-Gharbi et al., 2012). In (Makaremi

and Ahamdi, 2009), the authors have chosen a

uniform partitioning which gives the better accuracy

among a limited set of bin numbers. Hence this

requires a high computational cost and cannot

guarantee the optimal choice of bin number. To

overcome this problem, we propose different rules

that have already been proved to be useful in other

applications (Legg et al., July 2007)(Hacine-Gharbi

et al., 2012).

Let us use an N-sample dataset with standard

deviationσ. Sturges (ST) proposed a bin number

k1log



N (Sturges, 1926) while Scott (SC)

proposed a bin width Δ3.5σ

√



⁄

(Scott, 1979).

Freedman and Diaconis (FD) proposed the bin width

computation as Δ2.IQRX

√



⁄

, where the term

IQR stands for the interquartile range (Freedman and

Diaconis, 1981).

A novel approach for deriving the number of

bins has been previously proposed in (Hacine-

Gharbi et al., 2012) and (Hacine-Gharbi et al.,

2013). In (Hacine-Gharbi et al., 2012), the bin

number is chosen in such a way that the bias of MI

and entropy estimates is zero. In (Hacine-Gharbi et

al., 2013), the optimum number of bins is given by a

mean squared error minimization procedure of the

estimates. For the two discrete random variables X

and Y with standard deviation σ



and σ



respectively, and with extents



and A



respectively, and a correlation coefficientρ, the Low

Mean Square Error (LMSE) strategy for the

histogram based estimation of



X;Y



in eq. (1)

gives the following bin number (Hacine-Gharbi et

al., 2013):

kround





14

√

L

with L





.



































(2)

3 ESTIMATION OF MUTUAL

INFORMATION BASED ON

HOLISTIC AND LOCAL

METHODS

Generally, the face recognition methods can be

classified into two categories: holistic matching

methods and local matching methods. The first

holistic approach attempts to identify faces using the

whole face region, while other approaches use local

regions as inputs to a recognition system (Khan,

Javed and Anjum, 2005). In (Ruiz-del-Solar and

Navarrete, 2005) comparative studies of different

options in a holistic method have been discussed.

The options are related to the projection operation of

the faces which is helpful for data size reduction, the

matching criterion for the similarity measure and the

classification method employed. In (Zou and Nagy,

2007), the authors have investigated a comparative

study between local matching methods. The methods

can be divided into three important steps: 1)

alignment and partitioning, 2) feature extraction, and

ICPRAM2014-InternationalConferenceonPatternRecognitionApplicationsandMethods

778

3) combination / classification. The first step aims at

aligning each face into a common coordinate system

using affine transforms (translation, rotation and

scaling) that produce similar faces. The purpose is to

make the classification process insensitive to any

variation of the face pause conditions. This step is

often difficult. The face partitioning is then achieved

for local comparisons. The second step is the feature

extraction procedure that consists both of extracting

pertinent information from the data and reducing the

number of parameters for the classifier. The third

step is the classifier that works with the previous

input features. The local features can be combined

together before classification or considered as input

for individual classifiers which results are combined

for the final class assignment.

Regarding the last step, measures of similarity

between images are mandatory. This task can be

achieved using MI measures. Using this tool, the

recognition of an unknown facial image can be

achieved by searching for the class c (person) that

maximizes the MI between the unknown image and

each train image. Following this procedure, the

authors in (Pluim, Maintz and Viergever, 2003) have

mentioned that the MI measure can be estimated

globally, on the entire image, or locally, on a sub-

image. In particular in (Makaremi and Ahamdi,

2009), the MI has been estimated in local regions.

Each image is horizontally divided into two sub-

images (left and right sides) with an overlap. Next,

each side is divided into smaller overlapping strips.

So, each image of class c is represented by a set of

strips defined as(Makaremi and Ahamdi, 2009):



S

,



,S

,



for1 ,

where S

,



andS

,



are the ith strips of the right

and the left sides of the image respectively and M is

the number of strips. Moreover, the mean intensity

of each strip is displaced to zero and its histogram is

estimated. The MI between an unknown and

reference image is estimated as the average of the

MI between strips at the similar location. Also

mutual information between images with shifts has

been considered to avoid comparing strips that not

belong to the same face part.

Firstly, we investigate in this paper the choice of

bin number to avoid the binning problem in face

recognition. Secondly we investigate the MI based

recognition system with holistic and local approach.

A comparison between the two approaches is

therefore carried out with various image partitioning

and different parameter value choices in the

estimation algorithms.

4 EXPERIMENTAL RESULTS

In this section, we present different experiences in

which we discuss the problem of bin number choice

for histogram based approach of MI estimation.

Moreover, we present from these experiences the

performance of holistic and local methods. To

achieve these studies, we have chosen the Olivetti

Research Laboratory (ORL) database which consists

of 40 subjects each having 10 images (The Database

of Faces, 2002). This 400 images database has been

used in many scientific works as reference database

for face recognition (Makaremi and Ahamdi,

2009),(Khan, Javed and Anjum, 2005). The

subjects’ images have been taken with front view

(with tolerance for some side movement), varying

the lighting, facial expressions (open/closed eyes,

smiling/not smiling) and facial details (glasses/no

glasses) (The Database of Faces, 2002) (see figure

1).

Figure 1: Examples of images of Faces of 4 subjects (ORL

Database (AT&T Laboratories Cambridge,(The Database

of Faces, 2002)))

For the construction and testing of the

recognition system, one image is used for the

training stage and the nine other ones are used for

the testing stage. This experience is repeated ten

times: each time takes into account the following

image in the database for all the subjects as a new

training image. The accuracies are then computed as

the mean value of ten accuracy estimates.

In the two next sections we discuss the problem

of bin number choice in the cases of local method

and holistic (global) method respectively.

4.1 Face Recognition Accuracy using

Local Method

In order to study the effect of the choice of bin

number on system’s performance, we have chosen a

particular partitioning of the image as shown in

figure 2. In a first experience we have divided

horizontally the image into two sub-images (left and

OntheBinNumberChoiceofJointHistogramEstimationAppliedtoMutualInformationbasedFaceRecognition

779

right sides). Next, each side is divided vertically on

M regions (strips) with no overlapping avoiding

redundancy between strips.

Once the partition is fixed, the question of the

bin number choice that gives the better accuracy

now arises. One method consists in testing a bin

number choice within a large set of choices and to

evaluate the accuracy of the face recognition system

for each choice. This technique requires an

exhaustive search yielding to high computation

costs. To overcome this problem we propose a few

bin number estimation rules that exist in literature:

Scott (SC), Sturges (ST), Fredman (FD), Low Mean

Square Error (LMSE) rules respectively.

Figure 2: Images partitioning.

Table 1 displays the accuracy rates vs the bin

number (BN) and the number of strips (M) while

table 2 displays the accuracy rates considering four

rules that estimate the bin number with different

numbers of strips M.

Table 1: Accuracy rates as a function of the Number of

Bins (NB) and the number of strips (M) in the case of the

local method

2 5 10 20 30 50 100

1 56.88 64.00 63.11 63.03 62.39 61.27 55.88

2 53.11 59.06 58.67 58.61 58.06 56.64 49.44

4 51.06 55.97 55.86 54.64 54.75 54.17 42.03

8 47.19 51.31 51.81 51.81 52.28 50.11 25.92

16 43.28 47.00 48.17 48.81 48.72 40.41 18.41

The following remarks and explanations can be

made from Table 1:

 Even if optimum values exist in Table 1, the

accuracy rate remains approximately the same

(with small variations, less than 2 points) when the

NB changes from 5 to 30.

 The accuracy of the local method using 2

bins with smaller strips size is better than the

method using 100 bins. This can be explained by

the insufficient number of samples that cause

possible empty bins in the case of 100 bins.

 The accuracy rate decreases when the number

of strips M grows, whatever the BN value. This

can be firstly explained by the local strategy that

greatly suffers from misalignments that may

appear between images.

Actually, searching for the optimal alignment is a

very difficult task that has not been performed in

this study. Secondly, the combination of the local

image features (MI between image strips) or the

combination of the local classifiers (one classifier

per image component or feature) may influence the

results. We investigate in this study the average

value of the features, whereas other combination

exist (Borda counting, majority voting…) (Zou

and Nagy, 2007).

 The results show a higher robustness of the

automatic vs fixed bin number choice when the

regions become smaller.

Table 2: Accuracy rates for four bin number estimation

rules: SC, ST, FD, LMSE with different numbers of strips

(M)

Rule

SC ST FD LMSE

1 64.36 63.06 65.19 64.83

2 61.19 59.06 62.58 61.11

4 57.94 55.81 58.89 59.14

8 54.64 51.69 55.03 54.61

16 51.33 48.17 51.72 51.25

The Table 2 shows the gain that can be obtained

with an efficient automatic bin number searching

procedure. Indeed, the best accuracy rates given by

the automatic estimation of the BN is 3 points

superior to the fixed BN case, comparatively, except

for the case M=1. The ST method gives the poorer

results since this method only takes into account the

number of bins N in the BN calculation while the

other methods take into account other statistical

information about the data (variance for SC, IQR for

FD and variance and correlation for LMSE).

Actually, the ST method is equivalent to taking a

fixed BN for all the histograms to be estimated.

4.2 Face Recognition Accuracy using

Holistic Method

Table 3 displays the accuracy rates vs the bin

number (BN) and the number of strips (M) in the

holistic case while table 4 displays the accuracy

rates considering four rules that estimate the bin

number with different numbers of strips M.

Table 3 shows similar results comparatively to

those obtained in the local case (Table 1) with

slightly better results when the NB increases for the

holistic method.

ICPRAM2014-InternationalConferenceonPatternRecognitionApplicationsandMethods

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Table 3: Accuracy rate (ACC) as a function of the Number

of Bins (NB) in the case of holistic (global) method.

BN 2 5 10 20 30 50 100

ACC 56.63 63.63 64.36 64.61 64.52 64.19 61.19

Table 4: Accuracy rate for the four bin number estimation

rules: SC, ST, FD, LMSE in the case of holistic (global)

method

Rule SC ST FD LMSE

Accuracy rate 66.11 64.52 66.52 65.36

Table 4 shows that the FD method gives the best

results with 2 points more with respect to the best

result obtained by exploring different fixed BN

values displayed in Table 3. This proves that

searching for an optimal NB can be attractive

practically. The results also show that better

accuracy recognition rates can be achieved with data

driven bin number choices rather than fixed bin

numbers. Indeed, the rule proposed by Sturges only

depends on the sample number N and this is

equivalent to a fixed bin number since all the images

have the same size (112x92 samples). The bin

number choice for the other approaches is data

driven since the numbers either depend on the

standard deviation (SC), the IQR (FD) and the

correlation coefficient and standard deviation

(LMSE).

Since the LMSE method gives an optimal BN

that depends on the correlation parameter between

the data (the correlation is not taken into account

with the three other methods SC, ST, FD), the

accuracy rate for this method should be better since

intra class images are highly correlated. However,

the LMSE method assumes normal distribution of

the data. This hypothesis is not fulfilled with the

pixel distribution of the image strips of the full

image.

However, it should be noticed that the

computational cost for the LMSE method is far

lower than the FD method.

5 CONCLUSIONS

In many works in the literature dealing with face

recognition, local methods are shown to be

preferable than global ones (Hermosillo and

Faugeras, 2001). However, many databases need

alignment procedures. This task is often difficult to

achieve. In this paper, we show that without the

alignment procedure, holistic methods are better

than local ones with the ORL database using MI

based similarity measures.

The automatic BN searching procedure on the data

tended to improve the accuracy rate in a recognition

faces system based on MI histogram estimations.

The searching procedure is based on criteria

which are sensitive to the statistical properties of the

data. Adapting the criterion to the exact properties of

the data constitutes a perspective of this study.

Other works will be investigated using more

“difficult” datasets including effects of illumination

variation, pose, and facial expressions. A deeper

comparison of local and global approaches will be

carried out using such databases (e.g. the extended

Yale face database B).

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