The reduction of wavelet transform coefficients which represent the face in
the image is the most significant problem due to the redundancy of features
which are not required in Discrete Wavelet Transform (DWT) and these features
in turn draws a negative effect on the classification process. This problem
refers to the properties of wavelet functions (families) which are used in particular
application to produce the coefficients to represent the features. In the case
of face classification, some of these coefficients dont have face information
that leads to increase the error rate of classification. For this problem, its
important to reduce the coefficients by choosing those coefficients that contain
face information and ignoring the remaining.
Wavelet analysis, as opposed to Fourier analysis, offers extra freedom as the
selection of atom of the transform inferred from the analysis of wavelet is
left to the user. Moreover, according to the objectives of wavelet processing
(Misiti et al., 2006), they prefer the continuous
transform to the discrete transform, if the redundancy is useful to analyze
the signal and make opposite choice, for signal compression. In the latter case,
it is preferable to use the filters with wavelets whereas in the former case
almost any zero integral function is appropriate. The Haar base (Karimi
et al., 2006) which appeared at the beginning of the last century
was replaced by its successors, Gaussian Morlet wavelets (Dan
and Wentao, 2009), Meyer wavelets (Xia and Suter, 1996)
and Daubechies wavelets (Vonesch et al., 2007).
Among the new wavelets transforms, Daubechies wavelets are the most widely used.
Besides, numerous wavelets regularly appear in the books and are made available
in specialized software application. Construction of new wavelets was very intense
in the first ten years after their introduction but recently, it has become
less regular and bears on increasing specific goals, often associated with limited
In facial images, the features extracted by wavelet function are used to classify
the face. In the study of Ghosal et al. (2009),
Gabor wavelets are used to extract the features for face recognition but dealing
with all Gabor features is consider as a time consuming process. To resolve
this problem, they proposed the use of Random Forests variable importance
computation feature. This computation feature is used to compute the most important
Gabor features. This in turn, reduces the feature space by more than hundreds
of features and speeds up the classification process. Eltoukhy
et al. (2009) proposed computer aided diagnosis system based on
wavelet analysis multi-level decomposition using three types of wavelet function
(db 8, sym 8 and bior 3.7) to extract the features and Euclidean Distance Method
(EDM) as classifier to increase diagnostic accuracy. They also solve the problem
of coefficients redundancy by selecting the biggest hundred coefficients
from each level of decomposition and then passing these coefficients to EDM
In this study, wavelet analysis multi-level of decomposition is used to transform
the facial images by using Symlet wavelets in order to produce features coefficients
and then the proposed statistical model is applied to these coefficients to
reduce the redundancy and deleting all coefficients that dont have information.
Finally, the last stage uses EDM as a simple classifier. Also the comparison
between the method in (Eltoukhy et al., 2009)
when it was used for face classification and the proposed method which uses
two different databases and the result showed the increase in classification
accuracy up to 40%.
Symlets (Guo-Sheng et al., 2006) form a family
of almost similar wavelets proposed by Daubechies adjusting the construction
of dbN. From the symmetry, the other properties of the two families are similar.
An example of Symlets of order 2 to 8 (sym 1 is simply the Haar wavelet) is
represented in Fig. 1.
From, Misiti et al. (2006), the idea of construction
consists of re-using the m0 function introduced for dbN, consider
| m0 (ω) |2 as function W of the variable z = eiω.
We can factorize W in various manners in the form of W (z) = U (z) U (z-1),
since the roots of W with module different from 1 go in pair: if z1
is a root then z1-1 is also a root.
|| Symlets: sym 2
By constructing U so that its roots are all of module<1, we construct the
Daubechies wavelets dbN. The filter U has minimal phase. Another option, attained
by optimizing factorization so that the filter U has almost linear phase and
produces much more symmetric filter: the symlets.
The Euclidean Distance Method (EDM): Euclidean Distance Method (EDM)
is the simplest classifier and easy in implementing as it was explained by Eltoukhy
et al. (2009). For each class, the class core vector is the mean
of 50% of the class vectors as in Eq. 1. For a new feature
vector, the distances between feature vector and the class core vectors a re
calculated using Eq. 2. The system automatically classifies
the feature vector in the class for which the distance obtained is the smallest.
where, Vj is the coefficient vector for each training image, j is
the index of vector N is the number of images used for training, Dist is the
calculated distance between the tested image and every core vector, k is the
length of vectors. Vitest the feature vector of face to
be classified and Vicore is the vector core of each class.
The facial images that are used in testing the effectiveness of the proposed
system are decomposed into four levels of decomposition using sym 8. The resulting
tables showed the distribution of used face images over different classes. For
each experiment, a class core vector is calculated for each class using Eq.
1 and then all the features extracted from the ORL database and YALE database
are used in the testing phase including those used to produce the class core
vectors. Equation 2 is used to measure the distance between
the coefficient vector and each available class core vectors.
THE PROPOSED STATISTICAL MODEL
Suppose, m1, m2 and m3 are the mean of class1,
class2 and class3, respectively and mT is mean
total of all classes. Three matrices are as shown in Fig. 2.
From the Fig. 2 it can be seen that the coefficients extracted
from the classes are good for classification process that leads to increase
|| For three classes without overlapping
|| For three classes with overlapping
From Fig. 3, it can be seen that the coefficients extracted
from the classes are not good for classification process due to the overlapping
between the classes that leads to increase the probability of error.
Now from Fig. 3, use the following formula:
where, mT is the total mean, mi is class mean, i is index
of the class and n the number of classes.
So the variance of mi is:
where, ni is the number of the features in class i.
The metric obtained from the Fig. 3 is not efficient for
features extraction of wavelet transform due to overlapping between classes.
So we will use another metric from Fig. 2 as follow:
where, Vari variance of the class i.
Now the way to select the desired features coefficients will be as follow:
If Var_mod≤1 delete all coefficients belong to this column and they did
not take in our consideration, otherwise we keep it.
To calculate the error probability we will take this example:
Suppose we have two classes as shown in Fig. 4:
From each class of any wavelet coefficients we can approximate it to follow
Chosen Feature Extraction of class1 ~ Gaussian (m1, σ1)
Chosen Feature Extraction of class2 ~ Gaussian (m2, σ2)
where, x is the random variable, jth is feature and i is class index.
Using our classifier, the probability of error to classify the variable X is:
||Histogram of the features coefficients of two classes
RESULTS AND DISCUSSION
The trend of low frequency coefficients of face images reduce the coefficients
number and minimize the effect of varying facial expressions, shadow and variance
in pose. These low frequency coefficients of ten images for randomly chosen
15 individuals out of a total of 40 individuals in ORL database of each class
are retained as feature vector for classification while eleven images were chosen
for each of the 15 individuals in YALE database. For matching and classification
purpose in feature space, a natural choice is Euclidian distance metric:
The simulation result show that if we take the threshold to be equal to 1,
the accuracy can be improved from 88% up to 99.33% on ORL database and from
63.03% up to 87.88% on YALE database.
||Face images in the Yale face database (Top) and the ORL face
Olivetti research laboratory face database: The presented statistical
model is evaluated on the Olivetti Research Laboratory face database (ORL) (http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html).
This database contains 10 different images for each of 40 people. The images
of the same person are taken at different times, under slightly varying lighting
conditions and different facial expressions. Some images have been captured
with and without glasses. The head of the individuals in the images is slightly
tilted or rotated.
Yale university database: The second database is YALE database which
is established by Yale University. Eleven images of 15 persons with total of
165 images for our experiment. The images of each person are with different
facial expression or configuration: like center-light, with/without glass, happy,
sad, left-light with/without glass, normal, right-light, sad and sleepy, surprised
and wink. Few examples of these images are shown in Fig. 5.
In previous experiments the idea of Eltoukhy et al.,
(2009) for selecting the biggest coefficients from the each level of decomposition
were tested. During the previous tests it was found that the Symelt wavelets
were the best wavelets for face classification. The selection of Symlet wavelets
is based on their properties (Guo-Sheng et al., 2006).
The main problem in method (Eltoukhy et al., 2009)
is that it is sensitive to variation in pose, illumination and expressions.
The selection of the biggest coefficients doesnt represent the important
face features and has an impact on the classification process in general. In
the experiments, distribution of the features was given more attention and two
statistical methods were used to select the significant features based on this
distribution. Therefore, the effectiveness of the classification process is
less sensitive to any type of variation.
||Comparison between the biggest coefficients method and the
new two variance methods
|| The accuracy with different threshold values with YALE database
|| The accuracy with different threshold values with ORL database
In Table 1 the comparison between the method by Eltoukhy
et al. (2009) when it is used for face recognition (classification)
and the proposed methods has been given. The simulation result shows that accuracy
improvement has been achieved from 88 up to 99.33% on ORL database and from
63.03 up to 88.48% on YALE database.
The method in the first row takes the biggest 200 coefficients for classification
process then the result shows that this method is not efficient for representing
The two methods proposed in row 2 and 3 in Table 1 resulted
in the increase of classification accuracy when they are applied on the same
two databases and results were close to each other.
Table 2 and 3 show the optimum threshold
to reduce features extracted by Symlet wavelets. In the simulation, the original
features extracted by symlets wavelets were 125440 for YALE database and 22848
for ORL database. But this number of coefficients has been significantly reduced
to the values indicated in Table 2 and 3
for different threshold limits by applying the statistical model. In Table
2, it can be seen that accuracy achieved in row 4 is the highest compared
with the rest. In Table 3, the result shows that the accuracy
values remained at 99.33% until a threshold value of 1.7 and reduced down to
98.67% for threshold value of 2.
In Table 4 and 5 other values of threshold
have been tested to ensure that the optimum threshold values found in Table
2 and 3 is the optimum value. In addition, it can be seen
that the highest threshold values were chosen for the test and the result indicates
the decrease in the classification accuracy.
||The accuracy with other different threshold values with YALE
||The accuracy with other different threshold values with ORL
In this study wavelet analysis of multi-level of decomposition is used to transform
the facial images by using Symlet8 wavelets in order to produce features
coefficients and two new statistical models were applied to these coefficients
to reduce the redundancy ignoring all coefficients for which threshold value
are lower. At last, EDM was used as a simple classifier. The results showed
that, the two new variances increased the accuracy up to 40% which is more than
the accuracy obtained when using the biggest wavelet coefficients. Besides,
the simplest classifier affects the classification accuracy. From the final
results, it was found that the optimum threshold values were (0.065) for the
first method and (1.2) for the second method.
Moreover, works are underway to further increase the classification accuracy
by using new statistical methods.