
Iranian Journal of Electrical & Electronic Engineering, Vol. 4, Nos. 1 & 2, Jan. 2008 48
a =P q =s |q =s , 1 i,j N , 0 a 1
t
t+1
(1)
N
ij
j=1
∑
(2)
• B={b
j
(k)} is the observation symbol probability
matrix, where:
b (k)=P o =v |q =s , 1 j N, 1 k M
t t
j k j
(3)
• π={π
1
,π
2
,…,π
N
} is the initial state distribution,
where:
1
i i
(4)
Using shorthand notation HMM is defined as following
triple:
(5)
As said before HMMs generally work on sequences of
symbols called observation vectors, while an image
usually is represented by a simple 2D matrix. In the case
of using a one dimensional HMM in face recognition
problems, the recognition process is based on a frontal
face view where the facial regions like hair, forehead,
eyes, nose and mouth come in a natural order from top
to bottom. In this paper we divided image faces into
seven regions which each is assigned to a state in a left
to right one dimensional HMM. Figure 1 shows the
mentioned seven face regions.
Figure 2 shows equivalent one-dimensional HMM
model for a partitioned image into seven distinct regions
like figure 1.
Fig. 1 Seven regions of face coming from top to down in
natural order.
Fig. 2 A one dimensional HMM model with seven states for a
face image with seven regions.
The main advantage of the model above is its simple
structure and small number of parameters to adjust.
In previous researches, the eyebrows and chin regions
weren’t considered and the models have five states.
Also the regions were featured using gray levels or
some transforms like DCT and wavelets. We use SVD
coefficients. The next section is a brief description of
the SVD.
2.2 Singular Value Decomposition
The Singular Value Decomposition (SVD) has been an
important tool in signal processing and statistical data
analysis. Singular values of given data matrix contain
information about the noise level, the energy, the rank
of the matrix, etc. As singular vectors of a matrix are the
span bases of the matrix, and orthonormal, they can
exhibit some features of the patterns embedded in the
signal. SVD provides a new way for extracting
algebraic features from an image.
A singular value decomposition of a m×n matrix X is
any function of the form:
X=U
(6)
where U(m×m) and V(m×m) are orthogonal matrix, and
Σ is and m×n diagonal matrix of singular values with
components
ij
and
ii
. Furthermore, it
can be shown that there exist non-unique matrices U
and V such that
1 2
. The columns of the
orthogonal matrices U and V are called the left and right
singular vectors respectively; an important property of
U and V is that they are mutually orthogonal [21].
The main theoretical property of SVD relevant to face
image recognition is its stability on face image. Singular
values represent algebraic properties of an image [22].
So because of these reasons and some experimental
results, we find out that SVD is a robust feature
extraction technique for face images.
3 The Proposed System
3.1 Filtering
Most of the face recognition systems commonly use
preprocessing to improve their performance. In the
proposed system as the first step, we use a specific filter
which directly affects the speed and recognition rate of
the algorithm. Order-statistic filters are nonlinear spatial
filters. Their operations are as follows; a sliding window
moves from left to right and top to down with steps of