despite large changes in the visual stimulus due to viewing conditions, expression, aging, and distractions such as glasses or changes in hairstyle or facial hair.
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Eigenfaces Recognition Technique for Verifying Noisy Facial Images S.M. Ali and Hiba Saadoon Ghanem Remote Sensing Research Unit, College of Science, University of Baghdad,
:اﻟﺨﻼﺻﺔ ﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ﺘﻘﻨﻴﺔ ﺍﻟﻭﺠﻭﻩ ﺍﻟﺫﺍﺘﻴﺔ ﻟﺘﺸﺨﻴﺹ ﺼﻭﺭ ﻟﻭﺠﻭﻩ ﺒﺸﺭﻴﺔ، ﻓﻲ ﻫﺫﺍ ﺍﻟﺒﺤﺙ ﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ﻁﺭﻴﻘﺔ ﺍﻟﻤﺭﻜﺒﺎﺕ.ﻤﺘﺭﺩﻴﺔ ﺒﺈﻀﺎﻓﺔ ﻜﻤﻴﺎﺕ ﻤﺨﺘﻠﻔﺔ ﻤﻥ ﺍﻟﻀﻭﻀﺎﺀ
ﻟﻭﻭﻑ ﺍﻟﺭﻴﺎﻀﻴﺔ ﻟﺤﺴﺎﺏ-" ﺍﻟﻤﻌﺘﻤﺩﺓ ﻋﻠﻰ ﺘﺤﻭﻴﻠﺔ ﻜﺎﺭﻫﻨﻭﻥPCA" ﺍﻷﺴﺎﺴﻴﺔ - ﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ﻤﻌﻴﺎﺭ ﻗﻴﻡ ﻤﻌﺩل.ﺍﻟﻤﺘﺠﻬﺎﺕ ﺍﻟﺫﺍﺘﻴﺔ ﻟﻠﺼﻭﺭ ﺍﻟﻤﺴﺘﺨﺩﻤﺔ ﻓﻲ ﺍﻻﺨﺘﺒﺎﺭ ﺍﻟﺨﻁﺄ ﺍﻷﺩﻨﻰ ﻟﺤﺴﺎﺏ ﺩﺭﺠﺔ ﺍﻟﺘﺸﺎﺒﻪ ﺒﻴﻥ ﻤﺠﻤﻭﻋﺔ ﺍﻟﺼﻭﺭ ﺍﻟﻤﺤﻔﻭﻅﺔ ﻓﻲ-ﻤﺭﺒﻊ
ﺃﻨﻭﺍﻉ ﻤﺨﺘﻠﻔﺔ ﻤﻥ ﺍﻟﻀﻭﻀﺎﺀ.ﺍﻟﺒﻴﺎﻨﺎﺕ ﻭﺒﻴﻥ ﺍﻟﺼﻭﺭ ﺍﻟﻤﻁﻠﻭﺏ ﺘﺸﺨﺼﻴﻬﺎ-ﻗﺎﻋﺩﺓ
ﻤﻠﺤﻴﺔ ﻭ ﺒﻬﺎﺭﺍﺘﻴﺔ( ﻭﺒﻘﺩﺭﺍﺕ ﻤﺨﺘﻠﻔﺔ ﺘﻡ ﺇﻀﺎﻓﺘﻬﺎ ﻋﻠﻰ ﺍﻟﺼﻭﺭ، ﻤﻨﺘﻅﻤﺔ،)ﻜﺎﻭﺴﻴﺔ ﺃﺜﺒﺘﺕ ﻫﺫﻩ.ﺍﻟﻤﺨﺘﺒﺭﺓ ﻟﺤﺴﺎﺏ ﻓﺎﻋﻠﻴﺔ ﺍﻟﺨﻭﺍﺭﺯﻤﻴﺔ ﺍﻟﻤﺴﺘﺨﺩﻤﺔ ﻓﻲ ﻋﻤﻠﻴﺔ ﺍﻟﺘﺸﺨﻴﺹ
ﺍﻟﺨﻭﺍﺭﺯﻤﻴﺔ ﻗﺩﺭﺘﻬﺎ ﺍﻟﺠﻴﺩﺓ ﻋﻠﻰ ﺘﺸﺨﻴﺹ ﺼﻭﺭ ﺍﻟﻭﺠﻭﻩ ﻟﻘﺩﺭﺍﺕ ﻤﺘﻔﺎﻭﺘﺔ ﻤﻥ .ﺍﻟﻀﻭﻀﺎﺀ
Abstract: In this paper, the eigenfaces recognition method is used to verify noisy images of human faces. The Principal Component Analysis “PCA” which is based on the implementation of the Karhunen-Loeve “KL” transformation is used to compute the Eigenvectors of the test’s images. The similarity between the preserved test’s faces (as Database record) and the input test’s face is performed by utilizing the Mean-Square-Error “MSE” criterion. Different amount of noises (Gaussian, Uniform, Saltand-Pepper) have been added on the test’s image face and compared with preserved faces to deduce the verification reliability of the utilized recognition method. 107
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1. Introduction: The face is our primary focus of attention in social intercourse, playing a major role in conveying identity and emotion. We can recognize thousands of faces learned throughout our life time and identify familiar faces at a glance even after years of separation. This skill is quite robust, despite large changes in the visual stimulus due to viewing conditions, expression, aging, and distractions such as glasses or changes in hairstyle or facial hair. As a consequence, the visual processing of human faces has fascinated philosophers and scientists for centuries, including figure such as Aristotle and Darwin [1]. Computational models of face recognition, in particular, are interesting because they can contribute not only to theoretical insights but also to practical applications. Computers that recognize faces could be applied to a wide variety of problems, including criminal identification, security systems, image and film processing, and human-computer interaction [2]. 2. Face Recognition using Eigenfaces: Eigenfaces approach is a principal component analysis method, in which a small set of characteristic pictures are used to describe the variation between face images. As a general view, this algorithm extracts the relevant information of an image and encodes it as efficiently as possible. For this purpose, a collection of images from the same person is evaluated in order to obtain the variation. The goal is to find out the eigenvectors of the covariance matrix of the distribution, spanned by a training set of face images. Later every face image is represented by a linear combination of these eigenvectors [1, 3]. 3. Principal Component Analysis (PCA): The principal component analysis is one of the most successful techniques that have been used in image recognition and compression. The idea behind using PCA for face recognition is to express the large 1D vector of pixels constructed from 2-D facial image into the compact principal component of the feature space. This is called Eigen-Space projection, which can be calculated by identifying the eigenvectors of the 108
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covariance matrix derived from a set of facial images (vectors). The mean advantage gained from the principal component analysis is that: it can be used to compress the data size, by reducing the number of dimensions, without much loss of information [4]. 4. Noise Models Noise is any undesired information on image. The principal sources of noise in digital images arise during image acquisition (digitization) and/or transmission [5]. In this paper the effects of adding different amount of noise (i.e. Gaussian, Uniform, and Salt-Pepper) on the eigenfaces recognition reliability will be studied; 4.1 The Gaussian Noise: Because of its mathematical tractability in both the spatial and frequency domains, Gaussian (also called normal) noise model is the most frequent considered model in most image processing implementation. The Probability-Density-Function “PDF” of the Gaussian random variable isgiven by [5]: ρ ( x) =
1
( x− m) 2
e
2σ 2
(1) 2πσ 2 Where x is the gray level, m is the mean, and σ is the standard deviation. 4.2 The Uniform Noise: With the uniform distribution, the gray level values of the noise are evenly distributed across a specific range (e.g. 0 → 255 for 8-bits image), The PDF of uniform noise is given by [5]:
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1 if a ≤ x ≤ b b − a ρ ( x) = 0 Otherwise The mean of this density function is given by : 1 µ= b−a and its variance by
(2)
(b − a ) 2 12 Where: x is the gray level, a and b are any range values. σ2 =
4.3 The Impulse Salt-and-Pepper Noise: In this noise model there are only two possible values; i.e. a and b. The probability of each is typically less than 0.1, with numbers greater than 0.1, the noise dominating the image signal. The PDF of this noise model is given by [6]; ρ a , if x = a ρ ( x) = ρ b , if x = b (3) 0, Otherwise Where: x is the gray level, a is the starting range (appear dark dot), while b is the ending range (appear light dot), and ρa& ρb are the pepper and salt probabilities, respectively. 5. Computation of the Eigenface: Once we acquire a database of facial images [as shown in figure (1)], we can running forward to determine their eigenface, as follows
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(1)
(6 )
(2)
(3)
(7)
(8)
(4)
(9)
(5)
(10)
(11)
Figure (1): Training set of eleven facial images, each of 256×256 pixels size. A two dimensional image f(x, y) can be considered as a one dimensional vector of N2 elements, for images of N×N pixels. These vectors can be arranged in an array, as column vectors, each one after the other; as given below:
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image1 image2 f (1) f2 (1) 1 f1 (2) f2 (2) f2 (3) f1 (3) . . . . f ( N) ( f 2 N) 1 . . . . . . 2 2 f1 ( N ) f2 ( N )
. . . . . . . . imagem . . . . . . . . fm (1) . . . . . . . . f m ( 2) . . . . . . . . fm (3) . . . . . . . . . . . . . . . . . . . . . . . . . . fm ( N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . fm ( N 2 )
The average face of the set is defined by: −
m
f (k) = ∑ fi (k),
for k = 1,2,..., N 2
i =1
(5)
This average face can be rearranged again as a 2D function, shown in figure (2);
Figure (2): The average face of the training set. The average reduced image can now be computed by subtracting the average face elements from the corresponding training faces elements; given by: m
−
Ri (k ) = ∑ [ fi (k) − f (k )], i =1
112
for k = 1,2,..., N 2
(6)
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Where Ri(k) referred as eigenfaces; each can be reformatted back into 2D image function given by fR(x, y); as shown in figure (3). The covariance matrix of the mean reduced matrix can now be computed as:
Cm, m = RmT , N 2 RN 2 , m
(7)
Where: “T” represents the matrix transposition. As can be deduced by glancing to eq. (7) above, the covariance matrix is a 2D array of dimensionality equal to the number of the training images.
(1)
(6)
(2)
(3)
(7)
(4)
(8)
(5)
(9)
(10)
(11)
Figure (3): Mean reduced eigenfaces of the training set of images. 113
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The covariance “C” is a real symmetric matrix has m eigenvalues “λi” and eigenvectors “Vi”; illustrated by: CVi = λiVi , for i = 1,2,..., m
(5)
The eigenvectors for the set of eigenfaces have been computed and listed in Table-1. Note, the eigenvectors elements have been reduced to only 3 decimal values. Now if a new face test image is entered to be verified as to be existed as one of the trained facial images or not, the training set matrix becomes as; image1 f (1) 1 f1 (2) f1 (3) . . f ( N) 1 . . . 2 f1 ( N )
image2 f 2 (1)
Test image f m+1 (1) f m (2 ) f m+1 ( 2) f m (3) f m+3 (3) . . . . fm ( N ) f m+1 ( N ) . . . . . . f m ( N 2 ) f m+1 ( N 2 )
. . . . . . . imagem . . . . . . . f m (1)
f 2 ( 2) f 2 (3) .
. . . . . . . . . . . . . . . . . . . . .
. f2 ( N )
. . . . . . . . . . . . . .
. .
. . . . . . . . . . . . . .
. . . . . . . . f 2 (N 2 ) . . . . . . .
Example of the test image as one of the trained set is shown in figure (4).
Figure (4): Test image represent the first image in the training set [ordered as (12)]. Following the same procedures as being done for the previous training set, the eigenvectors for the new training set of images will be as given in Table-2. The similarity between the verifying face and the training set can now be represented by the minimum distance test given by [6]: 114
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m
Min{MSEk = MIN{∑ (Vk ,i −VM +1,i ) 2 }, for k = 1,2,......m i =1
Utilizing the above relationship, the Min {MSE} is obvious [see Table-2] has been found (≅ 3.1156×10-15) between V1 and V12; i.e. the test facial face represents the first in the training set. In an attempt to see the validity of this algorithm on a test image for an existed person but with different view, we have selected the facial image shown in figure (5); set. also representing the first in training
Figure (5): the test image, different view of the 1st in training set. Again, the Min {MSE} has been found to be between the 1st and last eigenvectors; i.e. =1.8197. 6. The Effect of Noise: The effect of the noise’s models discussed in section(4) above on has been tested by adding different amount of them on an existed facial image in the training set and counting the Min{MSE}. Examples of the tested noisy images (the amount of the added noises and Min {MSE} for each are mentioned below them) are shown in figure (6).
Salt and pepper ρa and Salt and pepper ρa and ρb = 0.06 2.3187 ρb = 0.02 0.77505
Gaussian Gaussian σ2=600, µ=0 µ=0 2.2668 2.4493
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Uniform σ2=5500, σ2=4200, µ=0 2.576
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Figure (6): different amount of the added noise and its min{MSE} Conclusion: The validity of the principal component eigenvector recognizing algorithm has been studied and discussed against various amount of noise powers, and for different noises models. It has been found that; for lowpower of additive noises (i.e. Uniform, Gaussian, Impulse Salt-Pepper), the algorithm run adequately and identify the exact facial images. Further works may be required to test the validity of the presented recognizing algorithm for changes in image scaling and for rotations (with little angels). It is our opinion that; the presented technique could be applied for other types of images; e.g. fingerprints, eye’s-iris, hands, and might be for voices signals also. References: 1) 2)
3) 4) 5) 6) 7)
Turk M. and pentland A., "eigenfaces For Recognition", journal of cognitive neuroscience, Vol. 3, No. 1, 1991 Turk M and Pentland A., "Face recognition using eigenfaces", Proc. IEEE Computer Society Conf. Computer Vision and Pattern Recognition, pp. 586-591, 1991 Jorge O., "Face Recognition Techniques", Project report no.533, university of Wisconsin Lyon D. and Vincent N.," Interactive Embedded Face Recognition", Journal of object technology, vol.8, no. 1, 2009 Mahmood Mohamed Ali ,"image enhancement using wavelet transform", M.Sc. thesis, university of technology,2005 R.C. Gonzalez, and R.E. Woods, “Digital image processing,” 2nd Edition, Prentice Hall, 2001. Dangeti S., "Denoising Techniques-A comparison", M.SC. Thesis, Louisiana state university and agricultural and mechanical college, 2003
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