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Face Recognition Using Fuzzy Integral and Wavelet Decomposition Method Keun-Chang Kwak, Member, IEEE, and Witold Pedrycz, Fellow, IEEE
Abstract—In this paper, we develop a method for recognizing face images by combining wavelet decomposition, Fisherface method, and fuzzy integral. The proposed approach is comprised of four main stages. The first stage uses the wavelet decomposition that helps extract intrinsic features of face images. As a result of this decomposition, we obtain four subimages (namely approximation, horizontal, vertical, and diagonal detailed images). The second stage of the approach concerns the application of the Fisherface method to these four decompositions. The choice of the Fisherface method in this setting is motivated by its insensitivity to large variation in light direction, face pose, and facial expression. The two last phases are concerned with the aggregation of the individual classifiers by means of the fuzzy integral. Both Sugeno and Choquet type of fuzzy integral are considered as the aggregation method. In the experiments we use n-fold cross-validation to assure high consistency of the produced classification outcomes. The experimental results obtained for the Chungbuk National University (CNU) and Yale University face databases reveal that the approach presented in this paper yields better classification performance in comparison to the results obtained by other classifiers. Index Terms—Classifier aggregation, face databases, face recognition, Fisherface method, fuzzy integral, wavelet decomposition.
I. INTRODUCTORY COMMENTS
F
ACE recognition is one of the most interesting and challenging areas in computer vision and pattern recognition. The popular approaches for face recognition are the eigenface and Fisherface method. The eigenface method, or principal component analysis (PCA), is the most well-known method for face recognition [1]. Each of them comes with some advantages but is not free from limitations and drawbacks when cast in the setting of face recognition. PCA is a popular approach in image processing and communication theory that is quite often referred to as a Karhunen–Loeve (KL) transformation. The PCA approach exhibits optimality when it comes to dimensionality reduction. However, it is not ideal for classification purposes as it retains unwanted variations occurring due to diversified lighting and facial expression [2]. To overcome this problem, Manuscript received July 20, 2003; revised November 30, 2003. This work was supported by the Postdoctoral Fellowship Program of the Korea Science and Engineering Foundation (KOSEF), the Canada Research Chair (CRC) Program, the Natural Sciences and Engineering Research Council (NSERC), and the Alberta Software Engineering Research Consortium (ASERC). This paper was recommended by Associate Editor I. Bloch. K.-C. Kwak is with the Department of Electrical Engineering, Chungbuk National University, Cheongju, 361-763 Korea. W. Pedrycz is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, T6G 2G7 Canada and also with the Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSMCB.2004.827609
proposed was an enhancement known as a Fisherface method, or Fisher’s linear discriminant (FLD), linear discriminant analysis (LDA) [2]. This statistically motivated method maximizes the ratio of the determinant of between-class scatter matrix and within-class scatter matrix and in this sense attempts to involve information about classes of the patterns under consideration. In general, this method is used in conjunction with the PCA where the PCA technique first projects the set of images to a lower-dimensional space so that the resulting within-class scatter matrix, to be used by the FLD, becomes nonsingular. FLD is capable of forming well-separated classes in a low-dimensional subspace, even under severe variation in lighting and facial expressions. There are various enhancements to the generic form of the FLD technique such as enhanced FLD [3], direct LDA [4], kernel LDA [5], and uncorrelated discriminant transformation [6]. Furthermore, other methods of face recognition have been researched including PCA mixture model [7], most discriminating feature (MDF) [8], independent component analysis (ICA) representation [9], and elastic bunch graph matching [10], to name a few representative directions. It is also worth mentioning a host of methods arising in the realm of computational intelligence (CI) [11], [12] in which we witness a diversity of synergistic links emerging between neural networks, fuzzy sets, and evolutionary computing. On the other hand, the recent trend of approaches in face recognition involves wavelet-based methods. In the previous works, the wavelet transform has been applied to image processing and texture classification with an objective to carry out a comprehensive multiresolution decomposition. The goal of the wavelet decomposition was to realize cleaning and compressing signals, or images that is the part of compression, denoising, and feature detection in images. Li [13] presented a novel combination of wavelet techniques and eigenfaces method to extract features for face recognition. Lai [14] presented a new method for holistic face representation using spectroface combined with the wavelet transform and a Fourier transform. Chien [15] performed face recognition using discriminant waveletfaces and nearest feature classifiers. All these studies [13]–[15] exploited only an approximation image among the four subimages being available. This choice was primarily motivated by an observation that this image is the best approximation of the original image within the lower-dimensional space and contains the highest energy content within the four subimages available. On the other hand, Sergent [16] found that the low-frequency band and high-frequency components band performed different roles in the classification task. The low-frequency components contribute to the global description, while the high-frequency components contribute to the finer details required in the identifica-
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tion task. Taking this into account, we consider approximation images as well as the three detailed images including auxiliary information. In this study, we are concerned with Fisherface method based on four subimage sets decomposed by wavelets. The fusion of the individual classifiers is realized through fuzzy integration with fuzzy integral being employed in this construct [17], [18]. The ability of the fuzzy integral to combine the results of multiple sources of information has been researched in various application areas such as pattern recognition [19]–[24], image processing [25]–[27], and biometrics [28], [29]. In problems of an online handwriting character recognition pattern, Cho [19], [20] proposed a method of combining the results coming from multiple neural networks serving as individual classifiers. Chiang [21] developed hybrid fuzzy-neural systems for handwritten word recognition using self-organizing feature map and Choquet fuzzy integral. Gader [24] used fuzzy integral for information fusion when dealing with three types of classifiers (segmentation-based model, hidden Markov model (HMM), and fuzzy HMM) for infrared imagery object recognition. In the area of image processing, Pham [25] constructed an alternative solution to image restoration by fusing multiple image filters (mean filter, Sobel filter, and adaptive Wiener filter). Frigui [27] developed interactive image retrieval by constructing dissimilarity measure based upon the notion of the fuzzy integral. In the area of biometrics, Pham [28] proposed a method of similarity normalization for speaker verification. Mirhosseini [29] studied a feature-based matching approach to face recognition involving finding correlation between facial features such as eyes, mouth, and nose. This paper is organized in the following manner. Section II provides the theory of fuzzy measure and fuzzy integral. Section III describes the wavelet decomposition and shows how the four subimage sets are developed. In Section IV, we provide the summary of the well-known Fisherface techniques. Section V describes the technique of face recognition realized by means of fuzzy integral and wavelet decomposition. Section VI covers simulation results for the two face databases available at Chungbuk National University (CNU) and Yale University [30]. Finally, concluding comments are covered in Section VII.
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Fig. 1. Plot of db1 in the Daubechies family wavelets.
1) 2) In case of equality:
; . satisfies the following in-
(1) Therefore, the belief measure can be defined as follows: (2) is a complement of The plausibility measure folwhere lows the duality condition and is defined as follows: 1) ; 2) . is represented as follows: In case of (3) Therefore, the plausibility measure satisfies the relationship: (4) The relation between belief measure and plausibility measure is captured through the following expression: (5) B. Fuzzy Measure
II. FUZZY MEASURE AND FUZZY INTEGRAL Fuzzy integrals are nonlinear functionals defined with respect to fuzzy measure. In this section, we briefly review the properties of belief and plausibility measure, fuzzy measure, and fuzzy integrals of two types (Sugeno and Choquet). Our objective is also to contrast these two constructs of fuzzy integrals. A. Belief and Plausibility Measure Fuzzy measure links directly to the fundamental concepts used in evidence theory [31] such as belief and plausibility function. In what follows we briefly review a special subclass of belief and plausibility measure as originally introduced by Shafer be a finite set and let [31]. Let denote the family of all subsets of . The belief function is defined as follows: defined on
is called a fuzzy measure if A set function the following conditions are satisfied: 1) boundary conditions: ; , if and ; 2) monotonicity: , if is 3) continuity: an increasing sequence of measurable sets. Starting from this definition, Sugeno [17] introduced a -fuzzy measure that satisfies the following addiso-called tional property: (6) and , and for some . for all Evidently when , the -fuzzy measure becomes a or , standard probability measure. When the -fuzzy measure becomes a belief measure and plausibility
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Fig. 2. Plots of Daubechies wavelet D4 (db2, db4, db6, and db8).
measure, respectively, [32]. In general, the value of can be determined owing to the boundary condition of the -fuzzy mea, hence, the value of is sure. This condition reads as determined by solving the following: (7) Put it equivalently (8) where , and is the value of the fuzzy density function. The solution can be easily obtained; obviously . Thus the we are interested in the unique root greater than calculations of the fuzzy integral with respect to a -fuzzy measure can be realized once we provided with the values of the density function available for the individual points. The values can be interpreted as the degrees of importance (relevance) of the corresponding sources of information . C. Fuzzy Integral The fuzzy integral of function h computed over of the function h with respect to a fuzzy measure g is defined in the form (9) When the values of
are ordered in the decreasing sequence, , the Sugeno fuzzy integral is
computed as follows: (10) where denotes a subset of elements of assumed by the the universe of discourse. The values of fuzzy measure over the corresponding subsets of elements can be determined recursively in the form (11) (12) It has been shown that (10) is not a proper extension of the usual Lebesgue integral. In other words, when the measure is additive the above expression does not return the integral in the Lebesgue sense. In order to overcome this drawback, Murofushi
and Sugeno [18] proposed a so-called Choquet fuzzy integral computed in the following manner: (13)
III. WAVELET DECOMPOSITION In this section, we utilize how can utilize the two-dimensional (2-D) wavelet analysis to efficiently decompose an image. First, we convert RBG color image to its grayscale version using a 2-D discrete wavelet transform. Afterwards, we perform a single level wavelet decomposition of the resulting image. This decomposition generates coefficient matrices of the one-level approximation and horizontal, vertical, and diagonal details, respectively. From the obtained coefficients, we construct the approximation and three detailed images via the high-pass and low-pass filtering realized with respect to the column vectors and the row vectors of array pixels. In this manner, we can repeatedly perform a multilevel wavelet decomposition, such as two-level, three-level, and so forth. In this paper, we use the most known Daubechies (db1), along with wavelet D4 (db2, db4, db6, and db8) [33]. The names of the Daubechies family wavelets are denoted as dbN, where N stands for the order of the wavelet. The db1 wavelet is the same as the Haar wavelet, refer to Fig. 1. The other Daubechies wavelet D4 is shown in Fig. 2. Fig. 3 shows the architecture of the 2-D wavelet decomposition realized at level 1. Here, H and L represent the high-pass denotes the and low-pass filter, respectively. The symbol downsampling by 2. As shown in Fig. 3, in this way we can ob. In the previous works tain four subimages image is used to carry out face recognition; [13]–[15], the the choice of the is dictated by its best performance among the four subimages occurring at the same level. In our study, we as well as the three reperform the face recognition using , and (which, in our opinion, maining subimages include information that is unavailable in the image). As a matter of fact, there is a strong experimental evidence behind this selection. Fig. 4 visualizes the original image and the subimages decomposed by wavelet transform at level 1, 2, and 3, respectively. IV. FISHERFACE METHOD The well-known approach coming under the name of Fisherface is insensitive to large variation. It is worth stressing
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Fig. 3.
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Architecture of 2-D wavelet decomposition occurring at level 1.
training set of N face images by fine the covariance matrix as follows:
. We de-
(14) (15) Then, the eigenvalues and eigenvectors of the covariance matrix R are calculated, respectively. Let contain the r eigenvectors corresponding to the r largest eigenvalues. For a set of original face images Z, their corresponding can be obtained reduced feature vectors by projecting Z into the PCA-transformed space according to the following relationship: (16) The second processing stage is based on the use of the FLD and can be described as follows. Consider c classes in the problem involving N samples. Let the between-class scatter matrix be defined as (17) Fig. 4. Original image and wavelet decomposition at each level. (a) Original image. (b) Wavelet decomposition at level 1. (c) Wavelet decomposition at level 2. (d) Wavelet decomposition at level 3.
that by maximizing the ratio of between-class scatter matrix and within-class scatter matrix, FLD produces well separated classes in a low-dimensional subspace, even under severe variation in lighting and facial expressions. In what follows, we briefly describe the Fisherface method. This method consists of two stages. The first stage is used to project face pattern from a high-dimensional image space into a lower-dimensional space using PCA method. The second stage is performed by the FLD known as class-specific method that finds the optimal projection where the optimality is sought from the classification standpoint. Therefore, we can perform by first projecting the image set to a lower-dimensional space using PCA so that the resulting within-class scatter matrix becomes nonsingular before computing the optimal projection. array containing levels of Let a face image be a 2-D may be conintensity of the individual pixels. An image veniently considered as a vector of dimension . Denote the
is the number of samples in ith class and where being the mean of class mean of all samples, with within-class scatter matrix is defined in the form
is the . The (18)
where is the covariance matrix of class . The optimal projection matrix is chosen as a matrix with orthonormal columns that maximizes the ratio of the determinant of the between-class scatter matrix of the projected samples to the determinant of the within-class scatter matrix of the projected samples, i.e., [2]
(19) is the set of generalized eigenvecwhere and corresponding to the tors (discriminant vectors) of largest generalized eigenvalues , i.e., (20)
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Fig. 5.
Face representation as a linear combination of feature vector and Fisherface images.
Fig. 6.
Overall architecture of face recognition using wavelet decomposition, Fisherface, and fuzzy integral.
where the rank of is or less because it is the sum of “c” matrices of rank one or less. Thus, the upper bound on the . As a result, the feature vectors values of “m” is equal to for any face images can be calculated as follows:
linear combination of the feature vector and Fisherface images in ith face image is shown in Fig. 5.
(21)
In what follows, we combine the wavelet decomposition, Fisherface method, and fuzzy integral into a single coherent classification platform. The architecture of the overall face recognition system using the proposed method is shown in Fig. 6. In Fig. 6, some of the specifics of the classifier (especially when it comes to the dimensionality of the architecture) relate to CNU face database to be used in Section VI. The CNU face database contains 100 face images from ten individuals and the total number of images for each person is equal to 10. The experiment is repeated for each of the ten possible choices using tenfold cross-validation technique [7], [36]. We partitioned the set of all images into ten segments each of which has one image per person for all people. Then, each segment contains ten images. We used nine segments for the
To complete classification of a new face image , we compute a Euclidean distance between a given image and a pattern in the training set that is (22) Note that this distance is computed on a basis of and where these two are the FLD-transformed feature vectors of and , respectively. Noticeably, the distance face image information obtained by Fisherface method is effectively used in the following section. Also, we perform Fisherface method based on four subimage sets decomposed by wavelet transform, respectively. The face representation coming as a
V. FACE RECOGNITION BY FUZZY INTEGRAL AND WAVELET DECOMPOSITION
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values follows:
of the kth class in ith classifier are as (24)
where is the number of samples in kth class . Step 4) Aggregate the output of each classifier and the degree of importance of each classifier using the mechanisms of the fuzzy integral using either (10) or (13). The class with the highest value is declared to be the output of the classifier. The fuzzy used in the classifier can be either densities estimated subjectively [34] or obtained from the training data [19], [35]. Here we follow the two approaches and contrast the results. The computations based on the training data are carried out as of follows: Fig. 7. Four subimages obtained by wavelet decomposition at level 3. (a) Approximation. (b) Horizontal detail. (c) Vertical detail. (d) Diagonal detail.
training and evaluated the performance on the test image set. This process is repeated for each of the ten possible choices. Step 1) Perform the wavelet decomposition for the training image set (images of 90 faces) to extract the intrinsic features of the patterns. We obtain the decomposed four image sets as shown in Fig. 6. Here, we obtained four sets of 90 80 60 pixel face images by wavelet decomposition at level 3. The example of four subimages obtained by the three-level wavelet decomposition is shown in Fig. 7. Step 2) Use the Fisherface method combining PCA with LDA for the decomposed four subimage sets, respectively. Here, the upper bound on the number of eigenvectors is 80 (N-c). The number of discrimilargest gennant vectors corresponding to the eralized eigenvalues is 9. The feature vectors of the training image set and a given test image are obtained by using (21). The values of Euclidean distance are computed by using the feature vectors produced from the training image set and a given test image using (22). Step 3) Generate the membership grades based on the distance information produced in the previous phase (Step 2). The membership grades can be determined in many different ways. Here, we follow the method introduced in [23] where their calculations use the distances between the test image and those images existing in the training set, namely (23) , i is the number where of classifier and j is the index of the training face denotes an average distance between all image. is the Euclidean distance values in ith classifier, distance of feature vector between a given test image and j’th training image in ith classifier. The output
(25) denotes the classification rate in [0, 1] where of each classifier for the training set of images. These values are obtained by leave-one-out techis the subjective nique for the training set; weight value. In this case, the values of are experimentally selected through a trial and error method. If we use only fuzzy densities obtained from the training data, the performance of several classifiers may show a similar classification rate because we have not gained any assurance that the performance obtained from the training data would lead to similar results for the test data (as a matter of fact, we could have ended up with the far worse results). Moreover, because the classifier operating on the approximation image (compressed image) set generally shows the best recognition results than three classifiers constructed from the set of three detail images, we need to use the subjective weight value. Hence, we performed the experiments and vary the other as follows; we fix weight values with an interval of 0.1 from 0 to , we can perform the 0.5. If conventional Fisherface method based only on the , compressed image set. If this method may bring out worse performance than that of each classifiers due to an equal weight values. assumes any On the other hand, if intermediate value coming from this interval, this architecture carries out classification that is inherently based on the fusion combining approximation image set and the detailed image sets.
VI. EXPERIMENTS This section reports on the comprehensive set of experiments and draws conclusions as to the performance of the individual approaches.
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Some face images of CNU face database.
Fig. 9. Fisherface images obtained by each classifier. (a) Fisherface images for classifier1. (b) Fisherface images for classifier2. (c) Fisherface images for classifier3. (d) Fisherface images for classifier4.
A. Chungbuk National University Face Databases The CNU database contains 100 face images coming from ten individuals, see Fig. 8. In essence, we are dealing with a ten-class problem. The total number of images for each person is ten. They vary in face pose and exhibit a substantial level of light variation. The size of each original image is 640 480. The wavelet decomposition was completed at level 3. Each image was digitized and presented by a 80 60 pixel (db1) array whose values of the gray levels ranged in between 0 and 255. In the following, we evaluate the performance of face recognition through a tenfold cross-validation, which helps us establish sound confidence intervals as to the recognition rates and
produce better reliability of the results [7], [36]. We first project -dimenthe image space from -dimensional space into sional feature space and then compute the discriminant and are and , respecvectors (the rank of tively). We selected 80 eigenvectors and nine discriminant vectors. Fig. 9 shows Fisherface images obtained by the Fisherface method for the four classifiers. , and we use relationship (25) to Let produce the values of the fuzzy densities. Here, weight values of each classifier obtained by the experimental trial and error as deand . scribed before as follows We used the most known Daubechies(db1) and wavelet D4 for the wavelet decomposition. To help understand the performance of fuzzy integral, Table I shows the results obtained by fuzzy
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TABLE I RESULTS OBTAINED BY FUZZY INTEGRAL FOR TENTH SEGMENT IMAGE SET
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COMPARISON
OF
TABLE III RECOGNITION RATES AND STANDARD DEVIATION BY FUZZY INTEGRAL
TABLE II COMPARISON OF RECOGNITION RATES FOR TENTH SEGMENT IMAGE SET
integral for tenth segment image set. Here, the output class is determined as class with maximum value of the fuzzy integral. Table II contains the comparison of recognition rates for the tenth segment image set among test image set. Finally, when using the db1 and D4, the mean and standard deviation of the classification accuracy are listed in Table III. Noticeably, the proposed method shows better performance than other well-known classifiers, such as eigenface method and Fisherface method. We also found that the fuzzy fusion method lead to the better performance in comparison to the case we rely only on the approximation image (i.e., the compressed image). Graphically, Fig. 10 includes the pertinent plots that help visualize the differences. In the case of eigenface method, we found the recognition rate to be 84.6% which is somewhat low. This could be attributed to the fact that PCA retains unwanted variations due to lighting and facial expression. In the case of Fisherface method, we noticed substantial improvement and the recognition rate of 95.4% is mainly due to the fact that this method is insensitive to large variation in light direction, face pose, and facial expression. On the other hand, the recognition rates by Sugeno fuzzy integral and Choquet fuzzy integral are equal to 98% and 98.4%, respectively. B. Yale Face Databases The Yale face database contains 165 face images of 15 individuals, as shown in Fig. 11. There are 11 images per subject, one for each facial expression or configuration: center-light, glasses/no glasses, happy, normal, left-light, right-light, sad, sleepy, surprised, and wink. The size of each original image is 243 320. In this experiment we used the face image (243 207) cropped and resized to remove the background
Fig. 10. Comparison of recognition rates obtained for several families of wavelets.
information. The wavelet decomposition was applied at level 2. Each image was digitized and presented by a 61 52 pixel (db1) array whose gray levels ranged between 0 and 255. The experiments are completed in the same format as for the previous database. We evaluated the performance of the classifier via eleven-fold cross-validation. We first projected the -dimensional image space into -dimensional PCAdiscriminant transformed space and then computed the vectors as described in Section IV. We selected 135 eigenvectors and 14 discriminant vectors. Fig. 11 shows some of these images. The classification results are reported in the same manner as for the previous experiment. The classification rates both mean values and standard deviations for the db1 and wavelet family D4 are reported in Table IV. In all cases the superiority of the approach presented here becomes evident. The plots of the classification rates are covered in Fig. 12. As before, PCA method is the weakest classifier (again for the same reasons as observed for the previous database). On the other hand, the Fisherface method yields 96.60% classification rate whereas the integration using Sugeno fuzzy integral and Choquet fuzzy integral brings us close to 100% producing 99.11% and 99.24%, respectively.
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Fig. 11.
Some face images of the Yale face database.
COMPARISON
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TABLE IV RECOGNITION RESULTS AND STANDARD DEVIATION FOR THE FUZZY INTEGRAL
The classification experiments were carried out on the commonly available face databases from the CNU and Yale. Consequently, we were able to reduce sensitivity caused by varying illumination and viewing conditions associated with the original image. REFERENCES
Fig. 12. Comparison of recognition rates obtained for different families of wavelets.
VII. CONCLUDING COMMENTS We have discussed the fusion (aggregation) of multiple classifiers for the face recognition problem realized with the aid of fuzzy integration (via the Sugeno and Choquet fuzzy integral) of outcomes of the individual classifiers. It has been experimentally demonstrated that the aggregation of the classifiers operating on four subimage sets generated by wavelet decomposition leads to better classification results (when compared to the conventional techniques of the eigenface and Fisherface method).
[1] M. Turk and A. Pentland, “Face recognition using eigenfaces,” in Proc. IEEE Conf. Computer Vision Pattern Recognition, 1991, pp. 586–591. [2] P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp. 711–720, July 1997. [3] C. Liu and H. Wechsler, “Gabor feature based classification using the enhanced fisher linear discriminant model for face recognition,” IEEE Trans. Image Processing, vol. 11, pp. 467–476, Apr. 2002. [4] H. Yu and J. Yang, “A direct LDA algorithm for high-dimensional data with application to face recognition,” Pattern Recognit., vol. 34, pp. 2067–2070, 2001. [5] L. Juwei, K. N. Plataniotis, and A. N. Venetsanopoulos, “Face recognition using kernel direct discriminant analysis algorithms,” IEEE Trans. Neural Networks, vol. 14, pp. 117–126, Jan. 2003. [6] Z. Jin, J. Y. Yang, Z. S. Hu, and Z. Lou, “Face recognition based on the uncorrelated discriminant transformation,” Pattern Recognit., vol. 34, pp. 1405–1416, 2001. [7] H. C. Kim, D. Kim, and S. Y. Bang, “Face recognition using the mixture-of-eigenfaces method,” Pattern Recognit. Lett., vol. 23, pp. 1549–1558, 2002. [8] D. L. Swets and J. Weng, “Using discriminant eigenfeatures for image retrieval,” IEEE Trans. Pattern Anal. Machine Intell., vol. 18, pp. 831–836, Aug. 1996. [9] M. S. Bartlett, J. R. Movellan, and T. J. Sejnowski, “Face recognition by independent component analysis,” IEEE Trans. Neural Networks, vol. 13, pp. 1450–1464, Nov. 2002. [10] L. Wiskott, J. M. Fellous, N. Kruger, and C. Malsburg, “Face recognition by elastic graph matching,” IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp. 775–779, July 1997. [11] M. J. Er, S. Wu, and H. L. Toh, “Face recognition with radial basis function (RBF) neural networks,” IEEE Trans. Neural Networks, vol. 13, pp. 697–710, May 2002. [12] C. Liu and H. Wechsler, “Evolutionary pursuit and its application to face recognition,” IEEE Trans. Pattern Anal. Machine Intell., vol. 22, pp. 570–582, June 2000. [13] B. Li and Y. Liu, “When eigenfaces are combined with wavelets,” Knowl.-Based Syst., vol. 15, pp. 343–347, 2002. [14] J. H. Lai, P. C. Yuen, and G. C. Feng, “Face recognition using holistic Fourier invariant features,” Pattern Recognit., vol. 34, pp. 95–109, 2001. [15] J. T. Chien and C. C. Wu, “Discriminant waveletfaces and nearest feature classifiers for face recognition,” IEEE Trans. Pattern Anal. Machine Intell., vol. 24, pp. 1644–1649, Feb. 2002. [16] J. Sergent, “Microgenesis of face perception,” in Proc. Aspects Face Processing. Dordrecht, The Netherlands, 1986. [17] M. Sugeno, “Fuzzy measures and fuzzy integrals—A survey,” in Fuzzy Automata and Decision Processes, M. M. Gupta, G. N. Saridis, and B. R. Gaines, Eds. Amsterdam, The Netherlands: North Holland, 1977, pp. 89–102.
KWAK AND PEDRYCZ: FACE RECOGNITION USING FUZZY INTEGRAL
[18] T. Murofushi and M. Sugeno, “An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure,” Fuzzy Sets Syst., vol. 29, pp. 201–227, 1988. [19] S. B. Cho and J. H. Kim, “Multiple network fusion using fuzzy logic,” IEEE Trans. Neural Networks, vol. 6, pp. 497–501, March 1995. [20] , “Combining multiple neural networks by fuzzy integral for robust classification,” IEEE Trans. Syst., Man, Cybern., vol. 25, pp. 380–384, Feb. 1995. [21] J. H. Chiang and P. D. Gaber, “Hybrid fuzzy-neural systems in handwritten word recognition,” IEEE Trans. Fuzzy Syst., vol. 5, pp. 497–510, Nov. 1997. [22] J. H. Chiang, “Choquet fuzzy integral-based hierarchical networks for decision analysis,” IEEE Trans. Fuzzy Syst., vol. 7, pp. 63–71, Feb. 1999. , “Aggregating membership values by a Choquet-fuzzy-integral [23] based operator,” Fuzzy Sets Syst., vol. 114, pp. 367–375, 2000. [24] P. D. Gader, M. A. Mohamed, and J. M. Keller, “Fusion of handwritten word classifiers,” Pattern Recognit. Lett., vol. 17, pp. 577–584, 1996. [25] T. D. Pham, “An image restoration by fusion,” Pattern Recognit., vol. 34, pp. 2403–2411, 2001. [26] T. D. Pham and H. Yan, “Color image segmentation using fuzzy integral and mountain clustering,” Fuzzy Sets Syst., vol. 107, pp. 121–130, 1999. [27] H. Frigui, “Interactive image retrieval using fuzzy sets,” Pattern Recognit. Lett., vol. 22, pp. 1021–1031, 2001. [28] T. Pham and M. Wagner, “Similarity normalization for speaker verification by fuzzy fusion,” Pattern Recognit., vol. 33, pp. 309–315, 2000. [29] A. R. Mirhosseini, H. Yan, K. M. Lam, and T. Pham, “Human face image recognition: An evidence aggregation approach,” Comput. Vis. Image Understand., vol. 71, no. 2, pp. 213–230, 1998. [30] Yale Face Database [Online]. Available: http://cvc.yale.edu/projects/ yalefaces/yalefaces.html [31] G. Shafer, A Mathematical Theory of Evidence. Princeton, NJ: Princeton Univ. Press, 1976. [32] G. Banon, “Distinction between several subsets of fuzzy measures,” Fuzzy Sets Syst., vol. 5, pp. 291–305, 1981. [33] I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inform. Theory, vol. 36, pp. 961–1005, Sept. 1990. [34] X. Li, Z. Q. Liu, and K. M. Leung, “Detection of vehicles from traffic scenes using fuzzy integrals,” Pattern Recognit., vol. 35, pp. 967–980, 2002. [35] H. Tahani and J. M. Keller, “Information fusion in computer vision using the fuzzy integral,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 733–741, May/June 1990.
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[36] B. Moghaddam, “Principal manifolds and probabilistic subspaces for visual recognition,” IEEE Trans. Pattern Anal. Machine Intell., vol. 24, pp. 780–788, June 2002.
Keun-Chang Kwak (M’04) received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Chungbuk National University (CNU), Cheongju, South Korea, in 1996, 1998, and 2002, respectively. From 2002 to 2003, he worked as a Researcher in the Brain Korea 21 Project Group, CNU. He is currently a Postdoctoral Fellow in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. His research interests include biometrics, computational intelligence, fuzzy modeling, and pattern recognition.
Witold Pedrycz (F’99) is a Professor and Chair in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada. He is also a Canada Research Chair in computational intelligence. He is actively pursuing research in computational intelligence, fuzzy modeling, knowledge discovery and data mining, fuzzy control including fuzzy controllers, pattern recognition, knowledge-based neural networks, telecommunication networks, relational computation, and software engineering. He has published numerous papers in this area. He is also an author of seven research monographs covering various aspects of computational intelligence and software engineering. Dr. Pedrycz has been a member of numerous program committees of IEEE conferences in the area of fuzzy sets and neurocomputing. He currently serves as an Associate Editor of the IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B and the IEEE TRANSACTIONS ON FUZZY SYSTEMS.
