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  • 1. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 1 COMPARISON OF MAHALANOBIS AND MANHATTAN DISTANCE MEASURES IN PCA BASED FACE RECOGNITION Smt. Mahananda D. Malkauthekar1 , Smt. Shubhangi D. Sapkal2 1 Asst. Prof. in M.C.A. Dept., Govt. Engg. College, Karad, Maharashtra, India, 2 Asst. Prof. in M.C.A. Dept., Govt. Engg. College, Aurangabad, Maharashtra, India, ABSTRACT With the growth of information technology there is a greater need of high security, so biometric authentication systems are gaining importance. Face recognition is more used because it’s easy and non intrusive method during acquisition procedure. Here PCA algorithm is used for the feature extraction. Distance metric or matching criteria is the main tool for retrieving similar images from large image databases for the above category of search. Two distance measures, such as the Manhattan Distance, Mahalanobis Distance have been proposed in the literature for measuring similarity between feature vectors. In content-based image retrieval systems, Manhattan distance and Euclidean distance are typically used to determine similarities between a pair of image. Here facial images of three subjects with different expression and angles are used for classification. Experimental results are compared and the results show that the Mahalanobis distance performs better than the Manhattan Distance. Keywords: Covariance Matrix, Distance Measures, Eigenvectors, FERET Database, Image Classification, PCA. I. INTRODUCTION The task of identifying objects and features from image data is central in many active research fields. In this paper I address the inherent problem that a single object may give rise to many possible images, depending on factors such as the lighting conditions, the pose of the object, and its location and orientation relative to the camera [3]. Principal component analysis (PCA) based systems are used often [1][16]. In this paper I have studied 2 distance measures on the FERRET database to see the performance of the principal component analysis (PCA) based face recognition system [8]. Here the Mahalanobis distance and Manhattan distance is employed to measure the INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING & TECHNOLOGY (IJCET) ISSN 0976 – 6367(Print) ISSN 0976 – 6375(Online) Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME: www.iaeme.com/ijcet.asp Journal Impact Factor (2014): 8.5328 (Calculated by GISI) www.jifactor.com IJCET © I A E M E
  • 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 2 similarity between original data and reconstructed data. The proposed classification algorithm for face recognition has been evaluated under varying illumination and poses using standard face databases [9]. Results shows that the Mahalanobis distance gives the better performance than the Manhattan distance in overall face recognition rate. But the images of changed complexions are recognized better by Manhattan distance as compared to the Mahalanobis distance. II. DISTANCE MEASURES Let X, Y is Eigen feature vectors of length n. Then we can calculate the following distances between these feature vectors. 1. The Mahalanobis distance. It is a very useful way of determining the "similarity" of a set of values from an "unknown: sample to a set of values measured from a collection of "known" samples. One of the main reasons the Mahalanobis distance method is used is that it is very sensitive to inter-variable changes in the training data. In addition, since the Mahalanobis distance is measured in terms of standard deviations from the mean of the training samples, the reported matching values give a statistical measure of how well the spectrum of the unknown sample matches (or does not match) the original training spectra. Mahalanobis, distance (Johnson and Wichern, 1998) from x to µ, can be written[3]. Where µk are the mean and xi is the input vector of attributes where Σ is the covariance matrix given by and the individual covariance values of Σ are computed from the outer product sum given by[5]. 2. Manhattan Distance It is also called the L1 distance. If u= (x1, y1) and v= (x2, y2) are two points, then the Manhattan distance between u and v is given by MH (a, b) =ല x1 – x2 ല + ല y1 – y2 ല
  • 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 3 Instead of two dimensions, if the points have n- dimensions, such as 1 2(x ,x , ,x )na = K and 1 2(y , y , , y )nb = K then, eq. 3 can be generalized by defining the Manhattan distance between a and b as[2] [3]. MH (a, b) =ല x1 – x2 ല + ല y1 – y2 ല + …+ ല xn – yn ല 1 | | n i i i x y = = −∑ III. SYSTEM DEVELOPMENT 1. Resizing Size of the images is fixed. Original image size is (768*512*3), which is changed to (60*60). RGB images are converted to grayscale images. Facial images of 3 subjects (9 images for each person, 9*3=27 images) with different expression and angles are used for classification shown below [11] [12]. Fig. 1 Three different persons are used and 9 images of each person. Front faces for each class are used for training and all images of faces with different angles and expressions are used for testing [9] 2. Manhattan distance 2.1. Class1 The Manhattan distance of front face of class1 (Person1) with all images of three classes is shown in table 1, Where I1 to I9 are the images and P1, P2, P3 are the persons. Table1 P1 P2 P3 I1 0 0.0073 0.038 I2 0.0037 0.012 0.0034 I3 0.0018 0.0152 0.0645 I4 0.0044 0.0048 0.0208 I5 0.0049 0.0158 0.0083 I6 0.0075 0.0092 0.019 I7 0.0123 0.0078 0.0636 I8 0.0075 0.0162 0.014 I9 0.0095 0.0039 0.0505
  • 4. International Journal of Computer Engineering and Technology (IJCET ISSN 0976 - 6375(Online), Volume 5, Issue 2.2. Class2 The Manhattan distance of front face class2 (Person2) with all images of three classes is calculated and shown in table 2. I1 I2 I3 I4 I5 I6 I7 I8 I9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 4 Graph1 ce of front face class2 (Person2) with all images of three classes is Table 2 P1 P2 P3 I1 0.0073 0 0.0307 I2 0.0045 0.0139 0.0162 I3 0.0078 0.0062 0.0376 I4 0.002 0.0191 0.0028 I5 0.0106 0.0221 0.0181 I6 0.0243 0.0119 0.041 I7 0.026 0.0079 0.0072 I8 0.0186 0.0675 0.0022 I9 0.0347 0.0148 0.0467 Graph 2 P1 P2 P3 I1 I2 I3 I4 I5 I6 I7 I8 I9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 P1 P2 P3 I1 I2 I3 I4 I5 I6 I7 I8 I9 ), ISSN 0976-6367(Print), ce of front face class2 (Person2) with all images of three classes is
  • 5. International Journal of Computer Engineering and Technology (IJCET ISSN 0976 - 6375(Online), Volume 5, Issue 3.3. Class3 The Manhattan distance of front face of class3 (Person3) with all images of three classes is calculated and shown in table 3. I1 I2 I3 I4 I5 I6 I7 I8 I9 3. Mahalanobis distance 3.1 Class1 The Mahalanobis distance of front face of class1 (Person shown in table 1, Where I1 to I9 are the images and P1, P2, P3 are the persons. I1 I2 I3 I4 I5 I6 I7 I8 I9 0 0.5 1 1.5 2 2.5 3 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 5 The Manhattan distance of front face of class3 (Person3) with all images of three classes is Table 3 P1 P2 P3 I1 0.038 0.0307 0 I2 0.0379 0.0242 0.1084 I3 0.0474 0.0037 0.0222 I4 0.0446 0.0156 0.1024 I5 0.0958 0.1231 0.1815 I6 0.1231 0.1495 0.2211 I7 0.204 0.0899 0.5564 I8 0.3923 0.2937 0.2066 I9 1.4487 2.7268 1.8823 Graph 3 The Mahalanobis distance of front face of class1 (Person4) with all images of three classes is shown in table 1, Where I1 to I9 are the images and P1, P2, P3 are the persons. Table 4 P1 P2 P3 I1 0 0.6122 0.3321 I2 1.361 1.9732 1.6931 I3 0.6562 1.2684 0.9883 I4 0.9482 1.5603 1.2803 I5 0.4505 0.1617 0.1184 I6 0.1862 0.7983 0.5183 I7 0.7641 1.3762 1.0961 I8 0.4634 1.0755 0.7954 I9 0.6567 1.2688 0.9888 0 0.5 1 1.5 2 2.5 3 P1 P2 P3 I1 I2 I3 I4 I5 I6 I7 I8 I9 ), ISSN 0976-6367(Print), The Manhattan distance of front face of class3 (Person3) with all images of three classes is ) with all images of three classes is
  • 6. International Journal of Computer Engineering and Technology (IJCET ISSN 0976 - 6375(Online), Volume 5, Issue 3.3. Class2 The Mahalanobis distance of front face of class2 (Person2 shown in table 5, Where I1 to I9 are the images and P1, P2, P3 are the persons. I1 I2 I3 I4 I5 I6 I7 I8 I9 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 6 Graph 4 distance of front face of class2 (Person2) with all images of three classes is , Where I1 to I9 are the images and P1, P2, P3 are the persons. Table 5 P1 P2 P3 I1 0.6122 0 0.2801 I2 0.5983 1.2104 0.9303 I3 0.0696 0.5426 0.2625 I4 1.9263 2.5384 2.2583 I5 1.4615 0.8494 1.1295 I6 0.475 1.0871 0.8071 I7 0.4415 1.0536 0.7735 I8 1.0629 1.6751 1.395 I9 0.112 0.7242 0.4441 Graph 5 P1 P2 P3 I1 I2 I3 I4 I5 I6 I7 I8 I9 0 0.5 1 1.5 2 2.5 3 P1 P2 P3 I1 I2 I3 I4 I5 I6 I7 I8 I9 ), ISSN 0976-6367(Print), all images of three classes is
  • 7. International Journal of Computer Engineering and Technology (IJCET ISSN 0976 - 6375(Online), Volume 5, Issue 3.4. Class3 The Mahalanobis distance of front face of class3 (Person3 shown in table 5, Where I1 to I9 are th I1 I2 I3 I4 I5 I6 I7 I8 I9 IV. EXPERIMENTAL ANALYSIS The results of the methods, compared. First the recognition rate for all three classes with different types of images is calculated by Manhattan distance method and then result is compared with the recognition rate obtained by Mahalanobis distance method for all three classes for the same types of images as used in distance method. The following tables and graphs show the co 1. Recognition rate using Manhattan distance The table 7 shows the recognition rate obtained by images are used 27. 4 images are facing frontally, 17 images are angle changed, 4 are of changed expressions and 2 images are of changed complexions [ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 7 distance of front face of class3 (Person3) with all images of three classes is shown in table 5, Where I1 to I9 are the images and P1, P2, P3 are the persons [14]. Table 6 P1 P2 P3 I1 0.3321 0.2801 0 I2 0.9631 1.5753 1.2952 I3 0.8397 1.4518 1.1717 I4 0.2214 0.8336 0.5535 I5 0.0154 0.6276 0.3475 I6 0.637 1.2492 0.9691 I7 0.4197 1.0319 0.7518 I8 0.6413 1.2535 0.9734 I9 0.2902 0.9024 0.6223 Graph 6 EXPERIMENTAL ANALYSIS The results of the methods, the Mahalanobis distance and the Manhattan distance are compared. First the recognition rate for all three classes with different types of images is calculated distance method and then result is compared with the recognition rate obtained by distance method for all three classes for the same types of images as used in distance method. The following tables and graphs show the comparative study. Manhattan distance The table 7 shows the recognition rate obtained by Manhattan distance method. Here total images are used 27. 4 images are facing frontally, 17 images are angle changed, 4 are of changed expressions and 2 images are of changed complexions [13]. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 P1 P2 P3 I1 I2 I3 I4 I5 I6 I7 I8 ), ISSN 0976-6367(Print), ) with all images of three classes is distance and the Manhattan distance are compared. First the recognition rate for all three classes with different types of images is calculated distance method and then result is compared with the recognition rate obtained by distance method for all three classes for the same types of images as used in Manhattan method. Here total images are used 27. 4 images are facing frontally, 17 images are angle changed, 4 are of changed
  • 8. International Journal of Computer Engineering and Technology (IJCET ISSN 0976 - 6375(Online), Volume 5, Issue Manhattan Distance Measure Recognition rate by Manhattan distance with Front faces No. of images 4 Recognized Images 2 Recognition rate 50 2. Recognition rate using Mahalanobis The table 8 shows the recognition rate obtained by total images are used 27. 4 images are facing frontally, 17 images are angle changed, 4 are of changed expressions and 2 images are of changed complexions [13 Mahalanobis Distance Measure Recognition rate by Front faces No. of images 4 Recognized Images 3 Recognition rate 75 0 10 20 30 40 50 60 International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 8 Table 7 Recognition rate by Manhattan distance with dimensionality reduction Front Changed angle Changed Changed Expressions Complexion 17 4 2 8 0 1 47.05 0 50 Graph 7 lanobis distance shows the recognition rate obtained by Mahalanobis distance method. Here 7. 4 images are facing frontally, 17 images are angle changed, 4 are of s are of changed complexions [13]. Table 8 Recognition rate by Mahalanobis distance with dimensionality reduction ont faces Changed angle Changed Changed Expressions Complexion 4 17 4 2 3 9 0 0 75 52.94 0 0 Expressions Complexion No. of images Recognized Images Recognition rate ), ISSN 0976-6367(Print), Recognition rate by Manhattan distance with Changed Complexion distance method. Here 7. 4 images are facing frontally, 17 images are angle changed, 4 are of distance with Changed Complexion
  • 9. International Journal of Computer Engineering and Technology (IJCET ISSN 0976 - 6375(Online), Volume 5, Issue 3.Comparison Table 9 shows the comparative study of two methods Manhattan distance. Comparison Recognition rate by Mahalanobis distance and Manhattan Front faces Changed angle PCA test with Manhattan distance Measure 50 0 PCA test with Mahalanobis distance Measure 75 52.94 Graph 9 shows that the Mahalanobis distance gives the better performance than the Manhattan distance in overall face recognit recognized better by Manhattan distance as compared to the Mahalanobis distance. . 0 10 20 30 40 50 60 70 80 Front faces Changed angle International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 9 Graph 8 Table 9 shows the comparative study of two methods Mahalanobis distan Table 9 Recognition rate by Mahalanobis distance and Manhattan Distance Measure using PCA Changed angle Changed Angle Changed Complexion Total Recognised Images Percentage 0 47.05 50 11 52.94 0 0 12 Mahalanobis distance gives the better performance than the Manhattan distance in overall face recognition rate. But the images of changed complexions are recognized better by Manhattan distance as compared to the Mahalanobis distance. Expressions Complexion Changed angle Changed Changed No. of images Recognized Images Recognition rate ), ISSN 0976-6367(Print), distance and the Recognition rate by Mahalanobis distance and Manhattan Percentage 40.74 44.44 Mahalanobis distance gives the better performance than the ion rate. But the images of changed complexions are
  • 10. International Journal of Computer Engineering and Technology (IJCET ISSN 0976 - 6375(Online), Volume 5, Issue V. CONCLUSION In this paper it is observed that distance method. It has recognized total distance) has recognized total 11(40.74) an improvement in the overall performance VI. FUTURE SCOPE The drawback of the Mahalanobis distance is the equal adding up of the variance normalized squared distances of the features. In the case of noise free signals this leads to the best possible performance. But if the feature is distorted by noise, due feature can have such a high value that it covers the information provided by the other features and leads to a misclassification. Therefore, to find classification procedures which are more robust to noise we have to find a distance measure which gives less weight to the noisy features and more weight to the clean features. This can be reached by comparing the different input features to decide which feature should be given less weight or being excluded and which weight. [3][5][15]. REFERENCES Journal Papers [1]. Mini Singh Ahuja1, Sumit Chhabra2, “ Recognition”, International Journal of Enterprise Computing and Business Systems ISSN (Online): 2230-8849, Vol. 1 Issue 2 July 2011. [2]. Thiyagarajan, Arulselvi & Sainarayanan, “ with Different Distance Measures”, 4, Issue 6. [3]. Supriya Kapoor, Shruti Khanna, Rahul Bhatia, and Mahalanobis distance”, Vol. 7, No. 2, 2010. 01020304050607080 Frontfaces Recognition rate by Mahalanobis distance and Manhattan Distance Measure using PCA International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 10 Graph 9 In this paper it is observed that the Mahalanobis distance gives the better results as t has recognized total 12(44.44%) images out of 27 while L1 norm (Manhattan 11(40.74) images out of 27. Hence using Mahalanobis distance overall performance. The drawback of the Mahalanobis distance is the equal adding up of the variance normalized squared distances of the features. In the case of noise free signals this leads to the best possible performance. But if the feature is distorted by noise, due to the squaring of the distances, a single feature can have such a high value that it covers the information provided by the other features and leads to a misclassification. Therefore, to find classification procedures which are more robust to e to find a distance measure which gives less weight to the noisy features and more weight to the clean features. This can be reached by comparing the different input features to decide which feature should be given less weight or being excluded and which feature should have more Mini Singh Ahuja1, Sumit Chhabra2, “Effect of Distance Measures in PCA Recognition”, International Journal of Enterprise Computing and Business Systems ISSN Vol. 1 Issue 2 July 2011. Thiyagarajan, Arulselvi & Sainarayanan, “Statistical Models for Face Recognition System with Different Distance Measures”, International Journal of Image Processing (IJIP), Volume Shruti Khanna, Rahul Bhatia, “Facial Gesture recognition using correlation Vol. 7, No. 2, 2010. Changedangle ChangedAngle Changed… TotalRecognised… Percentage Recognition rate by Mahalanobis distance and Manhattan Distance Measure using PCA PCA test with Manhattan distance Measure PCA test with Mahalanobi s distance Measure ), ISSN 0976-6367(Print), distance gives the better results as Manhattan images out of 27 while L1 norm (Manhattan Mahalanobis distance there is The drawback of the Mahalanobis distance is the equal adding up of the variance normalized squared distances of the features. In the case of noise free signals this leads to the best possible to the squaring of the distances, a single feature can have such a high value that it covers the information provided by the other features and leads to a misclassification. Therefore, to find classification procedures which are more robust to e to find a distance measure which gives less weight to the noisy features and more weight to the clean features. This can be reached by comparing the different input features to decide feature should have more Effect of Distance Measures in PCA-Based Face Recognition”, International Journal of Enterprise Computing and Business Systems ISSN Statistical Models for Face Recognition System International Journal of Image Processing (IJIP), Volume Facial Gesture recognition using correlation
  • 11. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print), ISSN 0976 - 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME 11 [4]. Vytautas Perlibakas,” Distance measures for PCA-based face recognition”, Pattern Recognition Letters 25 (2004) 711–724, 2003. [5]. S. Sural, G. Qian and S. Pramanik “A Comparative Analysis of Two Distance Measures in Color Image Databases” IEEE International Conference on Image processing, vol.1, pp I-401 - I-404, 2002. [6]. S.Chitra, Dr.G.Balkrishanan, “A Survey of Face Recognition Feature Extraction Process of Dimensionality Reduction Techniques”, Journal of Theoretical and Applied Information Technology, Vol. 36, No.1. Theses [7]. Wendy S. Yambor, Bruce A. Draper, J. Ross Beveridge, “Analyzing PCA-based Face Recognition Algorithms: Eigenvector Selection and Distance Measures”, Computer Science Department, Colorado State University, July 1, 2000. [8]. Wendy S. Yambor, “Analysis of PCA-Based and Fisher Discriminant-Based Image Recognition Algorithms”, Technical Report CS 00-103, July 2000. [9]. Yangfeng Ji, Tong Lin, Hongbin Zha,” Mahalanobis Distance Based Non-negative Sparse Representation for Face Recognition”, Key Laboratory of Machine Perception (Ministry of Education), School of EECS, Peking University, Beijing 100871, China. Proceedings Papers [10]. Vadivel, A. K. M. S. S. A., A. K. Majumdar, and Shamik Sural. "Performance comparison of distance metrics in content-based image retrieval applications." In Proc. of Internat. Conf. on Information Technology, Bhubaneswar, India, pp. 159-164. 2003. [11]. Mrs. M. D. Malkauthekar,”Classifying Facial Images”, IEEE International conf. ICETECT11 Kanyakumari , Tamilnadu, 2011. [12]. Mrs. M. D. Malkauthekar and Mrs. S. D. Sapkal, “Face Recognition with Fourier Descriptors and Fisher Discriminant Analysis”, International Conference on Biometrics Technologies and Applications, August 29, 2008, Pragati Maidan, New Delhi, India. [13]. Mrs. M. D. Malkauthekar, Mrs. S. D. Sapkal and Ms. S.N. Kakarwal “Analysis of Classification Methods of Face Images using PCA and Fisher-Based Algorithms”, ICACT- 2008, 26-28 Dec 2008 Hyderabad. [14]. Mrs. M. D. Malkauthekar and Mrs. S. D. Sapkal, “Experimental Analysis of Facial Images using Fisher Discriminant and PCA” IEEE sponsored IACC 2008, Patiala. [15]. Mrs. M. D. Malkauthekar, Mrs. S. D. Sapkal and Mr. V.P. Kshirsagar, “Face Recognition Using Nearest Neighbor Method”, ICSCI-2008, Hyderabad. [16]. Erum Naz, Umar Farooq, Tabbasum Naz, “Analysis of Principal Component Analysis-Based and Fisher Discriminant Analysis-Based Face Recognition Algorithms”, IEEE--ICET 2006 2nd International Conference on Emerging Technologies.