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 &
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ISSN 0976 – 6367(Print)
ISSN 0976 – 6375(Online)
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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
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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
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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
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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
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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
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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

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ISSN 0976 - 6375(Online), Volume 5, Issue 5, May (2014), pp. 01-11 © IAEME
11
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