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The Effect of Distance Measures on the Recognition Rates of PCA
and LDA Based Facial Recognition Philip Miller, Jamie Lyle
Digitial Image Processing Clemson Universtiy
{pemille, jlyle}@clemson.edu
Abstract Many components affect the success of a facial
recognition system. While some research attempts
to improve on PCA or LDA algorithms, an often
overlooked component is the distance measure. In
this paper we show that the choice of distance
measure greatly affects the recognition rate.
Experiments are performed using the FRGC and
FERET face databases. Recognition rates of ten
distance measures are compared. There is an
inconsistency of performance for each distance
measure across each algorithm and face database.
This shows that being able to determine the best
distance measure before running the recognition
algorithm will make the recognition system more
successful.
1. Introduction Increasing the effectiveness of facial recognition
systems is important to biometrics researchers.
There are a few ways to accomplish this.
Improvements could be made to the matching
algorithm like PCA and LDA. A better training set
can be used. This paper looks at different distance
measures.
Distance measures are the last component of facial
recognition. Images are projected into an
eigenspace or fisherspace and represented as
vectors. The distance between the vectors of two
images is the similarity of the images.
For each recognition algorithm used with each
data set, one distance measure will be superior for
the experiment as a whole. That same distance
measure will not be superior for all experiments. A
single subject is not guaranteed to be matched
best with the superior distance measure either.
Therefore, the need exists for adaptive distance
measure selection.
In section 2 and 3 we will discuss the PCA and LDA
algorithms respectively. Section 4 will give an
explanation of each of the distance measures used
in the experiments. The face images used will be
described in section 5. The results of the
experiments are discussed in section 6. Finally, the
paper ends with our conclusion in section 7.
2. Eigenfaces Eigenfaces is the approach proposed by Turk and
Pentland in [1]. In this approach, face images are
projected into a lower dimensional space using
Principal Component Analysis (PCA). Each image is
represented by a vector of weights needed to
reconstruct the image. Although, in order to
simplify the problem and reduce data storage,
some information is lost in the algorithm and it
may not be possible to completely reconstruct the
images.
The system must first be initialized using a training
set of face images. These images should be
preprocessed to show only the face in order to
reduce recognizing the background or a particular
hairstyle. The eyes in the images should also be in
the same location for normalization. The first step
is to find the mean of all the training images π1,
π2 ,β¦, ππwhere ππ is a column vector. The mean can
be found by the equation Ο =1
π Ti
Mi=1 . The mean
is then subtracted from all training images
ππ = ππ β π for π = 1 π‘π π. The covariance matrix
of the images is found to be πΆ = π΄π΄π , where π΄ =
[π1 π2 β¦ππ]. If the face images started off being
of size π by π, then πΆ is a matrix of size π2 by π2.
This is a large matrix and it is not feasible to solve
for the eigenvalues of πΆ. Since the number of
images is less than the dimensions there will only
be π β 1 meaningful eigenvectors as opposed
to π2. So instead of solving for the eigenvectors of
πΆ, we can solve for the eigenvectors of πΏ,
where πΏ = π΄ππ΄. πΏ is a matrix of size π by π, so
solving for the eigenvectors will find
π eigenvectors. Choosing the desired number of
eigenvectors of πΏ (from the largest eigenvalues),
build the matrix π. Following the equation, π =
π΄π, the eigenfaces of the system are the columns
of π [1].
A face image π is projected into the face space (or
PCA space) by the operation Ξ© = ππ(T β Ο). Ξ© is
a vector composed of the weights describing the
contribution of each eigenface in representing the
input image. To recognize the face, a distance
measure π· is used to find the distance between Ξ©
and Ξ©π , for each subject , where Ξ©π is the face
image of subject π projected into PCA space. If the
minimum of these distances is below the
threshold, then the new face is classified as
belonging to subject π [2].
3. Fisherfaces Fisherfaces is the face recognition approach
proposed by Belhumeur et al. in [3]. Where
Eigenfaces is based on PCA, the Fisherfaces
approach is based on Fisherβs Linear Discriminant
Analysis (LDA). Fisherfaces seeks to maximize the
ratio of between-class scatter to within-class
scatter. Eigenfaces maximizes total scatter across
all images, which makes it susceptible to lighting
variations. Fisherfaces seeks to reduce errors due
to lighting variations by maximizing the ratio
mentioned above.
First let the between-class scatter matrix be
defined as ππ΅ = ππ ππ β π (ππ β π)πππ=1 where π
is the number of classes or subjects, ππ is the mean
image of subject ππ , and ππ is the number of
training images for subject ππ . Also let the within-
class scatter be defined as ππ = (π₯π βπ₯ππππ
ππ=1
ππ)(π₯πβππ)π. If ππ is nonsingular, the
eigenvectors can be found by solving the
equation ππ΅ β πππππ€π = 0. If ππ is singular, the
images can be reduced by running PCA and then
LDA (Fisherface) can be performed [2]. Building π
from the eigenvectors found, the images can be
projected into LDA space by ππππ . Recognition of
a face image follows much the same as in the
Eigenfaces approach [2].
4. Distance Measures Distance measures are used to compute the
difference between two vectors. The CSU Face
Identification Evaluation System includes many
common distance measures that are used to
compute the similarity between two images. Some
of them are studied in this paper. A few
uncommon ones were also implemented. In the
definitions of the distance measures in the
following subsections, let π’ and π£ be vectors
representing arbitrary images in PCA or LDA space
[4].
In order to compute distances in Mahalinobis
space vectors π’ and π£ must be transformed.
Remember that the set of eigenvalues, ππ , from
PCA are the sample variances, ππ2, along the
dimensions represented by the eigenvectors. In
Malahinobis space, sample variances along those
dimensions are one. Let π and π be the vectors in
Mahalinobis space corresponding to π’ and π£. The
vectors are related though the following equations
[4].
ππ =π’π
ππ (1)
ππ =π£π
ππ (2)
Equations 3 through 9 are explained in [4].
Equation 10 is explained in [5]. Equation 11 and 12
are explained in [6].
4.1 CityBlock
π·πΆππ‘π¦π΅ππππ π’, π£ = π’π β π£π
π
(3)
4.2 Euclidean
π·πΈπ’πππππππ π’, π£ = (π’π β π£π)2
π
(4)
4.3 Correlation
π·πΆπππππππ‘πππ π’, π£
= π’π β π’ (π£π β π£ )π
(π β 1) π’π β π’ 2
π
π β 1 π£π β π£ 2
π
π β 1
(5)
4.4 Covariance
π·πΆππ£πππππππ π’, π£ = π’ππ π£π
π’π2
π π£π2
π
(6)
4.5 Mahalinobis CityBlock
π·ππβπΏ1 π’, π£ = ππ β ππ
π
(7)
4.6 Mahalinobis Euclidean
π·ππβπΏ2 π’, π£ = (ππ β ππ)2
π
(8)
4.7 Mahalinobis Cosine
π·ππβπΆππ πππ π’, π£ = π β π
π π (9)
4.8 Hellinger
π·π»ππππππππ π’, π£ = ( π’π β π£π )2
π
(10)
4.9 Canberra
π·πΆπππππππ π’, π£ = π’π β π£π
π’π + π£π π
(11)
4.10 Czekanowski
π·πΆπ§ππππππ€π ππ π’, π£ = 2 β min(π’π , π£π)π
π’π + π£ππ
(12)
5. Data The Facial Recognition Grand Challenge Data is a
set of multi-modal biometric data collected at the
University of Notre Dame. The experiments used
only the subset of 2D frontal face images with
neutral facial expression and controlled lighting.
We used 3939 images from 193 subjects for
training and 18370 images from 563 subjects for
the probe/gallery set. Because the data was
collected on the Notre Dame campus, a large
majority of the images are of people 18 to 22 years
of age [7].
The Facial Recognition Technology (FERET)
Database was created on the behest of the
Defense Advanced Research Projects Agency and
National Institute of Standards and Technology.
The experiments used 600 images from 300
subjects for training and 1536 images from 680
subjects for the probe/gallery set [8].
6. Results Experiments were run using PCA and LDA based
facial recognition with the FRGC and FERET
databases. These experiments produced distances
for each face from each of the other faces. From
these distances, cumulative match characteristic
(CMC) curves were created to show the success of
each distance measure as a whole. Figures 1
through 4 are the CMC curves for the experiments.
Figure 1: PCA using FERET
Figure 2: LDA using FERET
Figure 3: PCA using FRGC
Figure 4: LDA using FRGC
In Figure 1, the top curve is MahCosine. In Figure 2,
the top curves are both Correlation and
Covariance. In Figure 3, the top curve is MahCosine
and Czekanowski. In Figure 4, the top curves are
both Correlation and Covariance.
For PCA using the FERET database, the top curve,
MahCosine, gave a top rank image to subject
match in all but 204 images. When MahCosine did
not produce a rank of 1 for an image the best
ranking distance measure was recorded in table 1.
Ties were given to MahCosine. If any of the other
distance measures tied for best rank they all
received credit. There is some overlap. If the best
matching distance measure was a rank 1 match it
was also recorded.
Best Match Rank 1 Match
CityBlock 33 19
Euclidean 38 20
Correlation 35 20
Covariance 33 21
MahL1 26 12
MahL2 36 18
MahCosine 51 -
Hellinger 7 7
Canberra 6 2
Czekanowski 54 28
Table 1: Non Rank 1 MahCosine best match
0.8
0.9
1
0 200 400 600
CityBlockCorrelationCovarianceEuclideanMahCosineMahL1MahL2CanberraCzekanowskiHellinger
0.8
0.9
1
0 200 400 600
CityBlockCorrelationCovarianceEuclideanMahCosineMahL1MahL2CanberraCzekanowskiHellinger
0.8
0.9
1
0 200 400 600
CityBlockCorrelationCovarianceEuclideanMahCosineMahL1MahL2CanberraCzekanowskiHellinger
0.8
0.9
1
0 200 400 600
CityBlockCorrelationCovarianceEuclideanMahCosineMahL1MahL2CanberraCzekanowskiHellinger
7. Conclusions MahCosine and Czekanowski performed best with
PCA while Correlation and Covariance performed
best with LDA. This does not mean, though, that
these distance measures should be used in all
circumstances and ignore the others. Certain
individuals were matched better with a distance
measure other than the top one for the system.
A system that uses different distance measures for
each image will perform better than a system that
only uses one. In the case of PCA based facial
recognition using the FERET Database, 153 images
would have been matched better using a mixed
distance measure approach.
This data proves that a need exists for a system
that uses different distance measures for each
image. Further research is required into how to
accomplish this task.
8. References [1] M. Turk and A. Pentland, Eigenfaces for
Recognition, Journal of Cognitive Neuroscience, vol. 3, no. 1, pp. 71-86, 1991.
[2] Lecture slides from Cpsc 881-03 βBiometricsβ at Clemson University with Dr. Damon Woodard. βAppearance based facial recognitionβ. Summary of PCA and LDA. http://people.clemson.edu/~woodard/protected/Lecture21.pdf
[3] P.N. Belhumeur, J.P. Hespanha and D.J. Kriegman, Eigenfaces vs. Fisherfaces: recognition using class-specific linear projection, IEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 7, pp. 711-720, 1997.
[4] Beveridge, R., Bolme, D., Teixerira, M., and Draper, B., 2003, The CSU Face Identification Evaluation System User's Guide: Version 5.0, Colorado State University.
[5] Abdi, H., 2007, Distance, in Salkind, N., ed., Encyclopedia of Measurement and Statistics, Sage Publications , Thousand Oaks, CA.
[6] Androutsos, D., Plataniotiss, K.N., and Venetsanopoulos, A.N., 1998, Distance measures for color image retrieval, Image Processing, 1998. ICIP 98. Proceedings. 1998 International Conference on, vol.2, no., pp.770-774 vol.2, 4-7 Oct 1998.
[7] Phillips, P.J., Flynn, P.J., Scruggs, T., Bowyer, K.W., Jin Chang, Hoffman, K., Marques, J., Jaesik Min, Worek, W., 2005, Overview of the face recognition grand challenge, Computer Vision and Pattern Recognition. CVPR 2005. IEEE Computer Society Conference on , vol.1, no., pp. 947-954 vol. 1, 20-25 June 2005.
[8] Rallings C., Thrasher M., Gunter C., Phillips P.J., Wechsler H., Huang J., and Rauss P.J., 1998, The FERET database and evaluation procedure for face-recognition algorithms, Image and Vision Computing, Volume 16, Number 5, 27 April 1998, pp. 295-306(12).