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We present symbolic kernel discriminant analysis (symbolic KDA) for face recognition in the framework of symbolic data analysis. Classical KDA extracts features, which are single-valued in nature to represent face images. These single-valued variables may not be able to capture variation of each feature in all the images of same subject; this leads to loss of information. The symbolic KDA algorithm extracts most discriminating nonlinear interval-type features which optimally discriminate among the classes represented in the training set. The proposed method has been successfully tested for face recognition using two databases, ORL database and Yale face database. The effectiveness of the proposed method is shown in terms of comparative performance against popular face recognition methods such as kernel Eigenface method and kernel Fisherface method. Experimental results show that symbolic KDA yields improved recognition rate.

Kernel principal
component analysis (KPCA) [

In this paper, new appearance based method is proposed in the framework of symbolic data analysis, namely, symbolic KDA for face recognition, which is a generalization of the classical KDA to symbolic objects. In the first step, we represent the face images as symbolic objects (symbolic faces) of interval type variables. Each symbolic face summarizes the variation of feature values through the different images of the same subject. It also drastically reduces the dimension of the image space without losing a significant amount of information.

In the second step, we apply symbolic KDA algorithm to extract interval type nonlinear
discriminating features. According to this algorithm, in the first phase, we apply kernel
function to symbolic faces, as a result a pattern in the original input space is mapped into a
potentially much higher dimensional feature vector in the feature space, and then performs in
the feature space to choose subspace dimension carefully. In the second phase, we extract
interval type nonlinear discriminating features, which are robust to variations due to
illumination, orientation and facial expression. Finally, minimum distance classifier with
symbolic dissimilarity measure [

The remainder of this paper is organized as follows: In Section

Let

We denote

We represent the

Let us consider the matrix

The

Let

The nonlinear mapping,

Substituting (

Since, each
symbolic face

The lower bound features of each symbolic face

The interval-type
features

The proposed symbolic KDA method is experimented with the
face images of the ORL and Yale databases. The effectiveness of proposed method is shown
in terms of comparative performance against two face recognition methods. In particular, we
compared our algorithm with kernel Eigenface
[

We assess the feasibility and performance of the proposed symbolic KDA on the face
recognition task using ORL database. The ORL face database is composed of 400 images
with ten different facial views that represent various expressions and orientations for each
of the 40 distinct subjects as shown in Figure

Some typical images of one subject of ORL database.

Our goal is to find appropriate kernel function and corresponding optimal kernel parameters (i.e., the order of the polynomial kernel and the width of the Gaussian kernel) for our proposed method. The experimental results shows that the order of the polynomial kernel should be three and the width of Gaussian kernel should be four for proposed symbolic KDA with respect to a minimum distance classifier.

After determining the optimal kernel parameters, we set out to select the dimension of
discriminant subspace with respect to two different kernels. Table

Optimal parameters corresponding to each method with respect to two different kernels.

Method | Polynomial kernel | Gaussian kernel | ||
---|---|---|---|---|

Order | Subspace dimension | Width | Subspace dimension | |

Kernel Eigenfaces | 1 | 44 | 7 | 47 |

Kernel fisherfaces | 3 | 35 | 5 | 44 |

Symbolic KDA | 3 | 18 | 4 | 24 |

After selection of optimal parameters and optimal subspace for each method with respect to
different kernels, all three methods are reevaluated using same set of training and testing
samples. The average recognition rates for the best case are shown in Table

Comparison of symbolic KDA method using optimal parameters.

Kernel Eigenface | Symbolic KDA | Kernel Fisherface | |
---|---|---|---|

Polynomial kernel | 55.61 | 91.86 | 84.95 |

Gaussian kernel | 58.05 | 89.39 | 81.57 |

In order to examine, whether symbolic KDA is statistically significant and better than other
methods in terms of its recognition rate. We evaluate the experimental results presented in
Table

The receiver operating characteristic (ROC)
curve in Figure

The ROC performance of proposed symbolic KDA Method, Kernel Eigenface Method, Kernel Fisherface Method and Eigenface Method.

The experiments were
conducted using Yale database to evaluate the excellence of the symbolic KDA
for frontal face recognition under different nondark backgrounds. The Yale face
database consists of a total of 165 images obtained from 15 different people,
with 11 images from each person. Figure

Some typical images of one subject of Yale face database.

The experiments were conducted using two different kernels, namely, polynomial kernel and Gaussian kernel. The order of the polynomial kernel should be 2 and the width of Gaussian kernel should be four for proposed symbolic KDA with respect to a minimum distance classifier.

After finding optimal kernel parameter
(degree 2) for the symbolic KDA method, the experiments were conducted to find
optimal subspace for proposed symbolic KDA, kernel Fisherface, and kernel Eigenface
method. The recognition rates, training time, and optimal subspace dimension for
Kernel Fisherface, Kernel Eigenface, and symbolic KDA are listed in Table

Comparison of classification performance using Yale face database.

Methods | kernel Fisherfaces | kernel Eigenfaces | symbolic KDA |
---|---|---|---|

Recognition rates (%) | 87.15 | 83.28 | 89.00 |

Training time (seconds) | 48.346 | 68.751 | 38.324 |

Feature dimension | 42 | 54 | 15 |

In this paper, we propose a novel symbolic KDA method for face recognition. Symbolic data representation of face images using interval-type features are desirable facial features to cope up with the variations due to illumination, orientation, and facial expression changes. The feasibility of the symbolic KDA has been tested successfully on frontal face images of ORL and Yale databases. Experimental results show that symbolic KDA method with polynomial kernel leads to improved recognition rate at reduced computational cost.

The proposed symbolic KDA has many advantages compared to other popular appearance-based methods. The drawback of other appearance-based methods is that in order to recognize a face seen from a particular pose and under a particular illumination, the face must have been previously seen under the same conditions. The symbolic KDA overcomes this limitation by representing the faces by interval-type features so that even the faces seen previously in different poses, orientations, and illuminations are recognized. Another important merit is that we can use more than one probe image with inherent variability of a face for face recognition, this yields improved recognition rate. This is clearly evident from the experimental results. We observe from the experimental results that the proposed symbolic KDA method yields improved recognition rate in terms of time and feature reduction compare to other kernel-based methods.

The main drawback of our proposed symbolic KDA method is that pose variation is limited up to 20 degree orientation and the performance of proposed method decreases on face images with pose variation greater than 20 degree orientation. The proposed method did not achieve 100% accuracy, this is due to the fact that while constructing the symbolic faces, there may be chance of misalignment of coordinates of eyes, mouth, and nose because of different facial expressions in training images. It can be observed in experimental results obtained using Yale face database, which contains face images with different facial expressions. Moreover, the performance of the proposed symbolic KDA method decreases on images with more variation in facial expressions of Yale face database compared to performance on images with less variation in facial expressions of ORL face database.

The authors are indebted to the referees for their helpful comments and suggestions, which improved the earlier version of the paper.