Distance measures for PCA-based face recognition
Introduction
Principal component analysis (PCA) or Karhunen–Loeve transform (KLT)-based face recognition method was proposed in (Turk and Pentland, 1991). It was studied by computer scientists (Moon and Phillips, 1998; Yilmaz and Gokmen, 2001; Navarrete and Ruiz-del-Solar, 2001, Navarrete and Ruiz-del-Solar, 2002) and psychologists (Abdi et al., 1995; Hancock et al., 1996), used as a baseline method for comparison of face recognition methods (Moghaddam and Pentland, 1998; Phillips et al., 2000) and implemented in commercial applications (Viisage, 2001). Using PCA we find a subset of principal directions (principal components) in a set of training faces. Then we project faces into this principal components space and get feature vectors. Comparison is performed by calculating the distance between these vectors. Usually comparison of face images is performed by calculating the Euclidean distance between these feature vectors. Sometimes the angle-based distance is used. Mathematical formulation of this recognition method is presented in the next section. Although there exist many other distance measures, we were able to find only few attempts to create, compare and use other distance measures (Navarrete and Ruiz-del-Solar, 2002; Phillips et al., 1997, Phillips et al., 2000) in order to achieve better recognition results.
In this article we compare recognition performance of 14 distance measures including Euclidean, angle-based, Mahalanobis and their modifications. Also we propose modified sum square error-based distance and modified Manhattan distance measures. The experiments showed, that the proposed distance measures were among the best distance measures with respect to different characteristics of the biometric systems. For comparison we used the following characteristics of the biometric systems: equal error rate (EER), first one recognition rate, area above cumulative match characteristic (CMC), area below receiver operating characteristic (ROC), percent of images that we need to extract in order to achieve 100% cumulative recognition.
Section snippets
PCA-based face recognition
In this section we will describe Karhunen–Loeve transform (KLT)-based face recognition method, that is often called principal component analysis (PCA) or eigenfaces. We will present only main formulas of this method, whose details could be found in (Groß, 1994).
Let be N-element one-dimensional image and suppose that we have r such images (j=1,…,r). A one-dimensional image-column X from the two-dimensional image (face photography) is formed by scanning all the elements of the two-dimensional
Distance measures
Let , be eigenfeature vectors of length n. Then we can calculate the following distances between these feature vectors (Grudin, 1997; Yambor and Draper, 2002; Phillips et al., 1999, Phillips et al., 2000; Cekanavicius and Murauskas, 2002):
- (1)
Minkowski distance (Lp metrics)here p>0;
- (2)
Manhattan distance (L1 metrics, city block distance)
- (3)
Euclidean distance (L2 metrics)
- (4)
Squared euclidean distance (sum
Experiments and results
For experiments we used images from the AR (AR, 1998; Martinez and Benavente, 1998), Bern (1995), BioID (2001), Yale (1997), Manchester (1998), MIT (MIT, 1989; Turk and Pentland, 1991), ORL (ORL, 1992; Samaria and Harter, 1994), Umist (Umist, 1997; Graham and Allinson, 1998), FERET (Phillips et al., 1997) databases. From these databases we collected the database containing photographies of 423 persons (two images per person––one for learning and one for testing). In order to avoid recognition
Conclusions
In this publication we compared 14 distance measures and their modifications for principal component analysis-based face recognition method and proposed modified sum squared error (SSE)-based distance measure. Recognition experiments were performed using the database containing photographies of 423 persons. The experiments showed, that the proposed distance measure is among the first three best measures with respect to different characteristics of the biometric systems. The best recognition
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