A Comparison between Quaternion Euclidean Product Distance and Cauchy Schwartz Inequality Distance - Advances in Computational Science and Engineering - page 130

complex feature vector form that aims at obtaining a higher recognition effect
and a faster speed than serial fusion. According to the advantage above, a wavelet
decomposition feature based quaternion fusion with QEPD [1,2] for palmprint
or finger texture authentication was proposed. Also, another one with CSID [11]
migrated from face verification is implemented for palmprint or finger in this
paper. We propose a reasonable interpretation for the mathematical significance
of 4-feature parallel fusion by introducing quaternion prior to a discrimination by
such two distances respectively. Then we conduct a comparison on experimental
aspects for providing a conclusion which algorithm is better. The rest of paper
is organized as follows: section 2 introduces QEPD and CSID in turn. The next
section we propose our experimental comparison between such two distances for
handmetrics verification in details. After that, section 4 provides a conclusion.

2QEPDandCSID

2.1 Quaternion Euclidean Product Distance (QEPD)

We here make QEPD available as a discriminant distance. The relationship be-
tween QEP and quaternion modulus was discussed in [1,2]. Thus for an ar-
bitrary pixel corresponding to 4 separable wavelets decomposition sub-image,
consider two quaternions, which the former is from this pixel as the template,
P = a + bi + cj + dk and the latter is from the tester, in which Q = t + xi + yj +
zk . Ideally, if P

2 , where is a particular case of Quater-
nion Euclidean product. We can estimate the difference between the template
quaternion matrix and the tester one by QEPD D ( PP,|PQ|

= Q such that PP = |P |

) as a discriminant

distance. In which is an entry of the matrix with the modulus of QEP,

and the operator D is certain kind of distance, e.g. L 1 norm, L ∞ norm distance,
Euclidean distance etc. Such two matrices have the same size as the subimage
above. The reason use the modulus is that the multiplication of two different
quaternions is a quaternion so that it is impossible to compare directly with PP .
Because PQ is not a scalar as follows.

|PQ|

PQ =( a − bi − cj − dk )( t + xi + yj + zk )
=( at + bx + cy + dz )+( ax − bt − cz + dy ) i
+( ay + bz − ct − dx ) j +( az − by + cx − dt ) k

(1)

Therefore, the similarity between template and tester QEPD D ( PP,|PQ| )is
obtained by Euclidean distance between the matrix of the absolute value of the
equation (1) and that of template's modulus square |P |

2 .

2.2 Cauchy-Schwartz Inequality Distance (CSID)
(1) Autocorrelation for signal processing. Autocorrelation is used fre-
quently in signal processing for analyzing functions or series of values, such
as time domain signals [6]. We here discuss the case of discrete signal sequence,
defining

R xx ( j )=
n

x n x n−j (2)