Di Liu Dong-mei Sun Zheng-ding Qiu Institute of Information Science,

A Comparison in Handmetric between Quaternion Euclidean Product Distance and Cauchy Schwar

tz Inequality Distance

Di Liu Dong-mei Sun Zheng-ding Qiu

Institute of Information Science,

Beijing Jiaotong University, P.R.China 100044

Outline

• 1.Introduction

• 2.QEPD & CSID

• 3.Comparison by Experiment

• 4.Conclusion

1.Introduction

• This paper proposes a comparison in handmetrics between Quaternion Euclidean product distance (QEPD) and Cauchy-Schwartz inequality distance (CSID), where "handmetrics" refers to biometrics on palmprint or finger texture.

1.Introduction

• All two distances could be constructed by quaternion which was introduced for reasonable feature representation of physical significance, i.e. 4-feature parallel fusion. Simultaneously, such quaternion representation enables to avoid incompatibleness of multi-feature dimensionality of quaternion fusion.

1.Introduction

• We give a comparison on experimental aspects for providing a conclusion which algorithm is better.

2.QEPD & CSID

Quaternion Euclidean Product Distance (QEPD)

• Quaternion

is a non-commutative extension of complex numbers, which first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space [4].

| , , ,P a bi cj dk a b c d

Quaternion Euclidean Product Distance

• Quaternion properties2 2 2 1i j k ijk

, , ,

, ,

ij k ji k jk i

kj i ki j ik j

Conjugate P a bi cj dk

Modulus 2 2 2 2 0P PP a b c d

| , , ,Q t xi yj zk t x y z Assume

Quaternion Euclidean Product Distance

Equality , , ,P Q a t b x c y d z

Addition ( ) ( ) ( ) ( )P Q a t b x i c y j d z k

Multiplication ( ) ( )

( ) ( )

P Q at bx cy dz bt ax cz dy i

ct ay dx bz j dt az by cx k

Inner Product ,P Q at bx cy dz

Quaternion Euclidean Product QEPP PQ

Quaternion Euclidean Product Distance

• QEPD

Consider that when , QEP is equal to

, i.e. the square of modulus

Assume two quaternions P and Q, which the former is from an arbitrary pixel corresponding to 4 separable wavelets decomposition sub-image from template,

P Q2

PP P

Quaternion Euclidean Product Distance

Ideally, if such that , where is a particular case of QEP. We estimate the difference between the template quaternion P and the test one (Q) by QEPD,

where

P Q 2PP P

( ,| |)D PP PQ

( ) ( )

( ) ( )

PQ at bx cy dz ax bt cz dy i

ay bz ct dx j az by cx dt k

Cauchy-Schwartz inequality distance

• Autocorrelation for signal processing. Autocorrelation is used frequently in signal processing for analyzing functions or series of values, such as time domain signals [7].

Let x(t) as a continuous signal sequence, where denotes x(ti); x(tj) as the sequence at time ti; tj respectively. Define the autocorrelation in the case of serial signals in the equation 10.

Cauchy-Schwartz inequality distance

• In the case of discrete signal sequence, we define

as autocorrelation for

• This term also has many properties, e.g. symmetry, when x(n) is a real sequence,

Cauchy-Schwartz inequality distance

• when x(n) is a complex one

In addition, a quaternion wavelet feature representation for parallel fusion is viewed as a real signal sequence. Under a similar image acquisition condition, there is more or less a distinction among sample images. To this end, the images are treated as two dimensional stochastic signal.

Cauchy-Schwartz inequality distance

• Cauchy-Schwartz inequality distance. Consider two quaternions, which the forme

r is constructed by the same pixel corresponding to 4 separable wavelets decomposition coefficients from the template, and the latter from the tester, in which

If template and tester one belong to the same person, will be more similar with

Cauchy-Schwartz inequality distance

• than that from different persons, i.e.

modulus of is smaller than that from different persons. Thus we obtain

• Thus we view P and Q as real signal sequences in sense of discrete-time signals, i.e. transform quaternion P and Q into forms of and

Cauchy-Schwartz inequality distance

in which x(p) and x(q) are discrete sequences at time p and q respectively. According to the autocorrelation discussed above, the equation (12) can be rewritten as

• Where p = q - , time q can be viewed as a time delay to p. Now evolve the equation (12)

Cauchy-Schwartz inequality distance

• Replace (15) with (14), we obtain

• It is easily found that the left side of the equation above is larger than 0. To this end, it is accommodate to set this as the distance of the pixel for 4 sub-images with a reasonable physical significance.

Cauchy-Schwartz inequality distance

• Where k is number of pixel in such sub-images, n is the component number of discrete signal sequence, e.g. k = 4 because of the quaternion. Notice that , the sum of , evidently has a characteristic of

3.Comparison by Experiment

• Database

BJTU-HA biometric database, an inherited collection work by Institute of Information Science, Beijing Jiaotong University, is utilized for our palmprint and middle finger texture verification experiment. It contains totally 1,500 samples from 98 person's palmprint and middle finger with different illumination conditions.

QEPD computation

QEPD computation

QEPD computation

CSID computation

ROC curve for 4 distances

CONCLUSION

• This paper mainly proposes a comparison by two different discriminate distance, Quaternion Euclidean Product Distance (QEPD) and Cauchy-Schwartz Inequality Distance (CSID), in order to solve space incompatibleness and curse of dimensionality in a non-subspace means. From the experiments, we can safely provide two conclusions: (1) the algorithm of QEPD is better than that of CSID, (2) finger texture, this novel biometric is a better discriminative than traditional biometric palmprint for QEPD or CSID.

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That’s end!

Thank you!