Gabor Wavelets in Behavioral Biometrics

Introduction

Biometrics involves the development of statistical and mathematical methods applicable to data analysis problems in the biological sciences. More specifically, the term “biometrics” is derived from the Greek words bio (life) and metric (to measure). The sole purpose of biometrics is either identification or verification. This is performed based on two unique characteristics of human; the physiological uniqueness and the behavioral uniqueness. Physiological uniqueness is the characteristics that people are born with, which include fingerprint, face, and iris, etc. Behavioral uniqueness is the characteristics that people adapt to as they grow-up, which include gait, speech, etc. These are called behavioral, because uniqueness of these acts is expressed through behavior of an individual, for example, the way one walks (gait), and talks (speech).

Possibly the first known example of biometrics in practice was in India and China around the 15^th century, where children's palm prints and footprints was marked on paper with ink to distinguish the young children from one another and also to analyze their future. In the 1960’s, scientists started to use computer for automated face and speech recognition. Many applications have been developed and different approaches have been taken into account to further modify the existing systems. Among all approaches, biologically inspired computational methods such as Gabor Wavelets are more powerful in terms of optimality, as they seem to be in coherence with the physics of the natural world.

In 1946 using Schwarz inequality arguments, Dennis Gabor proved the “indeterminacy relations of quantum mechanics” (Gabor, 1946). He proved that a signal’s specificity, simultaneously in time and frequency is fundamentally limited by a lower bound on the product of its bandwidth and duration. Gabor referred to Heisenberg & Weyl's uncertainty-related-proofs for the natural world and derived the uncertainty relation for information. Based on this understanding he found the general family of signals that optimize this trade-off and thus achieve the theoretical lower limit of joint uncertainty in time and frequency. The so-called Gabor signals take the general form:

Here the complex notation describes the modulation product of a sine wave with arbitrary frequency ω and a Gaussian envelope of arbitrary duration σ occurring at epoch t₀. Gabor also proposed representing arbitrary signals by a pseudo expansion set of these elementary signals, which he termed “logons” in the information plane, indexed by all different frequencies of modulation and all different epochs of time.

The research on suitability of Gabor’s theory in computer vision fund its motivation in the discovery of a group of neuroscientists and psychologists who claimed by showing experimental results that in visual perception natural images are decomposed into Fourier-like spatial-frequency components (Campbell & Robson, 1968; Kulikowski & Bishop, 1981; Maffei & Fiorentini, 1973; Pollen et al., 1971). In 1980 Marcelja published a paper where Dennis Gabor’s “Theory of communications” was pointed out to all vision scientists by interpretations of cortical simple cell receptive-field profiles. Then subsequent research (Daugman, 1980; DeValois et al., 1982; Poleen & Ronner, 1981; Tootell et al., 1981) confirmed that usually two to five interleaved regions of simulative-restrictive influences weighted by a tapering envelope constitute the receptive-field profile of a simple cell, and Gabor signals with suitably chosen parameters invariably give a good fit to such spatial profiles.