Hand Biometrics Using High Order Zernike Moments

<Contributed to MeLi Wiki by Professor George Bebis, Department of Computer Science and Engineering, University of Nevada, Reno>

Hand Biometrics

Recently, there has been increased interest in developing biometrics-based authentication systems which has led to intensive research in fingerprint, face, hand, ear, and iris recognition . Each biometric has its strength and weakness depending on the application and its requirements. The geometry of the hand contains relatively invariant features of an individual. Hand-based authentication is among the oldest live biometrics-based authentication modalities. The existence of several hand-based authentication commercial systems and patents indicate the effectiveness of this type of biometric.

Hand shape can be easily captured in a relatively user friendly manner by using conventional CCD cameras. This technology is more acceptable by the public in daily life mainly because it lacks a close connection to forensic applications. However, hand-based verification systems are usually employed in small scale person authentication applications due to the fact that geometric features of the hand (e.g., finger length/width, area/size of the palm) are not as distinctive as fingerprint or iris features. Lately, there has been some interest lately in fusing different biometrics to increase system performance. The ease of use and acceptability of hand-based biometrics make hand shape a good candidate in these heterogeneous systems.

The majority of hand-based verification systems use geometric measurements. In these systems, the user is asked to place his/her hand on a surface and align it, with the help of some guidance pegs, on the surface. A mirror is usually used to obtain a side view of the hand using a single camera. The alignment operation simplifies the feature extraction process to a great extent and allows high processing speeds. In most cases, a few handcrafted geometric features (e.g. length, width and height of the fingers, thickness of the hand, aspect ratio of fingers and palm, etc.) are extracted, making it possible to construct a small template. Removal of pegs, to improve convenience, and use of more powerful feature extraction techniques to capture the shape of the hand more accurately represent promising research directions in this area. In this assignment, you will investigate using high-order Zernike moments for capturing hand geometry.

Zernike Moments

Moment functions of image intensity values are used to capture global features of the image in pattern recognition and image analysis . Among many moment based descriptors, Zernike moments have minimal redundancy, rotation invariance and robustness to noise; therefore they are used in a wide range of applications on image analysis, reconstruction and recognition. Zernike moments are based on a set of complex polynomials that form a complete orthogonal set over the interior of the unit circle . They are defined to be the projection of the image function on these orthogonal basis functions. The basis functions <math>V_{n,m}(x,y)</math> are given by:

<math> V_{n,m}(x,y)=V_{n,m}(\rho,\theta)=R_{n,m}(\rho)e^{jm\theta} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1) </math>

where n is a non-negative integer, m is non-zero integer subject to the constraints n-|m| is even and n<|m|, ρ is the length of the vector from origin to (x,y), <math>\theta</math> is the angle between vector ρ and the x-axis in a counter clockwise direction and <math>R_{n,m}(\rho)</math> is the Zernike radial polynomial. The Zernike radial polynomials,<math>R_{n,m}(\rho)</math>,are defined as:

<math> R_{n,m}(\rho)=\sum_{k=|m|,n-k=even}^{n}\frac{(-1)^{\frac{n-k}{2}}\frac{n+k}{2}!}{\frac{n-k}{2}!\frac{k+m}{2}!\frac{k-m}{2}!}\rho^{k}=\sum_{k=|m|,n-k=even}^{n}\beta_{n,m,k}\rho^{k} \;\;\;\;\;\;\;\;(2) </math>

Note that <math>R_{n,m}(\rho)=R_{n,-m}(\rho)</math>. The basis functions in equation 1 are orthogonal thus satisfy

<math> \frac{n+1}{\pi}\iint_{x^{2}+y^{2}\leqslant 1}V_{n,m}(x,y)V_{p,q}^{*}(x,y)=\delta _{n,p}\delta _{m,q} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3) </math>

where

<math> \delta_{a,b}=\begin{cases} 1 & \text{ if } a=b \\ 0 & \text{ if } otherwise \end{cases} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(4) </math>

The Zernike moment of order n with repetition m for a digital image function <math>f(x,y)</math> is given by :

<math> Z_{n,m}=\frac{n+1}{\pi}\sum\sum_{x^{2}+y^{2}\leq 1}f(x,y)V_{n,m}^{*}(x,y) \;\;\;\;\;\;\;\;(5) </math>

where <math>V_{n,m}^{*}(x,y)</math> is the complex conjugate of <math>V_{n,m}(x,y)</math>. To compute the Zernike moments of a given image, the image center of mass is taken to be the origin. The function <math>f(x,y)</math> can then be reconstructed by the following truncated expansion:

<math> \tilde{f}(x,y)=\sum_{n=0}^{N}\frac{C_{n,0}}{2}R_{n,0}(\rho)+\sum _{n=1}^{N}\sum_{m>0}(C_{n,m}cosm\theta+s_{n,m}sinm\theta)R_{n,m}(\rho) \;\;\;\;\;\;\;\;(6) </math>

where N is the maximum order of Zernike moments we want to use, Cn,m and <math>S_{n,m}</math> denote the real and imaginary parts of <math>Z_{n,m}</math> respectively.

Accurate and Efficient Computation of High Order Zernike Moments

Direct computation of Zernike moments is very expensive, limiting their use especially at high orders. There have been some efforts to reduce the computational cost by employing quantized polar coordinate systems, however, quantization introduces some error in high order Zernike moments. To preserve accuracy, a different approach has been proposed which does not use any form of coordinate transformation and employs arbitrary precision arithmetic . The computational complexity is reduced by detecting the common terms in Zernike moments with different order and repetition. Specifically, by substituting equations 2 and 1 in 5 and re-organizing the terms the Zernike moments can be calculated as follows:

<math> \begin{align} Z_{n,m} &=\frac{n+1}{\pi}\sum \sum_{x^{2}+y^{2}\leq 1}(\sum_{k=|m|}^{n}\beta_{n,m,k}\rho^{k})e^{-jm\theta}f(x,y)\\ &= \frac{n+1}{\pi}\sum_{k=|m|}^{n}\beta_{n,m,k}(\sum \sum_{x^{2}+y^{2}\leq 1}e^{-jm\theta}\rho^{k}f(x,y))\\ &= \frac{n+1}{\pi}\sum_{k=|m|}^{n}\beta_{n,m,k}\chi _{m,k} \;\;\;\;\;\;\;\;\;\;\;\;\;\;((7)

\end{align} </math>