Generalized Gabor filters for palmprint recognition

Abstract

Orientation-based coding approaches have recently been widely employed for face and palmprint recognition where generally, one starts with a set of Gabor filters to extract orientation information and then proceeds to code dominant orientations as features for each point of the palmprint. However, as the Gabor filter is developed to model two-dimensional receptive fields of simple cells in straits cortex, it might not be our best choice when dealing with curved and complex structures inherent in the palmprint texture. Motivated by this intuition, this paper shows that Gabor filters are a subset of a bigger family of filters which we refer to as generalized Gabor filter (GGF). Depending on the values of its parameters, a GGF takes a rather diverse shapes and orientations, which results in a potentially finer feature extraction capability. We show this improved capability by employing GGFs in the palmprint verification process. In applying our method, two different sub-banks of GGFs are defined for the orientation-based feature extraction of palmprints, and when compared with Gabor filters, it will be shown that GGFs have the upper hand in capturing orientation features. Furthermore, compared with the competitive code—one of the well-known orientation-based coding methods—the number of employed orientations is reduced to half. This would automatically compensate for a double usage of the filter banks, which otherwise could increase the time complexity of using GGFs. These ideas are further elaborated using a set of experiments on PolyU II and PolyU 2D/3D palmprint databases. The results show the preeminence of using GGFs both in terms of accuracy and efficiency.

Appendix

We divide Eq. (4) into two parts based on distributive law:

$$\begin{aligned} B\left( {x,y,x_{0} ,y_{0} ,\eta ,\theta ,\omega_{0} } \right) & = B_{1} + B_{2} = \frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left( { - \left( {x - x_{0} } \right)\sin \theta + \left( {y - y_{0} } \right)\cos \theta } \right)^{\eta } \\ & \quad \times \exp \left( { - \frac{{\omega_{0}^{2} }}{{8\kappa^{2} }}\left( {4\left( {\left( {x - x_{0} } \right)\cos \theta + \left( {y - y_{0} } \right)\sin \theta } \right)^{2} + \xi \left( { - \left( {x - x_{0} } \right)\sin \theta + \left( {y - y_{0} } \right)\cos \theta } \right)^{2} } \right)} \right) \\ & \quad \times \exp \left( {i\left( {\omega_{0} \left( {x - x_{0} } \right)\cos \theta + \omega_{0} \left( {y - y_{0} } \right)\sin \theta } \right)} \right) - \frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left( { - \left( {x - x_{0} } \right)\sin \theta + \left( {y - y_{0} } \right)\cos \theta } \right)^{\eta } \\ & \quad \times \exp \left( { - \frac{{\omega_{0}^{2} }}{{8\kappa^{2} }}\left( {4\left( {\left( {x - x_{0} } \right)\cos \theta + \left( {y - y_{0} } \right)\sin \theta } \right)^{2} + \xi \left( { - \left( {x - x_{0} } \right)\sin \theta + \left( {y - y_{0} } \right)\cos \theta } \right)^{2} } \right)} \right) \times \exp \left( { - \frac{{\kappa^{2} }}{2}} \right). \\ \end{aligned}$$

(13)

For the first term, the 2-D Fourier transform of the GG wavelet defined in Eq. (4) is given by

$$\begin{aligned} \bar{B}_{1} \left( {u,v } \right) & = \int \int \frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left( { - \left( {x - x_{0} } \right)\sin \theta + \left( {y - y_{0} } \right)\cos \theta } \right)^{\eta } \\ & \quad \times \exp \left( { - \frac{{\omega_{0}^{2} }}{{8\kappa^{2} }}\left( {4\left( {\left( {x - x_{0} } \right)\cos \theta + \left( {y - y_{0} } \right)\sin \theta } \right)^{2} + \xi \left( { - \left( {x - x_{0} } \right)\sin \theta + \left( {y - y_{0} } \right)\cos \theta } \right)^{2} } \right)} \right) \\ & \quad \times \left[ {\exp \left( {i\left( {u_{0} \left( {x - x_{0} } \right) + v_{0} \left( {y - y_{0} } \right)} \right)} \right)} \right]\exp \left( { - i\left[ {ux + vy} \right]} \right)dxdy \\ \end{aligned}$$

(14)

where $u_{0} = \omega_{0} \cos \theta$ and $v_{0} = \omega_{0} \sin \theta$. We rewrite the wavelet formula to include a coefficient, a Gaussian envelope, and a sinusoid carrier in $N$-dimensions.

$$\bar{B}_{1} \left( {u,v } \right) = K\left( {X - X_{0} } \right)^{\eta } \wp \left[ {A\left( {X - X_{0} } \right)} \right]S\left( {X - X_{0} } \right).$$

(15)

$K\left( {X - X_{0} } \right)^{\eta }$ (the coefficient), $\wp \left( z \right)$(the Gaussian envelope), and $S\left( z \right)$(the sinusoidal carrier) are defined as follows:

$$\wp \left( Z \right) = \exp \left( { - Z^{T} Z} \right)$$

(16)

$$S\left( Z \right) = \exp \left( { - jU_{0}^{T} Z} \right).$$

(17)

The Fourier transform of this function could be derived as:

$$\begin{aligned} \bar{B}\left( U \right) & = \mathop \int \nolimits_{ - \infty }^{ + \infty } K\left( {X - X_{0} } \right)^{\eta } \wp \left[ {A\left( {X - X_{0} } \right)} \right]\exp \left( {jU_{0}^{T} \left( {X - X_{0} } \right)} \right)\exp \left( { - jU^{T} X} \right)dX \\ & \quad = \exp \left( { - jU_{0}^{T} X_{0} } \right)\mathop \int \nolimits_{ - \infty }^{ + \infty } K\left( {X - X_{0} } \right)^{\eta } \wp \left[ {A\left( {X - X_{0} } \right)} \right]\exp \left( {jU_{0}^{T} X} \right)\exp \left( { - jU^{T} X} \right)dX \\ & \quad = \exp \left( { - jU_{0}^{T} X_{0} } \right)\mathop \int \nolimits_{ - \infty }^{ + \infty } K\left( {X - X_{0} } \right)^{\eta } \wp \left[ {A\left( {X - X_{0} } \right)} \right]\exp \left( { - j\left( {U - U_{0} } \right)^{T} X} \right)dX \\ \end{aligned}$$

(18)

let $\bar{X} = A\left( {X - X_{0} } \right).$ Then, we have $d\bar{X} = AdX$. So:

$$\bar{B}_{1} \left( U \right) = \frac{K}{{A^{\eta + 1} }}\exp \left( { - jU_{0}^{T} X_{0} } \right)\mathop \int \nolimits_{ - \infty }^{ + \infty } \bar{X}^{\eta } \wp \left[ {\bar{X}} \right]\exp \left[ { - j\left( {U - U_{0} } \right)^{T} \left( {A^{ - 1} \bar{X} + X_{0} } \right)} \right]d\bar{X}$$

(19)

$$\Rightarrow \bar{B}_{1} \left( U \right) = \frac{K}{{A^{\eta + 1} }}\exp \left[ { - j\left( {U - U_{0} } \right)^{T} X_{0} } \right]\exp \left( { - jU_{0}^{T} X_{0} } \right)\mathop \int \nolimits_{ - \infty }^{ + \infty } \bar{X}^{\eta } \wp \left[ {\bar{X}} \right]\exp \left[ { - j\left(A^{ - T} \left( {U - U_{0} } \right) \right)^{T} \bar{X}} \right]d\bar{X}.$$

(20)

Since(see [46]): $\mathop \int \nolimits_{ - \infty }^{ + \infty } x^{\eta } \exp \left( { - px^{2} + 2qx} \right)dx = \frac{1}{{2^{\eta - 1} p}}\sqrt {\frac{\pi }{p}} \frac{{d^{\eta - 1} }}{{dq^{\eta - 1} }}\left( {q\exp \left( {\frac{{q^{2} }}{p}} \right)} \right),\quad p > 0.$ We have $\bar{B}_{1} \left( U \right) = \frac{K}{{A^{\eta + 1} }}\exp \left[ { - j\left( {U - U_{0} } \right)^{T} X_{0} } \right]\exp \left( { - jU_{0}^{T} X_{0} } \right)\frac{1}{{2^{\eta - 1} }}\sqrt \pi \frac{{d^{\eta - 1} }}{{d\left( { - j\left(A^{ - T} \left( {U - U_{0} } \right)\right)^{T}/2} \right)^{\eta - 1} }}\left( {\left( { - j\left ( A^{ - T} \left( {U - U_{0} } \right) \right)^{T}/2} \right) \exp \left( {{\raise0.7ex\hbox{${\left(A^{ - 2T} \left( {U - U_{0} } \right)^{2}\right)^{T} }$} \!\mathord{\left/ {\vphantom {{A^{-T} \left( {U - U_{0} } \right)^{2} } 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)} \right).$

Here, $K = \frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left[ {0 1} \right]^{\eta } \left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]^{\eta }$, $A = C \cdot R$ where $C = \left[ {\begin{array}{*{20}c} {i\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} } & 0 \\ 0 & {i\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} } \\ \end{array} } \right]$ is the coefficient matrix and $R = \left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]$ is the rotation matrix. Since $R$ is a rotation:

$$A^{ - 1} = C^{ - 1} R^{ - T}$$

(21)

$$A^{ - T} = RC^{ - 1} = C^{ - 1} R$$

(22)

$$||A|| = ||C|| \cdot ||R|| = \frac{{\omega_{0}^{2} \sqrt \xi }}{{4\kappa^{2} }}$$

(23)

$$\begin{aligned} \Rightarrow A^{ - T} & = \left[ {\begin{array}{*{20}c} {i\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} } & 0 \\ 0 & {i\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right] = - \frac{1}{{\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} }}\left[ {\begin{array}{*{20}c} {i\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} } & 0 \\ 0 & {i\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right] = \\ & \quad - \frac{1}{{\frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}}}\left[ {\begin{array}{*{20}c} {i\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \cos \theta } & {i\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \sin \theta } \\ {i\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \sin \theta } & {i\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \cos \theta } \\ \end{array} } \right]. \\ \end{aligned}$$

(24)

Let $\varUpsilon \left( { A^{-T} \left( {U - U_{0} }\right) } \right) = \frac{{d^{\eta - 1} }}{{d\left( { - j\left(A^{ - T} \left( {U - U_{0} } \right)\right)^{T}/2} \right)^{\eta - 1} }}\left( {\left( { - j\left(A^{ - T} \left( {U - U_{0} } \right)\right)^{T}/2} \right)\exp \left( {{\raise0.7ex\hbox{${\left(A^{ - 2T} \left( {U - U_{0} } \right)^{2}\right)^{T} }$} \!\mathord{\left/ {\vphantom {{A^{ - 2T} \left( {U - U_{0} } \right)^{2} } 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)} \right).$ Since the GGF is a two-dimensional wavelet, we have

$$\bar{B}_{1} \left( {u,v} \right) = \frac{K}{{A^{\eta + 1} }}\frac{1}{{2^{\eta - 1} }}\sqrt \pi \exp \left[ { - j\left( {ux_{0} + vy_{0} } \right)} \right]\varUpsilon \left( {\left[ {\begin{array}{*{20}c} {u - u_{0} } \\ {v - v_{0} } \\ \end{array} } \right]} \right)$$

(25)

$$\begin{aligned} \bar{B}_{1} \left( {u,v} \right) & = \frac{{\frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left[ {\begin{array}{*{20}c} 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]^{\eta } }}{{\left( { - \frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}} \right)^{\eta + 1} }}\exp \left[ { - j\left( {ux_{0} + vy_{0} } \right)} \right] \\ & \quad \varUpsilon \left( { - \frac{i}{{\frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}}}\left[ {\begin{array}{*{20}c} {\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \cos \theta } & {\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \sin \theta } \\ { - \sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \sin \theta } & {\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \cos \theta } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u - u_{0} } \\ {v - v_{0} } \\ \end{array} } \right]} \right) \\ & \quad = \frac{{\frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left[ {\begin{array}{*{20}c} 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]^{\eta } }}{{\left( { - \frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}} \right)^{\eta + 1} }}\exp \left[ { - j\left( {ux_{0} + vy_{0} } \right)} \right]\varUpsilon \left( { - \frac{i}{{\frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}}}\left[ {\begin{array}{*{20}c} {\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \left( {u - u_{0} } \right)\cos \theta + \sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \left( {v - v_{0} } \right)\sin \theta } \\ { - \sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \left( {u - u_{0} } \right)\sin \theta + \sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \left( {v - v_{0} } \right)\cos \theta } \\ \end{array} } \right]} \right) \\ \end{aligned}$$

(26)

For the second term we have:

$$\begin{aligned} \bar{B}_{2} \left( U \right) & = \mathop \int \nolimits_{ - \infty }^{ + \infty } K\left( {X - X_{0} } \right)^{\eta } \wp \left[ {A\left( {X - X_{0} } \right)} \right]\exp \left( { - \frac{{\kappa^{2} }}{2}} \right)\exp \left( { - jU^{T} X} \right)dX \\ & \quad = \exp \left( { - \frac{{\kappa^{2} }}{2}} \right)\mathop \int \nolimits_{ - \infty }^{ + \infty } K\left( {X - X_{0} } \right)^{\eta } \wp \left[ {A\left( {X - X_{0} } \right)} \right]\exp \left( { - jU^{T} X} \right)dX \\ \end{aligned}$$

(27)

let $\bar{X} = A\left( {X - X_{0} } \right).$ Then, we have $d\bar{X} = AdX.$ So:

$$\bar{B}_{2} \left( U \right) = \frac{K}{{A^{\eta + 1} }}\exp\left( { - \frac{{\kappa^{2} }}{2}} \right)\mathop \int \nolimits_{ - \infty }^{ + \infty } \bar{X}^{\eta } \wp \left[ {\bar{X}} \right]\exp \left[ { - jU^{T} \left( {A^{ - 1} \bar{X} + X_{0} } \right)} \right]d\bar{X}$$

(28)

$$\Rightarrow \bar{B}_{2} \left( U \right) = \frac{K}{{A^{\eta + 1} }}\exp \left( { - \frac{{\kappa^{2} }}{2}} \right)\exp \left(-jU^{T}X_{0} \right)\mathop \int \nolimits_{ - \infty }^{ + \infty } \bar{X}^{\eta } \wp \left[ {\bar{X}} \right]\exp \left[ { - jA^{ - T} U\bar{X}} \right]d\bar{X}.$$

(29)

Since (see [46]) $\mathop \int \nolimits_{ - \infty }^{ + \infty } x^{\eta } \exp \left( { - px^{2} + 2qx} \right)dx = \frac{1}{{2^{\eta - 1} p}}\sqrt {\frac{\pi }{p}} \frac{{d^{\eta - 1} }}{{dq^{\eta - 1} }}\left( {q\exp \left( {\frac{{q^{2} }}{p}} \right)} \right),\quad p > 0.$We have $\bar{B}\left( U \right) = \frac{K}{{A^{\eta + 1} }}exp\left( { - \frac{{\kappa^{2} }}{2}} \right)\exp \left(-jU^{T}X_{0}\right)\frac{1}{{2^{\eta - 1} }}\sqrt \pi \frac{{d^{\eta - 1} }}{{d\left( { - j\left(A^{ - T} U\right)^{T}/2} \right)^{\eta - 1} }}\left( {\left( { - j\left(A^{ - T} U\right)^{T}/2} \right)\exp \left( {{\raise0.7ex\hbox{${\left(A^{ - 2T} U^{2} \right)^{T}}$} \!\mathord{\left/ {\vphantom {{A^{ - 2T} U^{2} } 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)} \right).$ Let $\phi \left( {U} \right) = \frac{{d^{\eta - 1} }}{{d\left( { - j\left(A^{ - T} U\right)^{T}/2} \right)^{\eta - 1} }}\left( {\left( { - j\left(A^{ - T} U\right)^{T}/2} \right)\exp \left( {{\raise0.7ex\hbox{${\left(A^{ - 2T} U^{2}\right)^{T} }$} \!\mathord{\left/ {\vphantom {{A^{ - 2T} U^{2} } 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right)} \right).$ Since the GGF is a two-dimensional wavelet, we have

$$\bar{B}_{2} \left( {u,v} \right) = \frac{K}{{A^{\eta + 1} }}\exp \left( { - \frac{{\kappa^{2} }}{2}} \right)\exp\left(-jU^{T}X_{0}\right)\frac{1}{{2^{\eta - 1} }}\sqrt \pi \phi \left( {A^{-T} \left[ {\begin{array}{*{20}c} u \\ v \\ \end{array} } \right]} \right)$$

(30)

$$\begin{aligned} \bar{B}_{2} \left( {u,v} \right) & = \frac{{\frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left[ {\begin{array}{*{20}c} 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]^{\eta } }}{{\left( { - \frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}} \right)^{\eta + 1} }}\exp \left( { - \frac{{\kappa^{2} }}{2}} \right)\exp\left(-j\left(ux_{0}+vy_{0}\right) \right)\frac{1}{{2^{\eta - 1} }}\phi \left( { - \frac{i}{{\frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}}}\left[ {\begin{array}{*{20}c} {\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \cos \theta } & {\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} \sin \theta } \\ { - \sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \sin \theta } & {\sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} \cos \theta } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} u \\ v \\ \end{array} } \right]} \right) \\ & \quad = \frac{{\frac{{\omega_{0} }}{{\sqrt {2\pi } \kappa }}\left[ {\begin{array}{*{20}c} 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]^{\eta } }}{{\left( { - \frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}} \right)^{\eta + 1} }}\exp \left( { - \frac{{\kappa^{2} }}{2}} \right)\left(-j\left(ux_{0}+vy_{0}\right)\right)\frac{1}{{2^{\eta - 1} }}\phi \left( { - \frac{i}{{\frac{{\sqrt \xi \omega_{0}^{2} }}{{4\kappa^{2} }}}}\left[ {\begin{array}{*{20}c} {\sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} u\cos \theta + \sqrt {\frac{{\omega_{0}^{2} \xi }}{{8\kappa^{2} }}} v \sin \theta } \\ { - \sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} u\sin \theta + \sqrt {\frac{{\omega_{0}^{2} }}{{2\kappa^{2} }}} v \cos \theta } \\ \end{array} } \right]} \right). \\ \end{aligned}$$

(31)