Face recognition based on gradient gabor feature and Efficient Kernel Fisher analysis

Abstract

In this paper, a new Gradient Gabor (GGabor) filter is proposed to extract multi-scale and multi-orientation features to represent and classify faces. Gradient Gabor filters combine the derivative of Gaussian functions and the harmonic functions to capture the features in both spatial and frequency domains to deliver orientation and scale information. The spatial positions are encoded through using Gaussian derivatives which allow it to provide more stable information. An Efficient Kernel Fisher analysis method is proposed to find multiple subspaces based on both GGabor magnitude and phase features, which is a local kernel mapping method to capture the structure information in faces. The experiments on two face databases, FRGC version 1 and FRGC version 2, are conducted to compare performances of the Gabor and GGabor features. The experiment results show that GGabor yield a powerful tool to model faces, and the Efficient Kernel Fisher classifier can improve the efficiency of the original Kernel Fisher Discriminant analysis method.

Appendix

The 1D Gradient Gabor filter has a form of

$$ G\psi (x) = \left( { - x\exp \left( { - 2\pi f_{0} xi} \right) + C} \right)\exp \left( { - \pi x^{2} } \right). $$

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It is known that the Fourier transform of a Gaussian function exp(−πx ²) has a Gaussian form:

$$ \mathop {G\Uppsi (f)}\limits^{\_\_\_\_} = \int\limits_{ - \infty }^{\infty } {\exp \left( { - \pi x^{2} } \right)} \exp ( - 2\pi ifx){\text{dx}} = \exp \left( { - \pi f^{2} } \right). $$

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The Fourier transform of the derivative of a Gaussian function −xexp(−πx ²) is

$$ \begin{aligned} \int\limits_{ - \infty }^{\infty } { - x\;\exp \left( { - \pi x^{2} } \right)} \exp ( - 2\pi ifx){\text{dx}} = & \int\limits_{ - \infty }^{\infty } {\exp \left( { - \pi x^{2} } \right)} {\frac{ - 1}{2\pi j}}\;{\frac{{{\text{d}}\;\exp ( - 2\pi ifx)}}{\text{df}}}\exp ( - 2\pi ifx){\text{dx}} \\ & = - if\exp \left( { - \pi f^{2} } \right) \\ \end{aligned} $$

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$$ \begin{aligned} \mathop {G\psi (f)}\limits^{\_\_\_\_\_} = & \int\limits_{ - \infty }^{\infty } {G\psi (x)} \exp ( - 2\pi ifx){\text{dx}} = - i\left( {f - f_{0} } \right) \\ \, & \exp \left( { - \pi \left( {f - f_{0} } \right)^{2} } \right) + C. \\ \end{aligned} $$

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By setting $\mathop {G\psi (0)}\limits^{\_\_\_\_\_} = 0$, then $C = if_{0} \exp \left( { - \pi f_{0}^{2} } \right).$

The 2D Gradient Gabor wavelet in Eq. 17 can be reformulated as

$$ G\psi_{u,v} (x,y) = - \left( {\cos \left( {\Upphi_{u} } \right)x + \sin \left( {\Upphi_{u} } \right)y} \right){\frac{{||k_{u,v} ||^{4} }}{{\sigma^{4} }}}\exp \left( {{{ - ||k_{u,v} ||^{2} } \mathord{\left/ {\vphantom {{ - ||k_{u,v} ||^{2} } {2\sigma^{2} \left( {x^{2} + y^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {2\sigma^{2} \left( {x^{2} + y^{2} } \right)}}} \right)\exp \left( {i\left( {k_{v} \cos \left( {\Upphi_{u} } \right)x + k_{v} \sin \left( {\Upphi_{u} } \right)y} \right)} \right). $$

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Then we define a DC-free Gradient Gabor wavelet as follows:

$$ \begin{aligned} G\psi_{u,v} (x,y) = & - {\frac{{||k_{u,v} ||^{4} }}{{\sigma^{4} }}}\left( {\left( {\cos \left( {\Upphi_{u} } \right)x + \sin \left( {\Upphi_{u} } \right)y} \right)\exp \left( {i\left( {k_{v} \cos \left( {\Upphi_{u} } \right)x + k_{v} \sin \left( {\Upphi_{u} } \right)y} \right)} \right) + C} \right) \\ \, & \exp \left( { - ||k_{u,v} ||^{2} /2\sigma^{2} \left( {x^{2} + y^{2} } \right)} \right) \\ \end{aligned} $$

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$$ \begin{aligned} \mathop {{\text{G}}\psi_{u,v} (f,\varphi )}\limits^{\_\_\_\_\_\_\_\_\_} = & - {\frac{{||k_{u,v} ||^{4} }}{{\sigma^{4} }}}\left( {{\frac{{\left( {\cos \left( {\Upphi_{u} } \right) + \sin \left( {\Upphi_{u} } \right)} \right)}}{{i\sigma^{2} }}}} \right)\left( {f - k_{v} \cos \left( {\Upphi_{u} } \right) + \varphi - k_{v} \sin \left( {\Upphi_{u} } \right)} \right) \\ \, & \, \exp \left( { - {\frac{\pi }{{\sigma^{2} }}}\left( {\left( {f - k_{v} \cos \left( {\Upphi_{u} } \right)} \right)^{2} + \left( {\varphi - k_{v} \sin \left( {\Upphi_{u} } \right)} \right)^{2} } \right)} \right) + C \\ \end{aligned} $$

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if $\mathop {G\psi_{u,v} (0,0)}\limits^{\_\_\_\_\_\_\_\_\_} = 0,$ then

$$ C = - \left( {{\frac{{\left( {\cos \left( {\Upphi_{u} } \right) + \sin \left( {\Upphi_{u} } \right)} \right)}}{{i\sigma^{2} }}}\left( {k_{v} \cos \left( {\Upphi_{u} } \right) - k_{v} \sin \left( {\Upphi_{u} } \right)} \right)\exp \left( { - {\frac{\pi }{{\sigma^{2} }}}\left( {k_{v} } \right)^{2} } \right)} \right). $$

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