In an image processing paper, it was explained that a 2D Gabor filter is constructed in the Fourier domain using the following formula:(adsbygoogle = window.adsbygoogle || []).push({});

$$ H(u,v)=H_R(u,v) + i \cdot H_I(u,v)$$

where H_{R}(u,v) and H_{I}(u,v) are the real and imaginary components, respectively. It also mentions that the real and imaginary components are calculated by the following two functions:

$$ H_R(u,v)=\frac{1}{2}G(u-u_0,v-v_0) + \frac{1}{2}G(u+u_0,v+v_0)$$

$$ H_I(u,v)=\frac{i}{2}G(u-u_0,v-v_0) + \frac{i}{2}G(u+u_0,v+v_0)$$

where G(u,v)=F{g(x,y)} and F{} denotes FT, and

$$ h(x,y)=g(x\prime, y\prime)\cdot e^{-i2\pi(u_0x+v_0y)} $$

$$g(x\prime,y\prime)=\frac{1}{2\pi\sigma_x\sigma_y} e^{-\frac{1}{2}\big[(\frac{x\prime}{\sigma_{x\prime}})^2+(\frac{y\prime}{\sigma_{y\prime}})^2\big]}$$

h(x,y) is the Gabor filter in the spatial domain with (u_{0}, v_{0}) as the center frequencies. I do know that the Fourier transform of a 1D Gaussian function f(x)=e^{-ax2 }is measured using the following functional:$$

\mathcal{F_x(e^{-ax^2})(k)}=\sqrt{\frac{\pi}{a}}e^{\frac{-\pi^2k^2}{a}}$$

My questions are 1) how can I calculate the Fourier transform for the 2D anisotropic Gaussian function g(x,y)? 2) why are there two spatial standard deviations (σ_{x}, and σ_{x'}) defined in the Gaussian functional?

Thank you in advance.

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# Fourier Transform of a 2D Anisotropic Gaussian Function

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