# Fourier Transform of a 2D Anisotropic Gaussian Function

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1. May 30, 2016

### Mahpak

In an image processing paper, it was explained that a 2D Gabor filter is constructed in the Fourier domain using the following formula:
$$H(u,v)=H_R(u,v) + i \cdot H_I(u,v)$$
where HR(u,v) and HI(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 (u0, v0) 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 (σx, and σx') defined in the Gaussian functional?

Note that you can do the separation $g(x,y) = g_x(x)g_y(y)$.