Fourier Transform of a 2D Anisotropic Gaussian Function

  • #1
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?

Thank you in advance.
 

Answers and Replies

  • #2
blue_leaf77
Science Advisor
Homework Helper
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1) how can I calculate the Fourier transform for the 2D anisotropic Gaussian function g(x,y)?
Note that you can do the separation ##g(x,y) = g_x(x)g_y(y)##.
 

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