Fourier Transform of a 2D Anisotropic Gaussian Function

The Fourier transform of a Gaussian function in one dimension is already given, so you can use that to find the Fourier transform of ##g_x(x)## and ##g_y(y)##. Then, use the fact that the Fourier transform of a product of functions is the convolution of their Fourier transforms to find the Fourier transform of ##g(x,y)##.2) why are there two spatial standard deviations (σx, and σx') defined in the Gaussian functional?The two spatial standard deviations, σx and σx', are used to control the anisotropic nature of the Gaussian function. They allow for the manipulation of the width and orientation of the Gaussian function, which can be useful in image processing tasks. Overall, the
  • #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.
 
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  • #2
Mahpak said:
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)##.
 

1. What is a 2D anisotropic Gaussian function?

A 2D anisotropic Gaussian function is a mathematical function that is commonly used in image processing and signal analysis. It describes a Gaussian distribution that is stretched or compressed along two different axes, allowing for asymmetry in the shape of the distribution.

2. How is a 2D anisotropic Gaussian function different from a regular Gaussian function?

In a regular Gaussian function, the distribution is symmetric and equal along all axes. However, in a 2D anisotropic Gaussian function, the distribution can be stretched or compressed along two different axes, allowing for more flexibility in representing data.

3. What is the Fourier Transform of a 2D anisotropic Gaussian function used for?

The Fourier Transform of a 2D anisotropic Gaussian function is used to analyze and process images or signals that have anisotropic properties. It can help identify patterns and features that may not be easily visible in the original data.

4. How is the Fourier Transform of a 2D anisotropic Gaussian function calculated?

The Fourier Transform of a 2D anisotropic Gaussian function is calculated using mathematical equations and algorithms. It involves converting the function from its spatial domain to its frequency domain, which allows for better analysis and processing of the data.

5. Can the Fourier Transform of a 2D anisotropic Gaussian function be reversed?

Yes, the Fourier Transform of a 2D anisotropic Gaussian function can be reversed to return the function to its original form. This is known as the inverse Fourier Transform and is commonly used in image reconstruction and signal processing.

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