<p>A 2D Fourier transform is performed by first doing a 1D Fourier transform on each row of the image, then taking the result and doing a 1D Fourier transform on each column.  Or vice versa; it doesn't matter.</p>

<p>Just as a 1D Fourier transform allows you to decompose a function into a sum of (1D) sine waves at various frequencies, a 2D Fourier transform decomposes a function as a sum of 2D sine waves.  These waves can have different frequencies along the x and y axes.  They generically have the form:</p>

<p>$$ \exp \bigl(i \cdot (k_x x + k_y y) \bigr) $$</p>

<p>where $k_x$ and $k_y$ are the frequencies along the $x$ and $y$ axes. These two values form a vector called the wave-vector.  In the spatial domain, the wave is oriented along the $(k_x, k_y)$ vector with a frequency along its axis of $\sqrt{k_x^2 + k_y^2}$.</p>

<p>Just as for the 1D Fourier transform, there exist both discrete and continuous versions.  The result of a discrete 2D Fourier transform is a matrix of complex amplitudes for a set of discrete $(k_x, k_y)$ values.  This is commonly visualized (like in the paper you linked to) as an image where the pixel at coordinates $(k_x, k_y)$ represents the amplitude of that wave-vector.</p>

<p>So, an annular shape in a 2D Fourier transform indicates rotational invariance of the distribution of frequencies (i.e. just as much amplitude for waves in every direction), with a narrow range of magnitudes (from the inside of the annulus to the outside).  In other words, the paper is using the Fourier transform to demonstrate that their noise is reasonably isotropic and band-limited.</p>

CC BY-SA 4.0
2018-10-24T17:07:50.870Z
I like how this is simpler than the u-v form of the equation.  There's a lot to be studied in DFT on how this is good and what can improve.
MisterGeeky
CC BY-SA 3.0
2015-08-25T22:04:13.903Z
2015-11-03T19:18:26.903Z
2015-11-03T19:18:26.903Z
Nathan Reed
<p>I understand how a 1D Fourier transform separates a signal into its component frequencies, but I'm having difficulty understanding how a 2D Fourier transform affects a 2D image.</p>

<p>From <a href="https://computergraphics.stackexchange.com/questions/283/is-all-grid-based-noise-inevitably-anisotropic" title="Is all grid based noise inevitably anisotropic?">another question</a>, <a href="https://computergraphics.stackexchange.com/users/196/john-calsbeek">John Calsbeek</a> linked to an <a href="http://www.cs.utah.edu/~aek/research/noise.pdf" rel="nofollow noreferrer">interesting paper about measuring the quality of noise functions</a>. This showed various noise functions and the Fourier transform of each.</p>

<p>Is this a discrete transform of the pixel data, or a continuous transform of the continuous interpolating function which is used to generate the noise at arbitrary points?</p>

<p>Is the annular shape analagous to taking 1D Fourier transforms of the line through the centre of the image at every possible angle? Or is the transform for each possible angle also measured across the whole 2D space rather than only along a line through the centre? I'm trying to get an intuitive feel for what changes in the input image correspond to what changes in the Fourier transform.</p>

CC BY-SA 3.0
2015-08-26T15:22:03.767Z
Just for future people's curiosity, you might want to make "another question" be a link to that question.
porglezomp
CC BY-SA 3.0
2015-08-26T15:26:13.020Z
@porglezomp that's a good point - done.
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<noise><fourier-transform>
How does a 2D Fourier Transform of an image work?
trichoplax is on Codidact now
# How does a 2D Fourier Transform of an image work?

(Asked by: username_0 on 2015-08-19)

I understand how a 1D Fourier transform separates a signal into its component frequencies, but I'm having difficulty understanding how a 2D Fourier transform affects a 2D image.

From [another question](https://computergraphics.stackexchange.com/questions/283/is-all-grid-based-noise-inevitably-anisotropic "Is all grid based noise inevitably anisotropic?"), [John Calsbeek](https://computergraphics.stackexchange.com/users/196/john-calsbeek) linked to an [interesting paper about measuring the quality of noise functions](http://www.cs.utah.edu/~aek/research/noise.pdf). This showed various noise functions and the Fourier transform of each.

Is this a discrete transform of the pixel data, or a continuous transform of the continuous interpolating function which is used to generate the noise at arbitrary points?

Is the annular shape analagous to taking 1D Fourier transforms of the line through the centre of the image at every possible angle? Or is the transform for each possible angle also measured across the whole 2D space rather than only along a line through the centre? I'm trying to get an intuitive feel for what changes in the input image correspond to what changes in the Fourier transform.

**Tags:** noise, fourier-transform

> **Comments:**
> **username_1**: Just for future people's curiosity, you might want to make "another question" be a link to that question.  
> **username_0**: @username_1 that's a good point - done.  

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## 1 answers

### Answer by username_2 on 2015-08-25. Score: 8 | Accepted answer

A 2D Fourier transform is performed by first doing a 1D Fourier transform on each row of the image, then taking the result and doing a 1D Fourier transform on each column. Or vice versa; it doesn't matter.

Just as a 1D Fourier transform allows you to decompose a function into a sum of (1D) sine waves at various frequencies, a 2D Fourier transform decomposes a function as a sum of 2D sine waves. These waves can have different frequencies along the x and y axes. They generically have the form:

$$ \exp \bigl(i \cdot (k\_x x + k\_y y) \bigr) $$

where $k\_x$ and $k\_y$ are the frequencies along the $x$ and $y$ axes. These two values form a vector called the wave-vector. In the spatial domain, the wave is oriented along the $(k\_x, k\_y)$ vector with a frequency along its axis of $\sqrt{k\_x^2 + k\_y^2}$.

Just as for the 1D Fourier transform, there exist both discrete and continuous versions. The result of a discrete 2D Fourier transform is a matrix of complex amplitudes for a set of discrete $(k\_x, k\_y)$ values. This is commonly visualized (like in the paper you linked to) as an image where the pixel at coordinates $(k\_x, k\_y)$ represents the amplitude of that wave-vector.

So, an annular shape in a 2D Fourier transform indicates rotational invariance of the distribution of frequencies (i.e. just as much amplitude for waves in every direction), with a narrow range of magnitudes (from the inside of the annulus to the outside). In other words, the paper is using the Fourier transform to demonstrate that their noise is reasonably isotropic and band-limited.

> **Comments:**
> **username_3**: I like how this is simpler than the u-v form of the equation.  There's a lot to be studied in DFT on how this is good and what can improve.  

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