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Fourier Transforms (FFT)

Interpret Fourier Transforms (FFT)

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Visually Interpreting FFTs

The best way to gain an intuitive understanding of what an FFT image represents is with some visual examples.  The remainder of these pages assume a basic understanding of the relationship between 2D image files and their Fourier transforms.   Even if you already have an appreciation of the mathematics behind FFTs of 2D image files, you may still appreciate the information on this page.

Visually Interpreting Fourier Transforms

First, here is an image of mathematically generated noise with a Gaussian distribtution and its FFT:

Gaussian Noise        Gaussian Noise FFT
Gaussian Noise                                                                 Gaussian Noise FFT

Note that the FFT has exactly one bright pixel, directly in the center.  Unless you’re processing a completely black image, a 2D Fourier transform of an image file (where all pixels have positive values) will always have a bright pixel in the center.  That center pixel is called the DC term and represents the average brightness across the entire image.  Note also that this FFT has no visible patterns unlike the FFT of the microscope photograph we examined earlier.  This FFT has no visible patterns because the original image contains no patterns; it is comprised of purely random pixel values.

Below left is an image of a sin wave where black pixels represent the bottom of the sine wave, white pixels the top and the gray pixels in between represent the sloping areas of the curve.  On the right is the FFT of that image:

Sine wave (Frequency 2)        Sine wave (Frequency 2) FFT

2D Fourier transforms are always symmetrical.  The upper left quadrant is identical to the lower right quadrant and the upper right quadrant is identical to the lower left quadrant.  This is a natural consequence of how Fourier transforms work.

In the FFT image above right, the two dots on either side of the DC term encode a sine wave of frequency 2 with an amplitude that covers the entire dynamic range of the original 8-bit grayscale image.  The sine wave starts high at the left, goes down and back up in the middle (cycle 1) and then down and back up again (cycle 2) ending high at the right side of the image.  The sine wave image on the left was created by performing an inverse FFT function of the image on the right, but it could have been processed either way.  The two images contain exactly the same information.

In this next example, the sine wave has a frequency of four rather than two.  The only difference in the FFT is that the two bright dots surrounding the center pixel are two pixels further from the center.

Sine wave (Frequency 4)        Sine wave (Frequency 4) FFT

Extending that concept, here is a sine wave of frequency 26 and its FFT.  Note again that the only difference in the FFT is that the two bright pixels are further from the center.

Sine wave (Frequency 26)        Sine wave (Frequency 26) FFT
This concept works equally well for horizontal and diagonal patterns:

Horizontal sine wave (Frequency 4)        Horizontal sine wave (Frequency 4)

Diagonal sine wave (Frequency 8)        Diagonal sine wave (Frequency 8) FFT

In each case a line connecting the dots in the FFT is perpendicular to the pattern the dots encode.

Complex patterns and FFTs

Where FFTs really get interesting is when multiple patterns are combined.  The examples below were all created by adding the FFT images and then performing an inverse FFT to create the spatial image on the left:

Interfering sine waves        Interfering sine waves FFT

In the example above, vertical sine waves of three different frequencies have been combined.  The interference patterns of the sine waves create the light and dark bands in the image on the left.  If you extend this concept of adding sine waves of higher and higher frequency you eventually end up with a horizontal line across middle of the FFT image and one bright vertical band through the middle of the original image as all the frequencies cancel each other out except for the top of the sine wave with frequency one.

Sine waves (All Frequencies)        Sine waves (All Frequencies) FFT

This adding of FFT images can be performed with any FFT images, not just those with vertical sin waves.  Here are some more complex examples adding images with sine waves in various orientations.  If you take a few minutes to examine the patterns created by each pair of symmetrical dots in the FFT (frequency domain) you may be able to visualize the resulting pattern in the spatial image:

Sine wave interference        Sine wave interference FFT
Vertical sine wave (Freq 2) plus
Diagonal sine wave (Freq 12)

Sine wave interference        Sine wave interference FFT
Same as above plus…
Vertical sine wave (Freq 26)

Here is one last example. 
Sine wave interference        Sine wave interference FFT

The image above left is a simple combination of horizontal and vertical sine waves.  Its FFT is shown to the right.  We then decrease the contrast of the original image and add random noise to create the image below and its resulting FFT to the right. 

Sine wave interference + Noise        Sine wave interference + Noise FFT

Note that the diamond pattern is still visible in the FFT.  This is precisely because the original sinusoidal patterns are still present in the processed image, even though their relative contribution to the overall image has been significantly reduced.  There are a few additional bright pixels in the FFT surrounding the original diamond pattern.  These pixels represent higher frequency, but lower amplitude, sine waves present in the combined image.  The brightness of a pixel in the FFT is proportional to the amplitude of the sine wave it encodes.  Higher frequency sine waves (indicated by bright pixels further from the center of the FFT) represent sharper edges between dark and light regions in the original image.  In this case those higher frequency sine waves are likely the result of taking an FFT of an 8-bit image. 

This “noisy” image illustrates that patterns in the original image, whether visible to the eye or not, will become apparent in the FFT of the image.

Continue to Interpreting Read Noise Frame FFTs >>

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