What does an FFT of a photo show, and why do lines or star-like patterns appear?

Asked 3/27/2012

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I’m learning how to analyze images with a 2D Fast Fourier Transform (FFT). In an FFT display of a photograph, the low spatial frequencies appear near the center and the higher frequencies toward the edges or corners. How should I interpret this result? Specifically, what does the bright center represent, and why might an FFT show a horizontal bright line through the middle or a star/sun-like pattern of rays?

Originally by Photography Stack Exchange contributor. Source · Licensed CC BY-SA 4.0

Photography Stack Exchange contributor

14y ago

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You have a function of the spatial coordinates (x, y), the coordinates of the original image. Suppose, for clarity, that we are talking about a value from 0 to 255 for each (x, y) point in your original image. The transform is a function, again from 0 to 255, of the momentum coordinates (k1, k2) . The point (0, 0) - the sun - corresponds to the intensity of the constant part of the original function. Don't think, for a moment, to the fact that it represents an image, think of it like... a 2d bar chart or something like that. The constant is the average over the (periodically arranged) image. As you progress from the center you are sampling at higher frequencies (with sinusoidal and cosinusoidal function of increasing frequency). Given the spatial resolution of the details of your original image, you can see that the corners (high k1 frequency, high k2 frequency) are black (that is, the intensity of the transfor is low), and the central zone, lighter, correesponds to the "typical" spatial lenght of the details of your image. If you had took a picture of a more regular object (a grid?) you would have found a "typical" k corresponding to your "typycal" lenght (for example, this is the process that is used in physics to reconstructs the features of cristals).

The central line corresponds to the average values along the y direction for the various sampling frequencies along the x direction. It is roughly constant: this means that the average value of the image along the short side, independently of the frequency of sampling along the long side, is the same. This should be because the image exhibits a symmetry (the horizon) with a single feature (the girl) in a very concentrated region of space. It is relatively bright because the average value is influenced by the sky, which is mostly uniform and bright.

As an exercise, you could try to take a picture of a single/a few light object against a dark background and compare the results.

Originally by user5032. Source · Licensed CC BY-SA 4.0

user5032

14y ago

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A 2D FFT converts an image from spatial coordinates (x, y) into spatial-frequency coordinates. The bright center is the zero/low-frequency region: it represents the constant or slowly varying part of the image, closely related to the average brightness and broad tonal structure. As you move away from the center, you get higher spatial frequencies, which correspond to finer detail and sharper transitions.

A bright line through the center usually means the image contains strong structure or repeated variation in the perpendicular direction. For example, a horizontal line in the FFT indicates strong vertically varying content or repeated horizontal features in the photo’s tones/details.

A star- or sun-like FFT pattern appears when the image has strong edges, textures, or repeated structures in several preferred directions. Natural images often contain directional detail, so energy in the frequency domain can concentrate along rays rather than being evenly spread.

To understand this well, it helps to first learn the basic 1D Fourier transform, then extend that idea to 2D image processing.

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