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Every pattern is stacked sine waves
Building stripes from waves
Take a pattern of stripes like a zebra crossing. Built from waves, the first ingredient is a flat gray: the average contrast, a wave that never repeats, which engineers call the DC level. Add a wave with one cycle across the image, then two, and nothing zebra-like appears yet. At three cycles a massive contribution arrives, and the stripes start to show.
Amplitude and phase
Each wave has an amplitude and a phase, and the phase fixes how its peaks and troughs line up with the others. Getting the phases right is what makes the components add up in the right places. As more components go in, the image sharpens. JPEG compression works this way, storing an image as the frequencies that contribute to it.
Running it backward
The same decomposition runs in reverse: start with a finished pattern and ask which frequencies built it. It works on a pattern in time like a musical note as well as on one in space like an image, and physicists use it everywhere from the positions of atoms in a crystal lattice to the cosmic microwave background.
How does JPEG use waves to shrink images?
JPEG splits an image into blocks of 8 by 8 pixels and applies the discrete cosine transform, a close relative of the Fourier transform, to each block. The fine, high-frequency components matter least to the human eye, so the encoder stores them coarsely or drops them. The standard was published in 1992. The more of those components are discarded, the smaller the file and the blurrier the result.
Why can any pattern be built from sine waves?
Joseph Fourier proposed in 1822, in his work on how heat flows through solids, that a function could be written as a sum of sines and cosines. Mathematicians spent the following decades working out exactly which functions it holds for. For the signals and images met in practice, the sum matches the original as more waves are added. Sines and cosines are convenient because each one is a pure frequency.
How do crystals reveal atoms through Fourier math?
When X-rays scatter off a crystal, the pattern of bright spots on a detector is the Fourier transform of how electrons are arranged in the crystal. Crystallographers measure the spot intensities and work backward to the positions of the atoms. Detectors record only intensity and not phase, which leaves a gap that crystallographers call the phase problem. Methods for filling that gap earned several Nobel Prizes in chemistry.
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