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Seeing between the Pixels

Brian Hayes

Lost in Frequency Space

Another alternative to pixels replaces an image with its spectrum.

The idea of a spectrum is most familiar in the context of one-dimensional signals, such as the sound of a flute or the light of a star. In these instances the signal is a graph of amplitude as a function of time, and the spectrum of the signal is a graph of power or intensity as a function of frequency. The conversion from the time domain to the frequency domain relies on Joseph Fourier's remarkable discovery that any periodic waveform, no matter how complex, can be constructed as the sum of simple sine and cosine waves.

Fourier analysis can be applied to a two-dimensional image by measuring "spatial frequencies" along both the x and y axes. Think of a photograph of a ladder leaning against a picket fence. If you scan across the image horizontally, recording the brightness of each point as you go, you get a spectrum with an energy peak at the frequency corresponding to the spacing of the fence pickets; when you scan vertically, the strongest peak is at a lower spatial frequency, that of the ladder rungs. The complete spectrum of an image records the contributions of all spatial frequencies, from zero up to the maximum resolution of the image data.

A form of Fourier analysis commonly used in image processing is called the discrete cosine transform. The procedure begins by dividing the image into square patches of, say, eight by eight pixels. For each such patch there are 64 discrete Fourier components, representing all possible spatial frequencies and phases along the x and y axes. Any eight-by-eight patch of the original image can be reconstructed by adding up various weighted combinations of these basic frequency elements. The zero-frequency component gives the average brightness of the patch. If the patch has gradual variations in brightness over its entire width or height, then some of the lower frequency components will be strongly represented in the spectrum. Sharp edges and fine lines in the image emphasize the high-frequency components.

Fourier analysis yields yet another method of image compression. It might seem at first that nothing is gained by converting to frequency space, since an eight-by-eight patch with 64 pixels also has 64 Fourier components. But in most images not all the Fourier components are equally important. For example, in a patch of nearly uniform color only the zero-frequency element has a significant weight, and the rest of the coefficients can be thrown away. This situation is common enough to make Fourier compression worthwhile. Indeed, the discrete cosine transform is the main compression method in JPEG images.

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