spatialfrequency

Spatial frequency describes how rapidly image brightness changes across space. It is measured in cycles per unit distance, with common units including cycles per degree of visual angle in vision science and cycles per millimeter or per image in imaging systems. Spatial frequency characterizes the level of detail present in a scene.

In two dimensions, spatial frequency is part of the Fourier description of an image. Each frequency component corresponds to a sinusoidal pattern with a specific frequency and orientation. Low spatial frequencies reflect broad, smooth variations, such as overall shading or large objects, while high spatial frequencies correspond to fine details, edges, and textures. The distribution of energy across frequencies can be represented by a power spectrum or magnitude spectrum.

The Fourier transform maps spatial patterns to the frequency domain. The 2D spectrum shows how much energy is present at each frequency and orientation. The modulation transfer function (MTF) describes how a system—an optical lens, sensor, or display—attenuates or preserves different spatial frequencies, influencing perceived sharpness and contrast. Filters such as low-pass, high-pass, or band-pass operators modify the frequency content to achieve denoising, compression, edge enhancement, or texture analysis.

In practice, accurate representation of spatial details requires appropriate sampling according to the Nyquist criterion: the sampling rate must be at least twice the highest spatial frequency present; undersampling leads to aliasing and distortion. Spatial frequency is a foundational concept in image processing, vision science, and related fields, linking perceptual performance, imaging system design, and computational analysis.