Spatial frequency
In mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, and engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

) of the structure repeat per unit of distance. The SI
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

 unit of spatial frequency is cycles
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

 per meter. In image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

 applications, spatial frequency is often expressed in units of cycles per millimeter and sometimes incorrectly in units of line pairs per millimeter (incorrect since a line pair may describe a square wave but not a sinusoidal wave).

In wave mechanics, the spatial frequency is related to the wavelength by the formula

Likewise, the wave number
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...

  is related to spatial frequency and wavelength by

Visual perception

In the study of visual perception
Visual perception
Visual perception is the ability to interpret information and surroundings from the effects of visible light reaching the eye. The resulting perception is also known as eyesight, sight, or vision...

, sinusoidal
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

A grating is any regularly spaced collection of essentially identical, parallel, elongated elements. Gratings usually consist of a single set of elongated elements, but can consist of two sets, in which case the second set is usually perpendicular to the first...

s (images containing bands of white and black in a regular pattern) are frequently used to probe the capabilities of the visual system. In these stimuli, spatial frequency is expressed as the number of cycles per degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...

 of visual angle
Visual angle
The visual angle is the angle a viewed object subtends at the eye, usually stated in degrees of arc.It also is called the object's angular size....


Different spatial frequencies convey different information about the appearance of a stimulus. High spatial frequencies represent abrupt spatial changes in the image, such as edges, and generally correspond to configural information and fine detail. Low spatial frequencies, on the other hand, represent global information about the shape, such as general orientation and proportions.
In the general population of adults, the threshold for spatial frequency discrimination is about 7%. It is often poorer in dyslexic individuals.

Sinusoidal gratings and Michelson equation

There is an important quantitative concept related to spatial frequency, known as the Michelson equation:

In layman's terms, this is the ratio of the crest-trough distance to the maximum thereof, which is twice the average. One degree on the visual field represents four cycles on the wave pattern.
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