Spatial frequency

In mathematics, physics, and engineering, 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) of the structure repeat per unit of distance. The SI unit of spatial frequency is cycles per meter. In image processing 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 $\nu$ is related to the wavelength $\lambda$ by the formula

$\nu ={\frac {1}{\lambda }}.$

Likewise, the wave number $k$ (e.g. in radians per meter) is related to spatial frequency and wavelength by

$k=2\pi \nu ={\frac {2\pi }{\lambda }}.$

Visual perception

In the study of visual perception, sinusoidal gratings (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 of visual angle.

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 featural information and fine detail. Low spatial frequencies, on the other hand, represent global information about the shape, such as general orientation and proportions. Rapid and specialised perception of faces is known to rely more on low spatial frequency information. 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:

${\frac {I_{\mathrm {max} }-I_{\mathrm {min} }}{I_{\mathrm {max} }+I_{\mathrm {min} }}}.$

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.

See also

• Fourier analysis
• Superlens

External links

• "Tutorial: Spatial Frequency of an Image". Hakan Haberdar, University of Houston. Retrieved March 2012. {{cite web}}: Check date values in: |accessdate= (help)
• Kalloniatis, Michael; Luu, Charles (2007). "Webvision: Part IX Psychophysics of Vision. 2 Visual Acuity, Contrast Sensitivity". University of Utah. Retrieved July 2009. {{cite web}}: Check date values in: |accessdate= (help)
• "The Contrast Sensitivity Function, Prepared by Peter Wenderoth". Macquarie University Department of Psychology. Retrieved May 2009. {{cite web}}: Check date values in: |accessdate= (help)