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| {{merge|Spatial frequency|wavenumber|date=June 2013}}
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| 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 [[Sine wave|sinusoidal components]] (as determined by the [[Fourier transform]]) of the structure repeat per unit of distance. The [[SI]] unit of spatial frequency is [[rotation|cycles]] per [[meter]]. In [[image processing|image-processing]] applications, spatial frequency is often expressed in units of cycles per millimeter or equivalently line pairs per millimeter.
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| In wave mechanics, the spatial frequency is commonly denoted by <math>\xi</math><ref>SPIE Optipedia article: [http://spie.org/x34301.xml "Spatial Frequency"]</ref>
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| or sometimes <math>\nu</math>, although the latter is also used<ref>As in e.g. [[Planck constant|Planck's formula]].</ref>
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| to represent temporal [[frequency]]. It is related to the [[wavelength]] <math>\lambda</math> by the formula
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| :<math>\xi = \frac{1}{\lambda}.</math>
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| Likewise, the [[wavenumber|angular wave number]] <math>k</math>, measured in radians per meter, is related to spatial frequency and wavelength by
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| :<math>k = 2 \pi \xi = \frac{2 \pi}{\lambda}.</math>
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| ==Visual perception==
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| In the study of [[visual perception]], [[sine-wave|sinusoidal]] [[grating]]s are frequently used to probe the capabilities of the [[visual system]]. In these [[Stimulus (physiology)|stimuli]], spatial frequency is expressed as the number of cycles per [[degree (angle)|degree]] of [[visual angle]]. Sine-wave gratings also differ from one another in amplitude (the magnitude of difference in intensity between light and dark stripes), and angle.
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| === Spatial-frequency theory ===
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| The spatial-frequency theory refers to the theory that the [[visual cortex]] operates on a code of spatial frequency, not on the code of straight edges and lines hypothesised by Hubel and Wiesel.<ref> Martinez, L. M., & ALonzo, J. M. (2203). Complex receptive fields in primary visual cortex. ''The Neuroscientist, 9'', 317-331</ref><ref> De Valois, R. L., & De Valois, K. K. (1988). ''Spatial vision''. New York: Oxford University Press.</ref> In support of this theory is the experimental observation that the visual cortex neurons respond even more robustly to sine-wave gratings that are placed at specific angles in their [[Receptive field#Visual system|receptive fields]] than they do to edges or bars. Most neurons in the primary visual cortex respond best when a sine-wave grating of a particular frequency is presented at a particular angle in a particular location in the visual field <ref>Issa, N. P., Trepel, C., & Stryker, M. P. (2000). Spatial frequency maps in cat visual cortex. ''The Journal of Neuroscience, 20'', 8504-8514</ref> | |
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| The spatial-frequency theory of vision is based on two physical principles:
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| # Any visual stimulus can be represented by plotting the intensity of the light along lines running through it.
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| # Any curve can be broken down into constituent sine waves by [[Fourier analysis]].
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| The theory states that in each functional module of the visual cortex, Fourier analysis is performed on the receptive field and the neurons in each module are thought to respond selectively to various orientations and frequencies of sine wave gratings. When all of the visual cortex neurons that are influenced by a specific scene respond together, the perception of the scene is created by the summation of the various sine-wave gratings. One is generally not aware of the individual spatial frequency components since all of the elements are essentially blended together into one smooth representation. However, computer-based filtering procedures can be used to deconstruct an image into its individual spatial frequency components.<ref> Blake, R. and Sekuler, R., ''Perception'', 3rd ed. Chapter 3. ISBN 978-0072887600 </ref> Research on spatial frequency detection by visual neurons complements and extends previous research using straight edges rather than refuting it.<ref> Pinel, J. P. J., ''Biopsychology'', 6th ed. 293-294. ISBN 0-205-42651-4 </ref>
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| Further research shows that 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.<ref>{{cite journal |author=Bar M |title=Visual objects in context |journal=Nat. Rev. Neurosci. |volume=5 |issue=8 |pages=617–29 |date=Aug 2004 |pmid=15263892 |doi=10.1038/nrn1476 |quote=<br/>[http://www.nature.com/nrn/journal/v5/n8/box/nrn1476_BX2.html Box 2: Spatial frequencies and the information they convey] }}</ref> Rapid and specialised perception of faces is known to rely more on low spatial frequency information. <ref>{{cite journal |author=Awasthi B, Friedman J, Williams, MA| title=Faster, stronger, lateralized: Low spatial frequency information supports face processing|journal=Neuropsychologia| volume=49 |issue=13 |pages=3583–3590 |year=2011 |doi=10.1016/j.neuropsychologia.2011.08.027}}</ref> In the general population of adults, the threshold for spatial frequency discrimination is about 7%. It is often poorer in dyslexic individuals.<ref>{{cite journal |author=Ben-Yehudah G, Ahissar M |title=Sequential spatial frequency discrimination is consistently impaired among adult dyslexics |journal=Vision Res. |volume=44 |issue=10 |pages=1047–63 |date=May 2004 |pmid=15031099 |doi=10.1016/j.visres.2003.12.001 }} </ref>
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| ===Sinusoidal gratings and Michelson equation===
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| There is an important quantitative concept related to spatial frequency, known as the [[Albert Michelson|Michelson]] equation:
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| :<math>\frac{I_\mathrm{max}-I_\mathrm{min}}{I_\mathrm{max}+I_\mathrm{min}}.</math>
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| 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.<ref>"Vision" McGraw-Hill Encyclopedia of Science & Technology, vol. 19, p.292 1997</ref>
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| ==See also==
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| *[[Fourier analysis]]
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| *[[Superlens]]
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| *[[Visual perception]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{cite web
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| | url = http://www.haberdar.org/Spatial-Frequency-of-an-Image-Tutorial.htm
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| | title = Tutorial: Spatial Frequency of an Image
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| | publisher = Hakan Haberdar, University of Houston
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| | accessdate = March 2012
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| }}
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| * {{cite web
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| | url = http://webvision.med.utah.edu/KallSpatial.html#contrast
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| | title = Webvision: Part IX Psychophysics of Vision. 2 Visual Acuity, Contrast Sensitivity | publisher = University of Utah
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| | first1 = Michael | last1 = Kalloniatis | first2 = Charles | last2 = Luu
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| | year = 2007 | accessdate = July 2009
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| }}
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| [[Category:Mathematical physics]]
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