# Spatial frequency

Green Sea Shell image
Spatial frequency representation of the Green Sea Shell image
Image and its spatial frequencies: Magnitude of frequency domain is logarithmic scaled, zero frequency is in the center. Notable is the clustering of the content on the lower frequencies, a typical property of natural images.

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 m. In image-processing applications, spatial frequency is often expressed in units of cycles per mm or equivalently line pairs per mm.

In wave mechanics, the spatial frequency is commonly denoted by ${\displaystyle \xi }$[1] or sometimes ${\displaystyle \nu }$, although the latter is also used[2] to represent temporal frequency. It equals the reciprocal of wavelength ${\displaystyle \lambda }$,

${\displaystyle \xi ={\frac {1}{\lambda }}.}$

Likewise, the angular wave number ${\displaystyle k}$, measured in rad per m, is related to spatial frequency and wavelength by

${\displaystyle k=2\pi \xi ={\frac {2\pi }{\lambda }}.}$

## Visual perception

In the study of visual perception, sinusoidal gratings 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. Sine-wave gratings also differ from one another in amplitude (the magnitude of difference in intensity between light and dark stripes), and angle.

### Spatial-frequency theory

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 on the basis of early experiments on V1 neurons in the cat.[3][4] 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 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.[5] (However, as noted by Teller (1984),[6] it is probably not wise to treat the highest firing rate of a particular neuron as having a special significance with respect to its role in the perception of a particular stimulus, given that the neural code is known to be linked to relative firing rates. For example, in color coding by the three cones in the human retina, there is no special significance to the cone that is firing most strongly – what matters is the relative rate of firing of all three simultaneously. Teller (1984) similarly noted that a strong firing rate in response to a particular stimulus should not be interpreted as indicating that the neuron is somehow specialized for that stimulus, since there is an unlimited equivalence class of stimuli capable of producing similar firing rates.)

The spatial-frequency theory of vision is based on two physical principles:

1. Any visual stimulus can be represented by plotting the intensity of the light along lines running through it.
2. Any curve can be broken down into constituent sine waves by Fourier analysis.

The theory (for which empirical support has yet to be developed) 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.[7] 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. (This procedure, however, does not address the problem of the organization of the products of the summation into figures, grounds, and so on. It effectively recovers the original (pre-Fourier analysis) distribution of photon intensity and wavelengths across the retinal projection, but does not add information to this original distribution. So the functional value of such a hypothesized procedure is unclear. Some other objections to the "Fourier theory" are discussed by Westheimer (2001) [8]). 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.[9] Research on spatial frequency detection by visual neurons complements and extends previous research using straight edges rather than refuting it.[10]

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. M. Bar (2004) has proposed that low spatial frequencies represent global information about the shape, such as general orientation and proportions.[11] Rapid and specialised perception of faces is known to rely more on low spatial frequency information.[12] In the general population of adults, the threshold for spatial frequency discrimination is about 7%. It is often poorer in dyslexic individuals.[13]

## References

1. ^ SPIE Optipedia article: "Spatial Frequency"
2. ^ As in e.g. Planck's formula.
3. ^ Martinez LM, Alonso JM (2003). "Complex receptive fields in primary visual cortex". Neuroscientist. 9 (5): 317–31. doi:10.1177/1073858403252732. PMC 2556291. PMID 14580117.
4. ^ De Valois, R. L.; De Valois, K. K. (1988). Spatial vision. New York: Oxford University Press.
5. ^ Issa NP, Trepel C, Stryker MP (2000). "Spatial frequency maps in cat visual cortex". The Journal of Neuroscience. 20 (22): 8504–8514. doi:10.1523/JNEUROSCI.20-22-08504.2000. PMC 2412904. PMID 11069958.
6. ^ Teller, D. "Linking Propositions"
7. ^ Barghout, Lauren (2014). Vision: How Global Perceptual Context Changes Local Contrast Processing (Ph.D. Dissertation 2003). Updated for Computer Vision Techniques. Scholars' Press. ISBN 978-3-639-70962-9.
8. ^ Westheimer, G. "The Fourier Theory of Vision"
9. ^ Blake, R. and Sekuler, R., Perception, 3rd ed. Chapter 3. ISBN 978-0-072-88760-0
10. ^ Pinel, J. P. J., Biopsychology, 6th ed. 293–294. ISBN 0-205-42651-4
11. ^ Bar M (Aug 2004). "Visual objects in context". Nat. Rev. Neurosci. 5 (8): 617–29. doi:10.1038/nrn1476. PMID 15263892. S2CID 205499985.
Box 2: Spatial frequencies and the information they convey
12. ^ Awasthi B, Friedman J, Williams MA (2011). "Faster, stronger, lateralized: Low spatial frequency information supports face processing". Neuropsychologia. 49 (13): 3583–3590. doi:10.1016/j.neuropsychologia.2011.08.027. PMID 21939676. S2CID 10037045.
13. ^ Ben-Yehudah G, Ahissar M (May 2004). "Sequential spatial frequency discrimination is consistently impaired among adult dyslexics". Vision Res. 44 (10): 1047–63. doi:10.1016/j.visres.2003.12.001. PMID 15031099. S2CID 12605281.