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 the reciprocal metre (m−1),[1] although cycles per meter (c/m) is also common.
, expressed in radian per metre (rad/m), is related to ordinary wavenumber and wavelength by
In these stimuli, spatial frequency is expressed as the number of cycles per degree of visual angle.
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.
[4][5] 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.
[6] (However, as noted by Teller (1984),[7] 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: The theory (for which empirical support has yet to be developed) states that in each functional module of the visual cortex, Fourier analysis (or its piecewise form [8]) 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.
(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.
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.
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.
[13] Rapid and specialised perception of faces is known to rely more on low spatial frequency information.
[14] In the general population of adults, the threshold for spatial frequency discrimination is about 7%.
Two dimensional k-space has been introduced into MRI as a raw data storage space.
It is very common that the raw data in k-space shows features of periodic functions.
An MRI raw data matrix is composed of a series of phase-variable spin-echo signals.
Due to the presence of the gradient G, the spatial information r is encoded onto the frequency
It becomes a periodic function of k with r as the k-space frequency but not as the "spatial frequency", since "spatial frequency" is reserved for the name of the periodicity seen in the real space r. The k-space domain and the space domain form a Fourier pair.
The spatial frequency information, which might be of interest to some MRI engineers, is not easily seen in the space domain but is readily seen as the data points in the k-space domain.