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 the reciprocal metre (m^-1),^[1] although cycles per meter (c/m) is also common. In image-processing applications, spatial frequency is often expressed in units of cycles per millimeter (c/mm) or also line pairs per millimeter (LP/mm).

In wave propagation, the spatial frequency is also known as wavenumber. Ordinary wavenumber is defined as the reciprocal of wavelength ${\displaystyle \lambda }$ and is commonly denoted by ${\displaystyle \xi }$^[2] or sometimes ${\displaystyle \nu }$:^[3]

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

Angular wavenumber ${\displaystyle k}$, expressed in radian per metre (rad/m), is related to ordinary wavenumber 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, such as contrast sensitivity. 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), orientation, and phase.

Spatial frequency in MRI

When spatial frequency is used as a variable in a mathematical function, the function is said to be in k-space. Two dimensional k-space has been introduced into MRI as a raw data storage space. The value of each data point in k-space is measured in the unit of 1/meter, i.e. the unit of spatial frequency.

It is very common that the raw data in k-space shows features of periodic functions. The periodicity is not spatial frequency, but is temporal frequency. An MRI raw data matrix is composed of a series of phase-variable spin-echo signals. Each of the spin-echo signal is a sinc function of time, which can be described by

$${\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin \omega _{\mathrm {r} }t}{\omega _{\mathrm {r} }t}}}$$

Where

$${\displaystyle \omega _{\mathrm {r} }=\omega _{\mathrm {0} }+{\bar {\gamma }}rG}$$

Here ${\displaystyle {\bar {\gamma }}}$ is the gyromagnetic ratio constant, and ${\displaystyle \omega _{\mathrm {0} }}$ is the basic resonance frequency of the spin. Due to the presence of the gradient G, the spatial information r is encoded onto the frequency ${\displaystyle \omega }$. The periodicity seen in the MRI raw data is just this frequency ${\displaystyle \omega _{\mathrm {r} }}$, which is basically the temporal frequency in nature.

In a rotating frame, ${\displaystyle \omega _{\mathrm {0} }=0}$, and ${\displaystyle \omega _{\mathrm {r} }}$ is simplified to ${\displaystyle {\bar {\gamma }}rG}$. Just by letting ${\displaystyle k={\bar {\gamma }}Gt}$, the spin-echo signal is expressed in an alternative form

$${\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin rk}{rk}}}$$

Now, the spin-echo signal is in the k-space. 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. Two pieces of information are found in each domain, the spatial information and the spatial frequency information. The spatial information, which is of great interest to all medical doctors, is seen as periodic functions in the k-space domain and is seen as the image in the space domain. 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.

See also

• Fourier analysis
• Superlens
• Visual perception
• Fringe visibility
• Reciprocal space

References

External links

• "Tutorial: Spatial Frequency of an Image". Hakan Haberdar, University of Houston. Retrieved 22 March 2012.
• Kalloniatis, Michael; Luu, Charles (2007). "Webvision: Part IX Psychophysics of Vision. 2 Visual Acuity, Contrast Sensitivity". University of Utah. Retrieved 19 July 2009.