Wikipedia
K-space can refer to:
- Another name for the spatial frequency domain of a spatial Fourier transform
- Reciprocal space, containing the reciprocal lattice of a spatial lattice
- Momentum space, or wavevector space, the vector space of possible values of momentum for a particle
- k-space (magnetic resonance imaging)
- Another name for a compactly generated space in topology
- K-space (functional analysis) is an F-space such that every twisted sum by the real line splits
- K-Space (band), a British-Siberian music ensemble
k-space is a formalism widely used in magnetic resonance imaging introduced in 1979 by Likes and in 1983 by Ljunggren and Twieg.
In MRI physics, k-space is the 2D or 3D Fourier transform of the MR image measured. Its complex values are sampled during an MR measurement, in a premeditated scheme controlled by a pulse sequence, i.e. an accurately timed sequence of radiofrequency and gradient pulses. In practice, k-space often refers to the temporary image space, usually a matrix, in which data from digitized MR signals are stored during data acquisition. When k-space is full (at the end of the scan) the data are mathematically processed to produce a final image. Thus k-space holds raw data before reconstruction.
k-space is in spatial frequency domain. Thus if we define k and k such that
k = γ̄GmΔt
and
k = γ̄nΔGτ
where FE refers to frequency encoding, PE to phase encoding, Δt is the sampling time (the reciprocal of sampling frequency), τ is the duration of G, γ̄ (gamma bar) is the gyromagnetic ratio, m is the sample number in the FE direction and n is the sample number in the PE direction (also known as partition number), the 2D- Fourier Transform of this encoded signal results in a representation of the spin density distribution in two dimensions. Thus position (x,y) and spatial frequency (k, k) constitute a Fourier transform pair.
Typically, k-space has the same number of rows and columns as the final image and is filled with raw data during the scan, usually one line per TR (Repetition Time).
An MR image is a complex-valued map of the spatial distribution of the transverse magnetization M in the sample at a specific time point after an excitation. Conventional qualitative interpretation of Fourier Analysis asserts that low spatial frequencies (near the center of k-space) contain the signal to noise and contrast information of the image, whereas high spatial frequencies (outer peripheral regions of k-space) contain the information determining the image resolution. This is the basis for advanced scanning techniques, such as the keyhole acquisition, in which a first complete k-space is acquired, and subsequent scans are performed for acquiring just the central part of the k-space; in this way, different contrast images can be acquired without the need of running full scans.
A nice symmetry property exists in k-space if the image magnetization M is prepared to be proportional simply to a contrast-weighted proton density and thus is a real quantity. In such a case, the signal at two opposite locations in k-space is:
S( − k, − k) = S(k, k)
where the star () denotes complex conjugation. Thus k-space information is somewhat redundant then, and an image can be reconstructed using only one half of the k-space, either in the PE (Phase Encode) direction saving scan time (such a technique is known as half Fourier or half scan) or in the FE (Frequency Encode) direction, allowing for lower sampling frequencies and/or shorter echo times (such a technique is known as half echo). However, these techniques are approximate due to phase errors in the MRI data which can rarely be completely controlled (due to imperfect static field shim, effects of spatially selective excitation, signal detection coil properties, motion etc.) or nonzero phase due to just physical reasons (such as the different chemical shift of fat and water in gradient echo techniques).
In mathematics, more specifically in functional analysis, a K-space is an F-space V such that every extension of F-spaces (or twisted sum) of the form
$$0\rightarrow \mathbb R\rightarrow X\rightarrow V\rightarrow 0. \,\!$$
is equivalent to the trivial one
$$0\rightarrow \mathbb R\rightarrow \mathbb R\times V \rightarrow V\rightarrow 0. \,\!$$
where $\mathbb R$ is the real line.
K-Space are a British- Siberian experimental electroacoustic improvisation music ensemble comprising Scottish percussionist Ken Hyder, English multi-instrumentalist Tim Hodgkinson, and Siberian percussionist and throat singer Gendos Chamzyryn. The trio was formed in Tuva, Siberia in 1996. They have played in concerts in Asia and Europe, and released four CDs, including Infinity (2008), which was a new type of CD that is different every time it is played.
In a review of K-Space's second album, Going Up (2004), François Couture of AllMusic described their music as a mixture of "psychedelic shamanism" and "the strangest Krautrock you ever heard".