K-space (MRI)

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 ${\displaystyle k_{\mathrm {FE} }}$ and ${\displaystyle k_{\mathrm {PE} }}$ such that

and

where FE refers to frequency encoding, PE to phase encoding, ${\displaystyle \Delta t}$ is the sampling time (the reciprocal of sampling frequency), ${\displaystyle \tau }$ is the duration of G_PE, ${\displaystyle {\bar {\gamma }}}$ (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 (${\displaystyle k_{\mathrm {FE} }}$, ${\displaystyle k_{\mathrm {PE} }}$) constitute a Fourier transform pair.

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