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 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 and such that and where FE refers to frequency encoding, PE to phase encoding, is the sampling time, is the duration of GPE, is the gyromagnetic ratio, m is the sample number in the FE direction and n is the sample number in the PE direction, the 2D-Fourier Transform of this encoded signal results in a representation of the spin density distribution in two dimensions. Thus position and spatial frequency 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. An MR image is a complex-valued map of the spatial distribution of the transverse magnetization Mxy in the sample at a specific time point after an excitation. Conventional qualitative interpretation of Fourier Analysis asserts that low spatial frequencies contain the signal to noise and contrast information of the image, whereas high spatial frequencies contain the information determining the image resolution. This is the basis for advanced scanningtechniques, 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 Mxy 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: 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 direction saving scan time or in the FE direction, allowing for lower sampling frequencies and/or shorter echo times. However, these techniques are approximate due to phase errors in the MRI data which can rarely be completely controlled or nonzero phase due to just physical reasons.
