K-space (magnetic Resonance Imaging)
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magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to generate pictures of the anatomy and the physiological processes inside the body. MRI scanners use strong magnetic fields, magnetic field gradients, and ...
(MRI), the ''k''-space or ''
reciprocal space Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray diffraction, X-ray and Electron diffraction, electron diffraction as well as the Electronic band structure, e ...
'' (a mathematical space of
spatial frequencies 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 tran ...
) is obtained as the 2D or 3D
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of the image measured. It was introduced in 1979 by Likes and in 1983 by Ljunggren and Twieg. In MRI physics, complex values are sampled in ''k''-space during an MR measurement in a premeditated scheme controlled by a ''
pulse sequence In Fourier transform NMR spectroscopy and imaging, a pulse sequence describes a series of radio frequency pulses applied to the sample, such that the free induction decay is related to the characteristic frequencies of the desired signals. Afte ...
'', 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''. It can be formulated by defining ''
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength) ...
s'' k_\mathrm and k_\mathrm for "frequency encoding" (FE) and "phase encoding" (PE): :k_\mathrm=\bar G_\mathrmm\Delta t :k_\mathrm=\bar n\Delta G_\mathrm \tau where \Delta t is the sampling time (the reciprocal of sampling frequency), \tau is the duration of ''G''PE, \bar (''gamma bar'') is the
gyromagnetic ratio In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI u ...
, ''m'' is the sample number in the FE direction and ''n'' is the sample number in the PE direction (also known as ''partition number''). Then, the 2D-
Fourier Transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of this encoded signal results in a representation of the spin density distribution in two dimensions. Thus position (''x'',''y'') and spatial frequency (k_\mathrm, k_\mathrm) 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''xy in the sample at a specific time point after an excitation. Conventional qualitative interpretation of
Fourier Analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo ...
asserts that low spatial frequencies (near the center of ''k''-space) contain the signal to noise and
contrast Contrast may refer to: Science * Contrast (vision), the contradiction in form, colour and light between parts of an image * Contrast (statistics), a combination of averages whose coefficients add up to zero, or the difference between two means * ...
information of the image, whereas high spatial frequencies (outer peripheral regions of ''k''-space) contain the information determining the
image resolution Image resolution is the level of detail of an image. The term applies to digital images, film images, and other types of images. "Higher resolution" means more image detail. Image resolution can be measured in various ways. Resolution quantifies ...
. 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''xy 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_\mathrm,-k_\mathrm) = S^*(k_\mathrm,k_\mathrm) \, where the star (^*) denotes
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. Thus ''k''-space information is somewhat redundant; an image can be reconstructed using only one half of the ''k''-space. Such is in either 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). MRI ''k-space'' is related to
NMR Nuclear magnetic resonance (NMR) is a physical phenomenon in which atomic nucleus, nuclei in a strong constant magnetic field are disturbed by a weak oscillating magnetic field (in the near and far field, near field) and respond by producing ...
''time-domain''Ernst RR, Bodenhausen G and Wokaun A (1987), ''Principles of nuclear magnetic resonance in one and two dimensions'', Oxford University Press. in all aspects, both being used for raw data storage. The only difference between the MRI ''k-space'' and the NMR ''time domain'' is that a gradient ''G'' is present in MRI data acquisition, but is absent in NMR data acquisition. As a result of this difference, the NMR ''FID'' signal and the MRI ''spin-echo'' signal take different mathematical forms: : \text=M_\mathrmcos(\omega_\mathrmt)exp(-t/T_\mathrm) and : \text= M_\mathrmsin(\omega_\mathrmt)/(\omega_\mathrmt) where : \omega_\mathrm=\omega_\mathrm + \bar{\gamma} rG Due to the presence of the gradient ''G'', the ''spatial information r'' (not the ''spatial frequency information k'') is encoded onto the frequency \omega, and at the same time the ''time-domain'' is renamed as ''k-space''.


References


Further reading

*McRobbie D., et al. ''MRI, From picture to proton.'' 2003 *Hashemi Ray, et al. ''MRI, The Basics'' 2ED. 2004. Magnetic resonance imaging