In mathematics, the Gibbs phenomenon (also known as ringing artifacts, named after the American physicist J. Willard Gibbs) is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function f behaves at a jump discontinuity: the nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit.
In MRI, artifactual ripples parallel to abrupt and intense changes are caused by the Fourier transformation.

See Gibbs Artifact, Truncation Artifact, Ringing Artifact Reduction.

An integral is a mathematical object that can be interpreted as an area or a generalization of area. A number computed by a limiting process in which the domain of a function, often an interval or planar region, is divided into arbitrarily small units, the value of the function at a point in each unit is multiplied by the linear or areal measurement of that unit, and all such products are summed (summation in the limit). In MRI for example this mathematical function is used in the Fourier transformation.

Usual shape of the lines in a NMR spectrum, characterized by a central peak with long tails; proportional to 1/[(1/T2)2 + (f - fo)2], where f is frequency and fo is the frequency of the peak (i.e., central resonance frequency). A Lorentzian function is the Fourier transformation of a decaying exponential.

(NMR) Nuclear Magnetic Resonance is a physical phenomenon of the magnetic property of nuclei, which have a positive nuclear spin quantum number.
Under the influence of an external static magnetic field this nuclei will precess about the direction of the magnetic field with an angular frequency (Larmor frequency). Through absorption and emission of RF energy (gradients, RF coils) at the resonance frequency (Larmor equation) and the processing of this raw data by the Fourier transformation - physical, chemical, electronic, and structural information about molecules can be obtained (NMR Magnetic Resonance Spectroscopy, Magnetic Resonance Imaging).

In parallel MR imaging, a reduced data set in the phase encoding direction(s) of k-space is acquired to shorten acquisition time, combining the signal of several coil arrays. The spatial information related to the phased array coil elements is utilized for reducing the amount of conventional Fourier encoding.
First, low-resolution, fully Fourier-encoded reference images are required for sensitivity assessment. Parallel imaging reconstruction in the Cartesian case is efficiently performed by creating one aliased image for each array element using discrete Fourier transformation. The next step then is to create an full FOV image from the set of intermediate images. Parallel reconstruction techniques can be used to improve the image quality with increased signal to noise ratio, spatial resolution, reduced artifacts, and the temporal resolution in dynamic MRI scans.
Parallel imaging algorithms can be divided into 2 main groups:
Image reconstruction produced by each coil (reconstruction in the image domain, after Fourier transform): SENSE (Sensitivity Encoding), PILS (Partially Parallel Imaging with Localized Sensitivity), ASSET.
Reconstruction of the Fourier plane of images from the frequency signals of each coil (reconstruction in the frequency domain, before Fourier transform): GRAPPA.
Additional techniques include SMASH, SPEEDER™, IPAT (Integrated Parallel Acquisition Techniques - derived of GRAPPA a k-space based technique) and mSENSE (an image based enhanced version of SENSE).