Quick Overview
Please note that there are different common names for this artifact.

A data truncation artifact may occur when the interface between high and low signal intensities is encountered in one imaging plane. The 2D-FT techniques transform the MR signal to spatial intensity image data with frequency and phase information encoding each axis in the plane of the scan. This artifact is found in both frequency and phase axes. Artifactual ripples adjacent to edges in an image or sharp features in a spectrum, caused by omission of higher frequency terms in Fourier transformation, particularly with the use of zero filling to replace unsampled higher frequencies.
Complex shapes are specified by series of sine and cosine waves of various frequencies, phase and amplitude. Some shapes are more difficult to encode than others. The most difficult shapes to represent with Fourier series of terms are waveforms with instantaneous transitions, tissue discontinuities or edges. The low-frequency components of the series describe the overall shape of the step function. Higher frequency components are needed to describe the corners if the step function more accurately. If not enough samples are taken, these areas cannot be accurately represented. The truncation of the infinite data series results in a ringing artifact because of the inability to accurately approximate this tissue discontinuity with a shorter truncated data set. Therefore, the ringing that occurs at all tissue boundaries on MR is called truncation artifact.

This problem can be easily resolved by taking more samples - a higher acquisition matrix and/or a smaller FOV. See Gibbs Artifact and Gibbs Phenomenon.

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.

Multiplication of acquired MR data by a function smoothly tapering off at higher spatial frequencies so as to reduce ringing artifacts near edges in the corresponding image or spectrum due to truncation and Gibbs phenomenon. It is a form of filtering.

Quick Overview
Please note that there are different common names for this MRI artifact.

The Gibbs or ringing artifact appears as a series of lines in the MR image parallel to abrupt and intense changes in the object at this location. This artifact does not occur visibly on smooth objects. This artifact is caused by the Gibbs phenomenon, an overshoot or ringing of Fourier series occurring at discontinuities.
In the spinal cord, a small syrinx can be simulated by the Gibbs phenomenon. Gibbs artifacts are also seen in other regions, for example the brain//skull interface.
Fine lines visible in an image may be due to undersampling of the high spatial frequencies, respectively incomplete digitization of the echo.
With more encoding steps the Gibbs artifacts is less intense and narrower. Therefore, e.g. the artifact is more intense in the 256 point dimension of a 256x512 acquisition matrix.

This problem can only be resolved by smoothing filters (LanczosSigmaFactor, 2-D Exponential Filtering, Gegenbauer Reconstruction etc.) or with a higher acquisition matrix and/or a smaller FOV, to smooth the object.

See also Gibbs Phenomenon and Apodization.

The multiplication of acquired MR data by a function smoothly tapering off at higher spatial frequencies so as to reduce ringing artifacts near edges in the corresponding image or spectrum due to truncation and Gibbs phenomenon. It is a form of filtering.