# Truncation (Gibb's) artifact

## Origin

Truncation or Gibbs artifacts appear as parallel lines adjacent to high-contrast interfaces, due to the reconstruction of the image by a Fourier transform from a finite sampled signal.

For example, if we consider the Fourier transform of a sinusoidal waveform, it corresponds to a Dirac function (with 2 symmetric spikes, one with a positive frequency and the other one with the same frequency value but negative). The Fourier transform of a square waveform corresponds to a sinus cardinal function with an infinite number of frequencies.

If we restrict this sinus cardinal function and take only its central part (finite sample), the inverse Fourier transform of that sample results in a square waveform corrupted by variable undershoot and overshoot oscillations. The larger the sample is, the more frequency components it contains, the better the approximation will be with shorter and less ample oscillations will be.
As k-space is composed of finite samplings of the MR signal, the Fourier series is truncated and artifacts will appear at the interfaces between highly-contrasted tissues. The number of samples is defined by the matrix size in both the frequency and phase-encode directions. As it depends on the properties of the Fourier transform used for image reconstruction, the truncation artifacts occur in both the frequency and phase-encode directions. However the truncation artifacts are best seen when the number of samples is the lowest, which is usually in the phase-encode direction.

## Remedies

Increasing the matrix size reduces the truncation artifact. However it results in a decrease of both the voxel size and the signal to noise ratio. As a consequence, even if truncation artifacts are still present, they can be masked by the image noise.