See Aliasing Artifact.

Oversampling is the increase in data to avoid aliasing and wrap around artifacts. Aliasing is the incorrectly mapping of tissue signal from outside the FOV to a location inside the FOV. This is caused by the fact, that the acquired k-space frequency data is not sampled density enough.
Oversampling in frequency direction, done by increasing the sampling frequency, prevents this aliasing artifact. The proper frequency based on the sampling theorem (Shannon sampling theorem/Nyquist sampling theorem) must be at least twice the frequency of each frequency component in the incoming signal. All frequency components above this limit will be aliased to frequencies between zero and half of the sampling frequency and combined with the proper signal information, which creates the artifact. Oversampling creates a larger field of view, more data needs to be stored and processed, but this is for modern MRI systems not a real problem. Oversampling in phase direction (no phase wrap), to eliminate wrap around artifacts, by increasing the number of phase encoding steps, results in longer scan/processing times.

The data acquisition beyond the FOV in phase encoding direction, with doubling the number of acquisitions and the scan time.

See also Oversampling and Aliasing Artifact.

Parts of the body, which extend beyond the Field of view in the phase encoding direction may still be visible in the image as artifacts.
Please note that aliasing in the frequency direction is also possible but can be eliminated in the same manner or by applying frequency specific filters to the received signal.

See Aliasing Artifact.

Doubling the sampling points in frequency encoding direction without expanding the scan time. The additional part is discarded after reconstruction.

See also Oversampling and Aliasing Artifact.