Filtering deletes components of the signal, high or low frequencies, band-pass, analog or digital. Whatever pattern or algorithms can be defined for data decimation.
Low pass filtering attenuates high frequency data and passes low frequency data. The reconstructed image will look a little blurrier, but nearly similar to the original image. The blurring is caused by the fact that the high spatial frequencies are lost, which contain information about edges in the image.
High pass filtering attenuates low frequencies and passes high frequencies. Most of the objects and contrast of the original image are lost in the reconstructed image, but the edges are clearly visible because high frequency data has been preserved.

The navigator technique measures with an additional quick MR prepulse the position, of e.g. the diaphragm before data collecting. Similar respiratory conditions of the patient can be identified and used to synchronize image data acquisition so that respiration induced image blurring is minimized by either respiratory ordered phase encoding or respiratory gating.
The prepulse sequence images a small area perpendicular to the structure, which is moving. The contrast of the interface between the diaphragm and the lung should be high to permit easy automatic detection. After data acquisition, the position of the interface is automatically recorded and imaging data are only accepted when the position of the interface falls within a range of prespecified values.
This technique has the advantage of greater accuracy than other respiratory gating (therefore used for coronary angiography) and has no need for additional sensing MRI equipment, as the MR system itself provides it.

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

This artifact is caused by movements of the patient or organic processes taking place in the body of the patient. The artifact appears as bright noise, repeating densities or ghosting in the phase encoding direction.

There are different solutions for reduction of phase encoded motion artifacts.
Cardiac and respiratory gating, breath holding, sedation of the patient, presaturation pulses for flow artifacts (e.g. arterial pulsation, breathing), fast imaging sequences, etc.

See also Motion Artifact, Ghosting Artifact, Motion Compensation Pulse Sequences and Artifact Reduction - Motion.

(SMART) Movement of the chest and abdominal wall also causes blurring and ghosting in the images due to variation of signal during data collection. Using SMART averaging is one way of reducing these artifacts but is not always possible or effective enough.

Volumetric imaging is a 3D technique where all the MRI signals are collected from the entire tissue sample and imaged as a whole entity, therefore providing a high signal to noise ratio. The acquisition of isotropic voxels or thin slices with high spatial resolution allows to create multiplanar reconstructions in all planes; a compensation for the usually longer scan time. The acquisition time can be reduced by parallel imaging technique.
New T2 weighted variants of 3D sequences (FSE-XETA, T2-SPACE, VISTA) have been introduced that differ from conventional FSE sequences. An echo train containing up to 200 echoes obtained at a minimum echo spacing allows very fast acquisition. A flip angle modulation (flip angle sweep - FAS) during the FSE readout carries magnetization as long as possible to avoid blurring and provide optimal signal at the effective echo time. This type of imaging is well suited for brain and spine MRI procedures.
Newer T1 weighted variants include Liver Acquisition with Volume Acquisition (LAVA) and T1W High Resolution Isotropic Volume Examination (THRIVE), which have advantages for dynamic breath hold imaging in liver and abdominal examinations.

See also Volume Imaging, 3 Dimensional Imaging.