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

Aliasing is an artifact that occurs in MR images when the scanned body part is larger than field of view (FOV). As a consequence of the acquired k-space frequencies not being sampled densely enough, whereby portions of the object outside of the desired FOV get mapped to an incorrect location inside the FOV. The cyclical property of the Fourier transform fills the missing data of the right side with data from behind the FOV of the left side and vice versa. This is caused by a too small number of samples acquired in, e.g. the frequency encoding direction, therefore the spectrums will overlap, resulting in a replication of the object in the x direction.
Aliasing in the frequency direction can be eliminated by twice as fast sampling of the signal or by applying frequency specific filters to the received signal.
A similar problem occurs in the phase encoding direction, where the phases of signal-bearing tissues outside of the FOV in the y-direction are a replication of the phases that are encoded within the FOV. Phase encoding gradients are scaled for the field of view only, therefore tissues outside the FOV do not get properly phase encoded relative to their actual position and 'wraps' into the opposite side of the image.

Use a larger FOV, RFOV or 3D Volume, apply presaturation pulses to the undesired tissue, adjust the position of the FOV, or select a small coil which will only receive signal from objects inside or near the coil. The number of phase encoding steps must be increased in phase direction, unfortunately resulting in longer scan times.
When this is not possible it can be corrected by oversampling the data. Aliasing is eliminated by Oversampling in frequency direction. No Phase Wrap (Foldover Suppression) options typically correct the phase encoding by doubling the field of view, doubling the number of phase encodes (to keep resolution constant) and halving the number of averages (to keep scan time constant) then discarding the additional data and processing the image within the desired field of view (but this is more time consuming).
Tissue outside this doubled area can be folded nevertheless into the image as phase wrap. In this case combine more than 2 number of excitations / number of signal averages with foldover suppression.
See also Aliasing, Foldover Suppression, Oversampling, and Artifact Reduction - Aliasing.

The partial Fourier technique is a modification of the Fourier transformation imaging method used in MRI in which the symmetry of the raw data in k-space is used to reduce the data acquisition time by acquiring only a part of k-space data.
The symmetry in k-space is a basic property of Fourier transformation and is called Hermitian symmetry. Thus, for the case of a real valued function g, the data on one half of k-space can be used to generate the data on the other half.
Utilization of this symmetry to reduce the acquisition time depends on whether the MRI problem obeys the assumption made above, i.e. that the function being characterized is real.
The function imaged in MRI is the distribution of transverse magnetization Mxy, which is a vector quantity having a magnitude, and a direction in the transverse plane. A convenient mathematical notation is to use a complex number to denote a vector quantity such as the transverse magnetization, by assigning the x'-component of the magnetization to the real part of the number and the y'-component to the imaginary part. (Sometimes, this mathematical convenience is stretched somewhat, and the magnetization is described as having a real component and an imaginary component. Physically, the x' and y' components of Mxy are equally 'real' in the tangible sense.)
Thus, from the known symmetry properties for the Fourier transformation of a real valued function, if the transverse magnetization is entirely in the x'-component (i.e. the y'-component is zero), then an image can be formed from the data for only half of k-space (ignoring the effects of the imaging gradients, e.g. the readout- and phase encoding gradients).
The conditions under which Hermitian symmetry holds and the corrections that must be applied when the assumption is not strictly obeyed must be considered.
There are a variety of factors that can change the phase of the transverse magnetization:
Off resonance (e.g. chemical shift and magnetic field inhomogeneity cause local phase shifts in gradient echo pulse sequences. This is less of a problem in spin echo pulse sequences.
Flow and motion in the presence of gradients also cause phase shifts.
Effects of the radio frequency RF pulses can also cause phase shifts in the image, especially when different coils are used to transmit and receive.
Only, if one can assume that the phase shifts are slowly varying across the object (i.e. not completely independent in each pixel) significant benefits can still be obtained. To avoid problems due to slowly varying phase shifts in the object, more than one half of k-space must be covered. Thus, both sides of k-space are measured in a low spatial frequency range while at higher frequencies they are measured only on one side. The fully sampled low frequency portion is used to characterize (and correct for) the slowly varying phase shifts.
Several reconstruction algorithms are available to achieve this. The size of the fully sampled region is dependent on the spatial frequency content of the phase shifts. The partial Fourier method can be employed to reduce the number of phase encoding values used and therefore to reduce the scan time. This method is sometimes called half-NEX, 3/4-NEX imaging, etc. (NEX/NSA). The scan time reduction comes at the expense of signal to noise ratio (SNR).
Partial k-space coverage is also useable in the readout direction. To accomplish this, the dephasing gradient in the readout direction is reduced, and the duration of the readout gradient and the data acquisition window are shortened.
This is often used in gradient echo imaging to reduce the echo time (TE). The benefit is at the expense in SNR, although this may be partly offset by the reduced echo time. Partial Fourier imaging should not be used when phase information is eligible, as in phase contrast angiography.

See also acronyms for 'partial Fourier techniques' from different manufacturers.

The FEER method was the first clinically useful flow quantification method using phase effects, from which all spin phase related flow quantification techniques currently in use are derived.
In this sequence a gradient echo is measured after a gradient with flow compensation. The measured signal phase should be zero for all pixels. A deviation from gradient symmetry by shifting the gradient ramp slightly away from the symmetry condition will impart a defined phase shift to the magnetization vectors associated with spins from pixels with flow.
Slight stable variations in the magnetic field across the imaging volume will prevent the phase angle from being uniformly zero throughout the volume in the flow-compensated image. The first image (acquired without gradient shift) serves as reference, defining the values of all pixel phase angles in the flow (motion) compensated sequence. Ensuing images with gradient phase shifts imparted in each of the 3 spatial axes will then permit measurement of the 3 components of the velocity vector v = (vx, vy, vz) by calculating the respective phases px, py and pz by simply subtracting the pixel phases measured in the compensated image from the 3 images with a well defined velocity sensitization.
The determination of all 3 components of the velocity vector requires the measurement of 4 images.
The phase quantification requires an imaging time four times longer than the simple measurement of a phase image and associated magnitude image. If only one arbitrary flow direction is of interest, it suffices to acquire the reference image plus one image velocity sensitized in the arbitrary direction of interest.

See also Flow Quantification.

Quantification relies on inflow effects or on spin phase effects and therefore on quantifying the phase shifts of moving tissues relative to stationary tissues.
With properly designed pulse sequences (see phase contrast sequence) the pixel by pixel phase represents a map of the velocities measured in the imaging plane. Spin phase effect-based flow quantification schemes use pulse sequences specifically designed so that the phase angle in a pixel obtained upon measuring the signal is proportional to the velocity. As the relation of the phase angle to the velocity is defined by the gradient amplitudes and the gradient switch-on times, which are known, velocity can be determined quantitatively on a pixel-by-pixel basis. Once, this velocity is known, the flow in a vessel can be determined by multiplying the pixel area with the pixel velocity. Summing this quantity for all pixels inside a vessel results in a flow volume, which is measured, e.g. in ml/sec.
Flow related enhancement-based flow quantification techniques (entry phenomena) work because spins in a section perpendicular to the vessel of interest are labeled with some radio frequency RF pulse. Positional readout of the tagged spins some time T later will show the distance D they have traveled.
For constant flow, the velocity v is obtained by dividing the distance D by the time T : v = D/T. Variations of this basic principle have been proposed to measure flow, but the standard methods to measure velocity and flow use the spin phase effect.
Cardiac MRI sequences are used to encode images with velocity information. These pulse sequences permit quantification of flow-related physiologic data, such as blood flow in the aorta or pulmonary arteries and the peak velocity across stenotic valves.

General MRI of the abdomen can consist of T1 or T2 weighted spin echo, fast spin echo (FSE, TSE) or gradient echo sequences with fat suppression and contrast enhanced MRI techniques. The examined organs include liver, pancreas, spleen, kidneys, adrenals as well as parts of the stomach and intestine (see also gastrointestinal imaging). Respiratory compensation and breath hold imaging is mandatory for a good image quality.
T1 weighted sequences are more sensitive for lesion detection than T2 weighted sequences at 0.5 T, while higher field strengths (greater than 1.0 T), T2 weighted and spoiled gradient echo sequences are used for focal lesion detection. Gradient echo in phase T1 breath hold can be performed as a dynamic series with the ability to visualize the blood distribution. Phases of contrast enhancement include the capillary or arterial dominant phase for demonstrating hypervascular lesions, in liver imaging the portal venous phase demonstrates the maximum difference between the liver and hypovascular lesions, while the equilibrium phase demonstrates interstitial disbursement for edematous and malignant tissues.
Out of phase gradient echo imaging for the abdomen is a lipid-type tissue sensitive sequence and is useful for the visualization of focal hepatic lesions, fatty liver (see also Dixon), hemochromatosis, adrenal lesions and renal masses. The standards for abdominal MRI vary according to clinical sites based on sequence availability and MRI equipment. Specific abdominal imaging coils and liver-specific contrast agents targeted to the healthy liver tissue improve the detection and localization of lesions.
See also Hepatobiliary Contrast Agents, Reticuloendothelial Contrast Agents, and Oral Contrast Agents.

For Ultrasound Imaging (USI) see Abdominal Ultrasound at Medical-Ultrasound-Imaging.com.