Partial Fourier Techniques

What is partial Fourier imaging?  

Partial Fourier imaging techniques are reconstruction methods in which data from as little as one-half of k-space is used to generate an entire MR image.  How can this be possible?

This amazing result derives from the fact that some of the information in k-space is redundant. Provided no phase errors occur during data collection, k-space possesses a peculiar mirrored property known as conjugate (or Hermitian) symmetry. 

Conjugate symmetry applies to pairs of points (like P and Q) that are located diagonally from each other across the origin of k-space. If the data at P is the complex number [a+bi], the data at Q is immediately known to be P's complex conjugate, [a−bi]. 

Conjugate symmetry of P and Q. If the data for one is known, the other can be calculated.

By an advanced but straightforward argument, conjugate symmetry can be shown to exist whenever a Fourier transform is performed on any real-valued function. In more concrete terms related to 2D MR imaging, the conjugately symmetric points represent corresponding data acquired on the rising and trailing tails of two echoes obtained with opposite phase encoding steps. In other words, the signal intensity of a point on the rising portion of an echo obtained using a positive phase-encode step is the complex conjugate of that on the downward portion of another echo obtained using the corresponding negative phase-encode step.

Conjugate (Hermitian) symmetry of k-space. Mirror image locations across the origin of k-space have real components of the same sign but imaginary components of the opposite sign.

The practical result of conjugate symmetry is that, in theory, only half of k-space data needs to be collected and the other half can be estimated. This can be translated into a reduction in imaging time, reduction in minimum echo time, or both. 

All image data sets contain some phase errors, and therefore the conjugate symmetry approximations are not perfect.  The sources of these phase errors include the usual "suspects": Bo inhomogeneity, susceptibility effects, eddy currents, physiologic motion, and spatial variations in transmit RF uniformity or surface coil sensitivity. As commercially implemented, therefore, partial Fourier techniques require sampling of slightly more than half the lines of k-space (typically about 60% for routine imaging, more for echo-planar imaging).  These extra lines are then used to generate phase correction maps of k-space, allowing a more accurate prediction of missing values.  

Two types of partial Fourier imaging are used in common practice and are available as options on all major brands of scanners.  These are known generically as "read conjugate symmetry" and "phase conjugate symmetry" respectively. These techniques are described more completely in the next two Q&As.

Advanced Discussion (show/hide)»

A current popular method used for partial Fourier estimation is known as homodyne reconstruction. This technique involves sequential application of a two filters to the acquired k-space data. The first (a high-pass filter) doubles the amplitude of this data and then discards the imaginary part of the image after the Fourier transform. The second (low-pass) Homodyne filter creates a "correction image" from a small set of data acquired symmetrically around the center of k-space. The phase of this "correction image" is subtracted from the phase of the first (high-pass) filtered image image before discarding the imaginary part of the image.

In echo-planar imaging (EPI) echoes acquired late after the RF-excitation pulse will have different phase than those occurring early. This is a source of additional phase errors and makes phase estimation more difficult. For EPI often 6/8 to 7/8 of k-space must be sampled in partial Fourier techniques to accurately estimate the remaining portion.

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References
     Feinberg DA, Hale JD, Watts JC et al.  Halving MR imaging time by conjugation: demonstration at 3.5 kG.  Radiology 1986; 161:527-531.
     MacFall JR, Pelc NJ, Vavrek RM.  Correction of spatially dependent phase shifts for partial Fourier imaging.  Magn Reson Imaging 1988; 6:143-145. 
    McGibney G, Smith MR, Nichols ST, Crawley A. Quantitative evaluation of several partial Fourier reconstruction algorithms used in MRI. Magn Reson Med 1993;30:51-59
     Williams LR. Symmetry. Lecture Notes for Computer Science 530, University of New Mexico, 2011. Available at http://www.cs.unm.edu/~williams/cs530/symmetry.pdf

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Related Questions
     How does phase-conjugate symmetry work? Why is it used? 
     What is read conjugate symmetry (fractional echo) imaging? Why would one only want to sample part of an echo? 

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