This blog post discusses the resolution loss when applying partial-Fourier imaging in GE-EPI in the presence of strong T2*-decay.

I found that that when I was aiming for high-resolutions, it is beneficial to refrain from the application of partial Fourier (PF) imaging as much as possible. For the long readout durations at high-resolutions and the fast T2/T2*-decay at high field strengths results in even stronger blurring of partial-Fourier.

### Example 1 of the blurring with and without partial Fourier imaging

### Example 2 of the blurring with and without partial Fourier imaging

## The magnitude point spread function is an uninterpretable measure of spatial resolution

Many of the MRI education books suggest that a center-out T2* weighting like in SE-EPI and with partial Fourier imaging would theoretically result in a sharper point spread function (PSF) (Jesmanowicz et al., MRM, 1998; Hyde et al., MRM, 2001; Hetzer et al., MRM, 2011). This really goes against my experience of real-live data (examples above) and hence, I tried to implement my own simulations below. Based on those simulations, I conclude that the magnitude PSF may be sharper for partial Fourier imaging, however it is not an adequate measure of the spatial resolution. The effective spatial resolution is better without partial Fourier imaging.

I tired to investigate the effect of *T _{2}^{*}*-attenuation during the EPI readout theoretically in two ways:

- conventional magnitude PSF estimation, where only one voxel is assumed to differ from zero (Delta function in panel A). The corresponding signal distribution in
*k*-space is then weighted with the*T*-decay for full_{2}^{*}*k*-space GE-EPI and half (PF) GE-EPI (panel B), respectively (dotted: mirrored in GE-EPI). The back-transformation into image space provides the PSF (panel C) and the phase distribution (panel D) of the EPI signal. - Additionally, I did the same estimation for a more realistic situation resembling edges/contrast: namely, that only one voxel has a signal different from a finite background signal (panel E). This example is introduced to show the different effect of the complex PSF compared to the magnitude PSF. The same
*T*-decay was introduced as described above (panel F) and the corresponding_{2}^{*}*T*-blurring in image space was calculated (panel G/H)._{2}^{*}

In addition to these theoretical simulations, I tried to investigate the *T _{2}^{*}*-blurring

**. Two different readout strategies were used to estimate the different**

*in-vivo**T*-blurring: full

_{2}^{*}*k*-space acquisition (readout window = 110 ms) and half

*k*-space acquisition (readout window = 60 ms) with the same TE. In order to correct for phase inhomogeneities in the half

*k*-space acquisition, 8% of the

*k*-space lines were acquired symmetrically, across the center of

*k*-space.

Conventional estimation of the PSF associated with* T _{2}^{*}*-blurring (one voxel ≠ 0; Fig. A) and full

*k*-space GE-EPI gives a larger FWHM compared to PF half

*k*-space EPI PSF (Fig. C). Note that due to the step in the phase distribution (Fig. D), the real and the imaginary part of the PSF have a narrower FWHM in full

*k*-space GE-EPI than in PF half

*k*-space EPI (Figs. I/J). For the adapted way of estimating

*T*-blurring (Fig. E), the magnitude PSF of full

_{2}^{*}*k*-space GE-EPI has a narrower FWHM than that of PF half

*k*-space EPI (Fig. G). This is a result of positive and negative interference of signal from adjacent voxels depending on their phase.

The experimental data confirm that full *k*-space acquisition appears to give better spatial specificity in the phase encoding direction than PF half *k*-space acquisition (Figs. M/N). Intracortical anatomical layers are blurred in the phase-encoding direction (white arrows), but not in the read direction (black arrows) for the PF half *k*-space GE-EPI (Fig. M). The discriminability of the intra-cortical layer-dependent signal peaks (Figs. O and P) confirms this (green arrows). Correspondingly, during a finger tapping task, the functional profiles delineated layer-dependent responses with higher effective resolution for the full *k*-space GE-EPI scheme compared to the case where *T _{2}^{*}*-attenuation is symmetric in

*k*-space (Figs. Q/R).

## Conclusion

The conventional usage of the width of the magnitude PSF to estimate *T _{2}^{*}*-blurring can be misleading. In order to take into account that MRI produces complex signals, which can add up constructively or destructively, the PSF must be considered in the complex domain (Figs. I-L). I show here that full

*k*-space GE-EPI acquisition has a higher effective resolution than partial-Fourier

*k*-space GE-EPI.

## My intuitive explanation for the blurring with PF

It is well described that outer k-space lines represent the high spatial frequencies and the k-space lines close to the k-space center represent the smooth spatial frequencies. Hence, when partial Fourier is applied, the outer k-space lines are under represented and the image will look blurrier. Without partial Fourier imaging, one side of k-space is has a relatively large signal and the other side of k-space has a relatively low signal due to T2*-decay. Hence, the suppression of the later-acquired outer k-space lines is accounted for by the large signal of the firstly acquired outer k-space lines. The result is a sharper signal of a out-center-out k-space acquisition trajectory compared to partial Fourier imaging, even though the FWHM of the magnitude point spread function is larger.

## The reconstruction method matters

At high resolutions, the EPI protocol is often limited by too long TEs. Hence, it is not easily possible to simply refrain from the application of PF imaging. In a good fraction of my experiments, I also need to apply it to some extent. In those cases, when the application of PF imaging is not avoidable, special care should be taken to reconstruct the partly acquired k-space with the appropriate reconstruction algorithm.

### How to change the reconstruction in the sequence code:

##### Zero-filling

There are a couple of algorithms already implemented from SIEMENS. The default in fMRI sequences is the algorithm of “Zero filling”. You can find this in the sequence code as “None”:

`pSeqExpo->setPCAlgorithm (SEQ::PC_ALGORITHM_NONE);`

This option has the worst blurring. However, because of the blurring, it also has the highest SNR. I would assume that this is the reason, why it is set as default.

##### Margosian

You can also choose the algorithm “MARGOSIAN”. In the sequence code this can be set as follows:

`pSeqExpo->setPCAlgorithm (SEQ::PC_ALGORITHM_MARGOSIAN);`

With MARGOSIAN, the missing k-space lines are not just zero-filled. Instead, they are filled with the complex conjugate of the symmetric acquired k-space lines. This algorithm results in less blurring. However, I find it very unstable and sensitive to phase-inconsistencies and B0-inhomogeneities. I also found it to result in additional void artifacts. The resulting tSNR is very low.

##### Submatrix

You can also choose the algorithm “SUBMATRIX”. In the sequence code this can be set as follows:

`pSeqExpo->setPCAlgorithm (SEQ::PC_ALGORITHM_SUBMATRIX);`

I don’t really understand what it does. But the final results are not much better than with Zero-filling.

##### POCS

Another option is POCS (projection onto convex sets). In my opinion, this is the best option. In the sequence code it look as follows:

`pSeqExpo->setPCAlgorithm (SEQ::PC_ALGORITHM_POCS_PE);`

or for 3D readouts:

`pSeqExpo->setPCAlgorithm (SEQ::PC_ALGORITHM_POCS_3D);`

The working principle it very similar to MAGROSIAN. In my understanding, the only difference is that it accounts for B0-inhomogeneities by allowing an asymmetric k-space and filing in the missing k-space lines iteratively.

This method is quite stable against phase inconsistencies.

However, the default number of POCS iterations is 2! This is a very low number and should be increased. In most cases, the POCS algorithm converges to a stable solution in 2-4 iterations. I always set it to 8. This increases the reconstruction time by a few milliseconds and is completely worth it.

I don’t know how to change the number of POCS iterations in the sequence code. However, I know how it can be set in xbuilder after the registration of the participant as described in the figures.

## Amount of partial-Fourier

Of course, it the amount PF-induced blurring depends on the amount of omitted k-space lines. In my experience, partial Fourier imaging with a factor of 7/8 in combination with POCS reconstruction can recover most of the high-resolution information.

### Comparison of zero-filling and POCS filling

### Comparison of PF in GE-EPI and SE-EPI

In when PF is applied in GE-EPI, the outer k-space lines are suppressed. This is similar in spin echo EPI. The only differente is that in SE-EPI the T2-decay results in an asymmetric k-space weighing onto of the additional symmetric T2* weighing. My simulations below suggest that the resulting blurring in SE-EPI is not as bad as the application of PF.

## Alternative measured of blurriness

Since the magnitude point-spread function is an elusive measure of the blurriness of an image, it tried to look at alternative measures. An Alternative measure of the effective resolution is the distance that two signal sources of the voxel size need to have to be separable as two individual objects. As the animations below show, with the application of PF, the point sources need to be further apart to be separable as two individual objects compared to full k-space imaging.

## Partial Fourier blurring without strong T2*

When zero filling is applied, the application of partial Fourier imaging can result in image blurring even with fast readouts (with minimal T2*-decay). This effect has is discussed in great detail from practiCal fMRI in this blog post. In this post, he also discusses other advantages and disadvantages of PF imaging, including the increased SNR and amplified void artifacts.

## Quotes from the experts:

- Using partial-Fourier is just like taking drugs; You know it’s bad for you and that you shouldn’t use it. But when you sit at the scanner you suddenly get the urge to use it. And think: I will use it just the once, I promise. Jon Polimeni, April 26th 2017, Hawaii.
- With slow gradients and fast decay, partial-Fourier is the only way to have enough signal. Its better to have a blurry image than no image at all. David Feinberg, Februar 26th 2018, Berkeley.
- Its completely expected that when you omit the acquisition of outer k-space lines that represent the high spatial frequencies the effective resolution will be decreased. I never trusted the magnitude PSF anyway. Larry Wald, Februar 13th 2014, Leipzig.

## Further reading

The basics of Partial Fourier imaging are explained on mriquestions.com and MR-Tip.com and CBS-harvard.

Both the application of partial-Fourier sampling and the application of parallel imaging increase the acquisition speed and reduce the echo time. Hence, it is often advised to only use one of the two. practiCal fMRI has an excellent blog post about the comparison of partial Fourier and GRAPPA here.

We had an abstract about the inadequate interpretation of the magnitude point-spread-function at ISMRM in 2015 (see poster here). This story was continued from Denis Chaimow and Amir Shmuel with this BioRXiv paper.

## Acknowledgments:

I learned all about partial Fourier imaging in discussion with Ben Poser, Valentin Kemper, Peter Koopmans, and Franz Patzig. I want to thank them for their input.

Excellent review. Thanks for detailing what for many are intuitions.

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Great stuff! Another old blog post on effects of smoothing on temporal SNR, and enhanced regional dropout, for Siemens default partial Fourier reconstruction compared to full k-space: https://practicalfmri.blogspot.com/2013/08/the-experimental-consequences-of-using.html.

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Awesome, thanks for the link. Great to see how pF can can also result in some blurring, even without strong T2* effects. I included it into the post now.

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