Fig 1.
A diagram of extension of the PSF data from a single (for example, forward) PSF reference data: (a) measured forward 3D PSF data set, distortion-free (top) and distorted (right) reference images calculated respectively by Eqs. 2 and 3, (front) 2D PSF image in y-s plane at selected fixed x; (b) an expansion image from a small portion of the 2D PSF image in (a); (c) procedure of inversion of the 2D PSF image along the y-direction in the s-coordinates with respect to the diagonal line; (d) mirrored 2D PSF image; (e) an extended PSF data corresponding to the reverse PSF data.
Finally, two EPI-PSF kernels for distortion correction (DiCo) in the both EPIs are calculated from the forward and extended PSF data.
Fig 2.
Distorted SE-EPI images of the phantom (a) and human brain (c) and the corresponding correlation maps of the EPI-PSF profiles (b and d).
The EPI-PSF profiles in the forward and extended reverse 3D PSF spaces were used for the calculation and collected at four different regions for the intensity comparisons, such as compressed (c-i), lowest correlation (c-ii), stretched (c-iii), and non-distorted regions (c-iv) of the human brain (c) and normalized by the largest value. In these figures, dashed and solid lines stand for forward and reverse PE scheme, respectively. The corresponding cross correlation (Corr.) value between the selected profiles is shown in each figure.
Fig 3.
Distortion correction with forward (row I) and reverse (row II) SE-EPIs and their differences (row III): (a and b) distorted and non-distorted reference (Ref.) images calculated from the 3D PSF data, (c) distorted EPI image, (d and e) the distortion-corrected EPI images corrected by the EPI-PSF kernels calculated from the corresponding measured (Mea.) and extended (Ext.) PSF data with forward (row I) and reverse (row II) PE polarity, respectively.
(I-f and II-f) subtracted images between (d) and (e), and (III-f) a subtracted image between (I-d) and (II-e), which corresponds to the proof of high quality result obtained by the proposed method.
Fig 4.
Forward (F) and reverse (R) GE- (I first row), SE- (I second row), and DW-EPIs (I third row) and FA maps (I bottom row) without and with DiCo.
An identical slice was chosen for SE- and DW-EPIs and corresponding FA maps. Weighted combination (WC) of the two distortion-corrected images was carried out using the proposed weighting maps (n = 2) and a final FA map was calculated based on final SE- and DW-EPIs. Green contours calculated from the SE-reference image (SE-ref.) were overlaid onto the enlarged images (II) of the anterior regions of the full FOV images (I). In the enlarged images of GE-EPIs, two blood vessels in the anterior regions of the GE-reference image (GE-ref) are used as landmarks to verify the correction fidelity.
Fig 5.
The EPI-PSF kernel based weighting maps and the ratio map in GE- (I) and SE-EPIs (II): two weighting maps, Wf(s) (a) and Wr(s) (b), are calculated respectively from the forward and reverse EPI-PSF kernel using Eq. 6 and the ratio (c) is shown in logarithmic scale using the equation loge(Wf(s)/Wr(s)).
Fig 6.
Selection of a suitable weighting exponent n for combination.
(a) Final FA and (b) corresponding relative SNR map with suitable n = 2 gained relative to the source scans; the line profile in (a) is taken along the white arrow; the red-contours mark areas corresponding to exactly same ROIs in where derivatives of pixel shifts are more than 0.5 pixel, (a-i) line profiles obtained from six FA maps, which include forward (F) and reverse (R) FA maps with DiCo and four final FA maps with different weighting exponents (n = 0, 1, 2, and 3), (a-ii) mean differences (in %) within red ROIs between five final FA maps with different weighting exponents (n = 0, 1, 2, 3, and 4) calculated using Eq. 8. Note that the mean difference of 16% was measured between FA maps obtained by simple (n = 0) and weighted averages (n = 1), which corresponds to n = 0 in (a-ii).
Fig 7.
The whole volume of SE-EPI sagittal images without (left) and with DiCo (middle) and corresponding color-coded FA maps (right): forward (F) PE, reverse (R) PE, and the weighted combination (WC) images are compared with the reference (Ref.) without distortions.
Yellow and red arrows point on completely recovered anatomical peculiarities in color-coded FA map (WC), but suboptimal recovery in color-coded FA maps (F and R).
Fig 8.
A reference image (A) without distortion, the corresponding final distortion-corrected FA maps with the proposed method (B), the original reversed gradient approaches using algorithms of Jacobian modulation (C) and least-squares restoration (D).
For comparison, a coronal (a), sagittal (b) and two axial images (c and d) were selected and white arrows point to the differences between FA maps.