Kazi Rafiqul Islam^{1} and Jingxin Zhang^{1}

^{1}School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, Australia

### Synopsis

Precise coordinates of trajectories are essential for image reconstruction of k-space data acquired from
non-rectangular trajectories, and measurement of the trajectories often requires prescan calibration that complicates the process. This abstract presents a simple
and effective method to estimate the coordinates of non-rectangular periodic trajectories from normal
scan data and demonstrates its efficacy in image reconstruction of in vivo scan
data acquired from ZIGZAG trajectory.

__Introduction__

In conjunction with proper
image reconstruction methods, K-space data acquisition using non-rectangular periodic
trajectories, eg ZIGZAG or sinusoidal wave pattern, can reduce the number of
phase encodings and hence accelerate data acquisition. Bunched phase encoding
(BPE)

^{1,2} is one of the examples of this kind. For an N x N image M, we
can use ZIGZAG or sinusoidal variation of phase direction gradient and
oversampling of k-space data during readout to acquire a k-space data matrix of
(N/R) x P, where R > 1 is the reduction factor and P/N is the oversampling
rate, with P ≥ NR. IFFT2 of the k-space data matrix gives the
aliased image D with N/R rows. The image M can be reconstructed by solving a complex
valued linear equation D = CM, where C is the aliasing coefficient matrix
constructed using the coordinates of the periodic trajectories. The image M can
be precisely reconstructed when the coordinates of the physically realized
trajectories are the same as the mathematically calculated ones.
In practice, however, the
physically realized trajectories are never the same as the calculated ones due
to gradient imprecision and field inhomogeneity, which results in poor image
quality. A fix to this problem is to measure the coordinates of the
trajectories and use them to construct C. Such measurements generally requires prescan
calibration that complicates the process and increases operation cost. To overcome this difficulty, we present a
simple and effective method to estimate the coordinates of non-rectangular
periodic trajectories from normal scan data.

__Method__

Fig
1 shows an example ZIGZAG k-space data acquisition, where the red dashed lines are the mathematically calculated
trajectory, x’s are the actual data positions. As shown by the black solid line in the
figure, in such acquisition, there is actually an underlying rectangular baseline
grid, with spacing P, which contains the baseline data points shown by the
black x’s. If the image signal of M is band
limited, the other data points, shown by the blue x’s, green x’s and red x’s can be regarded as the baseline data, the black x’s, shifted in the x and y directions

^{1}.
Thus we can use the baseline data points as reference
to estimate the x and y direction shifts
of other data points. In Fig. 1 example,
there are four sets of k-space data: black, blue, green and red. We use the
black x’s as reference and use the 2D cross-correlation technique

^{3 }to estimate the shifts of other data sets with
respect to the black x’s. With the estimated shifts, we can construct the estimated C matrix, C

_{est}, and use D = C

_{est}M to reconstruct the
image M. This method is developed based on two facts: i) The
coefficient matrix C is essentially determined by the shifts of the
non-baseline data points with respect to the baseline data points. ii) The 2D cross-correlation technique is an effective and reliable method for estimating the relative shifts
of 2D signals. It can be performed by convolution and also by Fourier transform
for fast computation.

__Experiments__

Experiments
were carried out on 2D in-vivo data from a spin-echo brain scan of a healthy
volunteer on Siemens Skyra 3T MRI scanner with 32-channel head coil (FOV: 240 mm,
Flip angle:10o, image matrix: 256×256). The k-space data
matrix was sampled on ZIGZAG trajectories with the reduction factor R = 2 and the
oversampling rate P/N = 8. The trajectory estimation method described above was
applied to the k-space data to obtain the coefficient C

_{est}, which is
used in D = C

_{est}M to reconstruct the image M.
For comparison, the image M was also constructed by solving D = CM, where C is constructed using the calculated coordinates of ZIGZAG trajectory.

__Results__

Confer Fig 2__Conclusion__

We have presented a simple method to estimate the coordinates
of non-rectangular periodic trajectories from normal scan data, and have used
it to reconstruct a high quality image from ZIGZAG acquired in vivo scan data
without prescan calibration. The result shows that the proposed method is effective.
The proposed method is applicable to general non-rectangular periodic
trajectories and compressed sensing BPE we proposed in ^{2}.### Acknowledgements

No acknowledgement found.### References

- Moriguchi H, Duerk JL. Bunched phase encoding
(BPE): A new fast data acquisition method in MRI. MRM. 2006;55(3):633-48.
- Jingxin Zhang, Kazi Rafiqul Islam, Kai Zhu. Compressed
sensing MRI using Bunched Phased Encoding. ISMRM 2017.
- Feng
Zhao, Qingming
Huang, Wen
Gao. Image Matching by Normalized
Cross-Correlation. ICASSP 2006.