Jingxin Zhang^{1,2}, Kazi Rafiqul Islam^{1}, and Kai Zhu^{1}

This abstract presents a novel method for compressed sensing (CS) MRI. This method combines the variable density random undersampling and iterative image reconstruction in CS-MRI with the regularly reduced bunched phase encoding (BPE) and linear equation based image reconstruction of BPE-MRI to further reduce data acquisition time and improve image quality of CS-MRI. Simulation results demonstrate the effectiveness and advantage of the presented method.

The proposed
method uses BPE generated by zigzag variation of phase direction gradient
and oversampling of k-space data during readout. Coupled with LEB reconstruction, BPE allows a regular reduction of factor R_{1 }> 1 along phase
direction. LEB
reconstruction^{2 }computes the image *M* using *D *= *CM*, where *C *is a
Fourier coefficient matrix constructed based
on
the sample coordinates, and *D* is
the bunched-sample matrix of k-space data acquired by BPE. The rows of *D* are in phase direction with dimension reduced by the factor R_{1} and consist of the aliased
data resulting from 1D IFFT along the rows. LEB
reconstruction can produce aliasing free image from the data of single receive coil without using coil sensitivity map. Besides BPE, the
proposed method uses an additional VD undersampling to randomly omit the regularly reduced BPEs by a factor R_{2} >
1. This amounts to randomly undersampling the columns of *D* by the factor R_{2} to
acquire a subset of *D*, *D _{S }*=

We conducted simulation with in-vivo
data to verify the proposed method. 2D
spin-echo brain scan of a
healthy volunteer was
performed on 3T Skyra (Siemens HealthCare, Erlangen,
Germany) using a 32-channel head coil (FOV: 240mm, Flip angle:
10^{o}, image matrix: 256 x 256)
and only one channel’s data were used in simulation. Informed consent was taken in
accordance with the institution’s
ethics policy. The original 256x256 k-space data were
interpolated to a 512 x 512 matrix,
with the added data points simulating the data points of underlying continuous k-space within the original sampling grid. The interpolated
data matrix were zigzag-sampled, with R_{1 }= 2, r = 2 (confer Fig 1), and then VD-undersampled along rows by the factor R_{2 }= 2 to obtain a 64x256 data matrix. This 64x256 data
matrix simulates the physically acquired data with total
reduction R = R_{1}R_{2 }= 4. It is used to form *D _{S}* for the iterative
CS reconstruction to compute $$$\hat{D}$$$, which is used in $$$\hat{D}$$$ =

Fig 2 and Fig 3 compare the performance of the proposed method and VD-CS method at reduction R = 4. The reconstructed images and difference images of both methods clearly show the superiority of the proposed method over VD-CS method. Quantitatively, the relative error for the proposed method is 0.0914, whereas it is 0.2170 for VD-CS method.

We have not compared the proposed method with the BPE alone method,^{} because the latter has been reported to perform poorly at R = 4.^{2}

1. Lustig M, Donoho DL, Santos JM, Pauly JM. Compressed Sensing MRI. IEEE Signal Processing Magazine. 2008;25(2):72-82.

2. Moriguchi H, Duerk JL. Bunched phase encoding (BPE): A new fast data acquisition method in MRI. Magnetic Resonance in Medicine. 2006;55(3):633-48.

Fig 1. Schematic diagram illustrating the proposed method for the k-space data acquisition and image reconstruction of an N x N image

Fig 2. Image reconstruction of proposed method at reduction R = 4. Left to right: FFT2 reconstructed reference image, image reconstructed by proposed method, corresponding difference image.

Fig 3. Image reconstruction of VD-CS at reduction R = 4. Left to right: FFT2 reconstructed reference image, image reconstructed by VD-CS, corresponding difference image.