Yang Song^{1}, Yida Wang^{1}, Xu Yan^{2}, Minxiong Zhou^{3}, Bingwen Hu^{1}, and Guang Yang^{1}

We used a reference-free model based on convolutional neural network (RF-CNN) to reconstruct the under-sampled magnetic resonance images. The model was trained without fully sampled image (FS) as the reference. We compared our model with the traditional compressed sensing reconstruction (CS) and the CNN model trained by FS. Mean square error and structure similarity were used to evaluate the model. Our RF-CNN model performed better than CS, but did not perform as good as usual CNN model.

We used the T1W image from MIDAS dataset (resolution=1x1x1 mm3)
in this work^{2}. We separated the dataset to three groups: training
data set (88 cases, 9888 slices), validation data set (4 cases, 432 slices),
and testing data set (5 cases, 540 slices). All images in the training dataset
were augmented with random zooming, shifting, rotating and shearing. Then we cropped
all images to the size of 192x192 and normalized them by subtracting mean value
and dividing by the standard deviation. A pseudo-randomly sampling (rate=40%)
mask was designed to under-sample the k-space data. We applied Fourier
transform (FT) on the images and extracted the k-space data according to the
sampling mask. We filled zero in the unsampled position in k-space to match the
sampling matrix size. Inverse FT was applied to get zero-filling reconstruction
(ZF). We showed the FS, sampling mask, and ZF in Figure 1.

We used a U-Net based 2D model named RF-CNN to reconstruct the MR
images (Figure 2)^{3}. We input ZF into the model and get the output of
U-Net based model to get the reconstructed image. Then FT was applied on the
output to get the reconstructed k-pace. We calculated the mean square error (MSE)
between the full sampled k-space and the corresponding reconstructed k-space
and used this value as the loss function of the model. However, the usual image-out
network used the MSE calculated between the reconstructed image and the FS
image as the loss function.

During the training, we used Adam algorithm to minimize the loss
function. Some tricks such as learning rate reducing and early stopping were
used to increase the efficiency of the training. All processes above were
implemented with TensorFlow 1.11^{4} and Python 3.5.

We compared the reconstruction accuracy among our
RF-CNN model, the image-out CNN model that trained by minimizing the MSE reference
to FS, and the compressed sensing (CS) reconstruction by Split Bregmam
Algorithm^{5.} We used MSE and structure similarity (SSIM) to quantify the
performance. Paired t-test was used for statistics.

We showed one slice of CS reconstruction, CNN reconstruction and RF-CNN reconstruction in Figure 3, respectively. CNN outperformed the CS and RF-CNN. The quality of RF-CNN is similar to the quality of CS reconstruction. We applied statistics on the MSE and SSIM of total 540 slices of 5 cases in testing data set. The mean value and the 95% confidence intervals (95% CIs) were calculated in Table 1. RF-CNN showed better reconstruction than the CS reconstruction (MSE: p<0.0001, SSIM p<0.001). The reconstruction time for each slice of CS is more than 1 second and that of RF-CNN is only about 10ms.

We also plotted the MSE of the validation data set against the iterative epoch during the training process in Figure 4. The training of RF-CNN converged more quickly than the standard CNN, while the iteration reach plateau the common CNN showed smaller MSE than RF-CNN.

1. Wang S, Su Z, Ying L, et al. Accelerating magnetic resonance imaging via deep learning[C]//Biomedical Imaging (ISBI), 2016 IEEE 13th International Symposium on. IEEE, 2016: 514-517.

2. Bullitt E, Zeng D, Gerig G, et al. Vessel tortuosity and brain tumor malignancy: A blinded study. Academic Radiology, 2005, 12:1232-1240

3. Ronneberger O, Fischer P, Brox T, U-Net: Convolutional Networks for Biomedical Image Segmentation, ArXiv, 2015: 1505.04597

4. Abadi M, Barham P, Chen J, et al. Tensorflow: a system for large-scale machine learning[C]//OSDI. 2016, 16: 265-283.

5. Goldstein T, Osher S. The split Bregman method for L1-regularized problems[J]. SIAM journal on imaging sciences, 2009, 2(2): 323-343.

Figure 1. The fully sampled image (a), sampling mask with sampling rate of
40% (b) and the under-sampled image with zero-filling (c).

Figure
2.
The architecture of the model. In each block, we convoluted twice with filter
size 3x3 and convoluted once with filter size 1x1. The number of filter is 32,
64, 128, and 256 from top to bottom in the architecture. We used Leak ReLU as
the activation function. Batch-Normalization was used before the activation.

Figure 3. The Reconstruction by CS (left), RF-CNN (middle)
and usual CNN (right) were shown in the top row. The corresponding
subtraction from reconstructions to FS were shown in the bottom row.

Figure
4.
The plot of MSE of training data set and validation set against the iterative
epochs. The MSE estimated by the training date set was marked to red (blue) for
the usual CNN (RF-CNN) model. The MSE estimated by the validation
date set was marked to green (black) for the usual CNN (RF-CNN) model.

Table 1. The MSE and SSIM estimated by CS, RF-CNN and
usual CNN compared with FS.