Guanxiong Luo^{1} and Peng Cao^{1}

^{1}The University of Hong Kong, Hong Kong, China

A deep neural network provides a practical approach to extract features from existing image database. For MRI reconstruction, we presented a novel method to take advantage of such feature extraction by Bayesian inference. The innovation of this work includes 1) the definition of image prior based on an autoregressive network, and 2) the method uniquely permits the flexibility and generality and caters for changing various MRI acquisition settings, such as the number of radio-frequency coils, and matrix size or spatial resolution.

With Bayes$$$'$$$ theorem, one could write the posterior as a product of likelihood and prior:$$f(\boldsymbol{x}|\boldsymbol{y}) = \frac{f(\boldsymbol{y}\mid\boldsymbol{x})g(\boldsymbol{x})}{f(\boldsymbol{y})} \propto f(\boldsymbol{y}\mid \boldsymbol{x} )\,g(\boldsymbol{x} )\label{eq:1},$$ where $$$f(\boldsymbol{y}\mid\boldsymbol{x})$$$ was probability of the measured k-space data $$$\boldsymbol{y}\in \mathbb{C}^M$$$ for a given image $$$\boldsymbol{x}\in \mathbb{C}^N$$$,where $$$N$$$ is the number of pixels and $$$M$$$ is the number of measured data points, and $$$g(\boldsymbol{x})$$$ was the prior model. The maximum a posterior estimation (MAP) could provide the reconstructed image $$$\hat{\boldsymbol{x}}$$$ , given by:$$\hat {\boldsymbol{x} }_{\mathrm {MAP} }(\boldsymbol{y})=\arg \max_{\boldsymbol{x}}f(\boldsymbol{x} \mid \boldsymbol{y})=\arg \max_{\boldsymbol{x}}f(\boldsymbol{y}\mid \boldsymbol{x} )\,g(\boldsymbol{x}). \tag{1}$$ The $$$n\times n$$$ image could be considered as an vectorized image $$$\mathit{\boldsymbol{x}} = (x^{(1)},...,x^{(n^2)})$$$, i.e., $$$x^{(1)}=x_{1,1}, x^{(2)}=x_{2,1},...,$$$ and $$$x^{(n^2)}=x_{n,n}$$$. The joint distribution of the image vector could be expressed as following :$$p(\boldsymbol{x};\boldsymbol{\pi,\mu,s, \alpha})=p(x^{(1)})\prod_{i=2}^{n^2} p(x^{(i)}\mid x^{(1)},..,x^{(i-1)}).$$$$$\boldsymbol{\pi,\mu,s,\alpha}$$$ were the parameters of mixture distribution for each pixel. Therefore, the network $$$\mathtt{NET}(\boldsymbol{x}, \Theta)$$$ shown in Figure 1(b), that predicts the distribution of all pixels of an image, was trained by$$\hat{\Theta} = {\underset {\Theta }{\operatorname {arg\,max}\ }}{ p(\boldsymbol{x}; \mathtt{NET}(\boldsymbol{x}, \Theta))},$$ where $$$\Theta$$$ was the trainable hyperparameters. When trained, the prior model $$$g(\boldsymbol{x})$$$ as $$g(\boldsymbol{x}) = p(\boldsymbol{x}; \mathtt{NET}(\boldsymbol{x}, \hat{\Theta})).\tag{2}$$ The measured k-space data $$$\boldsymbol{y}$$$ was given by$$\boldsymbol{y} = \mathbf{A} \boldsymbol{x} \ + \ \boldsymbol{\varepsilon},$$where $$$\mathbf{A}$$$ was the encoding matrix and $$$\boldsymbol{\varepsilon}$$$ was the noise. Substituting Eq. (2) into the log-likelihood for Eq. (1) yielded$$\hat{\boldsymbol{x}}_\mathrm{MAP}(\boldsymbol{y}) =\underset {\boldsymbol{x}}{\operatorname {arg\,max} }\,\log{f(\boldsymbol{y}\mid \boldsymbol{x} )} + \log{p(\boldsymbol{x}\mid \mathtt{NET}(\boldsymbol{x}, \hat{\Theta}))}. \tag{3}$$ The log-likelihood term $$${\log{f(\boldsymbol{y}\mid \boldsymbol{x} )}}$$$ had less uncertainties and was close to a constant with uncertainties from noise that was irrelevant and additive to $$$\boldsymbol{x}$$$. Hence, Eq. (3) can be rewritten as $${\hat {\boldsymbol{x} }}_{\mathrm {MAP} }(\boldsymbol{y}) = {\underset {\boldsymbol{x}}{\operatorname {arg\,max} }} \, \log{p(\boldsymbol{x}\mid \mathtt{NET}(\boldsymbol{x}, \hat{\Theta}))} \qquad \mathrm{s.t.} \quad \boldsymbol{y} = \mathbf{A} \boldsymbol{x} \ + \ \boldsymbol{\varepsilon}.$$ Then, the projected gradient method was used to solve above maximization, whose iterative steps were illustrated in Figure 1(c). The gradient was calculated by backpropagation.

For knee MRI, we used NYU FAST MRI reconstruction databases

Figure 2 shows the quantitative comparisons among the proposed method, GRAPPA, and compressed sensing

The same deep learning model has also been tested for various contrasts from other Cartesian k-space reconstruction experiments, thus there is no need to re-train the deep learning model for further studies.

The proposed method can reliably and consistently recover the nearly aliased-free images with relatively high acceleration factors. The reconstruction from a maximum of posterior showed the successful reconstruction of the detailed anatomical structures, such as vessels, cartilage, and membranes in-between muscle bundles.

In this work, the results demonstrated the successful reconstruction of high-resolution image (256 x 256 matrix) with low-resolution prior (trained with 128 x128 matrix), confirming the feasibility of reconstructing images of different sizes without the need for retraining the prior model.

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