Purpose: To develop an image space formalism of multi-layer convolutional neural networks (CNNs) for Fourier domain interpolation in MRI reconstructions and analytically estimate noise propagation during CNN inference. Theory and Methods: Nonlinear activations in the Fourier domain (also known as k-space) using complex-valued Rectifier Linear Units are expressed as elementwise multiplication with activation masks. This operation is transformed into a convolution in the image space. After network training in k-space, this approach provides an algebraic expression for the derivative of the reconstructed image with respect to the aliased coil images, which serve as the input tensors to the network in the image space. This allows the variance in the network inference to be estimated analytically and to be used to describe noise characteristics. Monte-Carlo simulations and numerical approaches based on auto-differentiation were used for validation. The framework was tested on retrospectively undersampled invivo brain images. Results: Inferences conducted in the image domain are quasi-identical to inferences in the k-space, underlined by corresponding quantitative metrics. Noise variance maps obtained from the analytical expression correspond with those obtained via Monte-Carlo simulations, as well as via an auto-differentiation approach. The noise resilience is well characterized, as in the case of classical Parallel Imaging. Komolgorov-Smirnov tests demonstrate Gaussian distributions of voxel magnitudes in variance maps obtained via Monte-Carlo simulations. Conclusion: The quasi-equivalent image space formalism for neural networks for k-space interpolation enables fast and accurate description of the noise characteristics during CNN inference, analogous to geometry-factor maps in traditional parallel imaging methods.

翻译：目的：建立多层卷积神经网络（CNN）在MRI重建中实现傅里叶域插值的图像域形式化方法，并解析估计CNN推理过程中的噪声传播。理论与方法：利用复值整流线性单元实现傅里叶域（亦称k空间）中的非线性激活，并将其表示为与激活掩膜逐元素相乘的操作。该操作可转化为图像域中的卷积运算。在k空间完成网络训练后，该方法可提供重建图像关于混叠线圈图像的导数代数表达式——这些线圈图像作为网络在图像域中的输入张量。由此可解析估计网络推理中的方差，并用于描述噪声特性。验证采用蒙特卡洛仿真及基于自动微分的数值方法。该框架在回顾性欠采样活体脑图像上进行了测试。结果：图像域推理结果与k空间推理结果几乎完全相同，相应定量指标已证实该结论。通过解析表达式获得的噪声方差图与蒙特卡洛仿真及自动微分方法所得结果高度吻合。与传统并行成像类似，该方法可充分表征噪声鲁棒性。科尔莫戈罗夫-斯米尔诺夫检验表明，蒙特卡洛仿真所得方差图中体素幅度呈高斯分布。结论：用于k空间插值的神经网络的准等价图像域形式化方法，可快速准确描述CNN推理过程中的噪声特性，类比于传统并行成像方法中的几何因子图。