  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.
