Image space formalism of k-space interpolation networks for analytical expression of noise characteristics

Motivation: Robust Artificial Neural Networks for k-space Interpolation (RAKI) exhibit superior image reconstructions compared to traditional Parallel Imaging. It is crucial to thoroughly characterize RAKI to gain insights into its functionality and stimulate further enhancements. Goal(s): Exploring how k-space interpolation with convolutional neural networks can be transformed into image domain to obtain an analytical description of noise characteristics. Approach: The nonlinear activation in k-space is expressed as elementwise multiplication. This can be transformed into convolution in image space. Results: The proposed image space formalism yields image reconstructions quasi-equivalent to k-space interpolation. The analytical expression of noise characteristics is in correspondence with Monte Carlo simulations. Impact: We propose an image space formalism for k-space interpolation with convolutional neural networks. This enables an analytical expression of the noise characteristics, analogous to g-factor maps in traditional parallel imaging methods.