Image space formalism of convolutional neural networks for k-space interpolation

Dawood P, Breuer F, Gram M, Homolya I, Jakob PM, Zaiß M, Blaimer M (2025)

Publication Type: Journal article

Publication year: 2025

Journal

Magnetic Resonance in Medicine Wiley

DOI: 10.1002/mrm.70002

Abstract

Purpose: Noise resilience in image reconstructions by scan-specific robust artificial neural networks for k-space interpolation (RAKI) is linked to nonlinear activations in k-space. To gain a deeper understanding of this relationship, an image space formalism of RAKI is introduced for analyzing noise propagation analytically, identifying and characterizing image reconstruction features and to describe the role of nonlinear activations in a human-readable manner. Theory and Methods: The image space formalism for RAKI inference is employed by expressing nonlinear activations in k-space as element-wise multiplications with activation masks, which transform into convolutions in image space. Jacobians of the de-aliased, coil-combined image relative to the aliased coil images can be expressed algebraically; thus, the noise amplification is quantified analytically (g-factor maps). We analyze the role of nonlinearity for noise resilience by controlling the degree of nonlinearity in the reconstruction model via the negative slope parameter in leaky ReLU. Results: The analytical g-factor maps correspond with those obtained from Monte Carlo simulations and from an auto differentiation approach for in vivo brain images. Apparent blurring and contrast loss artifacts are identified as implications of enhanced noise resilience. These residual artifacts can be traded against noise resilience by adjusting the degree of nonlinearity in the model (Tikhonov-like regularization) in case of limited training data. The inspection of image space activations reveals an autocorrelation pattern leading to a potential center artifact. Conclusion: The image space formalism of RAKI provides the means for analytical quantitative noise-propagation analysis and human-readable visualization of the effects of the nonlinear activation functions in k-space.

Authors with CRIS profile

Moritz Zaiß Professur für Multimodale Bildgebung in der klinischen Forschung

Involved external institutions

Julius-Maximilians-Universität Würzburg DE Germany (DE) Fraunhofer-Institut für Integrierte Schaltungen (IIS) DE Germany (DE) Universitätsklinikum Würzburg DE Germany (DE)

How to cite

APA:

Dawood, P., Breuer, F., Gram, M., Homolya, I., Jakob, P.M., Zaiß, M., & Blaimer, M. (2025). Image space formalism of convolutional neural networks for k-space interpolation. Magnetic Resonance in Medicine. https://doi.org/10.1002/mrm.70002

MLA:

Dawood, P., et al. "Image space formalism of convolutional neural networks for k-space interpolation." Magnetic Resonance in Medicine (2025).

BibTeX: Download