Physics-Informed DeepMRI: Bridging the Gap from Heat Diffusion to
  k-Space Interpolation
Zhuo-Xu Cui Congcong Liu Xiaohong Fan Chentao Cao Jing Cheng Qingyong Zhu Yuanyuan Liu Sen Jia Yihang Zhou Haifeng Wang Yanjie Zhu Jianping Zhang Qiegen Liu Dong Liang
In the field of parallel imaging (PI), alongside image-domain regularization
methods, substantial research has been dedicated to exploring $k$-space
interpolation. However, the interpretability of these methods remains an
unresolved issue. Furthermore, these approaches currently face acceleration
limitations that are comparable to those experienced by image-domain methods.
In order to enhance interpretability and overcome the acceleration limitations,
this paper introduces an interpretable framework that unifies both $k$-space
interpolation techniques and image-domain methods, grounded in the physical
principles of heat diffusion equations. Building upon this foundational
framework, a novel $k$-space interpolation method is proposed. Specifically, we
model the process of high-frequency information attenuation in $k$-space as a
heat diffusion equation, while the effort to reconstruct high-frequency
information from low-frequency regions can be conceptualized as a reverse heat
equation. However, solving the reverse heat equation poses a challenging
inverse problem. To tackle this challenge, we modify the heat equation to align
with the principles of magnetic resonance PI physics and employ the score-based
generative method to precisely execute the modified reverse heat diffusion.
Finally, experimental validation conducted on publicly available datasets
demonstrates the superiority of the proposed approach over traditional
$k$-space interpolation methods, deep learning-based $k$-space interpolation
methods, and conventional diffusion models in terms of reconstruction accuracy,
particularly in high-frequency regions.