  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.
