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.

翻译：在并行成像（PI）领域，除了图像域正则化方法外，大量研究致力于探索 k 空间插值。然而，这些方法的可解释性仍然是一个未解决的问题。此外，这些方法目前面临的加速限制与图像域方法相当。为了增强可解释性并克服加速限制，本文提出一个基于热扩散方程物理原理的统一 k 空间插值技术与图像域方法的可解释框架。在此基础上，提出一种新的 k 空间插值方法。具体而言，我们将 k 空间中高频信息衰减过程建模为热扩散方程，而重建低频频域中高频信息的过程可理解为逆热方程。然而，求解逆热方程是一个具有挑战性的逆问题。为解决这一挑战，我们修改热方程以符合磁共振并行成像物理原理，并采用基于评分的生成方法精确执行修改后的逆热扩散。最后，在公开数据集上的实验验证表明，所提出方法在重建精度上（尤其是高频区域）优于传统 k 空间插值方法、基于深度学习的 k 空间插值方法以及常规扩散模型。