This study introduces a frequency-domain, Hessian-free ray-Born inversion method for quantitative ultrasound tomography, building upon the Hessian-based approach presented in our previous work. Both methods model acoustic wave propagation using a ray-based approximation of the heterogeneous Green's function and iteratively solve the inverse problem in the frequency domain, progressing from low to high frequencies. In the previous study, each frequency subproblem is solved by iterative inversion of the Hessian matrix, which significantly increases computational costs. The present study addresses this limitation by diagonalizing the Hessian matrix through specific weighting, enabling a single-step inversion for each subproblem. This modification reduces computational expense by approximately an order of magnitude compared to the Hessian-based method, bringing its efficiency in line with radon-type, time-of-flight-based methods that use bent rays. Furthermore, by incorporating regularization directly into the forward operator and balancing computational efficiency with spatial resolution, the Hessian-free method achieves robust image reconstructions that are less sensitive to noise and inaccuracies in the initial model. For the ray-based approximation, this study introduces a paraxial ray-tracing system. Instead of independently tracing an auxiliary ray, the Jacobian of the ray is approximated by simultaneously tracing a paraxial ray alongside the linked ray. This approach improves computational efficiency while maintaining accuracy.

翻译：本研究在先前提出的基于Hessian矩阵方法的基础上，引入了一种频域无Hessian矩阵的射线Born反演方法，用于定量超声层析成像。两种方法均采用基于射线的异质格林函数近似来模拟声波传播，并在频域中从低频到高频迭代求解反问题。在先前研究中，每个频率子问题需通过迭代反演Hessian矩阵求解，这显著增加了计算成本。本研究通过特定加权对Hessian矩阵进行对角化处理，实现了每个子问题的单步反演，从而克服了这一局限。与基于Hessian矩阵的方法相比，此改进将计算成本降低了约一个数量级，使其计算效率与采用弯曲射线的Radon型走时层析方法相当。此外，通过将正则化直接融入正演算子，并在计算效率与空间分辨率之间取得平衡，无Hessian矩阵方法实现了鲁棒的图像重建，其对初始模型中的噪声和误差的敏感性更低。在射线近似方面，本研究引入了傍轴射线追踪系统。该方法不再独立追踪辅助射线，而是通过沿主射线同步追踪傍轴射线来近似射线的雅可比矩阵，在保持精度的同时提升了计算效率。