This study proposes a Hessian-inversion-free ray-born inversion approach for biomedical ultrasound tomography. The proposed approach is a more efficient version of the ray-born inversion approach proposed in [3]. Using these approaches, the propagation of acoustic waves are modelled using a ray approximation to heterogeneous Green's function. The inverse problem is solved in the frequency domain by iteratively linearisation and minimisation of the objective function from low to high frequencies. In [3], the linear subproblem associated with each frequency interval is solved by an implicit and iterative inversion of the Hessian matrix (inner iterations). Instead, this study applies a preconditioning approach on each linear subproblem so that the Hessian matrix becomes diagonalised, and can thus be inverted in a single step. Using the proposed preconditioning approach, the computational cost of solving each linear subproblem of the proposed ray-Born inversion approach becomes almost the same as solving one linear subproblem associated with a radon-type time-of-flight-based approach using bent rays. More importantly, the smoothness assumptions made for diagonalising the Hessian matrix make the image reconstruction more stable than the inversion approach in [3] to noise.

翻译：本研究提出一种无需黑塞矩阵逆的射线-玻恩反演方法，用于生物医学超声断层成像。该方法是文献[3]所提射线-玻恩反演方法的更高效版本。此类方法通过射线近似非均匀格林函数来建模声波传播过程。反问题在频域中通过迭代线性化及从低频到高频逐步最小化目标函数得以求解。文献[3]中，每个频率区间的线性子问题通过隐式迭代求逆黑塞矩阵（内迭代）进行处理。与此不同，本研究对每个线性子问题应用预处理策略，使得黑塞矩阵对角化，从而可通过单步求逆完成计算。采用所提出的预处理方法后，本射线-玻恩反演方法求解每个线性子问题的计算成本与基于弯曲射线的类拉东飞行时间法求解单个线性子问题的成本几乎相当。更重要的是，对角化黑塞矩阵所需的光滑性假设使得图像重建相比文献[3]的反演方法对噪声具有更强的稳定性。