  This study proposes a Hessian-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, and the inverse problem is solved in the
frequency domain via iteratively linearisation and minimisation of the
objective function from low to high frequencies. In [3], the linear subproblem
associated with each frequency set is solved via an implicit and iterative
inversion of the Hessian matrix (inner iterations). Instead, in this study,
each linear subproblem is weighted in a way in which the Hessian matrix be
diagonalised, and can thus be inverted in a single step. Using this
Hessian-free approach, the computational cost for solving each linear
subproblem becomes almost the same as solving one linear subproblem associated
with a radon-type time-of-flight-based approach using bent rays. This
computational cost is about an order of magnitude less than the equivalent
Hessian-based approach proposed in [3]. More importantly, the assumptions made
for diagonalising the Hessian matrix make the image reconstruction more robust
than the inversion approach in [3] to noise or initial guess.
