Analyzing Resolution and Model Uncertainties for Ultrasound Computed Tomography using Hessian Information

Characterizing the spatial resolution and uncertainties related to a tomographic reconstruction are crucial to assess its quality and to assist with the decision-making process. Bayesian inference provides a general framework to compute conditional probability density functions of the model space. However, analytic expressions and closed-form solutions for the posterior probability density are limited to linear inverse problems such as straight-ray tomography under the assumption of a Gaussian prior and data noise. Resolution analysis and uncertainty quantification is significantly more complicated for non-linear inverse problems such as full-waveform inversion (FWI), and sampling-based approaches such as Markov-Chain Monte-Carlo are often impractical because of their tremendous computational cost. However, under the assumption of Gaussian priors in model and data space, we can exploit the machinery of linear resolution analysis and find a Gaussian approximation of the posterior probability density by using the Hessian of the regularized objective functional. This non-linear resolution analysis rests on (i) a quadratic approximation of the misfit functional in the vicinity of an optimal model; (ii) the idea that an approximation of the Hessian can be built efficiently by gradient information from a set of perturbed models around the optimal model. The inverse of the preconditioned Hessian serves as a proxy of the posterior covariance from which space-dependent uncertainties as well as correlations between parameters and inter-parameter trade-offs can be extracted. Moreover, the framework proposed here also allows for inter-comparison between different tomographic techniques. Specifically, we aim for a comparison between tissue models obtained from ray tomography and models obtained with FWI using ultrasound data.