We develop a computational framework to quantify uncertainty in shear elastography imaging of anomalies in tissues. We adopt a Bayesian inference formulation. Given the observed data, a forward model and their uncertainties, we find the posterior probability of parameter fields representing the geometry of the anomalies and their shear moduli. To construct a prior probability, we exploit the topological energies of associated objective functions. We demonstrate the approach on synthetic two dimensional tests with smooth and irregular shapes. Sampling the posterior distribution by Markov Chain Monte Carlo (MCMC) techniques we obtain statistical information on the shear moduli and the geometrical properties of the anomalies. General affine-invariant ensemble MCMC samplers are adequate for shapes characterized by parameter sets of low to moderate dimension. However, MCMC methods are computationally expensive. For simple shapes, we devise a fast optimization scheme to calculate the maximum a posteriori (MAP) estimate representing the most likely parameter values. Then, we approximate the posterior distribution by a Gaussian distribution found by linearization about the MAP point to capture the main mode at a low computational cost.
