Convex ultrasound image reconstruction with log-Euclidean priors

Image reconstruction from noisy and incomplete observations is usually an ill-posed problem. A Bayesian framework may be adopted do deal with this such inverse task by well posing the reconstruction problem. In this approach, the ill poseness nature of the reconstruction is removed by minimizing a two-term energy function. The first term pushes the solution toward the data and the second regularizes the solution. A Bayesian algorithm for ultrasound image reconstruction and de-noising is proposed where an edge preserving prior is used to reduce the smoothing effect at the transitions. The prior distribution is based on log-Euclidean potential functions that are particular suitable in reconstruction problems under the constraint of positivity, that is, when the unknowns to be estimated should be positive, which is the case, where the noisy observations are modeled by a Rayleigh distribution. In this paper, the reconstruction procedure is formulated as the optimization of a convex function and a Newton method is adopted to obtain the minimizer. This strategy guarantees a convergence to the global minimum in a small number of iterations. Experimental results, using synthetic and real medical images are shown. The proposed method produces images where speckle noise is effectively suppressed and important clinical details (organ and tissue transitions) are preserved.