A new Bayesian approach is presented for extracting 2D object boundaries with measures of uncertainty. The boundaries are described by minimal closed sequences of segments and arcs, called mixed polygons. The sequence is minimal in the sense that it is able to describe all the geometrical properties of the boundary without being redundant. Based on geometrical measures evaluated on the object boundary model, a prior distribution is introduced in order to favor a mixed polygon with good geometrical properties, avoiding short sides, collinearity between segments, and so on. The estimation process is based on a two-stage procedure that combines reversible-jump MCMC (RJMCMC) and classic MCMC methods. The RJMCMC method is viewed as a model selection technique, and it is used to estimate the correct number of sides of the mixed polygon. The MCMC algorithm provides a sample of mixed polygons through which to evaluate the mixed polygon that best approximates the object boundary and its geometrical uncertainty. A convergence criterion for the RJMCMC method is provided. Copyright © 2011 John Wiley & Sons, Ltd.