Convex shape reconstruction from noisy ray probe measurements

Two algorithms for two-dimensional convex shape reconstruction from noisy ray probe measurements are developed and compared. Given a coordinate system located within the object, the data consists of a finite set of angles together with the corresponding radial distances to the boundary corrupted by additive noise. We first characterize when such data is consistent with some convex shape. The algorithms estimate the target shape by finding the consistent set of probe measurements that is closest to the original noisy data. A direct formulation leads to a quadratic minimization problem with nonlinear constraints. By applying a simple transformation, an alternative algorithm is developed that trades off performance for computational simplicity as it requires quadratic minimization with linear constraints. Both algorithms are successfully applied to a variety of shapes with substantial noise.