Convex set reconstruction using prior shape information

In this paper we present several algorithms for reconstructing 2D convex sets given support line measurements for which the angles are known precisely but the lateral displacements are noisy. We extend the algorithms given in a previous paper by explicitly incorporating prior information about the shape of the objects to be reconstructed. We develop the Scale-Invariant algorithms, which incorporate prior shape information by defining prior probabilities on support vectors, where a support vector is a vector formed from the lateral displacements of a particular set of support lines of an object. We also develop the Ellipse-Based algorithms, which either assume or jointly estimate the parameters of an ellipse, given prior distributions that favor ellipses. In order to relate the support vector prior probability to the expected shape of an object we develop a vector decomposition called the Size/Shape/Shift decomposition, which helps to provide insight into the detailed geometric relationship between support vectors and 2D convex objects. We then use the maximum a posteriori criterion to determine the specific form of the support vector estimator. The computations involve a quadratic programming optimization stage, which is used to determine one component of the decomposition, and either a line search or a conjugate gradient stage, which is used to determine the remaining components. The performance of the algorithms is demonstrated using simulated support line measurements of an ellipse.