Parametric Polytope Reconstruction, an Application to Crystal Shape Estimation

In situ imaging techniques are a promising direction for monitoring the distribution of crystal sizes and shapes during a crystallization process. Nevertheless, no tractable method yet exists for estimating complex crystal shapes. In this paper, an in situ imaging setup is presented and a novel algorithm for crystal shape estimation from a pair of images is presented. It is shown that such a shape estimation problem can be turned into parametric polytope reconstruction from projections. Based on results in polyhedral geometry, it is demonstrated that an accurate estimate of the crystal shape can be computed by solving a nonlinear least-squares problem built from samples in images and a prior model of the crystal. Effectiveness of the approach is proven on artificial and real images. Results show that very accurate estimations of crystal shapes can be obtained from well-chosen data points sampled on images.