Learning polytopes with fixed facet directions
- Katharina Jochemko (KTH Stockholm)
Geometric tomography is concerned with reconstructing shapes from geometric data such as volumes of sections and support function evaluations, a task that arises naturally in a variety of application areas, for example, robotics, computerized tomography and magnetic resonance imaging. In this talk we consider the task of reconstructing polytopes with fixed facet directions from finitely many (possibly noisy) support function evaluations. For fixed simplicial normal fan the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction. We show that under mild assumptions the least-squares estimate converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed. This is joint work with Maria Dostert.