Geometry of 3D Environments and Sum of Squares Polynomials
This work addresses spatial reasoning challenges for robotics and computer vision applications, presenting an incremental improvement through algebraic optimization techniques.
The paper tackles spatial reasoning problems in 3D environments, such as containing point clouds and computing distances between sets, by using sum of squares optimization to reduce these tasks to small semidefinite programs, with numerical experiments demonstrating feasibility in realistic scenarios.
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an obstacle) with convex or nearly-convex basic semialgebraic sets, computation of Euclidean distances between two such sets, separation of two convex basic semalgebraic sets that overlap, and tight containment of the union of several basic semialgebraic sets with a single convex one. We use algebraic techniques from sum of squares optimization that reduce all these tasks to semidefinite programs of small size and present numerical experiments in realistic scenarios.