Sum-of-squares geometry processing

Geometry of 3D Environments and Sum of Squares Polynomials

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 semalg...

The Sum of Two Squares

A 'sum of Squares' Theorem for Visibility Complexes a 'sum of Squares' Theorem for Visibility Complexes

We present a new and simpler method to implement in constant amortized time the ip operation of the so-called 'Greedy Flip Algorithm', an optimal algorithm to compute the visibility graph/complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses only the incidence structure of the visibility complex and only the primitive predicate which states tha...

Controls Applications of Sum of Squares Programming

We consider nonlinear systems with polynomial vector fields and pose two classes of system theoretic problems that may be solved by sum of squares programming. The first is disturbance analysis using three different norms to bound the reachable set. The second is the synthesis of a polynomial state feedback controller to enlarge the provable region of attraction. We also outline a variant of th...

The Sum of Digits Function of Squares

We consider the set of squares n2, n < 2k, and split up the sum of binary digits s(n2) into two parts s[<k](n 2) + s[≥k](n 2), where s[<k](n 2) = s(n2 mod 2k) collects the first k digits and s[≥k](n 2) = s(bn2/2kc) collects the remaining digits. We present very precise results on the distribution on s[<k](n 2) and s[≥k](n 2). For example, we provide asymptotic formulas for the numbers #{n < 2k ...