We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of $\mathbb{R}^d$ and a finitely supported measure. More precisely, the source measure is assumed to be supported on a simplex soup, i.e., on a union of simplices of arbitrary dimension between 2 and $d$. As in [F. Aurenhammer, F. Hoffman, and B. Aronov, Algorithmica, 20 (1998), pp. 61--76] we recast this optimal transport problem as the resolution of a nonlinear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram. We prove the convergence with linear speed of a damped Newton's algorithm to solve this nonlinear system. The convergence relies on two conditions: (i) a genericity condition on the point cloud with respect to the simplex soup, and (ii) a (strong) connectedness condition on the support of the source measure defined on the simplex soup. Finally, we apply our algorithm in ${R}^3$ to compute optimal transport plans between a measure supported on a triangulation and a discrete measure. We also detail some applications such as optimal quantization of a probability density over a surface, remeshing, or rigid point set registration on a mesh.

We present in this paper a generic and parameter-free algorithm to efficiently build a wide variety of optical components, such as mirrors or lenses, that satisfy some light energy constraints. In all of our problems, one is given a collimated or point light source and a desired illumination after reflection or refraction and the goal is to design the geometry of a mirror or lens which transports exactly the light emitted by the source onto the target. We first propose a general framework and show that eight different optical component design problems amount to solving a light energy conservation equation that involves the computation of visibility diagrams. We then show that these diagrams all have the same structure and can be obtained by intersecting a 3D Power diagram with a planar or spherical domain. This allows us to propose an efficient and fully generic algorithm capable to solve these eight optical component design problems. The support of the prescribed target illumination can be a set of directions or a set of points located at a finite distance. Our solutions satisfy design constraints such as convexity or concavity. We show the effectiveness of our algorithm on simulated and fabricated examples.