Approximating Functions on a Mesh with Restricted Voronoï Diagrams

We propose a method that computes a piecewise constant approximation of a
function defined on a mesh. The approximation is associated with the cells of a
restricted Voronoï diagram. Our method optimizes an objective function measuring
the quality of the approximation. This objective function depends on the
placement of the samples that define the restricted Voronoï diagram and their
associated function values. We study the continuity of the objective function,
derive the closed-form expression of its derivatives and use them to design a
numerical solution mechanism. The method can be applied to a function that has
discontinuities, and the result aligns the boundaries of the Voronoï cells with
the discontinuities. Some examples are shown, suggesting potential applications
in image vectorization and compact representation of lighting.