Interpolation and approximation for data living on manifold surfaces

摘要

Meshed surfaces are ubiquitous in digital geometry processing and computer graphics. The set of attributes associated with each vertex such as the vertex locations, curvature, temperature, pressure or saliency, can be recognized as data living on manifold surfaces. So interpolation and approximation for these data are of general interest. This paper presents two approaches for manifold data interpolation and approximation through the properties of Laplace-Beltrami operator (Laplace operator defined on a manifold surface). The first one is to use Laplace operator minimizing the membrane energy of a scalar function defined on a manifold. The second one is to use bi-Laplace operator minimizing the thin plate energy of a scalar function defined on a manifold. These two approaches can process data living on high genus meshed surfaces. The approach based on Laplace operator is more suitable for manifold data approximation and can be applied manifold data smoothing, while the one based on bi-Laplace operator is more suitable for manifold data interpolation and can be applied image extremal envelope computation. All the application examples demonstrate that our procedures are robust and efficient.