Extended Hermite Radial Basis Functions for Sparse Contours Interpolation

In this paper, we present an extended Hermite radial basis functions interpolant for surface reconstruction of sparse contours that allows for shape control with interactive constraints. Similar to the differential operator, the difference operator is used to construct gradient constraint and tangent constraint. Based on the theory of Hermite-Birkhoff interpolation, to construct more flexible geometry constraints, we incorporate the differential operator and difference operator to construct interpolation conditions in the interpolant. We construct some constraint rules to control the local trend of shape interactively. It is useful when the method interpolates sparse data that satisfies all the constraints but exhibits an undesirable trend of shape. For example, the interactive constraints can be used to fix holes or intersections for geometrically valid meshes. Regarding the geometry domain as a signed distance field of implicit function, we implement a constraint-based approach to interpolate the contours using the interpolant. The improved method can flexibly handle both parallel and non-parallel sparse contours. The numerical results of real geological and medical data show the robustness and performance of the extended Hermite radial basis functions interpolant.