Shape-preserving interpolation of irregular data by bivariate curvature-based cubic L1 splines in spherical coordinates

• Authors: John E. Lavery
• Venue: Computer Aided Geometric Design
• Year: 2005

We investigate C^1-smooth bivariate curvature-based cubic L"1 interpolating splines in spherical coordinates. The coefficients of these splines are calculated by minimizing an integral involving the L"1 norm of univariate curvature in four directions at each point on the unit sphere. We compare these curvature-based cubic L"1 splines with analogous cubic L"2 interpolating splines calculated by minimizing an integral involving the square of the L"2 norm of univariate curvature in the same four directions at each point. For two sets of irregular data on an equilateral tetrahedron with protuberances on the faces, we compare these two types of curvature-based splines with each other and with cubic L"1 and L"2 splines calculated by minimizing the L"1 norm and the square of the L"2 norm, respectively, of second derivatives. Curvature-based cubic L"1 splines preserve the shape of irregular data well, better than curvature-based cubic L"2 splines and than second-derivative-based cubic L"1 and L"2 splines. Second-derivative-based cubic L"2 splines preserve shape poorly. Variants of curvature-based L"1 and L"2 splines in spherical and general curvilinear coordinate systems are outlined.