A C̃2 spline quasi-interpolant for fitting 3D data on the sphere and applications

Abstract In this paper, a new local spline quasi-interpolant is constructed for fitting 3D data defined on a sphere-like surface S . After mapping the surface S to a rectangular domain, we use the tensor product of cubic polynomial B-splines and 2 π -periodic uniform algebraic trigonometric B-splines (UAT B-splines) of order four for constructing a new quasi-interpolant T . The use of UAT B-splines allows us to obtain an approximating surface which is almost of class C 2 and the approximation order is O ( h 4 ) . We show that this method is particularly well designed to render 3D closed surfaces and it has been successfully applied to reconstruct human organs such as the lung and left ventricle of the heart.