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

In [1], the authors have approached the sphere-like surfaces using the tensor product of an algebraic cubic spline quasi-interpolant with a 2π-periodic Uniform Algebraic Trigonometric B-splines (UAT B-splines) of order four. In this paper, we improve the results given in [1], by introducing a new quasi-interpolant based on the tensor product of an algebraic cubic spline quasi-interpolant with a periodic cubic spline quasi-interpolant, obtained by the periodization of an algebraic cubic spline quasi-interpolant. Our approach allows us to obtain an approximating surface which is of class C² and with an approximation order O(h⁴). 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 left ventricle of the heart.