Convexity and Solvability for Compactly Supported Radial Basis Functions with Different Shapes

It is known that interpolation with radial basis functions of the same shape can guarantee a non-singular interpolation matrix, whereas little is known when one uses various shapes. In this paper, we prove that a class of compactly supported radial basis functions are convex on a certain region; based on this local convexity and ready local geometrical property of the interpolation points, we construct a sufficient condition which guarantees diagonally dominant interpolation matrices for radial basis functions interpolation with various shapes. The proof is constructive and can be used to design algorithms directly. Real applications from 3D surface reconstruction are used to verify the results.