The ability of inverse interpolation to reach the data fitting goal depends on the accuracy of the forward interpolation operator. Forward interpolation is one of the classic problems in numerical analysis and has been studied extensively by generations of theoreticians and practitioners Fomel (1997b). The two simplest and most widely used methods are the nearest neighbor interpolation and linear interpolation. There are several approaches for constructing more accurate (albeit more expensive) linear forward interpolation operators: cubic convolution Keys (1981), local Lagrange, tapered sinc Harlan (1982), etc. Wolberg (1990) presents a detailed review of different conventional approaches.
Spline interpolation, based on representing the interpolated function by smooth piece-wise polynomials, has been in use for a long time de Boor (1978), but only recently Unser et al. (1993a,b) have discovered a way of implementing forward B-spline interpolation with an arbitrary order of accuracy in an efficient signal-processing fashion. The key idea is to implement the B-spline transform with recursive filtering. First, an efficient recursive filtering transforms regularly spaced data into spline coefficients, then the spline coefficients are interpolated onto irregular locations. B-spline interpolants exhibit a superior performance for any given order of accuracy in comparison with other methods of similar efficiency Thévenaz et al. (2000).
In this paper, I study the applicability of B-spline interpolation in the context of the inverse interpolation method. In the first section, I review the forward interpolation problem and confirm the observations of Thévenaz et al. (2000) about the superior performance of B-splines. The second section introduces a constructive method of creating discrete regularization operators from B-splines and helical filtering Claerbout (1998). The method performance is evaluated with a simple numerical test. In conclusion, I summarize the benefits of using B-splines for data regularization.