Inverse B - spline interpolation a

B-splines provide an accurate and efficient method for interpolating regularly spaced data. In this paper, I study the applicability of B-spline interpolation in the context of the inverse interpolation method for regularizing irregular data. Numerical tests show that, in comparison with lower-order linear interpolation, B-splines lead to a faster iterative conversion in under-determined problems and a more accurate result in over-determined problems. In addition, they provide a constructive method for creating discrete regularization operators from continuous differential equations. INTRODUCTION The problem of interpolating irregularly sampled data to regular grid (data regularization) can be recast as the inverse process with respect to interpolating regularly sampled data to irregular locations. Claerbout (1999) describes an iterative leastsquares optimization approach to data regularization. The optimization is centered around two goals. The first goal is to minimize the power of the residual difference between the observed and predicted data. The second goal is to style the solution according to some predefined regularization criterion. 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 e-mail: [email protected]