Inverse B-spline interpolation |

In the notation of Claerbout (1999), inverse interpolation amounts to a
least-squares solution of the system

where is a vector of known data at irregular locations , is a vector of unknown function values at a regular grid , is a linear interpolation operator of the general form (1), is an appropriate regularization (model styling) operator, and is a scaling parameter. In the case of B-spline interpolation, the forward interpolation operator becomes a cascade of two operators: recursive deconvolution , which converts the model vector to the vector of spline coefficients , and a spline basis construction operator . System (15-16) transforms to

We can rewrite (17-18) in the form that involves only spline coefficients:

After we find a solution of system (19-20), the model will be reconstructed by the simple convolution

This approach resembles a more general method of model preconditioning (Fomel, 1997a).

The inconvenient part of system (19-20) is the complex regularization operator . Is it possible to avoid the cascade of and and to construct a regularization operator directly applicable to the spline coefficients ? In the following subsection, I develop a method for constructing spline regularization operators from differential equations.

Inverse B-spline interpolation |

2014-02-15