B-spline regularization

As demonstrated in Chapter , B-splines provide an exceptionally accurate method of forward interpolation. In this section, I discuss how this choice of the forward operator affects the regularization part of the problem. In the case of B-spline interpolation, the forward operator $\bold{L}$ is a cascade of two operators: recursive deconvolution $\bold{B}^{-1}$, which converts the model vector $\bold{m}$ to the vector of spline coefficients $\bold{c}$, and a spline basis construction operator $\bold{F}$.System (-) transforms to

$$\begin{eqnarray} \bold{F B^{-1} m} & \approx & \bold{d}\;; \\ \epsilon \bold{D m} & \approx & \bold{0}\;.\end{eqnarray}$$ (10) (11)

We can rewrite (10-11) in the form that involves only spline coefficients:

$$\begin{eqnarray} \bold{F c} & \approx & \bold{d}\;; \\ \epsilon \bold{D B c} & \approx & \bold{0}\;. \end{eqnarray}$$ (12) (13)

After we find a solution of system (12-13), the model $\bold{m}$ will be reconstructed by the simple convolution  

$$\bold{m = B c}\;.$$ (14)

This approach is clearly just another version of model preconditioning.

The inconvenient part of system (12-13) is the complex regularization operator $\bold{D B}$. Is it possible to avoid the cascade of $\bold{B}$ and $\bold{D}$ and to construct a regularization operator directly applicable to the spline coefficients $\bold{c}$? The answer is positive. In the following subsection, I develop a method for constructing spline regularization operators from differential equations.