Inverse B-spline interpolation

Inverse Interpolation and Data Regularization

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

$$\mathbf{L m} \approx \mathbf{d}\;;$$ (15)

$$\epsilon \mathbf{A m} \approx \mathbf{0}\;,$$ (16)

where $\mathbf{d}$ is a vector of known data $f(x_i)$ at irregular locations $x_i$, $\mathbf{m}$ is a vector of unknown function values $f(n)$ at a regular grid $n$, $\mathbf{L}$ is a linear interpolation operator of the general form (1), $\mathbf{A}$ is an appropriate regularization (model styling) operator, and $\epsilon$ is a scaling parameter. In the case of B-spline interpolation, the forward interpolation operator $\mathbf{L}$ becomes a cascade of two operators: recursive deconvolution $\mathbf{B}^{-1}$, which converts the model vector $\mathbf{m}$ to the vector of spline coefficients $\mathbf{c}$, and a spline basis construction operator $\mathbf{F}$. System (15-16) transforms to

$$\mathbf{F B^{-1} m} \approx \mathbf{d}\;;$$ (17)

$$\epsilon \mathbf{A m} \approx \mathbf{0}\;.$$ (18)

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

$$\mathbf{W c} \approx \mathbf{d}\;;$$ (19)

$$\epsilon \mathbf{A B c} \approx \mathbf{0}\;.$$ (20)

After we find a solution of system (19-20), the model $\mathbf{m}$ will be reconstructed by the simple convolution

$$\mathbf{m = B c}\;.$$ (21)

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

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

Inverse B-spline interpolation