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Spline regularization

In many cases, the regularization (styling) condition originates in a continuous differential operator. For example, one can think of the gradient or Laplacian operator for regularizing smooth functions (Fomel, 2000b), plane-wave destructor for regularizing local plane waves (Fomel, 2000a), or the offset continuation equation for regularizing seismic reflection data (Fomel, 2000c).

Let us denote the continuous regularization operator by $D$. Regularization implies seeking a function $f(x)$ such that the least-squares norm of $D\left[f(x)\right]$ is minimum. Using the usual expression for the least-squares norm of continuous functions and substituting the basis decomposition (8), we obtain the expression

\left\Vert D\left[f(x)\right]\right\Vert =
\int \left(D...
...(\sum_{k \in K} c_k D\left[ \beta (x-k)\right]\right)^2\,dx\;.
\end{displaymath} (22)

The problem of finding function $f(x)$ reduces to the problem of finding the corresponding set of basis coefficients $c_k$. We can obtain the solution to the least-squares optimization by differentiating the quadratic objective function (22) with respect to the basis coefficients $c_k$. This leads to the system of linear equations
\sum_{k \in K} c_k \int D \left[\beta (x-k)\right]
D\left[\beta (x-j)\right] \,dx =
\sum_{k \in K} c_k d_{j-k} = 0\;,
\end{displaymath} (23)

d_j = \int D\left[\beta (x)\right] D\left[\beta (x-j)\right]\,dx\;.
\end{displaymath} (24)

Equation (23) is clearly a discrete convolution of the spline coefficients $c_k$ with the filter $d_j$ defined in equation (24). To transform the system (23) to a regularization condition of the form
\mathbf{D c} \approx \mathbf{0}\;,
\end{displaymath} (25)

we need to treat the digital filter $d_j$ as an autocorrelation and find its minimum-phase factor. Equation (25) replaces equation (20) in the inverse interpolation problem setting.

We have found a constructive way of creating B-spline regularization operators from continuous differential equations.

A simple regularization example is shown in Figure 28. The continuous operator $D$ in this case comes from the theoretical plane-wave differential equation. I constructed the auto-correlation filter $d_j$ according to formula (24) and factorized it with the efficient Wilson-Burg method on a helix (Sava et al., 1998). The figure shows three plane waves constructed from three distant spikes by applying an inverse recursive filtering with two different plane-wave regularizers. The left plot corresponds to using first-order B-splines (equivalent to linear interpolation). This type of regularizer is identical to Clapp's steering filters (Clapp et al., 1997) and suffers from numerical dispersion effects. The right plot was obtained with third-order splines. Most of the dispersion is suppressed by using a more accurate interpolation.

Figure 28.
B-spline regularization. Three plane waves constructed by 2-D recursive filtering with the B-spline plane-wave regularizer. Left: using first-order B-splines (linear interpolation). Right: using third-order B-splines.
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