Inverse B-spline interpolation |
Let us denote the continuous regularization operator by .
Regularization implies seeking a function such that the
least-squares norm of
is minimum. Using the usual
expression for the least-squares norm of continuous functions and
substituting the basis decomposition (8), we obtain
the expression
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 in this case comes from the theoretical plane-wave differential equation. I constructed the auto-correlation filter 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.
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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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Inverse B-spline interpolation |