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

where

Equation (23) is clearly a discrete convolution of the spline coefficients with the filter defined in equation (24). To transform the system (23) to a regularization condition of the form

we need to treat the digital filter 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 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.

sthree
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.
Figure 28. |
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Inverse B-spline interpolation |

2014-02-15