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In many cases, the regularization condition originates in a continuous
differential operator. I provide several examples of such differential
operators in Chapters and .

Let us denote the continuous regularization operator by *D*.
Regularization implies seeking a function *f*(*x*) 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 (), we obtain
the expression

| |
(87) |

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 () with
respect to the basis coefficients *c*_{k}. This leads to the system of
linear equations
| |
(88) |

where
| |
(89) |

Equation () is clearly a discrete convolution of the
spline coefficients *c*_{k} with the filter *d*_{j} defined in
equation (). To transform the system () to a
regularization condition of the form
| |
(90) |

we need to treat the digital filter *d*_{j} as an autocorrelation and
find its minimum-phase factor by spectral factorization. The
Wilson-Burg algorithm, described earlier, is an appropriate tool for
the task. Equation () replaces equation ()
in the inverse interpolation problem setting.
We have, thus, found a constructive way of creating B-spline
regularization operators from continuous differential equations.

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Stanford Exploration Project

12/28/2000