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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
| |
(15) |
The problem of finding function f(x) reduces to the problem of
finding the corresponding set of basis coefficients ck. We can
obtain the solution to the least-squares optimization by
differentiating the quadratic objective function (15) with
respect to the basis coefficients ck. This leads to the system of
linear equations
| |
(16) |
where
| |
(17) |
Equation (16) is clearly a discrete convolution of the
spline coefficients ck with the filter dj defined in
equation (17). To transform the system (16) to a
regularization condition of the form
| |
(18) |
we need to treat the digital filter dj 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 (18) replaces equation (13)
in the inverse interpolation problem setting.
We have, thus, found a constructive way of creating B-spline
regularization operators from continuous differential equations.
Next: Test example
Up: B-spline regularization
Previous: B-spline regularization
Stanford Exploration Project
12/30/2000