Spline regularization

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 $D\left[f(x)\right]$ is minimum. Using the usual expression for the least-squares norm of continuous functions and substituting the basis decomposition (), we obtain the expression  

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

$$\sum_{k \in K} c_k \int D \left[\beta (x-k)\right] D\left[\beta (x-j)\right] \,dx = \sum_{k \in K} c_k d_{j-k} = 0\;,$$ (88)

where  

$$d_j = \int D\left[\beta (x)\right] D\left[\beta (x-j)\right]\,dx\;.$$ (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  

$$\bold{D}_c \bold{c} \approx \bold{0}\;,$$ (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.