Homework 4

Helical derivative preconditioning

An alternative to the optimization problem (5) is the problem of minimizing $\vert\mathbf{x}\vert^2+\vert\mathbf{r}\vert^2$ under the constraint

$$\mathbf{F} \mathbf{P} \mathbf{x} + \epsilon \mathbf{r} = \mathbf{d}\;.$$ (6)

The model $\mathbf{m}$ is defined by $\mathbf{m}=\mathbf{P} \mathbf{x}$ , and the preconditioning operator $\mathbf{P}$ is related to the regularization operator $\mathbf{R}$ according to

$$\mathbf{P} \mathbf{P}^T = \left(\mathbf{R}^T \mathbf{R}\right)^{-1}\;.$$ (7)

The autocorrelation of the gradient filter $\mathbf{R}^T \mathbf{R}$ is the Laplacian filter, which can be represented as a five-point polynomial

$$L_2(Z_1,Z_2) = 4 - Z_1 - Z_1^{-1} - Z_2 - Z_2^{-1}\;.$$ (8)

To invert the Laplacian filter, we can put on a helix, where it takes the form

$$L_H(Z) = 4 - Z - Z^{-1} - Z^{N_1} - Z^{-N_1}\;,$$ (9)

and factor it into two minimum-phase parts $L_H(Z) = D(Z) D(1/Z)$ using the Wilson-Burg algorithm (, ). The factorization is tested in Figure 9, where the impulse response of the Laplacian filter gets inverted by recursive filtering (polynomial division) on a helix.

laplace
Figure 9. Impulse response of the five-point Laplacian filter (a) gets inverted by recursive filtering (polynomial division) on a helix. (b) Division by $D(Z)$ . (c) Division by $D(1/Z)$ . (d) Division by $D(Z) D(1/Z)$ .

inter1
Figure 10. Rainfall data interpolated using preconditioning with the inverse helical filter.

Figure 10 shows the interpolation result using conjugate-gradient optimization with equation (6) after 10 and 100 iterations. The corresponding correlation analysis is shown in Figure 11.

inter1-100-pred Figure 11. Correlation between interpolated and true data values for preconditioning with 100 iterations.

Homework 4