Homework 4

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## Helical derivative preconditioning

An alternative to the optimization problem (5) is the problem of minimizing under the constraint

 (6)

The model is defined by , and the preconditioning operator is related to the regularization operator according to

 (7)

The autocorrelation of the gradient filter is the Laplacian filter, which can be represented as a five-point polynomial

 (8)

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

 (9)

and factor it into two minimum-phase parts 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 . (c) Division by . (d) Division by .

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

Next: Shaping regularization Up: Spatial interpolation contest Previous: Gradient regularization

2014-10-21