Homework 4 |

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

The model is defined by , and the

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

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

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
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
.
Figure 9. |
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inter1
Rainfall data
interpolated using preconditioning with the inverse helical filter.
Figure 10. |
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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
Correlation between
interpolated and true data values for preconditioning with 100
iterations.
Figure 11. | |
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Homework 4 |

2014-10-21