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Factorization examples

The first simple example of helical spectral factorization is shown in Figure 1. A minimum-phase factor is found by spectral factorization of its autocorrelation. The result is additionally confirmed by applying inverse recursive filtering, which turns the filter into a spike (the rightmost plot in Figure 1.)

autowaves
autowaves
Figure 1.
Example of 2-D Wilson-Burg factorization. Top left: the input filter. Top right: its auto-correlation. Bottom left: the factor obtained by the Wilson-Burg method. Bottom right: the result of deconvolution.
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A practical example is depicted in Figure 2. The symmetric Laplacian operator is often used in practice for regularizing smooth data. In order to construct a corresponding recursive preconditioner, we factor the Laplacian autocorrelation (the biharmonic operator) using the Wilson-Burg algorithm. Figure 2 shows the resultant filter. The minimum-phase Laplacian filter has several times more coefficients than the original Laplacian. Therefore, its application would be more expensive in a convolution application. The real advantage follows from the applicability of the minimum-phase filter for inverse filtering (deconvolution). The gain in convergence from recursive filter preconditioning outweighs the loss of efficiency from the longer filter. Figure 3 shows a construction of the smooth inverse impulse response by application of the $ \mathbf{C} = \mathbf{P
P}^T$ operator, where $ \mathbf{P}$ is deconvolution with the minimum-phase Laplacian. The application of $ \mathbf{C}$ is equivalent to a numerical solution of the biharmonic equation, discussed in the next section.

laplac
laplac
Figure 2.
Creating a minimum-phase Laplacian filter. Top left: Laplacian filter. Top right: its auto-correlation (bi-harmonic filter). Bottom left: factor obtained by the Wilson-Burg method (minimum-phase Laplacian). Bottom right: the result of deconvolution.
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thin42
thin42
Figure 3.
2-D deconvolution with the minimum-phase Laplacian. Left: input. Center: output of deconvolution. Right: output of deconvolution and adjoint deconvolution (equivalent to solving the biharmonic differential equation).
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Next: Application of spectral factorization: Up: Method description Previous: Comparison of Wilson-Burg and

2014-02-15