Factorization examples

The first simple example of helical spectral factorization is shown in Figure 12. 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 12.)

 

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Figure 12 Example of 2-D Wilson-Burg factorization. From left to right: the input filter; its auto-correlation; the factor obtained by the Wilson-Burg method; the result of the deconvolution.

A practically useful example is depicted in Figure 13. The symmetric Laplacian operator is often used in practice for regularizing smooth data (see a more detailed discussion in Chapter ). In order to construct a corresponding recursive preconditioner, I factor the Laplacian auto-correlation (the biharmonic operator) using the Wilson-Burg algorithm. Figure 13 shows the resultant filter. The minimum-phase Laplacian filter has several times more coefficients that 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). As demonstrated by 2-D examples later in this chapter, the gain in convergence from recursive filter preconditioning outweighs the loss of efficiency from the longer filter. Figure 14 shows a construction of the smooth inverse impulse response by application of the $\bold{C} = \bold{P P}^T$operator, where $\bold{P}$ is deconvolution with the minimum-phase Laplacian. The application of $\bold{C}$ is equivalent to a numerical solution of the biharmonic equation, discussed in Chapter .

 

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Figure 13 Creating a minimum-phase Laplacian filter. From left to right: Laplacian filter; its auto-correlation; the factor obtained by the Wilson-Burg method (minimum-phase Laplacian); the result of the deconvolution.

 

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Figure 14 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).