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

Figure 12

A practically useful example is depicted in Figure . 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 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 shows a construction of the smooth inverse impulse response by application of the operator, where is deconvolution with the minimum-phase Laplacian. The application of is equivalent to a numerical solution of the biharmonic equation, discussed in Chapter .

Figure 13

Figure 14

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