In the one-dimensional case, one finite-difference representation of the squared Laplacian is as a centered 5-point filter with coefficients (1,-4,6,-4,1). On the same grid, the Laplacian operator can be approximated to the same order of accuracy with the filter (1/12,-4/3,5/2,-4/3,1/12). Combining the two filters in accordance with equation () and performing a spectral factorization with one of the standard methods Claerbout (1976, 1992), we can obtain a 3-point minimum-phase filter suitable for inverse filtering. Figure shows a family of one-dimensional minimum-phase filters for different values of the parameter .Figure demonstrates the interpolation results obtained with these filters on a simple one-dimensional synthetic. As expected, a small tension value () produces a smooth interpolation, but creates artificial oscillations in the unconstrained regions around sharp changes in the gradient. The value of leads to linear interpolation with no extraneous inflections but with discontinuous derivatives. Intermediate values of allow us to achieve a compromise: a smooth surface with constrained oscillations.

otens
One-dimensional minimum-phase
filters for different values of the tension parameter . The
filters range from the second derivative for to the first
derivative for .Figure 1 |

Figure 2

To design the corresponding filters in two dimensions, I define the finite-difference representation of operator () on a 5-by-5 stencil. The filter coefficients are chosen with the help of the Taylor expansion to match the desired spectrum of the operator around the zero spatial frequency. The matching conditions lead to the following set of coefficients for the squared Laplacian:

-1/60 | 2/5 | 7/30 | 2/5 | -1/60 |

2/5 | -14/15 | -44/15 | -14/15 | 2/5 |

7/30 | -44/15 | 57/5 | -44/15 | 7/30 |

2/5 | -14/15 | -44/15 | -14/15 | 2/5 |

-1/60 | 2/5 | 7/30 | 2/5 | -1/60 |

-1 | 24 | 14 | 24 | -1 |

24 | -56 | -176 | -56 | 24 |

14 | -176 | 684 | -176 | 14 |

24 | -56 | -176 | -56 | 24 |

-1 | 24 | 14 | 24 | -1 |

-1/360 | 2/45 | 2/45 | -1/360 | |

2/45 | -14/45 | -4/5 | -14/45 | 2/45 |

-4/5 | 41/10 | -4/5 | ||

2/45 | -14/45 | -4/5 | -14/45 | 2/45 |

-1/360 | 2/45 | 2/45 | -1/360 |

-1 | 16 | 16 | -1 | |

16 | -112 | -288 | -112 | 16 |

-288 | 1476 | -288 | ||

16 | -112 | -288 | -112 | 16 |

-1 | 16 | 16 | -1 |

Figure 3

Figure 4

Regarding the finite-difference operators as two-dimensional auto-correlations and applying the efficient Wilson-Burg method of spectral factorization described in Chapter , I obtain two-dimensional minimum-phase filters suitable for inverse filtering. The exact filters contain many coefficients, which rapidly decrease in magnitude at a distance from the first coefficient. For reasons of efficiency, it is advisable to restrict the shape of the filter so that it contains only the significant coefficients. Keeping all the coefficients that are 1000 times smaller in magnitude than the leading coefficient creates a 53-point filter for and a 35-point filter for , with intermediate filter lengths for intermediate values of . Keeping only the coefficients that are 200 times smaller that the leading coefficient, we obtain 25- and 16-point filters for respectively and .The restricted filters do not factor the autocorrelation exactly but provide an effective approximation of the exact factors. As outputs of the Wilson-Burg spectral factorization process, they obey the minimum-phase condition.

Figure 5

Figure shows the two-dimensional filters for different
values of and illustrates inverse recursive filtering, which
is the essence of the helix method
Claerbout (1998a,b, 1999). The case of leads
to the filter known as *helix derivative*
Claerbout (1999); Zhao (1999). The filter values are spread mostly in
two columns. The other boundary case () leads to a
three-column filter, which serves as the minimum-phase version of the
Laplacian. This filter has been shown previously in
Figure . As expected from the theory, the inverse
impulse response of this filter is noticeably smoother and wider than
the inverse response of the helix derivative. Filters corresponding
to intermediate values of exhibit intermediate properties.
Theoretically, the inverse impulse response of the filter corresponds
to the Green's function of equation (). The theoretical
Green's function for the case of is

(96) |

(97) |

In the next subsection, I illustrate an application of helical inverse filtering to a two-dimensional interpolation problem.

12/28/2000