Multidimensional recursive filters via a helix

Only a small nonstandardized portion of industrial codes in reflection seismology use inverse theory. In an effort to attack the N³, N=10³ exhaustion limitation of textbook inverse theory, while seeking an all-purpose preconditioner, I recently happened upon a discovery of a very fundamental nature with many applications that my students have begun working on. Now we shift gears completely.

This section introduces and expounds a basic principle that promises wide-ranging practical applications: A multiple-dimensional Cartesian space can very often be analyzed as a one-dimensional space. Many geophysical image and volumetric estimation problems appear to be multidimensional, but actually they can be handled much more simply. The idea is to use the helical coordinate system. This paper concentrates on explaining the helix idea and it concludes by pointing towards applications in 3-D migration, velocity estimation, and antialiasing.

To see the tip of the iceberg, consider this example: On a two-dimensional Cartesian mesh, the function $\left[ \begin{array} {rr} 1 & 1 \\ 1 & 1 \end{array} \right]$

has the autocorrelation $\left[ \begin{array} {rrr} 1 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 1 \end{array} \right]$.

Likewise, on a one-dimensional Cartesian mesh,

the function $\bold b = ( 1,1,0,0, \cdots, 0, 1,1)$

has the autocorrelation $\bold r = ( 1,2,1,0,\cdots,0,2,4,2,0,\cdots,1,2,1)$.

The numbers in the one-dimensional world are identical with the numbers in the two-dimensional world! This correspondence is no accident.

The correspondence between 2-D and 1-D is not merely a mapping between the 2-D and 1-D spaces and autocorrelations in those spaces. It is also a correspondence for filtering. Figure 5 shows some two-dimensional shapes that are convolved together. The left panel shows an impulse-response function, the center shows some impulses, and the right shows the superposition of responses.

 

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Figure 5 Two-dimensional convolution (as performed in one dimension).

A surprising, indeed amazing, fact is that Figure 5 was not computed with a two-dimensional convolution program. It was computed with a one-dimensional computer program. It could have been done with anybody's one-dimensional convolution program, either in the time domain or in the Fourier domain. This magical trick is done with the helical coordinate system.

The correspondences above are not limited to two dimensions. They also link multidimensional space to one-dimensional space. Although we can perform multidimensional filtering and spectral operations in one dimension, the gain is only conceptual. The cost is the same.

Breakthroughs arise, however, in concept and in computation when we inspect deconvolution, known to engineers as feedback filtering or recursive filtering, known to numerical analysts as back substitution, and known to physicists as finite differences for partial-differential equations (in Cartesian coordinates). Recursion is a powerful operation but it can be dangerous. With recursion, stability always needs to be considered. Multidimensional recursion would be devoid of power without stability theory and the useful techniques that go with it. Fortunately, the correspondence between one and many dimensions gives us the needed stability theory and techniques.