Examples of simple 2-D recursive filters

Figures 3 and 4 contain this 2-D filter  

$$\left[ \begin{array} {cc} 0 & -1/4 \\ 1 & -1/4 \\ -1/4 & -1/4 \end{array} \right]$$ (3)

Let us experiment using this 2-D filter as a recursive filter. In Figure 3 the input is shown on the left. This input contains two copies of the filter (3) near the top of the frame and some impulses near the bottom boundary. The second frame in Figure 3 is the result of deconvolution by the filter (3). Notice that deconvolution turns the filter itself into an impulse, while it turns the impulses into comet-like images. The use of a helix is evident by the comet images wrapping around the vertical axis.

 

wrap90
Figure 3 Illustration of 2-D deconvolution. Left is the input. Right is after deconvolution with the filter (3)

In Figure 4, many inputs are tested. Starting from the left are a low-pass blob, a Ricker wavelet, the filter (3) itself, and a couple impulses, one near the bottom boundary. The second frame shows deconvolution by the filter (3). The third frame compounds the second frame with an adjoint (reverse time) deconvolution. (Instead of blowing plumes to the right, it blows them to the left.) The fourth frame convolves the result with the original filter and its adjoint; and we see we are back where we started. No errors, no evidence remains of any of the boundaries where we have wrapped and truncated!

 

pdadj90
Figure 4 Recursive filtering backwards (leftward on the space axis) is done by the adjoint of 2-D deconvolution. Here we see that 2-D deconvolution compounded with its adjoint is exactly inverted by 2-D convolution and its adjoint.

In seismology we often have occasion to steer summation along beams. Such an impulse response is shown in Figure 5. Finally, I have long had an interest in filters that would destroy plane waves. The inverse of such a filter creates plane waves. A filter that creates two plane waves is illustrated in figure 6.

 

dip90
Figure 5 A simple low-order 2-D filter whose inverse times its inverse adjoint, is approximately a dipping seismic arrival.

 

waves90 Figure 6 A simple low-order 2-D filter whose inverse contains plane waves of two different dips. One of them is spatially aliased.