Examples of simple 2-d recursive filters

Figures 7 and 8 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 7 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 7 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.

 

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Figure 7 Illustration of 2-D deconvolution. Left is the input $\bold q$. Right is after deconvolution $\bold q/\bold F$ with the filter (3).

 

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Figure 8 Left is $(\bold q/\bold F)/\bold F'$ and right is $((\bold q/\bold F)/\bold F')\bold F'\bold F$.

Using the result of Figure 7 as an input, the first frame in Figure 8 results from backwards recursion, filtering backwards along the helix, the adjoint operator $\bold F'$.The adjoint blows energy leftward. Filtering forward and backwards has implemented a symmetrical smoothing operator $(1/\bold F)/\bold F'$.The last frame in Figure 8 shows that the two deconvolution smoothers can be inverted by convolution, convolving once with filter (3) and once with its adjoint (reverse on both axes). We see we are back where we started. No errors, no evidence remains of any of the boundaries where we have wrapped and truncated. Using the helix, we find that one-dimensional polynomial division is a rapidly invertible multidimensional transformation, one with many adjustable parameters.