Darkened squares depict a 2-D filter shaped like the Laplacian operator. The input data, the filter, and the output data are all on helical meshes all of which could be unrolled into linear strips. A compact 2-D filter like a Laplacian, on a helix is a sparse 1-D filter having long empty gaps.
Since the values output from filtering can be computed in any order, we can slide the filter coil over the data coil in any direction. The order that you produce the outputs is irrelevant. You could compute the results in parallel. We could, however, slide the filter over the data in the screwing order that a nut passes over a bolt. The screw order is the same order that would be used if we were to unwind the coils into one-dimensional strips and convolve them across one another. The same filter coefficients overlay the same data values if the 2-D coils are unwound into 1-D strips. We can do 2-D convolution with a 1-D convolution program.
Convolution creates an output qt from an input pt and a filter with
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Recursive filtering sometimes solves big problems with astonishing speed. It can propagate energy rapidly for long distances. Unfortunately, recursive filtering can also be unstable. The most interesting case, near resonance, is also near instability. There is an old textbook literature e.g. Claerbout or with extensive technology for recursive filtering in one dimension. The helix allows us to apply that technology to two (and more) dimensions. It is a wonderful insight. We could not previously do 2-D deconvolution because we had no stability theory for it. We cannot simply use polynomial division to undo the 2-D Laplacian operator for example, because the output will diverge. Poles and zeros tell us about 1-D stability but we don't have them for 2-D polynomials.
In 3-D we simply append one plane after another (like a 3-D Fortran array). It is easier to code than to explain or visualize a spool or torus wrapped with string, etc. Vector-valued time signals meaning multichannel time series () appear to be related to a scalar function of the space axes, but that relationship is not helpful here.