The helical coordinate

Multidimensional deconvolution breakthrough

Deconvolution (polynomial division) can undo convolution (polynomial multiplication). A magical property of the helix is that we can consider 1-D convolution to be the same as 2-D convolution. Consequently, a second magical property: We can use 1-D deconvolution to undo convolution, whether that convolution was 1-D or 2-D. Thus, we have discovered how to undo 2-D convolution. We have discovered that 2-D deconvolution on a helix is equivalent to 1-D deconvolution. The helix enables us to do multidimensional deconvolution.

Deconvolution is recursive filtering. Recursive filter outputs cannot be computed in parallel, but must be computed sequentially as in one dimension, namely, in the order that the nut screws on the bolt.

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 a large literature and extensive technology about recursive filtering in one dimension. The helix allows us to apply that technology to two (and more) dimensions. It is a huge technological breakthrough.

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.

The helical coordinate