The helical coordinate

THE MULTIDIMENSIONAL HELIX

Till now the helix idea was discussed as if it were merely a two-dimensional concept. Here we explore its multidimensional nature. Our main goal is to do multidimensional convolution with a one-dimensional convolution program. This allows us to do multidimensional deconvolution with a one-dimensional deconvolutional program which is ``magic'', i.e. many novel applications will follow.

We do multidimensional deconvolution with causal (one-sided) one-dimensional filters. Equation (7) shows such a one-sided filter as it appears at the end of a 2-D helix. Figure 13 shows it in three dimensions. The top plane in Figure 13 is the 2-D filter seen in equation (7). The top plane can be visualized as the area around the end of a helix. Above the top plane are zero-valued anticausal filter coefficients.

3dpef Figure 13. A 3-D causal filter at the starting end of a 3-D helix.

It is natural to ask, ``why not put the `1' on a corner of the cube?'' We could do that, but that is not the most general possible form. A special case of Figure 13, stuffing much of the volume with lots of zeros would amount to a `1' on a corner. On the other hand, if we assert the basic form has a `1' on a corner we cannot get Figure 13 as a special case. In a later chapter we'll see that we often need as many coefficients as we can have near the `1'. In Figure 13 we lose only those neighboring coefficients that 1-D causality requires.

Geometrically, the three-dimensional generalization of a helix is like string on a spool, but that analogy does not illuminate our underlying conspiracy, which is to represent multidimensional convolution and deconvolution as one-dimensional.

The helical coordinate