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 11 shows it in three dimensions. The top plane in Figure 11 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 11 A 3-D causal filter at the starting end of a 3-D helix.

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.