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 shows it in three dimensions. The top plane in Figure 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
A 3-D causal filter at the starting end of a 3-D helix.
Figure 13 |

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 , 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 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 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.

4/27/2004