The helical coordinate

  For many years it has been true that our most powerful signal-analysis techniques are in one-dimensional space, while our most important applications are in multi-dimensional space. The helical coordinate system makes a giant step towards overcoming this difficulty.

Many geophysical map estimation problems appear to be multidimensional, but actually they are not. To see the tip of the iceberg, consider this example: On a two-dimensional cartesian mesh, the function $\left[ \begin{array} {rr} 1 & 1 \\ 1 & 1 \end{array} \right]$

has the autocorrelation $\left[ \begin{array} {rrr} 1 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 1 \end{array} \right]$.

Likewise, on a one-dimensional cartesian mesh,

the function $\bold b = ( 1,1,0,0, \cdots, 0, 1,1)$

has the autocorrelation $\bold r = ( 1,2,1,0,\cdots,0,2,4,2,0,\cdots,1,2,1)$.

Observe the numbers in the one-dimensional world are identical with the numbers in the two-dimensional world. This correspondence is no accident.