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 $\begin{array} {\vert r\vert r\vert r\vert r\vert} \hline 0 & 0 & 0 &0 \\ \h... ... & 1 &0 \\ \hline 0 & 1 & 1 &0 \\ \hline 0 & 0 & 0 &0 \\ \hline\end{array}$

has the autocorrelation $\begin{array} {\vert r\vert r\vert r\vert} \hline 1 & 2 & 1\\ \hline 2 & 4 & 2\\ \hline 1 & 2 & 1 \\ \hline\end{array}$.

Likewise, on a one-dimensional cartesian mesh,

the function $\mathcal b = \begin{array} {\vert r\vert r\vert r\vert r\vert r\vert r\vert r\... ...rt r\vert r\vert r\vert} \hline 1&1&0&0& \cdots& 0& 1&1 \\ \hline\end{array}$

has the autocorrelation $\mathcal r = \begin{array} {\vert r\vert r\vert r\vert r\vert r\vert r\vert r\... ...rt r\vert} \hline 1&2&1&0&\cdots&0&2&4&2&0&\cdots&1&2&1 \\ \hline\end{array}$.

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