Causality in two-dimensions

Time-series analysis is rich with concepts that the helix now allows us to apply to many dimensions. First is the notion of an impulse function. Observe that an impulse function on the 2-D surface of the helical cylinder maps to an impulse function on the 1-D line of the unwound coil. An autocorrelation function that is an impulse corresponds both to a white (constant) spectrum in 1-D and to a white (constant) spectrum in 2-D. An autocorrelation of a typical two-dimensional data field will drop off with two-dimensional distance from the zero lag. On the one-dimensional helix, the autocorrelation gets re-energized when the lag is an integer multiple of the circumference of the helix. A causal filter in one dimension has a curious shape on the two-dimensional helix. I adopt the convention that the zero-lag response of the 1-D filter has the value ``1''. In one dimension, the causal filter has zeros before the ``1'' and various values after it. Supposing that nonzero filter coefficients lie within a short distance (two lags) from the ``1'', we can extract from the helix the 1-D causal filter and view it as a two-dimensional array  

$$\begin{array} {ccccc} \begin{array} {ccc} h & c & 0 \\ p ... ...& {\rm variable} &\quad +\quad& {\rm constrained}\end{array}$$ (6)

where a,b,c,...,u are adjustable coefficients. Thus we conclude that the 2-D analog of a 1-D causal filter has its abrupt beginning along the side of the 2-D filter.