(7) |

(8) |

(9) |

(10) |

Since
the autocorrelation of is
is a *second* derivative,
the operator must be something like a *first* derivative.
As a geophysicist, I found it natural to compare
the operator to by applying them to a local topographic map.
The result shown in
Figure 5
is that enhances drainage patterns whereas
enhances mountain ridges.

Figure 5

Our construction makes have the energy spectrum *k*_{x}^{2}+*k*_{y}^{2},
so the magnitude of its Fourier transform is .This rotationally invariant function in the Fourier domain
contrasts sharply with the nonrotationally invariant
function shape in (*x*,*y*)-space.
The difference must arise from the phase spectrum.
The factorization (10)
is nonunique in that causality
associated with the helix mapping
can be defined along either *x* or *y* axes;
thus the operator
(10)
can be rotated or reflected.

The operator has
curious similarities and differences
with the familiar gradient and divergence operators.
In two-dimensional physical space,
the gradient maps one field to *two* fields
(north slope and east slope).
The factorization of with the helix
gives us the operator that maps one field to *one* field.
Being a one-to-one transformation
(unlike gradient and divergence)
the operator is potentially invertible
by deconvolution (recursive filtering).

I have chosen the name^{}
``helix derivative''
or ``helical derivative'' for the operator .A telephone pole has a narrow shadow behind it.
The helix integral (middle frame of Figure 6)
and the helix derivative (left frame)
show shadows with an angular bandwidth approaching .Thus, is much less directional than either
or
.

Figure 6

This is where the story all comes together. One-dimensional theory, such as the Kolmogoroff spectral factorization, produces not merely a causal wavelet with the required autocorrelation. It produces a wavelet that is stable in deconvolution. Using in one-dimensional polynomial division, we can solve many formerly difficult problems very rapidly. Consider the Laplace equation with sources (Poisson's equation). Polynomial division and its reverse (adjoint) gives us which means that we have solved in Figure 6 by using polynomial division on a helix. Using the seven coefficients shown, the cost is fourteen multiplications (because we need to run both ways) per mesh point.

7/5/1998