The helical coordinate

HELIX LOW-CUT FILTER

Because the autocorrelation of $\bold H$ is $\bold H\T \ast \bold H = \bold R =-\nabla^2$ is a second derivative, the operator $\bold H$ must be something like a first derivative. As a geophysicist, I found it natural to compare the operator $\frac{\partial}{\partial y}$ with $\bold H$ by applying the helix derivative $\bold H$ to a local topographic map. The result shown in Figure 11 is that $\bold H$ enhances drainage patterns whereas $\frac{\partial}{\partial y}$ enhances mountain ridges.

helocut
Figure 11. Topography, helical derivative, slope south.

The operator $\bold H$ has curious similarities and differences with the familiar gradient and divergence operators. In 2-dimensional physical space, the gradient maps one field to two fields (north slope and east slope). The factorization of $-\nabla^2$ with the helix gives us the operator $\bold H$ that maps one field to one field. Being a one-to-one transformation (unlike gradient and divergence), the operator $\bold H$ is potentially invertible by deconvolution (recursive filtering).

I have chosen the name ``helix derivative'' or ``helical derivative'' for the operator $\bold H$ . A flag pole has a narrow shadow behind it. The helix integral (middle frame of Figure ) and the helix derivative (left frame) show shadows with an angular bandwidth approaching $180^\circ$ .

Our construction makes $\bold H$ have the energy spectrum $k_x^2+k_y^2$ , so the magnitude of the Fourier transform is $\sqrt{k_x^2+k_y^2}$ . It is a cone centered at the origin with there the value zero. By contrast, the components of the ordinary gradient have amplitude responses $\vert k_x\vert$ and $\vert k_y\vert$ that are lines of zero across the $(k_x,k_y)$ -plane.

The rotationally invariant cone in the Fourier domain contrasts sharply with the nonrotationally invariant helix derivative in $(x,y)$ -space. The difference must arise from the phase spectrum. The factorization (13) is nonunique because causality associated with the helix mapping can be defined along either $x$ - or $y$ -axes; thus the operator (13) can be rotated or reflected.

In practice, we often require an isotropic filter. Such a filter is a function of $k_r=\sqrt{k_x^2 + k_y^2}$ . It could be represented as a sum of helix derivatives to integer powers.

If you want to see some tracks on the side of a hill, you want to subtract the hill and see only the tracks. Usually, however, you do not have a very good model for the hill. As an expedient, you could apply a low-cut filter to remove all slowly variable functions of altitude. In Chapter we found the Sea of Galilee in Figure to be too smooth for viewing pleasure, so we made the roughened versions in Figure , a 1-dimensional filter that we could apply over the $x$ -axis or the $y$ -axis. In Fourier space, such a filter has a response function of $k_x$ or a function of $k_y$ . The isotropy of physical space tells us it would be more logical to design a filter that is a function of $k_x^2+k_y^2$ . In Figure 11 we saw that the helix derivative $\bold H$ does a nice job. The Fourier magnitude of its impulse response is $k_r=\sqrt{k_x^2 + k_y^2}$ . There is a little anisotropy connected with phase (which way should we wind the helix, on $x$ or $y$ ?), but it is not nearly so severe as that of either component of the gradient, the two components having wholly different spectra, amplitude $\vert k_x\vert$ or $\vert k_y\vert$ .

The helical coordinate