The helical coordinate |

It is nice having the 2-D helix derivative, but we can imagine even nicer 2-D low-cut filters. In 1-D, we designed a filter with an adjustable parameter, a cutoff frequency. In 1-D, we compounded a first derivative (which destroys low frequencies) with a leaky integration (which undoes the derivative at all other frequencies). The analogous filter in 2-D would be , which would first be expressed as a finite difference and then factored as we did the helix derivative.

helgal
Galilee roughened by gradient and
by helical derivative.
Figure 12. |
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We can visualize a plot of the magnitude of the 2-D Fourier transform of the filter equation (13). It is a 2-D function of and and it should resemble . The point of the cone becomes rounded by the filter truncation, so does not reach zero at the origin of the -plane. We can force it to vanish at zero frequency by subtracting .183 from the lead coefficient 1.791. I did not do that subtraction in Figure 12, which explains the whiteness in the middle of the lake. I gave up on playing with both and filter length; and now, merely play with the sum of the filter coefficients.

The helical coordinate |

2015-03-25