It is nice having the 2-D helix derivative, but we can imagine even nicer 2-D low-cut filters. In one dimension, equation () and () we designed a filters with an adjustable parameter, a cutoff frequency. We don't have such an object in 2-D so I set out to define one. It came out somewhat abstract and complicated, and didn't work very well, but along the way I found a simpler parameter that is very effective in practice. We'll look at it first.
Figure 10 Mammogram (medical X-ray). The cancer is the ``spoked wheel.'' (I apologize for the inability of paper publishing technology to exhibit a clear grey image.) The white circles are metal foil used for navigation. The little halo around a circle exhibits the impulse response of the helix derivative.
First I had a problem preparing Figure . It shows shows the application of the helix derivative to a medical X-ray. The problem was that the original X-ray was all positive values of brightness so there was a massive amount of spatial low frequency present. Obviously an x-derivative or a y-derivative would eliminate the low frequency, but the helix derivative did not. This unpleasant surprise arises because the filter in equation (11) was truncated after a finite number of terms. Adding up the terms actually displayed in equation (11), they sum to .183 whereas theoretically the sum of all the terms should be zero. From the ratio of .183/1.791 we can say that the filter pushes zero frequency amplitude 90% of the way to zero value. When the image contains very much zero frequency amplitude, this is not good enough. Better results could be obtained with more coefficients, and I did use more coefficients, but simply removing the mean saved me from needing a costly number of filter coefficients.
We can visualize a plot of the magnitude of the 2-D Fourier transform of the filter (11). It is a 2-D function of kx and ky and it should resemble .It does look like this even when the filter (11) has been truncated. The point of the cone becomes rounded and the truncated approximation of kr does not reach zero at the origin of the (kx,ky)-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 which explains the whiteness in the middle of the lake.
Now let us return to my more logical but less effective approach. I prepared a half dozen medical X-rays like Figure . The doctor brought her young son to my office one evening to evaluate the results. In a dark room I would show the original X-ray on a big screen and then suddenly switch to the helix derivative. Every time I did this, her son would exclaim ``Wow!'' The doctor was not so easily impressed, however. She was not accustomed to the unfamiliar image. Fundamentally, the helix derivative applied to her data does compress the dynamic range making weaker features more readily discernable. We were sure of this from theory and from various geophysical examples. The subjective problem was her unfamiliarity with our display. I found that I could always spot anomalies more quickly on the filtered display, but then I would feel more comfortable when I would discover those same anomalies also present (though less evident) in the original data. Thinking this through, I decided the doctor would likely have been more impressed had I used a spatial lowpass filter instead of the helix derivative. That would have left the details of her image (above the cutoff frequency) unchanged altering only the low frequencies, thereby allowing me to increase the gain.
In 1-D we easily make a low-cut filter by compounding a first derivative (which destroys low frequencies) with a leaky integration (which undoes the derivative at all other frequencies). We can do likewise with a second derivative. In Z-transform notation, we would use something like (-Z-1 + 2.00 - Z) / (-Z-1 + 2.01 - Z). (The numerical choice of the .01 controls the cutoff frequency.) We could use spectral factorization to break this spectrum into causal and anticausal factors. The analogous filter in 2-D is which could also be factored as we did the helix derivative. I tried it. I ran into the problem that my helix derivative operator had a practical built-in parameter, the number of coefficients, which also behaves like a cutoff frequency. If I were to continue this project, I would use expressions for directly in the Fourier domain where there is only one adjustable parameter, the cutoff frequency k0, and there is no filter length to confuse the issue and puff-up the costs.
A final word about the doctor. As she was about to leave my office she suddenly asked whether I had scratched one of her X-rays. We were looking at the helix derivative and it did seem to show a big scratch. What should have been a line was broken into a string of dots. I apologized in advance and handed her the original film negatives which she proceeded to inspect. ``Oh,'' she said, ``Bad news. There are calcification nodules along the ducts.'' So the scratch was not a scratch, but an important detail that had not been noticed on the original X-ray.
In preparing an illustration for here, I learned one more lesson. The scratch was small, so I enlarged a small portion of the mammogram for display. The very process of selecting a small portion followed by scaling the amplitude between maximum and minimum darkness of printer ink had the effect enhancing the visibility of the scratch on the mammogram itself. Now Figure shows it to be perhaps even clearer than on the helix derivative.
An operator for applying the helix filter is helderiv . helderivhelix-derivative filter