Today’s image uses a modification of the escape count to produce a continuous value that is used to set the color.

Recall that escape counting produces integer values, which creates color bands in the image.

So far, we have not done anything special for color smoothing. Even with the discrete valued iteration count color method, the color bands naturally disappear in a deep zoom. We re-scale the color palette, using both linear and non-linear methods, to get the optimal amount of color entropy. At some point, the color distance between color steps in the fractal calculation is less than the displayable color steps in the computer graphics hardware.

But re-scaling the palette does not work for no-zoom or low-zoom images. There are simply not enough color values. If scaling used to make each step small, then there is not enough color range, the image appears as a single color.

The first idea is to somehow make the escape count into a real number. If the leading edge of one band is n, and the next is n+1, then fill in the area in-between with real numbers between n and n+1. Once an orbit crosses the escape threshold, try to measure by how much. Then turn that over-shoot into a fraction between 0 and 1 and add it to the escape count. Personally, I tried several things, and I got some to work. My solutions were hacky and ugly. Someone else came up is this elegant answer: Continuous_(smooth)_coloring (Unfortunately the Wikipedia article does not say who developed this method. He or she certainly deserves some praise.)

I am not going to reproduce the math here. Wikipedia does a better job of that. Today’s picture shows that this method effectively removes the banding on low-zoom, low escape count, fractal images.