Smooth 1D Colors #12

fractal 25

Here is a small zoom. The accumulator colors method is again provides smooth colors. A variation is used to add some interest to the background, without distracting from the main fractal shapes.

The image center is -1.055+0.261i. It is the period 2/7 bulb off the main 1/2 bulb. After looking at a few pictures like this you soon learn how to read these fraction numbers off the image. Notice that there are seven branches radiating from the center, and that the longest is the second clockwise from the root. If you zoom out to the full view, you see the black area on the bottom is part of the main 1/2 bulb (nose or head).

In the top left corner, notice how one stalk reaches a pinch point the seems to grow again. This is true of all of the branches in the picture. It is almost a degenerate case, but applying the same fraction counting scheme as for the 2/7 branch point, we get 1/2. That tells us that all of these are growing off the main 1/2 bulb.

Smooth 1D Colors #11

Smooth Color Fractal 24

Today’s image uses the accumulator colors method with a variation in the accumulation function.

The log-log smooth coloring method is designed to be as close to escape count coloring as possible. There is very little room to get creative. The accumulator colors method abandons the idea that the integer portion of the smooth color value should match the escape count. Since something less than one is added to the accumulator at each step, the accumulator value is almost certainly less than the iteration escape count. This is not a big deal. In fact after applying linear or nonlinear re-scaling, even escape count coloring does not coincide with the escape count.

The only requirement for smooth coloring with the accumulator colors method is that the amount added at each step must be zero on the escape boundary. That provides a lot of freedom for variations.

Notice how a halo like effect is generated around the fractal. For the fractal purists out there, there is no mathematical significance to this variation. It is done purely for artistic effect.

Smooth 1D Colors #10

fractal 21

Here is another method to generate smooth colors.

I first learned about the log-log smooth coloring method  Continuous_(smooth)_coloring in the mid-1990s on the alt.fractals or was it sci.fractals news group. I was afraid of this method because the log function is slow. Actually, log-log is very efficient. But my fear of taking a performance hit was enough that I continued to use a different method that I call accumulator colors. My method turns out to be slower, but it has other advantages.

I assume you have some basic knowledge about generating fractal images, so I will jump right into the middle

Suppose z_0 is the start point, z_i is the orbit, and R is the escape radius. Color banding, happens when z_i gets close to R. The fractal function is continuous, but a small change in the starting point, z_0 can result in z_i landing on the “other side” of R, and thus getting assigned a different color.

Think of escape counting as adding 1 to an accumulator in the inner loop. The accumulator, acc, is set to 0 at the start of the iteration, and inside the inner loop we have

if( |z| < R ) { acc = acc + 1 }

This should make it obvious how a small change in |z| near R, results in a big change +1 in the result.

Now use this instead:

if( |z| < R ) { acc = acc + 1 - |z| / R }

The added value is between 0 and 1, and it is close to 0 when |z| is close to R. The resulting color is continuous.

The derivative of the function is not continuous. Often the result has a different kind of banding, where the rate of color change varies between bands. If the value is squared then we get a continuous derivative:

if( |z| < R ) { acc = acc + (1 - |z| / R)^2 }

https://fractalforums.org/fractal-mathematics-and-new-theories/28/smooth-1d-coloring/2753 has all the math details.

And https://fractalforums.org/image-threads/25/smooth-1d-coloring/2755 has many example images

This method has a rich set of variations.

Today’s image required some parameter twiddling to get a color range similar to the first log-log smooth color image.

Smooth 1D Colors #9

fractal 20

Here is another example of a low zoom fractal using Continuous_(smooth)_coloring. Wikipedia gives this method a name that is too generic. There are other smooth coloring methods. I am going to call this method the log-log color method. (The reason is obvious if you check the link.)

In earlier posts I used linear and nonlinear rescaling to tweak the colors on deep zoom fractals. Now, after applying the smooth color algorithm, palette rescaling can be used to tweak colors on low zoom fractals.

I am not going to demonstrate this continuous color method on a deep zoom fractal. Since this method is carefully crafted to match the escape count, up to a fractional amount there is no appreciable difference in the resulting images between the two methods.

I should also mention that the log-log method only works for degree-2 polynomials like z^2+c. It can be generalized to other polynomials, but often people overlook that. It does not generalize to non-polynomial formulas.

Tomorrow I will present a method that works everywhere.

Smooth 1D Colors #8

fractal 19

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.