---
slug: 2024/11/playing-with-fire-log-density
title: "Playing with fire: Tone mapping and color"
date: 2024-11-15 14:00:00
authors: [bspeice]
tags: []
---

So far, our `plot()` function has been fairly simple; map a fractal flame coordinate to a specific pixel,
and color in that pixel. This works well for simple function systems (like Sierpinski's Gasket),
but more complex systems (like the reference parameters) produce grainy images.

In this post, we'll refine the image quality and add color to really make things shine.

<!-- truncate -->

## Image histograms

:::note
This post covers sections 4 and 5 of the Fractal Flame Algorithm paper
:::

One problem with the existing chaos game is that we waste work
by treating pixels as a binary "on" (opaque) or "off" (transparent).
If the chaos game encounters the same location twice, nothing actually changes.

To demonstrate how much work is wasted, we'll render the reference image again.
However, we'll also count each time the chaos game encounters a pixel.
This gives us a kind of image "histogram":

import chaosGameHistogramSource from "!!raw-loader!./chaosGameHistogram"

<CodeBlock language="typescript">{chaosGameHistogramSource}</CodeBlock>

When the chaos game finishes, we find the pixel encountered most frequently.
Finally, we "paint" the image by setting each pixel's alpha value (transparency)
to the ratio of times encountered divided by the maximum:

import CodeBlock from "@theme/CodeBlock";

import paintLinearSource from "!!raw-loader!./paintLinear"

<CodeBlock language="typescript">{paintLinearSource}</CodeBlock>

import {SquareCanvas} from "../src/Canvas";
import FlameHistogram from "./FlameHistogram";
import {paintLinear} from "./paintLinear";

<SquareCanvas><FlameHistogram paint={paintLinear}/></SquareCanvas>

## Tone mapping

While using a histogram reduces the "graininess" of the image, it also leads to some parts vanishing entirely.
In the reference parameters, the outer circle is preserved, but the interior appears to be missing!

To fix this, we'll introduce the second major innovation of the fractal flame algorithm: [tone mapping](https://en.wikipedia.org/wiki/Tone_mapping).
This is a technique used in computer graphics to compensate for differences in how
computers represent brightness, and how people see brightness.

As a concrete example, high dynamic range (HDR) photography uses this technique to capture
nice images of scenes with a wide range of brightnesses. To take a picture of something dark,
you need a long exposure time. However, long exposures lead to "hot spots" (sections that are pure white).
By taking multiple pictures with different exposure times, we can combine them to create
a final image where everything is visible.

In fractal flames, this "tone map" is accomplished by scaling brightness according to the _logarithm_
of how many times we encounter a pixel. This way, "cold spots" (pixels the chaos game visits infrequently)
will still be visible, and "hot spots" (pixels the chaos game visits frequently) won't wash out.

<details>
    <summary>Log-scale vibrancy also explains fractal flames appear to be 3D...</summary>

    As mentioned in the paper:

    > Where one branch of the fractal crosses another, one may appear to occlude the other
    > if their densities are different enough because the lesser density is inconsequential in sum.
    > For example, branches of densities 1000 and 100 might have brightnesses of 30 and 20.
    > Where they cross the density is 1100, whose brightness is 30.4, which is
    > hardly distinguishable from 30.
</details>

import paintLogarithmicSource from "!!raw-loader!./paintLogarithmic"

<CodeBlock language="typescript">{paintLogarithmicSource}</CodeBlock>

import {paintLogarithmic} from './paintLogarithmic'

<SquareCanvas><FlameHistogram paint={paintLogarithmic}/></SquareCanvas>

## Color

Finally, we'll introduce the last innovation of the fractal flame algorithm: color.
By including a third coordinate ($c$) in the chaos game, we can illustrate the transforms
responsible for the image.

### Color coordinate

Color in a fractal flame is continuous on the range $[0, 1]$. This is important for two reasons:

- It helps blend colors together in the final image. Slight changes in the color value lead to
  slight changes in the actual color
- It allows us to swap in new color palettes easily. We're free to choose what actual colors
  each color value represents

We'll give each transform a color value ($c_i$) in the $[0, 1]$ range.
Then, at each step in the chaos game, we'll set the current color
by blending it with the previous color and the current transform:

$$
\begin{align*}
&(x, y) = \text{random point in the bi-unit square} \\
&c = \text{random point from [0,1]} \\
&\text{iterate } \{ \\
&\hspace{1cm} i = \text{random integer from 0 to } n - 1 \\
&\hspace{1cm} (x,y) = F_i(x,y) \\
&\hspace{1cm} (x_f,y_f) = F_{final}(x,y) \\
&\hspace{1cm} c = (c + c_i) / 2 \\
&\hspace{1cm} \text{plot}(x_f,y_f,c_f) \text{ if iterations} > 20 \\
\}
\end{align*}
$$

### Color speed

:::warning
Color speed as a concept isn't introduced in the Fractal Flame Algorithm paper.

It is included here because [`flam3` implements it](https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/variations.c#L2140),
and because it's fun to play with.
:::

Next, we'll add a parameter to each transform that controls how much it affects the current color.
This is known as the "color speed" ($s_i$):

$$
c = c \cdot (1 - s_i) + c_i \cdot s_i
$$

import mixColorSource from "!!raw-loader!./mixColor"

<CodeBlock language="typescript">{mixColorSource}</CodeBlock>

Color speed values work just like transform weights. A value of 1
means we take the transform color and ignore the previous color state.
A value of 0 means we keep the current color state and ignore the
transform color.

### Palette

Now, we need to map the color coordinate to a pixel color. Fractal flames typically use
256 colors (each color has 3 values - red, green, blue) to define a palette.
The color coordinate then becomes an index into the palette.

There's one small complication: the color coordinate is continuous, but the palette
uses discrete colors. How do we handle situations where the color coordinate is
"in between" the colors of our palette?

One way is to use a step function. In the code below, we multiply the color coordinate
by the number of colors in the palette, then truncate that value. This gives us a discrete index:

import colorFromPaletteSource from "!!raw-loader!./colorFromPalette";

<CodeBlock language="typescript">{colorFromPaletteSource}</CodeBlock>

<details>
    <summary>As an alternative...</summary>

    ...you could interpolate between colors in the palette.
    For example, `flam3` uses [linear interpolation](https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/rect.c#L483-L486)
</details>

In the diagram below, each color in our palette is plotted on a small vertical strip.
Putting the strips side by side shows the full palette used by the reference parameters:

import * as params from "../src/params"
import {PaletteBar} from "./FlameColor"

<PaletteBar height="40" palette={params.palette}/>

### Plotting

We're now ready to plot our $(x_f,y_f,c_f)$ coordinates. After translating from color coordinate ($c_f$)
to RGB value, add that to the image histogram:

import chaosGameColorSource from "!!raw-loader!./chaosGameColor"

<CodeBlock language="typescript">{chaosGameColorSource}</CodeBlock>

Finally, painting the image. With tone mapping, logarithms scale the image brightness to match
how it is perceived. With color, we use a similar method, but scale each color channel
by the alpha channel:

import paintColorSource from "!!raw-loader!./paintColor"

<CodeBlock language="typescript">{paintColorSource}</CodeBlock>

And now, at long last, a full-color fractal flame:

import FlameColor from "./FlameColor";

<SquareCanvas><FlameColor/></SquareCanvas>

## Summary

Tone mapping is the second major innovation of the fractal flame algorithm.
By tracking how often the chaos game encounters each pixel, we can adjust
brightness/transparency to reduce the visual "graining" of previous images.

Next, introducing a third coordinate to the chaos game makes color images possible,
the third major innovation of the fractal flame algorithm. Using a continuous
color scale and color palette adds a splash of color to our transforms.

The Fractal Flame Algorithm paper does go on to describe more techniques
not covered here. For example, Image quality can be improved with density estimation
and filtering. New parameters can be generated by "mutating" existing
fractal flames. And fractal flames can even be animated to produce videos!

That said, I think this is a good place to wrap up. We were able to go from
an introduction to the mathematics of fractal systems all the way to
generating full-color images. Fractal flames are a challenging topic,
but it's extremely rewarding to learn more about how they work.