Finish a first draft

blog/2024-11-15-playing-with-fire/2-transforms/index.mdx

@@ -180,9 +180,23 @@ import FlamePost from "./FlamePost";
 ## Final transforms
 
 Our last step is to introduce a "final transform" ($F_{final}$) that is applied
-regardless of which transform function we're using. It works just like a normal transform
+regardless of which transform the chaos game selects. It works just like a normal transform
 (composition of affine transform, variation blend, and post transform),
-but it doesn't change the chaos game state:
+but it doesn't change the chaos game state.
+
+With that in place, our chaos game algorithm changes slightly:
+
+$$
+\begin{align*}
+&(x, y) = \text{random point in the bi-unit square} \\
+&\text{iterate } \{ \\
+&\hspace{1cm} i = \text{random integer from 0 to } n - 1 \\
+&\hspace{1cm} (x,y) = F_i(x,y) \\
+&\hspace{1cm} (x_f,y_f) = F_{final}(x,y) \\
+&\hspace{1cm} \text{plot}(x_f,y_f) \text{ if iterations} > 20 \\
+\}
+\end{align*}
+$$
 
 import chaosGameFinalSource from "!!raw-loader!./chaosGameFinal"
 
@@ -198,4 +212,4 @@ Variations are the fractal flame algorithm's first major innovation over previou
 Blending variation functions and post/final transforms allows us to generate interesting images.
 
 However, the images themselves are grainy and unappealing. In the next post, we'll clean up
-the quality and add color.
+the image quality and add some color.

blog/2024-11-15-playing-with-fire/3-log-density/FlameColor.tsx

@@ -1,7 +1,7 @@
 import React, {useContext, useEffect, useMemo, useRef, useState} from "react";
 import * as params from "../src/params";
 import {PainterContext} from "../src/Canvas";
-import {colorFromPalette} from "./paintColor";
+import {colorFromPalette} from "./colorFromPalette";
 import {chaosGameColor, Props as ChaosGameColorProps, TransformColor} from "./chaosGameColor";
 
 import styles from "../src/css/styles.module.css";

blog/2024-11-15-playing-with-fire/3-log-density/chaosGameColor.ts

@@ -3,7 +3,9 @@ import {Props as ChaosGameFinalProps} from "../2-transforms/chaosGameFinal";
 import {randomBiUnit} from "../src/randomBiUnit";
 import {randomChoice} from "../src/randomChoice";
 import {camera, histIndex} from "../src/camera";
-import {colorFromPalette, paintColor} from "./paintColor";
+import {colorFromPalette} from "./colorFromPalette";
+import {mixColor} from "./mixColor";
+import {paintColor} from "./paintColor";
 
 const quality = 15;
 const step = 100_000;
@@ -13,51 +15,50 @@ export type TransformColor = {
     colorSpeed: number;
 }
 
-function mixColor(color1: number, color2: number, colorSpeed: number) {
-    return color1 * (1 - colorSpeed) + color2 * colorSpeed;
-}
-
 export type Props = ChaosGameFinalProps & {
     palette: number[];
     colors: TransformColor[];
     finalColor: TransformColor;
 }
 export function* chaosGameColor({width, height, transforms, final, palette, colors, finalColor}: Props) {
-    let currentColor = Math.random();
+    const imgRed = Array<number>(width * height).fill(0);
-    const red = Array<number>(width * height).fill(0);
+    const imgGreen = Array<number>(width * height).fill(0);
-    const green = Array<number>(width * height).fill(0);
+    const imgBlue = Array<number>(width * height).fill(0);
-    const blue = Array<number>(width * height).fill(0);
+    const imgAlpha = Array<number>(width * height).fill(0);
-    const alpha = Array<number>(width * height).fill(0);
 
     let [x, y] = [randomBiUnit(), randomBiUnit()];
+    let c = Math.random();
 
     const iterations = width * height * quality;
     for (let i = 0; i < iterations; i++) {
         const [transformIndex, transform] = randomChoice(transforms);
         [x, y] = transform(x, y);
+
+        // highlight-start
+        const transformColor = colors[transformIndex];
+        c = mixColor(c, transformColor.color, transformColor.colorSpeed);
+        // highlight-end
+
         const [finalX, finalY] = final(x, y);
 
         if (i > 20) {
             const [pixelX, pixelY] = camera(finalX, finalY, width);
             const pixelIndex = histIndex(pixelX, pixelY, width, 1);
 
-            if (pixelIndex < 0 || pixelIndex >= alpha.length)
+            if (pixelIndex < 0 || pixelIndex >= imgAlpha.length)
                 continue;
 
-            const transformColor = colors[transformIndex];
+            const colorFinal = mixColor(c, finalColor.color, finalColor.colorSpeed);
-            currentColor = mixColor(currentColor, transformColor.color, transformColor.colorSpeed);
-
-            const colorFinal = mixColor(currentColor, finalColor.color, finalColor.colorSpeed);
             const [r, g, b] = colorFromPalette(palette, colorFinal);
-            red[pixelIndex] += r;
+            imgRed[pixelIndex] += r;
-            green[pixelIndex] += g;
+            imgGreen[pixelIndex] += g;
-            blue[pixelIndex] += b;
+            imgBlue[pixelIndex] += b;
-            alpha[pixelIndex] += 1;
+            imgAlpha[pixelIndex] += 1;
         }
 
         if (i % step === 0)
-            yield paintColor(width, height, red, green, blue, alpha);
+            yield paintColor(width, height, imgRed, imgGreen, imgBlue, imgAlpha);
     }
 
-    yield paintColor(width, height, red, green, blue, alpha);
+    yield paintColor(width, height, imgRed, imgGreen, imgBlue, imgAlpha);
 }

blog/2024-11-15-playing-with-fire/3-log-density/chaosGameHistogram.ts

@@ -13,7 +13,9 @@ export type Props = ChaosGameFinalProps & {
 export function* chaosGameHistogram({width, height, transforms, final, paint}: Props) {
     let iterations = quality * width * height;
 
+    // highlight-start
     const histogram = Array<number>(width * height).fill(0);
+    // highlight-end
 
     let [x, y] = [randomBiUnit(), randomBiUnit()];
 
@@ -23,6 +25,7 @@ export function* chaosGameHistogram({width, height, transforms, final, paint}: P
         const [finalX, finalY] = final(x, y);
 
         if (i > 20) {
+            // highlight-start
             const [pixelX, pixelY] = camera(finalX, finalY, width);
             const hIndex = histIndex(pixelX, pixelY, width, 1);
 
@@ -31,6 +34,7 @@ export function* chaosGameHistogram({width, height, transforms, final, paint}: P
             }
 
             histogram[hIndex] += 1;
+            // highlight-end
         }
 
         if (i % step === 0)

blog/2024-11-15-playing-with-fire/3-log-density/colorFromPalette.ts

@@ -0,0 +1,4 @@
+export function colorFromPalette(palette: number[], colorIndex: number): [number, number, number] {
+    const paletteIndex = Math.floor(colorIndex * (palette.length / 3)) * 3;
+    return [palette[paletteIndex], palette[paletteIndex + 1], palette[paletteIndex + 2]];
+}

blog/2024-11-15-playing-with-fire/3-log-density/index.mdx

@@ -1,6 +1,6 @@
 ---
 slug: 2024/11/playing-with-fire-log-density
-title: "Playing with fire: Log-density and color"
+title: "Playing with fire: Tone mapping and color"
 date: 2024-11-15 14:00:00
 authors: [bspeice]
 tags: []
@@ -23,11 +23,17 @@ This post covers sections 4 and 5 of the Fractal Flame Algorithm paper
 To start, it's worth demonstrating how much work is actually "wasted"
 when we treat pixels as a binary "on" (opaque) or "off" (transparent).
 
-We'll render the reference image again, but this time, track each time
+We'll render the reference image again, but this time, count each time
-we encounter each pixel during the chaos game. When the chaos game finishes,
+we encounter a pixel during the chaos game. This gives us a kind of "histogram"
-find the pixel we encountered most frequently. Finally, "paint" the image
+of the image:
-by setting each pixel's transparency to ratio of times encountered
+
-divided by the maximum value:
+import chaosGameHistogramSource from "!!raw-loader!./chaosGameHistogram"
+
+<CodeBlock language="typescript">{chaosGameHistogramSource}</CodeBlock>
+
+When the chaos game finishes, find the pixel we encountered most frequently.
+Finally, "paint" the image by setting each pixel's alpha value (transparency)
+to the ratio of times encountered, divided by the maximum value:
 
 import CodeBlock from "@theme/CodeBlock";
 
@@ -39,22 +45,20 @@ import {SquareCanvas} from "../src/Canvas";
 import FlameHistogram from "./FlameHistogram";
 import {paintLinear} from "./paintLinear";
 
-<SquareCanvas><FlameHistogram quality={15} paint={paintLinear}/></SquareCanvas>
+<SquareCanvas><FlameHistogram paint={paintLinear}/></SquareCanvas>
 
-## Log display
+## Tone mapping
 
-While using a histogram to paint the image improves the quality,
+While using a histogram to paint the image improves the quality, it also leads to some parts vanishing entirely.
-it also leads to some parts vanishing entirely.
+In the reference parameters, the outer circle is preserved, but the interior appears to be missing!
-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 color, and how color is perceived by people.
+computers represent color, and how people see color.
 
 As a concrete example, high dynamic range (HDR) photography uses this technique to capture
 nice images of scenes with wide brightness ranges. To take a picture of something dark,
-you need a long exposure time. However, long exposures can lead to images that "wash out" and become pure white.
+you need a long exposure time. However, long exposures can lead to "hot spots" in images that are pure white.
 By taking multiple pictures using different exposure times, we can combine them to create
 a final image where everything is visible.
 
@@ -80,33 +84,134 @@ import paintLogarithmicSource from "!!raw-loader!./paintLogarithmic"
 
 import {paintLogarithmic} from './paintLogarithmic'
 
-<SquareCanvas><FlameHistogram quality={15} paint={paintLogarithmic}/></SquareCanvas>
+<SquareCanvas><FlameHistogram paint={paintLogarithmic}/></SquareCanvas>
 
 ## Color
 
-Finally, we'll spice things up with the last innovation introduced by
+Finally, we'll introduce the last innovation of the fractal flame algorithm: color.
-the fractal flame algorithm: color. By including a color coordinate
+By including a color coordinate ($c$) in the chaos game, we can illustrate the transforms
-in the chaos game, we can illustrate the transforms that are responsible
+responsible for each part of the image.
-for each part of an image.
+
+### Color coordinate
+
+Color in a fractal flame uses a range of $[0, 1]$. This is important for two reasons:
+
+- It helps blend colors together in the final image
+- It allows us to swap in new color palettes easily
+
+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.
+Similarly, a value of 0 means we keep the current color state and ignore the
+transform color.
 
 ### Palette
 
-Our first step is to define a color palette for the image. Fractal flames
+Now, we need to map the color coordinate to a pixel color. Fractal flames typically use
-typically use a palette of 256 colors that transition smoothly
+256 colors (each color has 3 values - red, green, blue) to define a palette.
-from one to another. In the diagram below, each color in our palette is plotted
+Then, the color coordinate becomes an index into the palette.
-on a small strip. Putting the strips side by side shows the palette for our image:
+
+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 also interpolate between colors in the palette.
+    For example: [`flam3` code](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 palette used by our reference image:
 
 import * as params from "../src/params"
 import {PaletteBar} from "./FlameColor"
 
 <PaletteBar height="40" palette={params.palette}/>
 
-### Color coordinate
+### 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 value 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. When using color, we 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 quality={15}/></SquareCanvas>
+<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 goes on to describe more techniques
+not covered here. Image quality can be improved with density estimation
+and filtering. New parameters can be generated by "mutating" existing
+fractal flames. 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.

blog/2024-11-15-playing-with-fire/3-log-density/mixColor.ts

@@ -0,0 +1,3 @@
+export function mixColor(color1: number, color2: number, colorSpeed: number) {
+    return color1 * (1 - colorSpeed) + color2 * colorSpeed;
+}

blog/2024-11-15-playing-with-fire/3-log-density/paintColor.ts

@@ -1,8 +1,6 @@
-export function colorFromPalette(palette: number[], colorIndex: number): [number, number, number] {
+// hidden-start
-    const paletteIndex = Math.floor(colorIndex * (palette.length / 3)) * 3;
+import {colorFromPalette} from "./colorFromPalette";
-    return [palette[paletteIndex], palette[paletteIndex + 1], palette[paletteIndex + 2]];
+// hidden-end
-}
-
 export function paintColor(
     width: number,
     height: number,