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flame coordinate to a specific pixel,\nand color in that pixel. This works well for simple function systems (like Sierpinski's Gasket),\nbut more complex systems (like the reference parameters) produce grainy images."]}),"\n",(0,t.jsx)(e.p,{children:"In this post, we'll refine the image quality and add color to really make things shine."}),"\n",(0,t.jsx)(e.h2,{id:"image-histograms",children:"Image histograms"}),"\n",(0,t.jsx)(e.admonition,{type:"note",children:(0,t.jsx)(e.p,{children:"This post covers sections 4 and 5 of the Fractal Flame Algorithm paper"})}),"\n",(0,t.jsx)(e.p,{children:'One problem with the current chaos game algorithm is that we waste work\nbecause pixels are either "on" (opaque) or "off" (transparent).\nIf the chaos game encounters the same pixel twice, nothing changes.'}),"\n",(0,t.jsx)(e.p,{children:'To demonstrate how much work is wasted, we\'ll count each time the chaos game\nvisits a pixel while iterating. This gives us a kind of image "histogram":'}),"\n","\n",(0,t.jsx)(i.Z,{language:"typescript",children:'// hidden-start\nimport { randomBiUnit } from "../src/randomBiUnit";\nimport { randomChoice } from "../src/randomChoice";\nimport { Props as ChaosGameFinalProps } from "../2-transforms/chaosGameFinal";\nimport { camera, histIndex } from "../src/camera";\n\nconst quality = 10;\nconst step = 100_000;\n// hidden-end\ntype Props = ChaosGameFinalProps & {\n paint: (\n width: number,\n height: number,\n histogram: number[]\n ) => ImageData;\n}\n\nexport function* chaosGameHistogram(\n {\n width,\n height,\n transforms,\n final,\n paint\n }: Props\n) {\n const pixels = width * height;\n const iterations = quality * pixels;\n\n // highlight-start\n const hist = Array<number>(pixels)\n .fill(0);\n\n const plotHist = (\n x: number,\n y: number\n ) => {\n const [pixelX, pixelY] =\n camera(x, y, width);\n\n if (\n pixelX < 0 ||\n pixelX >= width ||\n pixelY < 0 ||\n pixelY >= height\n )\n return;\n\n const hIndex =\n histIndex(pixelX, pixelY, width, 1);\n\n hist[hIndex] += 1;\n };\n // highlight-end\n\n let [x, y] = [\n randomBiUnit(),\n randomBiUnit()\n ];\n\n for (let i = 0; i < iterations; i++) {\n const [_, transform] =\n randomChoice(transforms);\n [x, y] = transform(x, y);\n const [finalX, finalY] = final(x, y);\n\n if (i > 20) {\n // highlight-start\n plotHist(finalX, finalY);\n // highlight-end\n }\n\n if (i % step === 0)\n yield paint(width, height, hist);\n }\n\n yield paint(width, height, hist);\n}'}),"\n",(0,t.jsx)(e.p,{children:'When the chaos game finishes, we find the pixel encountered most often.\nFinally, we "paint" the image by setting each pixel\'s alpha (transparency) value\nto the ratio of times visited divided by the maximum:'}),"\n","\n","\n",(0,t.jsx)(i.Z,{language:"typescript",children:"export function paintLinear(\n width: number,\n height: number,\n hist: number[]\n) {\n const img =\n new ImageData(width, height);\n\n let hMax = 0;\n for (let value of hist) {\n hMax = Math.max(hMax, value);\n }\n\n for (let i = 0; i < hist.length; i++) {\n const pixelIndex = i * 4;\n\n img.data[pixelIndex] = 0;\n img.data[pixelIndex + 1] = 0;\n img.data[pixelIndex + 2] = 0;\n\n const alpha = hist[i] / hMax * 0xff;\n img.data[pixelIndex + 3] = alpha;\n }\n\n return img;\n}"}),"\n","\n",(0,t.jsx)(r.ke,{children:(0,t.jsx)(p,{paint:x})}),"\n",(0,t.jsx)(e.h2,{id:"tone-mapping",children:"Tone mapping"}),"\n",(0,t.jsx)(e.p,{children:'While using a histogram reduces the "graining," it also leads to some parts vanishing entirely.\nIn the reference parameters, the outer circle is still there, but the interior is gone!'}),"\n",(0,t.jsxs)(e.p,{children:["To fix this, we'll introduce the second major innovation of the fractal flame algorithm: ",(0,t.jsx)(e.a,{href:"https://en.wikipedia.org/wiki/Tone_mapping",children:"tone mapping"}),".\nThis is a technique used in computer graphics to compensate for differences in how\ncomputers represent brightness, and how people actually see brightness."]}),"\n",(0,t.jsx)(e.p,{children:'As a concrete example, high-dynamic-range (HDR) photography uses this technique to capture\nscenes with a wide range of brightnesses. To take a picture of something dark,\nyou need a long exposure time. However, long exposures lead to "hot spots" (sections that are pure white).\nBy taking multiple pictures with different exposure times, we can combine them to create\na final image where everything is visible.'}),"\n",(0,t.jsxs)(e.p,{children:['In fractal flames, this "tone map" is accomplished by scaling brightness according to the ',(0,t.jsx)(e.em,{children:"logarithm"}),'\nof how many times we encounter a pixel. This way, "cold spots" (pixels the chaos game visits infrequently)\nare still visible, and "hot spots" (pixels the chaos game visits frequently) won\'t wash out.']}),"\n",(0,t.jsxs)(n,{children:[(0,t.jsx)("summary",{children:"Log-scale vibrancy also explains fractal flames appear to be 3D..."}),(0,t.jsx)(e.p,{children:"As mentioned in the paper:"}),(0,t.jsxs)(e.blockquote,{children:["\n",(0,t.jsx)(e.p,{children:"Where one branch of the fractal crosses another, one may appear to occlude the other\nif their densities are different enough because the lesser density is inconsequential in sum.\nFor example, branches of densities 1000 and 100 might have brightnesses of 30 and 20.\nWhere they cross the density is 1100, whose brightness is 30.4, which is\nhardly distinguishable from 30."}),"\n"]})]}),"\n","\n",(0,t.jsx)(i.Z,{language:"typescript",children:"export function paintLogarithmic(\n width: number,\n height: number,\n hist: number[]\n) {\n const img =\n new ImageData(width, height);\n\n const histLog = hist.map(Math.log);\n\n let hLogMax = -Infinity;\n for (let value of histLog) {\n hLogMax = Math.max(hLogMax, value);\n }\n\n for (let i = 0; i < hist.length; i++) {\n const pixelIndex = i * 4;\n\n img.data[pixelIndex] = 0; // red\n img.data[pixelIndex + 1] = 0; // green\n img.data[pixelIndex + 2] = 0; // blue\n\n const alpha =\n histLog[i] / hLogMax * 0xff;\n img.data[pixelIndex + 3] = alpha;\n }\n\n return img;\n}"}),"\n","\n",(0,t.jsx)(r.ke,{children:(0,t.jsx)(p,{paint:j})}),"\n",(0,t.jsx)(e.h2,{id:"color",children:"Color"}),"\n",(0,t.jsxs)(e.p,{children:["Now we'll introduce the last innovation of the fractal flame algorithm: color.\nBy including a third coordinate (",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"c"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"c"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",children:"c"})]})})]}),") in the chaos game, we can illustrate the transforms\nresponsible for the image."]}),"\n",(0,t.jsx)(e.h3,{id:"color-coordinate",children:"Color coordinate"}),"\n",(0,t.jsxs)(e.p,{children:["Color in a fractal flame is continuous on the range ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mo,{stretchy:"false",children:"["}),(0,t.jsx)(e.mn,{children:"0"}),(0,t.jsx)(e.mo,{separator:"true",children:","}),(0,t.jsx)(e.mn,{children:"1"}),(0,t.jsx)(e.mo,{stretchy:"false",children:"]"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"[0, 1]"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,t.jsx)(e.span,{className:"mopen",children:"["}),(0,t.jsx)(e.span,{className:"mord",children:"0"}),(0,t.jsx)(e.span,{className:"mpunct",children:","}),(0,t.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,t.jsx)(e.span,{className:"mord",children:"1"}),(0,t.jsx)(e.span,{className:"mclose",children:"]"})]})})]}),". This is important for two reasons:"]}),"\n",(0,t.jsxs)(e.ul,{children:["\n",(0,t.jsx)(e.li,{children:"It helps blend colors together in the final image. Slight changes in the color value lead to\nslight changes in the actual color"}),"\n",(0,t.jsx)(e.li,{children:"It allows us to swap in new color palettes easily. We're free to choose what actual colors\neach value represents"}),"\n"]}),"\n",(0,t.jsxs)(e.p,{children:["We'll give each transform a color value (",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsxs)(e.msub,{children:[(0,t.jsx)(e.mi,{children:"c"}),(0,t.jsx)(e.mi,{children:"i"})]})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"c_i"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,t.jsxs)(e.span,{className:"mord",children:[(0,t.jsx)(e.span,{className:"mord mathnormal",children:"c"}),(0,t.jsx)(e.span,{className:"msupsub",children:(0,t.jsxs)(e.span,{className:"vlist-t 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",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mo,{stretchy:"false",children:"["}),(0,t.jsx)(e.mn,{children:"0"}),(0,t.jsx)(e.mo,{separator:"true",children:","}),(0,t.jsx)(e.mn,{children:"1"}),(0,t.jsx)(e.mo,{stretchy:"false",children:"]"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"[0, 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A value of 1\nmeans we take the transform color and ignore the previous color state.\nA value of 0 means we keep the current color state and ignore the\ntransform color."}),"\n",(0,t.jsx)(e.h3,{id:"palette",children:"Palette"}),"\n",(0,t.jsx)(e.p,{children:"Now, we need to map the color coordinate to a pixel color. Fractal flames typically use\n256 colors (each color has 3 values - red, green, blue) to define a palette.\nThe color coordinate then becomes an index into the palette."}),"\n",(0,t.jsx)(e.p,{children:'There\'s one small complication: the color coordinate is continuous, but the palette\nuses discrete colors. How do we handle situations where the color coordinate is\n"in between" the colors of our palette?'}),"\n",(0,t.jsx)(e.p,{children:"One way to handle this is a step function. In the code below, we multiply the color coordinate\nby the number of colors in the palette, then truncate that value. This gives us a discrete index:"}),"\n","\n",(0,t.jsx)(i.Z,{language:"typescript",children:"export function colorFromPalette(\n palette: number[],\n colorIndex: number\n): [number, number, number] {\n const numColors = palette.length / 3;\n const paletteIndex = Math.floor(\n colorIndex * (numColors)\n ) * 3;\n return [\n palette[paletteIndex], // red\n palette[paletteIndex + 1], // green\n palette[paletteIndex + 2] // blue\n ];\n}"}),"\n",(0,t.jsxs)(n,{children:[(0,t.jsx)("summary",{children:"As an alternative..."}),(0,t.jsxs)(e.p,{children:["...you could interpolate between colors in the palette.\nFor example, ",(0,t.jsx)(e.code,{children:"flam3"})," uses ",(0,t.jsx)(e.a,{href:"https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/rect.c#L483-L486",children:"linear interpolation"})]})]}),"\n",(0,t.jsx)(e.p,{children:"In the diagram below, each color in the palette is plotted on a small vertical strip.\nPutting the strips side by side shows the full palette 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This time, we'll use a histogram\nfor each color channel (red, green, blue, alpha). 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With tone mapping, logarithms scale the image brightness to match\nhow it is perceived. With color, we use a similar method, but scale each color channel\nby the alpha channel:"}),"\n","\n",(0,t.jsx)(i.Z,{language:"typescript",children:"export function paintColor(\n width: number,\n height: number,\n red: number[],\n green: number[],\n blue: number[],\n alpha: number[]\n): ImageData {\n const pixels = width * height;\n const img =\n new ImageData(width, height);\n\n for (let i = 0; i < pixels; i++) {\n const scale =\n Math.log10(alpha[i]) /\n (alpha[i] * 1.5);\n\n const pixelIndex = i * 4;\n\n const rVal = red[i] * scale * 0xff;\n img.data[pixelIndex] = rVal;\n\n const gVal = green[i] * scale * 0xff;\n img.data[pixelIndex + 1] = gVal;\n\n const bVal = blue[i] * scale * 0xff;\n img.data[pixelIndex + 2] = bVal;\n\n const aVal = alpha[i] * scale * 0xff;\n img.data[pixelIndex + 3] = aVal;\n }\n\n return img;\n}"}),"\n",(0,t.jsx)(e.p,{children:"And now, at long last, a full-color fractal flame:"}),"\n","\n",(0,t.jsx)(r.ke,{children:(0,t.jsx)(b,{})}),"\n",(0,t.jsx)(e.h2,{id:"summary",children:"Summary"}),"\n",(0,t.jsx)(e.p,{children:'Tone mapping is the second major innovation of the fractal flame algorithm.\nBy tracking how often the chaos game encounters each pixel, we can adjust\nbrightness/transparency to reduce the visual "graining" of previous images.'}),"\n",(0,t.jsx)(e.p,{children:"Next, introducing a third coordinate to the chaos game makes color images possible,\nthe third major innovation of the fractal flame algorithm. Using a continuous\ncolor scale and color palette adds a splash of excitement to the image."}),"\n",(0,t.jsx)(e.p,{children:'The Fractal Flame Algorithm paper goes on to describe more techniques\nnot covered here. For example, image quality can be improved with density estimation\nand filtering. New parameters can be generated by "mutating" existing\nfractal flames. And fractal flames can even be animated to produce videos!'}),"\n",(0,t.jsx)(e.p,{children:"That said, I think this is a good place to wrap up. We went from\nan introduction to the mathematics of fractal systems all the way to\ngenerating full-color images. 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