28:null 29:["$","title",null,{"children":"Frechet Distance | Dylan Madisetti | Observable"}] 2e:T454,That star, or the largest point on the plot, is the Frechet distance! The Frechet distance is typically explained with a dog-walking analogy (what's the longest the leash needs to be to walk your dog given predetermined paths of you and your dog), but that analogy doesn't fully work to me. I hope this offers a intuitive way of understanding this metric, because the mathematical definition isn't very friendly either: ${tex.block`F(A,B) = \inf_{\alpha, \beta},\max_{t \in [0,1]}, \Bigg \{d \Big ( A(\alpha(t)), B(\beta(t)) \Big ) \Bigg \}`} After all the build up, the metric seems almost naive and reductive to me (there's so much information the metric ignores), but it's still useful! My favourite use of this metric is analyzing character strokes in teaching Hanzi characters: https://github.com/chanind/hanzi-writer A multidimensional version of the Frechet distance is used to compare distributions (assuming that they are Gaussian in nature), and is commonly used to evaluate the performance of Deep Learning generative models.2f:T1015,dijkstra = { //helper class for PriorityQueue class Node { constructor(val, priority) { this.val = val; this.priority = priority; } } class PriorityQueue { constructor() { this.values = []; } enqueue(val, priority) { let newNode = new Node(val, priority); this.values.push(newNode); this.bubbleUp(); } bubbleUp() { let idx = this.values.length - 1; const element = this.values[idx]; while (idx > 0) { let parentIdx = Math.floor((idx - 1) / 2); let parent = this.values[parentIdx]; if (element.priority >= parent.priority) break; this.values[parentIdx] = element; this.values[idx] = parent; idx = parentIdx; } } dequeue() { const min = this.values[0]; const end = this.values.pop(); if (this.values.length > 0) { this.values[0] = end; this.sinkDown(); } return min; } sinkDown() { let idx = 0; const length = this.values.length; const element = this.values[0]; while (true) { let leftChildIdx = 2 * idx + 1; let rightChildIdx = 2 * idx + 2; let leftChild, rightChild; let swap = null; if (leftChildIdx < length) { leftChild = this.values[leftChildIdx]; if (leftChild.priority < element.priority) { swap = leftChildIdx; } } if (rightChildIdx < length) { rightChild = this.values[rightChildIdx]; if ( (swap === null && rightChild.priority < element.priority) || (swap !== null && rightChild.priority < leftChild.priority) ) { swap = rightChildIdx; } } if (swap === null) break; this.values[idx] = this.values[swap]; this.values[swap] = element; idx = swap; } } } //Dijkstra's algorithm only works on a weighted graph. const pointDistance = (a, b) => { return ( (steps[0][a][0] - steps[1][b][0]) ** 2 + (steps[0][a][1] - steps[1][b][1]) ** 2 ); }; const adjacencyList = {}; const nodes = new PriorityQueue(); const distances = {}; const previous = {}; const path = []; //to return at end let frechet = 0; let frechet_time = 0; let smallest; //build up initial state let start = 0; let finish = {}; finish[[resolution - 1, resolution - 1]] = true; nodes.enqueue([0, 0], 0); distances[[0, 0]] = 0; let distance = []; // as long as there is something to visit while (nodes.values.length) { smallest = nodes.dequeue().val; if (finish[smallest]) { while (previous[smallest]) { path.unshift(smallest); distance.unshift(Math.sqrt(pointDistance(...smallest))); if (frechet < distance[0]) { frechet = distance[0]; frechet_time = distance.length; } smallest = previous[smallest]; } break; } if (smallest !== undefined) { for (const neighbor of [ [smallest[0] + 1, smallest[1]], [smallest[0], smallest[1] + 1] ]) { if (neighbor[0] >= resolution || neighbor[1] >= resolution) continue; // calculate new distance to neighboring node // console.log(distances, neighbor, "yikes"); let candidate = distances[smallest] + pointDistance(...neighbor); if ( distances[neighbor] === undefined || candidate < distances[neighbor] ) { //updating new smallest distance to neighbor distances[neighbor] = candidate; //updating previous - How we got to neighbor previous[neighbor] = smallest; //enqueue in priority queue with new priority nodes.enqueue(neighbor, candidate); } } } } path.unshift([0, 0]); distance.unshift(Math.sqrt(pointDistance(0, 0))); // distance.unshift(0); //distance = distance.slice(1).map((x, i) => x - distance[i]); return { path, distance, frechet: Math.max(...distance), frechet, frechet_time }; }30:T442,keyframes = { let keyframes = [Array(100), Array(100)]; let previous = [0, 0]; let count = 0; keyframes[0][0] = `0% {stroke-dashoffset: ${ controlPoints[0][controlPoints[0].length - 1].cumulative };}`; keyframes[1][0] = `0% {stroke-dashoffset: ${ controlPoints[1][controlPoints[1].length - 1].cumulative };}`; for (const point of dijkstra.path) { for (const i of [0, 1]) { if (point[i] != previous[i]) { /* let p = ((100 * point[i]) / resolution) | 0; let length = (controlPoints[i][controlPoints[i].length - 1].cumulative * (1 - count / (2 * resolution))) | 0; */ let p = ((100 * count) / (2 * resolution)) | 0; let length = (controlPoints[i][controlPoints[i].length - 1].cumulative * (1 - point[i] / resolution)) | 0; keyframes[i][p + 1] = `${p + 1}% {stroke-dashoffset: ${length};}`; } } previous = point; count++; } for (const i of [0, 1]) { keyframes[i] = keyframes[i].join(" "); } return keyframes; }31:T45a,//starting position controlPoints = { drawing; selection; let data = [[], []]; let datum = 0; for (let i = 0; i < 2; i++) { data[i] = doodles.q[i] .attr("d") .slice(1) .split(/L|,/) .map((a, b, c, d) => { let idx = (b / 2) | 0; if (b % 2 == 1) { let x = datum, y = parseFloat(a); //console.log(idx); if (idx == 0) { data[i].push({ x, y, dx: 0, dy: 0, length: 0, cumulative: 0 }); //console.log(data[0]); return; } let prev = data[i][idx - 1], dx = prev.x - x, dy = prev.y - y, length = (dx ** 2 + dy ** 2) ** 0.5; dx /= length; dy /= length; data[i].push({ x, y, dx, dy, length, cumulative: prev.cumulative + length }); return data[i][data[i].length - 1]; } else { datum = parseFloat(a); return; } }) .filter((a) => a && !isNaN(a.dx)); } return data; }32:T6e07,bunnies = [ { fill: "none", stroke: "blue", "stroke-width": "2", d: 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} ]35:[] 2c:{"id":"fa8022dea8830172","slug":"frechet-distance","title":"Frechet Distance","thumbnail":"76df1a21bf522cf1b6cb5b85fd8086271cd1eaefe738cfb625d4ab162fefb2c3","access_level":"public","create_time":"$D2022-11-24T15:39:21.267Z","update_time":"$D2024-01-09T19:38:43.273Z","view_time":null,"first_public_time":"$D2022-11-25T01:13:50.991Z","has_unpublished_edits":null,"fork":false,"stars":9,"starred":false,"owner":{"type":"team","id":"8880f12a08d5ccb7","login":"dmadisetti","name":"Dylan Madisetti","avatar_url":"https://avatars-next.observableusercontent.com/30767c701c28e63e28a14e0d59cbef6b4c88507040a73bc304f1a119abd852db","tier":"starter_2024"},"version":297,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"12ceb5d047bae1c3","login":"dmadisetti","name":"Dylan Madisetti","avatar_url":"https://avatars-next.observableusercontent.com/30767c701c28e63e28a14e0d59cbef6b4c88507040a73bc304f1a119abd852db","tier":null}],"members":[{"type":"individual","id":"12ceb5d047bae1c3","login":"dmadisetti","name":"Dylan Madisetti","avatar_url":"https://avatars-next.observableusercontent.com/30767c701c28e63e28a14e0d59cbef6b4c88507040a73bc304f1a119abd852db","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"lapsed":null,"subscribed":null,"forks":4,"collection":null,"type":"notebook","body":{"title":"Frechet Distance","cells":[{"id":0,"value":"# Frechet Distance\n\nBefore we get started, go ahead and make 2 similar (single line) drawings on top of each other by switching from green to blue. We'll be using this pair throughout the notebook to explain the 'Frechet distance'.\n\n> 😬 sorry there's not a ton of space draw on mobile. Also, there are some default sketches [inspired by picasso](https://www.pablopicasso.net/drawings/), to inspire you.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6,"value":"viewof drawing = doodles.node","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"which = Inputs.radio([\"green\", \"blue\"], {\n value: \"green\",\n label: html`Color`,\n format: (x) =>\n html`${x}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":103,"value":"viewof selection = Inputs.select(Object.keys(path_defaults), {\n label: html`Set to default`,\n values: Object.keys(path_defaults)\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"I asked you to make some similar drawings, but __how__ similar are they? If the drawings are similar, we should be able to synchronize the drawings in time. If the drawings are not similar this \"synchronization\" will be jumpy. Let's take a look at how synchronized your drawings are 😉.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":36,"value":"viewof redraw = Inputs.button(\"redraw\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":33,"value":"response = {\n redraw;\n selection;\n doodles;\n return html`\n`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":64,"value":"(how'd you do?)\n\nBut how is this done? Well, if we can take the distance of all the various points along a sketch, against every point of a different sketch. We can make an image or variation of a 'free-space diagram' from this comparison. Here, the position in the x direction indicates the amount drawn of a a sketch, and the y direction indicates the amount drawn of a different sketch, while the color indicates the distance between these path points. We off load this to the GPU because this is a lot of work:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":66,"value":"canvas = shader({\n width: 4 * resolution,\n height: 4 * resolution,\n visibility,\n uniforms: {\n steps: `vec2[${2 * resolution}]`,\n best: `ivec2[${2 * resolution}]`,\n path: \"bool\"\n }\n})`\n\n//$ {viridis}\n${turbo}\n\nvec3 color(ivec2 idx) {\n vec2 p = steps[idx.x];\n vec2 q = steps[${resolution} + int(idx.y)];\n\n float l = 2.0 * length(p.xy - q.xy)/float(iResolution.xy);\n return color_map(l);\n}\n\nvoid mainImage(out vec4 fragColor, in vec2 fragCoord) {\n ivec2 idx = ivec2(fragCoord / (iResolution.xy/${1.00001 * resolution}));\n ivec2 selected = best[idx.x + idx.y];\n if (idx.x == selected.x && path) {\n fragColor = vec4(vec3(1.0), 1);\n } else {\n fragColor = vec4((color(idx) +\n color(idx + ivec2(-1, 0)) +\n color(idx + ivec2(1, 0)) +\n color(idx + ivec2(0, -1)) +\n color(idx + ivec2(0, 1)))/5.0, 1);\n }\n}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":79,"value":"Pretty pretty huh? This is colored by the turbo colormap, where reds indicate large distances and blues smaller distances. The optimal timing is the path that goes from the bottom left corner to the top right corner and minimizes the distances it has to travel through. You can toggle this path with the checkbox below:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":86,"value":"viewof path = Inputs.checkbox([\"Show optimal path\"])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":94,"value":"Finally, we can plot the distances that our optimal path passes through","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":91,"value":"plot = Plot.plot({\n x: {\n label: \"time →\",\n axis: \"bottom\",\n ticks: 1,\n tickFormat: (t) =>\n [\"0\", \"end \" /*`${((5 * t) / (2 * resolution)).toFixed(2)}s`*/][\n (t > 0) | 0\n ]\n },\n y: { label: \"between drawing distance\" },\n marks: [\n Plot.lineY(\n dijkstra.distance.concat([\n dijkstra.distance[dijkstra.distance.length - 1],\n dijkstra.distance[dijkstra.distance.length - 1]\n ]),\n { strokeWidth: 1 }\n )\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":97,"value":"$2e","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":73,"value":"canvas.update({ steps: steps, best: dijkstra.path, path: path.length })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"import { Doodler } from \"@dmadisetti/doodler\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"import { turbo, viridis } from \"@flimsyhat/webgl-color-maps\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":71,"value":"import { shader } from \"@dmadisetti/uniform-shader\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":25,"value":"resolution = 250","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":46,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":50,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":41,"value":"steps = {\n let data = [[], []];\n for (let i = 0; i < 2; i++) {\n let last = controlPoints[i][controlPoints[i].length - 1],\n points = Array(resolution),\n total = last.cumulative,\n dt = total / resolution,\n running = 0,\n head = 0,\n path = controlPoints[i],\n remainder = dt,\n count = 1;\n points[0] = [path[head].x, path[head].y];\n while (running + dt < total - dt / 2) {\n let current = path[head];\n let x = current.x;\n let y = current.y;\n while (running + dt <= current.cumulative) {\n running += dt;\n x += current.dx * remainder;\n y += current.dy * remainder;\n remainder = dt;\n points[count++] = [x, y];\n }\n head++;\n remainder = current.cumulative - running;\n }\n if (!points[resolution - 1]) {\n points[resolution - 1] = [last.x, last.y];\n }\n data[i] = points.slice(0, resolution);\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"doodles = {\n selection;\n console.log(selection, path_defaults[selection]);\n\n return Doodler((q) => {\n let newer = q.pop();\n let older = q[(which.value == \"green\") | 0];\n if (newer != older) {\n newer.attr(\"stroke\", which.value);\n q[(which.value == \"green\") | 0] = newer;\n if (older) older.remove();\n }\n }, path_defaults[selection]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":105,"value":"path_defaults = {\n return {\n \"Picasso Penguin\": penguins,\n \"Picasso Dog\": dog,\n \"Bad Bunnies\": bunnies\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":177,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":273,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":194,"value":"doodles.export()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 2d:"$2c:body"