{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "ZKXDA3XF3Z"
    }
  ],
  "body": "\n## Summary\n\nThe cascade algorithm computes the [[scaling function|YH4BERB952]] of a wavelet from its filter coefficients by iterating the refinement equation.\nIt starts from a simple function, such as the indicator of $[0,1)$.\nEach step builds a new function from shrunken and shifted copies of the old one.\nFor good filters, such as the filters of the Daubechies wavelets, the iterates converge to the scaling function.\nMost compactly supported wavelets have no closed formula, so this is the usual way to draw them.\n\n## Explanations\n\n### L1 human\n\nA scaling function is defined by a self-similarity rule: it is a weighted sum of half-size copies of itself.\nThe cascade algorithm uses this rule as a recipe.\nStart with a box, apply the rule to get a new shape, then apply it again to the new shape, and so on.\nFor the Haar filter the box is already a fixed point: two half-width boxes side by side make the box again.\nFor Daubechies filters the shapes become finer and more jagged at each step and settle down to a fractal-looking curve.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "65T60",
      "42C40"
    ],
    "name": "Cascade algorithm",
    "same_as": {
      "wikidata": "Q5048176",
      "wikipedia": "Cascade algorithm"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Cascade_algorithm"
      }
    ],
    "tag": "VZK6ZAXWJR"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/VZK6ZAXWJR/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/VZK6ZAXWJR.md",
  "msc": [
    "65T60",
    "42C40"
  ],
  "name": "Cascade algorithm",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VZK/VZK6ZAXWJR.md",
  "status": null,
  "tag": "VZK6ZAXWJR",
  "uri": "m2://VZK6ZAXWJR",
  "url": "https://tamnd.github.io/math2/wiki/cascade-algorithm/",
  "writing": {
    "tag": "VZK6ZAXWJR",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}