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  "body": "\n## Summary\n\nThe discrete wavelet transform splits a sequence into a coarse part and a detail part.\nIt uses two filters followed by downsampling by $2$, and then repeats the split on the coarse part.\nFor filters from an orthonormal [[multiresolution analysis|1GHQMBR7DE]], each step is an orthogonal change of coordinates.\nSo the transform preserves energy and is easy to invert.\nThe outputs are the coefficients of a function in a [[wavelet|4PZ35C0GV3]] basis.\nThe [[fast wavelet transform|PDVYQAW887]] computes them in a number of steps proportional to the length of the signal.\nIt is the basis of wavelet compression, for example in the JPEG 2000 image standard.\n\n## Explanations\n\n### L1 human\n\nTake a list of numbers and group them in pairs.\nFor each pair, keep a scaled sum, which records the average, and a scaled difference, which records the detail.\nThen do the same with the list of sums, and again, until one number is left.\nFor a smooth signal most details are tiny, so one can drop them and still rebuild a good approximation.\nThis is the Haar case; other wavelets use longer filters in the same pattern.\n",
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