Markdown footnotes added in 2000/bmeyer
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@@ -215,7 +215,7 @@ In short --- running it through the preprocessor and the
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pretty-printer will give you something that looks slightly less
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like line noise and slightly more like a C program, but unless you
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are a true wizard, it is unlikely to gain you any insights into
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what is actually going on....\[0\]
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what is actually going on....[^0]
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### Bugs, Assumptions and TODO
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@@ -269,7 +269,7 @@ ordering. For each pixel, an expected value is calculated by creating a least
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squares predictor of order 12 based on all of the previously encoded/decoded
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pixels. However, the contribution of each pixel is weighted according to its
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[Manhattan-distance](https://en.wikipedia.org/wiki/Taxicab_geometry) from the
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current pixel, `D`. That weight used is `pow(0.8,D)`. \[1\] \[2\] \[3\]
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current pixel, `D`. That weight used is `pow(0.8,D)`. [^1] [^2] [^3]
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In a similar way, the prediction *errors* for all previous pixels
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are combined --- a weighted average (with weight `pow(0.7,D)`) of
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@@ -293,27 +293,27 @@ arithmetic decoder provides the information which half the actual
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value is in. In both cases, the interval borders are adjusted
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accordingly, the arithmetic coder/decoder updated, and the
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interval halving repeated until the precise value has been
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en/decoded\[4\].
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en/decoded[^4].
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[0] Yes, this could be seen as a challenge....
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[1] Actually, the weight is `pow(0.8,D)/s`, where `s` is the sigma
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[^0]: Yes, this could be seen as a challenge....
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[^1]: Actually, the weight is `pow(0.8,D)/s`, where `s` is the sigma
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used for encoding/decoding the particular previous pixel. Dividing
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by `s` is a rather arbitrary action justified only by the
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improvement in results. Mathematically, I can only justify
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dividing by `s^2`, or not dividing at all....
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[2] Obviously, these least squares predictors are not calculated
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[^2]: Obviously, these least squares predictors are not calculated
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from scratch for each pixel. Some clever reuse of earlier
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results allows `O(1)` calculation, rather than the
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`O(previous_rows*columns)` this description would suggest.
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[3] As anyone who has ever tried local least squares knows from
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[^3]: As anyone who has ever tried local least squares knows from
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bitter experience, there is a lot of numerical instability
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associated with them. So `glicbawls` adds a bias towards an
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averaging predictor to the equation system the least squares
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predictor is calculated from. The weight that is given to
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this bias is constantly adjusted throughout coding/decoding,
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thus justifying the 'a' in `glicbawls`.
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[4] In the case of near-lossless coding, the interval is not
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[^4]: In the case of near-lossless coding, the interval is not
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necessarily *halved*, but rather split at a convenient point
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(the split points are chosen in such a way as to minimise the
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expected number of bits needed for coding). Also, the
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