blog

commit

eaeb3345f772b0d6cd481343546eaf5a3869de0a

parent

3cae11aff1465d7317e82cfcc91118ac41112a7c

Author

Tobias Bengfort <tobias.bengfort@posteo.de>

Date

2026-06-28 08:09

contrast: use latex math

Diffstat

1 files changed, 38 insertions, 22 deletions

diff --git a/_content/posts/2022-09-10-contrast-algorithms/index.md b/_content/posts/2022-09-10-contrast-algorithms/index.md

@@ -17,7 +17,7 @@ claims that can be found throughout the internet.
   17    17 
   18    18 Let's start with some fractions.
   19    19 
   20    -1 -   **simple contrast**: `Ymax / Ymin`
   -1    20 -   **simple contrast**: $\frac{Y_{max}}{Y_{min}}$
   21    21 
   22    22     The authors do not have anything nice to say about this one:
   23    23 
@@ -25,31 +25,39 @@ Let's start with some fractions.
   25    25     > much higher dynamic range, and the logarithmic response characteristics
   26    26     > of the human eye.
   27    27 
   28    -1 -   **Weber Contrast**: `(Ymax - Ymin) / Ymin`
   -1    28 -   **Weber Contrast**: $\frac{Y_{max} - Y_{min}}{Y_{min}}$
   29    29 
   30    30     The authors provide a lot of background information on this one, but fail
   31    31     to mention one crucial point: This again is simple contrast, just with an
   32    32     offset of 1:
   33    33 
   34    -1         (Ymax - Ymin) / Ymin
   35    -1         = Ymax / Ymin - 1
   -1    34     $$
   -1    35     \frac{Y_{max} - Y_{min}}{Y_{min}}
   -1    36     = \frac{Y_{max}}{Y_{min}} - 1
   -1    37     $$
   36    38 
   37    -1 -   **Michelson Contrast**: `(Ymax - Ymin) / (Ymax + Ymin)`
   -1    39 -   **Michelson Contrast**: $\frac{Y_{max} - Y_{min}}{Y_{max} + Y_{min}}$
   38    40 
   39    41     Again they fail to mention a crucial detail: This is a Weber Contrast
   40    -1     that compares `Ymax` and the average of `Ymax` and
   41    -1     `Ymin`:
   -1    42     that compares $Y_{max}$ and the average of $Y_{max}$ and
   -1    43     $Y_{min}$:
   42    44 
   43    -1         (Ymax - Ymin) / (Ymax + Ymin)
   44    -1         = (2 * Ymax - (Ymax + Ymin)) / (Ymax + Ymin)
   45    -1         = (Ymax - (Ymax + Ymin) / 2) / ((Ymax + Ymin) / 2)
   46    -1         = (Ymax - Yavg) / Yavg
   -1    45     $$
   -1    46     \begin{aligned}
   -1    47     & \frac{Y_{max} - Y_{min}}{Y_{max} + Y_{min}} \\
   -1    48     =& \frac{2 \cdot Y_{max} - (Y_{max} + Y_{min})}{Y_{max} + Y_{min}} \\
   -1    49     =& \frac{Y_{max} - Y_{avg}}{Y_{avg}}
   -1    50     \end{aligned}
   -1    51     $$
   47    52 
   48    53     It can also easily be calculated from simple contrast:
   49    54 
   50    -1         (Ymax / Ymin - 1) / (Ymax / Ymax + 1)
   -1    55     $$
   -1    56     \frac{Y_{max} - Y_{min}}{Y_{max} + Y_{min}}
   -1    57     = \frac{\frac{Y_{max}}{Y_{min}} - 1}{\frac{Y_{max}}{Y_{max}} + 1}
   -1    58     $$
   51    59 
   52    -1 -   **WCAG 2.1**: `(Ymax + 0.05) / (Ymin + 0.05)`
   -1    60 -   **WCAG 2.1**: $\frac{Y_{max} + 0.05}{Y_{min} + 0.05}$
   53    61 
   54    62     The authors correctly observe:
   55    63 
@@ -75,15 +83,17 @@ together with two thresholds, we can call them *equivalent* if they give the
   75    83 same results.
   76    84 
   77    85 Mathematically speaking, we want the following to hold true for all color pairs
   78    -1 `(a, b)` and `(c, d)`:
   -1    86 $(a, b)$ and $(c, d)$:
   79    87 
   80    -1     f(a, b) < f(c, d) => g(a, b) < g(c, d)
   -1    88 $$
   -1    89 f(a, b) < f(c, d) \Rightarrow g(a, b) < g(c, d)
   -1    90 $$
   81    91 
   82    92 In other words, two contrast formulas are equivalent if there is a strictly
   83    93 monotonic map between them. I call such a map a "scaling".
   84    94 
   85    95 For example, simple contrast and Weber contrast are equivalent according to
   86    -1 this definition because `scale(x) = x - 1` is strictly monotonic. The same goes
   -1    96 this definition because $scale(x) = x - 1$ is strictly monotonic. The same goes
   87    97 for Michelson contrast.
   88    98 
   89    99 You could also have a stricter definition and require contrast formulas to be
@@ -95,8 +105,8 @@ will work with the weaker definition in this post.
   95   105 
   96   106 ## Lightness difference
   97   107 
   98    -1 The authors also mention lightness difference `Lmax -
   99    -1 Lmin` as a way to calculate contrast:
   -1   108 The authors also mention lightness difference $L_{max} -
   -1   109 L_{min}$ as a way to calculate contrast:
  100   110 
  101   111 > Instead of being based on luminance, which is not perceptually uniform (and
  102   112 > thus, the visual difference corresponding to a given luminance difference is
@@ -106,13 +116,19 @@ Lmin` as a way to calculate contrast:
  106   116 Using the CIELab model of lightness is one option, but there are others.
  107   117 For example, there is the [Weber-Fechner law][2]:
  108   118 
  109    -1     L = a * log(Y) + b
   -1   119 $$
   -1   120 L = a \cdot \log(Y) + b
   -1   121 $$
  110   122 
  111   123 So what happens if we build a lightness difference based on that definition?
  112   124 
  113    -1     (a * log(Ymax) + b) - (a * log(Ymin) + b)
  114    -1     = a * (log(Ymax) - log(Ymin))
  115    -1     = a * log(Ymax / Ymin)
   -1   125 $$
   -1   126 \begin{aligned}
   -1   127 & (a \cdot \log(Y_{max}) + b) - (a \cdot \log(Y_{min}) + b) \\
   -1   128 =& a \cdot (\log(Y_{max}) - \log(Y_{min})) \\
   -1   129 =& a \cdot \log(\frac{Y_{max}}{Y_{min}})
   -1   130 \end{aligned}
   -1   131 $$
  116   132 
  117   133 This turns out to be a scaled version of simple contrast. Even if you consider
  118   134 Weber-Fechner to be outdated, this is still a perceptual model. So both the