Changeset 1972 for www


Ignore:
Timestamp:
Nov 16, 2007, 2:34:25 AM (13 years ago)
Author:
Sam Hocevar
Message:
  • More about gamma correction. Fixed graphs.
Location:
www/study
Files:
6 added
4 edited

Legend:

Unmodified
Added
Removed
  • www/study/fig4-1-1.svg

    r1942 r1972  
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  • www/study/part4.html

    r1956 r1972  
    3333<h2> 4. Grayscale dithering </h2>
    3434
    35 <p> Generalising dithering to more than two colours is straightforward in the
    36 grayscale palette. Here are the results with 4×4 Bayer ordered dithering and
    37 with Floyd-Steinberg error diffusion: </p>
     35<p> At first sight, generalising dithering to three gray scales seems pretty
     36straightforward: just add gray 0.5 in the middle of the palette and dither
     37pixels in the [0, 0.5] range with black and gray, and pixels in the [0.5, 1]
     38range with gray and white. Here are two different results with 4×4 Bayer
     39ordered dithering and with Floyd-Steinberg error diffusion: </p>
    3840
    3941<p style="text-align: center;">
     
    4850</p>
    4951
    50 <p> Unfortunately the result is not as good as expected. The white pattern
    51 on Lenna’s cheeks is visually disturbing, and the whole image looks darker
    52 than with pure black-and-white dithering. But then, the previous dithering
    53 results looked a lot brighter than the original image. This is due to the
    54 output media’s <b>gamma</b>. </p>
     52<p> These are pretty much the images that imaging software such as The Gimp
     53would give (using “positioned” and “Floyd-Steinberg” dithering modes). </p>
     54
     55<p> Unfortunately the result is not as good as expected: the white pattern
     56on Lenna’s cheeks is visually disturbing, and there is a lot of 0.5 gray in
     57the image. Also, the whole image looks darker than with pure black-and-white
     58dithering, but these previous dithering results looked a lot brighter than
     59the original image anyway. </p>
     60
     61<p> All these issues have to do with the output media’s <b>gamma</b>. </p>
    5562
    5663<h3> 4.1. Introducing gamma </h3>
     
    7077<i>I = v<small><sup>γ</sup></small></i> where <i>v</i> is the coded colour
    7178value (between 0 and 1), <i>I</i> is the perceived colour intensity (between
    72 0 and 1) and <i>γ</i> is the gamma. Most modern computer systems use the
    73 sRGB gamma model close to the law with <i>γ = 2.2</i>. As can be seen, it is
    74 highly non-linear: </p>
     790% and 100%) and <i>γ</i> is the gamma. A pattern made of even-numbered
     800%-intensity pixels and 100%-intensity pixels has an intensity of 50% by
     81definition. But the corresponding grayscale</p>
     82
     83<p> Most modern computer systems use the sRGB gamma model close to the law
     84with <i>γ = 2.2</i>. As can be seen, it is highly non-linear: </p>
    7585
    7686<p style="text-align: center;">
    77   <img src="fig4-1-1.png" width="300" height="240" alt="introducing gamma" />
     87  <img src="fig4-1-1.png" width="460" height="256" alt="introducing gamma" />
    7888</p>
    7989
     
    8191pretty comprehensive essay</a> about why on a computer screen a 50% black and
    8292white pattern should be scaled down to a gray value of 0.73 instead of 0.5 and
    83 how major computer graphics software totally misses the point. </p>
     93how major computer graphics software totally misses the point. Conversely, it
     94clearly means that a gray value of 0.5 should not be emulated with a 50%
     95dither pattern. </p>
    8496
    85 <p> Conversely, it clearly means that a gray value of 0.5 should not be
    86 emulated with a 50% dither pattern. </p>
     97<p> The following figure shows the gamma curve for the naïve three-colour
     98grayscale gradient we saw above (red) compared to the two-colour gradient
     99(blue). Two major observations can be made: the new curve is far closer to
     100a perfect, linear gradient, and there is a singularity in the middle of
     101the curve, meaning a break in the gradient’s smoothness. </p>
     102
     103<p style="text-align: center;">
     104  <img src="fig4-1-2.png" width="460" height="256" alt="3-colour gamma" />
     105</p>
     106
     107<p> There are three possible ways to reduce the singularity and make the
     108gradient smoother and/or closer to the original colours: </p>
     109
     110<ul>
     111  <li> Choose a different middle gray value, for instance choosing gray 0.73
     112       will cancel the singularity and match the two-colour gradients we have
     113       been using so far. This is not always possible if the output palette
     114       is fixed. </li>
     115  <li> Don’t place the gray value at the middle of the gradient, for instance
     116       putting it around 25% itensity will again match the previous two-colour
     117       gradients. </li>
     118  <li> <b>Gamma-correct</b> input pixels before assigning them an output
     119       value. This ensures that the resulting gradient is perfectly linear.
     120       </li>
     121</ul>
     122
     123<h3> 4.2. Gamma correction </h3>
     124
     125<p> These are the results of gamma-correcting input pixels before doing
     126any computation on them, then using Floyd-Steinberg error diffusion: </p>
     127
     128<p style="text-align: center;">
     129  <img src="out4-2-1.png" width="256" height="256"
     130       class="inline" alt="Floyd-Steinberg, 2 colours, gamma-corrected" />
     131  <img src="grad4-2-1.png" width="32" height="256"
     132       class="inline" alt="Floyd-Steinberg, 2 colours, gamma-corrected gradient" />
     133  <img src="out4-2-2.png" width="256" height="256"
     134       class="inline" alt="Floyd-Steinberg, 3 colours, gamma-corrected" />
     135  <img src="grad4-2-2.png" width="32" height="256"
     136       class="inline" alt="Floyd-Steinberg, 3 colours, gamma-corrected gradient" />
     137</p>
    87138
    88139<!--
  • www/study/study.py

    r1969 r1972  
    426426        dest.setGray(x, y, 0.5)
    427427dest.writePng("pat4-1-1.png")
     428
     429# Output 4.2.1: gamma-corrected 2-colour Floyd-Steinberg
     430# Output 4.2.2: gamma-corrected 3-colour Floyd-Steinberg
     431def test42x(src, mat, threshold, name):
     432    (w, h) = src.size()
     433    dest = Image((w, h))
     434    lines = len(mat)
     435    rows = len(mat[0])
     436    offset = mat[0].index(-1)
     437    ey = [[0.] * (w + rows - 1) for x in range(lines)]
     438    for y in range(h):
     439        ex = [0.] * (rows - offset)
     440        for x in range(w):
     441            # Set pixel
     442            c = Gamma.CtoI(src.getGray(x, y)) + ex[0] + ey[0][x + offset]
     443            d = threshold(c)
     444            dest.setGray(x, y, Gamma.ItoC(d))
     445            error = c - d
     446            # Propagate first line of error
     447            for dx in range(rows - offset - 2):
     448                ex[dx] = ex[dx + 1] + error * mat[0][offset + 1 + dx]
     449            ex[rows - offset - 2] = error * mat[0][rows - 1]
     450            # Propagate next lines
     451            for dy in range(1, lines):
     452                for dx in range(rows):
     453                    ey[dy][x + dx] += error * mat[dy][dx]
     454        for dy in range(lines - 1):
     455            ey[dy] = ey[dy + 1]
     456        ey[lines - 1] = [0.] * (w + rows - 1)
     457    dest.writePng(name)
     458
     459mat = [[    0.,    -1, 7./16],
     460       [ 3./16, 5./16, 1./16]]
     461test42x(lenna256bw, mat, lambda x: x > 0.5, "out4-2-1.png")
     462test42x(gradient256bw, mat, lambda x: x > 0.5, "grad4-2-1.png")
     463test42x(lenna256bw, mat, lambda x: 0.25 * (x > 0.25) + 0.75 * (x > 0.75), "out4-2-2.png")
     464test42x(gradient256bw, mat, lambda x: 0.25 * (x > 0.25) + 0.75 * (x > 0.75), "grad4-2-2.png")
    428465
    429466##############################################################################
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