René Krieg
How to Convert a Bitmap to Polychromatic Light
c 2008, LightTrans GmbH, Jena
Copyright All rights reserved.
René Krieg
November 08, 2007
[email protected]
Contents
1.
The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.
False Color Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.
Real Color Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
H O W T O C O N V E R T A B I T M A P T O P O LY C H R O M AT I C L I G H T
1. The Problem
Since LightTrans VirtualLabTM represents light as electromagnetic
field(s) of certain wavelength(s), considerations have to be made about
an algorithm, able to transform a colored bitmap into polychromatic
light. That means, a mapping has to be found for the three image channels R, G, and B, and three two-dimensional fields with three di=erent
wavelengths.
There are two di=erent ways to solve this problem, providing di=erent results. They could be named "Real Color Representation" and "False
Color Representation" of the bitmap. While the latter keeps the portions
of the three color channels R, G, and B by simply equalizing with the
three wavelength channels only, the crucial constraint for the Real Color
conversion is the identicalness of the light view of the field to the original bitmap. Both procedures will be described in the following and are
outlined in the sketches below (Fig. 1 and Fig. 2).
Figure 1. Algorithm for the false color import of bitmaps. An arbitrary wavelength is mapped to each of the channels R,
G, and B, identifying the values of R, G, and B with the intensities of the fields.
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H O W T O C O N V E R T A B I T M A P T O P O LY C H R O M AT I C L I G H T
Figure 2. Algorithm for the real color import of bitmaps. The original color information is converted from the RGB color
space to a previously defined color space, spanned by three appropriate wavelengths.
For a detailed understanding of these considerations it’s essential to
know the LightTrans Supplement "How to Bring Natural Light on a Monitor" ([NLOM]). It deals with the problem of displaying light of certain
wavelength(s) on a computer monitor.
2. False Color Import
A false color import can be used if the colors shall be alienated or if the
colors do not have to be preserved.
The algorithm simply uses the value R as intensity for wavelength λ1 ,
the value G for wavelength λ2 and B for λ3 . That’s it.
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H O W T O C O N V E R T A B I T M A P T O P O LY C H R O M AT I C L I G H T
3. Real Color Import
Essentially, the import of a bitmap that should be shown in real colors
has to be the inversion of the wavelength-to-RGB algorithm described in
[NLOM]. The reason is, that at first the colors given in the bitmap RGB
color system ("source RGB") have to be transformed to a wavelength representation which can be handled in VirtualLabTM . The display routines
will convert them to the display RGB color space ("destination RGB").
That means for the import to do a conversion from the source RGB to a
new color space, defined by three wavelengths, which will be called "3L"
color space in the following. The three constituting wavelengths λ1 , λ2 ,
and λ3 can be chosen relatively free, the only condition to be fulfilled is
illustrated in Fig. 3: the triangle of displayable colors (called "gamut")
spanned by the positions of the three wavelengths in the chromaticity
diagram has to contain the gamuts of the used RGB spaces (usually:
source = sRGB, destination = LightTrans Color System) completely.
0.9
y
515
0.8
520 525
530
535
540
510
545
0.7
550
555
505
560
0.6
565
570
500
0.4
0.3
0.2
575
3L
0.5
580
585
590
595
600
605
610
620
635
700
sRGB
495
490
485
480
0.1
0.0
0.0
475
470
460
380
0.1
0.2
0.3
0.4
0.5
0.6
0.7 x 0.8
Figure 3. Gamut of the VirtualLabTM 3L color space (continuous line) in
relation to the standard RGB color space sRGB. For an explanation of the
background of this diagram see [NLOM], please. Source: Gernot Ho=mann
( ∼ ), modified by LightTrans.
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H O W T O C O N V E R T A B I T M A P T O P O LY C H R O M AT I C L I G H T
For the particular case that the destination RGB color space is defined by
three wavelengths λ R , λ G , λ B itself (i.e. if its primaries are pure colors on
the spectral colors curve), the wavelengths of the 3L color space should
be identical to them to avoid unnecessary out-of-gamut problems. This
usually applies to LightTrans VirtualLabTM , since it’s default color system for the display is the ’LightTrans Color System’1 . So the three 3L
wavelengths are identical to the spectral colors that define the LightTrans
Color System as primaries: λ1 = λ R , λ2 = λ G , and λ3 = λ B . They and
their chromaticities (according to the CIE31 measurement) are shown in
the following table:
λ(nm)
700
545
470
(x, y)
(0.7347, 0.2653)
(0.2658, 0.7243)
(0.1241, 0.0578)
The conversion of the ( R, G, B) triple of one bitmap pixel to the
3L color space will provide a triple of three weights (i1 , i2 , i3 ). These
correspond to the intensity values at this position in the three twodimensional fields as the destination of the import. The correct display
of this "monochromatic decomposition of polychromatic light" on the
monitor will be done by VirtualLabTM ’s light view routines, as described
in [NLOM].
The following algorithm is used to get the intensity triple for each
pixel:
1. Rescaling and linearizing:
⎛ ⎞ ⎛
⎞γ
r
R/255
⎜ ⎟ ⎜
⎟
⎝ g⎠ = ⎝ G/255⎠
b
B/255
2. Get the coordinates in the perceptional color space:
⎛ ⎞
⎛ ⎞
r
x
⎜ ⎟
⎜ ⎟
⎝ y ⎠ = TCIE · ⎝ g⎠
b
z
The conversion matrix TCIE is the inverse of the matrix described in
[NLOM] that is used for the transformation from perceptional color
space to RGB.
3. "Denormalizing", necessary to execute the inverse process of
1 Otherwise the following wavelengths are applied: (700 nm, 525 nm, 450 nm ) for ’Wide
Gamut’ and ’Pro Photo’, (611 nm, 535 nm, 465 nm ) else.
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H O W T O C O N V E R T A B I T M A P T O P O LY C H R O M AT I C L I G H T
wavelength-to-RGB:
⎛ ⎞ ⎛ ⎞
x
x
⎜ ⎟ ⎜ ⎟ ( x + y + z)
⎝y ⎠ = ⎝y⎠ ·
y
z
z
4. Get the intensities for the three wavelengths:
⎛ ⎞
⎛ ⎞
x
i1
⎜ ⎟
⎜ ⎟
⎝i2 ⎠ = T3L · ⎝ y ⎠ ,
i3
z
with
⎛
T3L
X ( λ1 )
⎜
= ⎝ Y ( λ1 )
Z ( λ1 )
X ( λ2 )
Y ( λ2 )
Z ( λ2 )
⎞ −1
X ( λ3 )
⎟
Y ( λ3 ) ⎠ .
Z ( λ3 )
The functions X (λ), Y (λ), and Z (λ) are the Color Matching Functions, measured by the Commission Internationale de l’Éclairage
(CIE), published in [CIE32], and cursory described in [NLOM].
5. Handle out-of-gamut errors. If at least one of the calculated intensities is less than zero, an out-of-gamut error occurred. That means,
the color of the pixel, according to the source RGB space, lies outside
the gamut of the 3L color space. In this case, all intensities have to
be shifted by subtracting the minimum value min(i1 , i2 , i3 ). To avoid
this kind of error a priori, the wavelengths must be chosen in a way
that the RGB triangle lies completely inside that of the 3L color space,
see explanation above.
As already mentioned, these steps has to be repeated for each pixel of
the bitmap, resulting in three fields of the wavelengths λ1 , λ2 , and λ3 ,
filled with the intensity values i1 , i2 , and i3 .
References
[CIE32] Commission Internationale de l’Eclairage Proceedings, Cambridge
University Press, Cambridge, 1932
[NLOM] How to Bring Natural Light on a Monitor, LightTrans Supplement, 2007.
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