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Colorspace Transforms |
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by Pascal Getreuer |
This package converts colors between sRGB, Y'PbPr,
Y'CbCr, JPEG-Y'CbCr, Y'UV, Y'IQ,
Y'DbDr, HSV,
HSL, HSI, CIE XYZ, CIE L*a*b* (CIELAB), CIE L*u*v* (CIELUV), and CIE L*ch
(CIELCH), and CIE CAT02 LMS. It can be used either as part of a C/C++ program or compiled as
a Matlab MEX function.
Contents
License (BSD)
Copyright © 2005–2010, Pascal Getreuer
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
- Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
- Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in
the documentation and/or other materials provided with the distribution.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS”
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.
Compiling
colorspace can be used either as part of a C/C++ program or compiled as
a Matlab MEX function.
To demonstrate colorspace for use in C programs, a small command line
program colorcalc is included. The program is compiled with GCC by
gcc colorcalc.c colorspace.c -lm -o colorcalc
This should produce a command line program colorcalc that converts
input sRGB values to other representations.
For use in Matlab,
colorspace is compiled as a MEX function by entering
mex colorspace.c
on the Matlab command console.
For MEX compiling to work, your system must have a C compiler and Matlab
must be configured to use it. For more information, see the help documentation for the mex command.
As an alternative to MEX, a pure M-code version colorspace.m is also included.
C Usage
First call GetColorTransform, specifying the source and destination
color spaces as "dest<-src" or "src->dest". Then call
ApplyColorTransform to perform the transform:
num S[3] = {173, 0.8, 0.5};
num D[3];
colortransform Trans;
if(!(GetColorTransform(&Trans, "HSI -> Lab")))
{
printf("Invalid syntax or unknown color space\n");
return;
}
ApplyColorTransform(Trans, &D[0], &D[1], &D[2], S[0], S[1], S[2]);
“num” is a typedef defined at the beginning of colorspace.h that may be set
to either double or float, depending on the application.
A colortransform may be applied any number of times. To transform the color representation
of an entire image, call GetColorTransform once and then call
ApplyColorTransform for each pixel.
Specific transformation routines can also be called directly. The following
converts an sRGB color to CIELAB and then back to sRGB:
num R = 0.85, G = 0.32, B = 0.5;
num L, a, b;
Rgb2Lab(&L, &a, &b, R, G, B);
Lab2Rgb(&R, &G, &B, L, a, b);
Generally, the calling syntax is
Foo2Bar(&B0, &B1, &B2, F0, F1, F2);
where (F0,F1,F2) are the coordinates of a color in space “Foo” and
(B0,B1,B2) are the transformed coordinates in space “Bar.” For any
transformation routine, its inverse has the syntax
Bar2Foo(&F0, &F1, &F2, B0, B1, B2);
The transform routines are consistently named with the first letter of a
color space capitalized with following letters in lower case and omitting
prime symbols. For example, Rgb2Ydbdr converts sRGB to Y'DbDr.
All transformations assume a two degree observer angle and a D65 illuminant.
The white point can be changed by modifying the WHITEPOINT_X, WHITEPOINT_Y,
WHITEPOINT_Z definitions at the beginning of colorspace.h.
List of transformation routines:
Rgb2Yuv(num *Y, num *U, num *V, num R, num G, num B)
Rgb2Ycbcr(num *Y, num *Cb, num *Cr, num R, num G, num B)
Rgb2Jpegycbcr(num *Y, num *Cb, num *Cr, num R, num G, num B)
Rgb2Ypbpr(num *Y, num *Pb, num *Pr, num R, num G, num B)
Rgb2Ydbdr(num *Y, num *Db, num *Dr, num R, num G, num B)
Rgb2Yiq(num *Y, num *I, num *Q, num R, num G, num B)
Rgb2Hsv(num *H, num *S, num *V, num R, num G, num B)
Rgb2Hsl(num *H, num *S, num *L, num R, num G, num B)
Rgb2Hsi(num *H, num *S, num *I, num R, num G, num B)
Rgb2Xyz(num *X, num *Y, num *Z, num R, num G, num B)
Xyz2Lab(num *L, num *a, num *b, num X, num Y, num Z)
Xyz2Luv(num *L, num *u, num *v, num X, num Y, num Z)
Xyz2Lch(num *L, num *C, num *h, num X, num Y, num Z)
Xyz2Cat02lms(num *L, num *M, num *S, num X, num Y, num Z)
Rgb2Lab(num *L, num *a, num *b, num R, num G, num B)
Rgb2Luv(num *L, num *u, num *v, num R, num G, num B)
Rgb2Lch(num *L, num *C, num *h, num R, num G, num B)
Rgb2Cat02lms(num *L, num *M, num *S, num R, num G, num B)
(Similarly for the inverse transformations.)
It is possible to transform between two arbitrary color spaces by first
transforming from the source space to sRGB and then transforming from
sRGB to the desired destination space. For transformations between CIE
color spaces, it is convenient to use XYZ as the intermediate space. This
is the strategy used by GetColorTransform and ApplyColorTransform.
MATLAB Usage
B = colorspace(S,A) converts the color
representation of image A where S is a string
specifying the conversion. S tells the source and
destination color spaces, S = 'dest<-src', or
alternatively, S = 'src->dest'. Supported color
spaces are
| 'RGB' | sRGB IEC 61966-2-1 |
| 'YPbPr' | Luma (ITU-R BT.601) + Chroma |
| 'YCbCr' | Luma + Chroma (digitized version of
Y'PbPr) |
| 'JPEG-YCbCr' | Luma + Chroma space used in JFIF JPEG |
| 'YUV' | NTSC PAL Y'UV Luma + Chroma |
| 'YIQ' | NTSC Y'IQ Luma + Chroma |
| 'YDbDr' | SECAM Luma + Chroma |
| 'HSV' or 'HSB' | Hue Saturation Value/Brightness |
| 'HSL' or 'HLS' | Hue Saturation Luminance |
| 'HSI' | Hue Saturation Intensity |
| 'XYZ' | CIE XYZ |
| 'Lab' | CIE L*a*b* (CIELAB) |
| 'Luv' | CIE L*u*v* (CIELUV) |
| 'LCH' | CIE L*C*H* (CIELCH) |
| 'CAT02 LMS' | CIE CAT02 LMS |
All conversions assume 2 degree observer and D65
illuminant. Color space names are case insensitive. When sRGB is
the source or destination, it can be omitted. For example
'yuv<-' is short for 'yuv<-rgb'.
Matlab uses two standard data formats for
sRGB: double data with intensities in the range 0 to 1, and uint8
data with integer-valued intensities from 0 to 255. colorspace
expects sRGB data to be scaled between 0 and 1, and only accepts double data.
If A is an M×3 array, like a colormap,
B will also have size M×3.
Typical Usage
How does one get color image data into Matlab?
The function imread imports most formats as a uint8 array of
size M×N×3, where the third dimension separates the R',G',
and B' color channels.
A = imread('boats.png');
A = double(A)/255;
subplot(2,2,1);
image(A);
axis image
(For images using palette indexing, imread instead returns
an array of color indices and a colormap; palette-based images require
other handling.) To view a color image, use image or
imshow. The image may either be a uint8 array with
intensities in the range [0,255] or a double array with the range
[0,1].
Once an sRGB array is loaded, colorspace can convert it
to another color representation. To convert to
Y'PbPr, for example, use
B = colorspace('YPbPr<-RGB',A);
Since the resulting array B is not in sRGB
representation, it no longer makes sense to visualize it as a single
color image, other than transforming it back to sRGB first. Instead,
view each of the channels B(:,:,1),
B(:,:,2), B(:,:,3) individually as gray-scale
images:
subplot(2,2,1);
imagesc(B(:,:,1));
colormap(gray(256));
axis image
title 'Y'''
subplot(2,2,3);
imagesc(B(:,:,2));
colormap(gray(256));
axis image
title P_b
subplot(2,2,4);
imagesc(B(:,:,3));
colormap(gray(256));
axis image
title P_r
To transform B back to sRGB, use colorspace again:
ARecovered = colorspace('RGB<-YPbPr',B);
Discrepancies
While much effort has been made to make colorspace accurate and in agreement with
standards, it is possible to see differences between colorspace and other color
transformation software. Potential sources for discrepancy are
- differences in how the components are scaled, for example, sRGB values scaled in [0,255]
vs. [0,1];
- different conventions for handling out-of-gamut colors;
- naming confusion over similar but distinct color spaces
(e.g., HSV vs. HSL vs. HSI and also Y'PbPr vs. Y'UV vs. Y'IQ);
- (for CIE spaces) differences in the gamma correction function, observer angle, or
white point.
Finally, although hopefully unlikely, it is possible that a discrepancy is due to a bug in
colorspace.
colorspace does have the property that transformation of an in-gamut color followed
by the inverse transformation accurately recovers the original color (see the Accuracy Test).
Transformations with colorspace assume a 2 degree observe, D65 illuminant,
and using the gamma correction function from Gamma Correction.
Beware that colorspace's transformations generally do not constrain colors to be in-gamut.
Particularly, transforming from another space to sRGB may obtain
R'G'B' values outside of the [0,1] range. In Matlab,
the result should be clamped to [0,1] before displaying:
image(min(max(B,0),1));
Challenges in Color Representation
Device-independent, quantitative description of color is a
surprisingly challenging problem. For example, four shades of gray
surrounded by black are perceived differently than the same four
shades surrounded by white [1].
Surrounding illumination affects the perceived color.
On black background, the lightest shade of gray seems to be almost
white. But on white background, the same shade appears significantly
darker. This discrepancy suggests that the perceived colors on a
monitor depend on the illumination of the surrounding room. Thus for
precise color description, color specifications include the intended
viewing conditions.
The intensity of a color is defined as the watts per unit
area rendered by the display device. Another problem is that even
under equal intensity, some colors are visually brighter than
others.
Intensity ≠ Visual Brightness.
To overcome this non-uniformity, many color spaces instead consider
luminance, a quantitative estimate of the perceived
brightness.
Different color representations try to overcome these problems,
with varying degrees of success. It is for this reason that there are
so many standard color representations.
Gamma Correction
CRT monitors have a nonlinear relationship between the input
voltages and the rendered intensities. To reproduce an image
accurately, the image is gamma-corrected in such a way that the
monitor displays the desired intensities.
In colorspace, the “RGB” space is sRGB. The sRGB space was
designed in 1996 for direct display on typical CRT monitors and standardized
in 1999 by International Electrotechnical Commission (IEC) as IEC 61966-2-1.
The “s” in sRGB is to mean “standard.” From
linear RGB values, the sRGB values are gamma-compensated by the formula
|
R' = 1.055 R1/2.4 − 0.055 |
if R ≤ 0.0031308, |
|
R' = 12.92 R, |
if R > 0.0031308, |
and similarly for G' and B' [4].
A standard notation is to denote R',G',B' quantities and derived
quantities with a prime ' to signify gamma-correction. Since
gamma-correction is already applied by digital cameras as standard
practice, most digital image data should be interpreted as R'G'B',
and not RGB.
Y'CbCr
and other Luma+Chroma Representations
The luma of a color is an estimate of brightness based on
gamma-corrected samples. Its definition (ITU-R Recommendation
BT.601-4) is
Y'601 = 0.299 R' + 0.587 G' + 0.114 B'.
This luma measure is (up to a scale factor) the Y' in
Y'PbPr, Y'CbCr,
JPEG-Y'CbCr, Y'UV, Y'IQ, and
Y'DbDr. The remaining two components in each of
these representations capture the chroma, the part of a color
independent of luma [2].
Y'PbPr
Given R', G', and B' in the range [0,1],
the Y'PbPr components are
|
| = |
| |
0.299 | 0.587 | 0.114 |
|
| −0.1687367 |
−0.331264 | 0.5 |
| 0.5 | −0.418688 |
−0.081312 |
| × |
|
with Y' in [0,1] and Pb, Pr in
[−0.5,0.5].
Y'CbCr
Y'CbCr, also called YCC, is a rescaling of
Y'PbPr such that component can be stored as
8-bit unsigned values. Given R', G', and B' in the range [0,1],
|
| = |
| + |
| |
65.481 | 128.553 | 24.966 |
|
| −37.797 |
−74.203 | 112.0 |
| 112.0 | −93.786 |
−18.214 |
| × |
|
with Y' in [16,235] and Cb, Cr in
[16,240].
JPEG-Y'CbCr
JPEG-Y'CbCr is another rescaling of
Y'PbPr, used in the JPEG image format,
|
| = |
| + |
| |
0.299 | 0.587 | 0.114 |
|
| −0.1687367 |
−0.331264 | 0.5 |
| 0.5 | −0.418688 |
−0.081312 |
| × |
|
with Y', Cb, Cr in [0,1].
HSV, HSL, and HSI
The Hue Saturation Value/Brightness (HSV/HSB) is an intuitive color
system, measuring the hue of a color as the angle on the HSV
color wheel, the saturation as the color's vibrancy, and the
color's value or approximate brightness.
HSV is related to sRGB by
| H = hexagonal hue angle | (0 ≤ H < 360), |
| S = C/V | (0 ≤ S ≤ 1), |
| V = max(R',G',B') | (0 ≤ V ≤ 1), |
where C = max(R',G',B') − min(R',G',B'). The hue angle H is computed on
a hexagon. The space is geometrically a hexagonal cone.
Conic representation of the HSV and HSL color spaces
(colorspace_demo.m).
The Hue Saturation Lightness (HSL or HLS) color space, has the same definition for color hue
as HSV. The other two components differ such that all colors tend to
white as lightness increases.
HSL is related to sRGB by
| H = hexagonal hue angle | (0 ≤ H < 360), |
| S = C/(1 - |2L - 1|) | (0 ≤ S ≤ 1), |
| L = (max(R',G',B') + min(R',G',B'))/2 | (0 ≤ L ≤ 1), |
where H and C are the same as in HSV. Geometrically, the space is a double hexagonal cone.
A third related space is Hue Saturation Intensity (HSI), which is popular in
computer vision. HSI is related to sRGB by
| H = polar hue angle | (0 ≤ H < 360), |
| S = 1 - min(R',G',B')/I | (0 ≤ S ≤ 1), |
| I = (R'+G'+B')/3 | (0 ≤ I ≤ 1). |
Unlike HSV and HSL, the hue angle H is computed on a circle rather than a hexagon.
The HSV, HSL, and HSI systems are ambiguous on whether components should
be based on RGB or gamma-corrected sRGB, and specify no white point.
When truly device-independent color reproduction is necessary, it is
better to use a CIE color space [2].
CIE Standard Color Spaces
In 1931, the Commission Internationale de L'Éclairage (CIE)
defined a standard color system for precise color reproduction called
XYZ. The XYZ color space has a linear relationship with
non-gamma-corrected RGB [2]:
|
| = |
| |
3.240479 | −1.53715 | −0.498535 |
|
| −0.969256 |
1.875992 | 0.041556 |
| 0.055648 | −0.204043 |
1.057311 |
| × |
|
The closely-related xyY space defines the chromaticity
coordinates,
|
| |
| y |
| | | x |
The CIE “tongue”: the region of all colors over
x and y (colorspace_ciedemo.m). |
In the figure, the U-shaped boundary is parameterized by light wavelength.
The triangular region corresponds to the sRGB space, the range of colors
that a typical computer monitor can display.
XYZ is the foundation of the L*a*b* (CIELAB), L*u*v* (CIELUV), and
L*ch color spaces. Let Xn,Yn,Zn be
the XYZ values of a reference white point. The white point in
colorspace is the standard D65 white point,
Xn = 0.950456, Yn =
1, Zn = 1.088754. The lightness,
denoted by L* in each of these spaces, is defined as
|
L* = 116 (Y/Yn)1/3 − 16, |
if Y/Yn > (6/29)3 |
|
L* = (Y/Yn − 4/29) 108/841,
| if Y/Yn ≤ (6/29)3 |
The white point has lightness 100, and provided 0 ≤ Y ≤
Yn, L* is in the range [0,100].
The other two components in each representation describe the
chromaticity. L*a*b* and L*u*v* both attempt to "perceptually
linearize" chromaticity, meaning that changes in color values
correspond to proportional changes in visual importance. L*ch is
L*a*b* with chromaticity expressed in polar coordinates.
Visualizations of the L*a*b* and L*u*v* color spaces
(colorspace_demo.m).
Accuracy Test
To verify the invertibility of the color transformations, this test
transforms sRGB data to a space, inverts, and compares with the
original data.
N = 1e5;
A = rand(N,3);
Space = {'YPbPr','YCbCr','JPEG-YCbCr','YDbDr','YIQ','YUV','HSV',...
'HSL','HSI','XYZ','Lab','Luv','LCH','CAT02LMS'};
fprintf('\n Transform RMSE Error Max Error\n\n');
for k = 1:length(Space)
B = colorspace([Space{k},'<-RGB'],A);
R = colorspace(['RGB<-',Space{k}],B);
RMSE = sqrt(mean((A(:) - R(:)).^2));
MaxError = max(abs(A(:) - R(:)));
fprintf(' RGB<->%-10s %9.2e %9.2e\n', ...
Space{k}, RMSE, MaxError);
end
Transform RMSE Error Max Error
RGB<->YPbPr 9.07e-017 4.44e-016
RGB<->YCbCr 1.06e-016 5.55e-016
RGB<->JPEG-YCbCr 1.06e-016 5.55e-016
RGB<->YDbDr 8.72e-017 4.44e-016
RGB<->YIQ 8.17e-017 4.44e-016
RGB<->YUV 6.99e-017 3.54e-016
RGB<->HSV 7.28e-017 1.22e-015
RGB<->HSL 8.06e-017 1.22e-015
RGB<->HSI 1.10e-016 7.77e-016
RGB<->XYZ 2.30e-016 6.36e-015
RGB<->Lab 1.09e-015 2.10e-014
RGB<->Luv 7.98e-016 2.00e-014
RGB<->LCH 1.11e-015 2.39e-014
RGB<->CAT02 LMS 8.29e-016 1.30e-014
Transformations are accurate to machine precision. The first
six spaces, being linearly related to sRGB, have higher accuracy
than the nonlinearly-related spaces.
References
This material is based upon work supported by the National
Science Foundation under Award No. DMS-1004694. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the author(s)
and do not necessarily reflect the views of the National Science Foundation.
