CIECAM02#CAT02

The inner circle is the stimulus, from which the tristimulus values should be measured in CIE XYZ using the 2° standard observer. The intermediate circle is the proximal field, extending out another 2°. The outer circle is the background, reaching out to 10°, from which the relative luminance (Y_b) need be measured. If the proximal field is the same color as the background, the background is considered to be adjacent to the stimulus. Beyond the circles which comprise the display field (display area, viewing area) is the surround field (or peripheral area), which can be considered to be the entire room. The totality of the proximal field, background, and surround is called the adapting field (the field of view that supports adaptation—extends to the limit of vision).{{cite book|title=Colorimetry: Understanding the CIE System|chapter=The Future of Colorimetry in the CIE: Color Appearance|publisher=Wiley Interscience|first=János|last=Schanda|isbn=978-0-470-04904-4|page=359|year=2007}}

When referring to the literature, it is also useful to be aware of the difference between the terms adopted white point (the computational white point) and the adapted white point (the observer white point).{{cite book|url=https://books.google.com/books?id=zgeuoz_muCIC&q=%22adapting+field%22+background+environment&pg=PA86 |title=Computational Colour Science Using MATLAB|first=Stephen|last=Westland|author2=Ripamonti, Caterina|publisher=John Wiley & Sons|isbn=0-470-84562-7|year=2004}} The distinction may be important in mixed mode illumination, where psychophysical phenomena come into play. This is a subject of research.

CIECAM02 defines three surround(ing)s – average, dim, and dark – with associated parameters defined here for reference in the rest of this article:{{cite conference|title=The CIECAM02 Color Appearance Model|book-title=IS&T/SID Tenth Color Imaging Conference|last=Moroney|first=Nathan|author2=Fairchild, Mark D.|author3=Hunt, Robert W.G.|author4=Li, Changjun|author5=Luo, M. Ronnier|author6=Newman, Todd|url=https://scholarworks.rit.edu/other/143/|location=Scottsdale, Arizona|date=November 12, 2002|publisher=The Society for Imaging Science and Technology|isbn=0-89208-241-0}}

• {{nobr|1=S_R = L_sw / L_dw}}: ratio of the absolute luminance of the reference white (white point) measured in the surround field to the display area. The 0.2 coefficient derives from the "gray world" assumption (~18%–20% reflectivity). It tests whether the surround luminance is darker or brighter than medium gray.
• F: factor determining degree of adaptation
• c: impact of surrounding
• N_c: chromatic induction factor

For intermediate conditions, these values can be linearly interpolated.

The absolute luminance of the adapting field, which is a quantity that will be needed later, should be measured with a photometer. If one is not available, it can be calculated using a reference white:

:$$

L_A = \frac{E_w}{\pi} \frac{Y_b}{Y_w} = \frac{L_W Y_b}{Y_w}

where Y_b is the relative luminance of background, the {{nobr|1=E_w = πL_W}} is the illuminance of the reference white in lux, L_W is the absolute luminance of the reference white in cd/m², and Y_w is the relative luminance of the reference white in the adapting field. If unknown, the adapting field can be assumed to have average reflectance ("gray world" assumption): {{nobr|1=L_A = L_W / 5}}.

Note: Care should be taken not to confuse L_W, the absolute luminance of the reference white in cd/m², and L_w the red cone response in the LMS color space.

{{cleanup section|reason=Sharpened LMS should be just called RGB per original definition, Fairchild book, and CAM16 doc. Might need an explanation about how it's a funky LMS instead of an additive light model though.|date=February 2021}}

1. Convert to the "spectrally sharpened" CAT02 LMS space to prepare for adaptation. Spectral sharpening is the transformation of the tristimulus values into new values that would have resulted from a sharper, more concentrated set of spectral sensitivities. It is argued that this aids color constancy, especially in the blue region. (Compare Finlayson et al. 94, Spectral Sharpening:Sensor Transformations for Improved Color Constancy)
2. Perform chromatic adaptation using CAT02 (also known as the "modified CMCCAT2000 transform").
3. Convert to an LMS space closer to the cone fundamentals. It is argued that predicting perceptual attribute correlates is best done in such spaces.
4. Perform post-adaptation cone response compression.

Given a set of tristimulus values in XYZ, the corresponding LMS values can be determined by the M_CAT02 transformation matrix (calculated using the CIE 1931 2° standard colorimetric observer). The sample color in the test illuminant is:

:$$

\begin{bmatrix}

L\\

M\\

S

\end{bmatrix}

=

\mathbf{M}_\mathit{CAT02}

\begin{bmatrix}

X\\

Y\\

Z

\end{bmatrix},\quad

\mathbf{M}_\mathit{CAT02}

=

\begin{bmatrix}

\;\;\,0.7328 & 0.4296 & -0.1624\\

-0.7036 & 1.6975 & \;\;\,0.0061\\

\;\;\,0.0030 & 0.0136 & \;\;\,0.9834

\end{bmatrix}

.

Once in LMS, the white point can be adapted to the desired degree by choosing the parameter D. For the general CAT02, the corresponding color in the reference illuminant is:

:$\backslash begin\{align\}$

L_c &= \Big(\frac{Y_w L_{wr}}{Y_{wr} L_w} D + 1-D\Big)L\\

M_c &=\Big(\frac{Y_w M_{wr}}{Y_{wr} M_w} D + 1-D\Big)M\\

S_c &= \Big(\frac{Y_w S_{wr}}{Y_{wr} S_w} D + 1-D\Big)S\\

\end{align}

File:ciecam02 degree of adaptation.svg

where the {{nobr|Y_w / Y_wr}} factor accounts for the two illuminants having the same chromaticity but different reference whites.{{cite journal|title=Chromatic Adaptation Transforms|first=Robert W. G.|last=Hunt|author2=Changjun Li |author3=M. Ronnier Luo |publisher=Wiley Interscience|doi=10.1002/col.20085|journal=Color Research & Application|date=February 2005|volume=30|quote=Chromatic adaptation transforms (CATs) have appeared in different forms. The reasons for these forms, and the relationships between them, are described. The factors governing which type of CAT should be used in different applications are explained|page=69|issue=1}} The subscripts indicate the cone response for white under the test (w) and reference illuminant (wr). The degree of adaptation (discounting) D can be set to zero for no adaptation (stimulus is considered self-luminous) and unity for complete adaptation (color constancy). In practice, it ranges from 0.65 to 1.0, as can be seen from the diagram. Intermediate values can be calculated by:

:$$

D = F \left( 1 - \textstyle{\frac{1}{3.6}} e^{-(L_A + 42) / 92} \right)

where surround F is as defined above and L_A is the adapting field luminance in cd/m².

File:ciecam02 luminance level adaptation factor.svg

In CIECAM02, the reference illuminant has equal energy {{nobr|1=L_wr = M_wr = S_wr = 100}}) and the reference white is the perfect reflecting diffuser (i.e., unity reflectance, and {{nobr|1=Y_wr = 100}}) hence:

:$\backslash begin\{align\}$

L_c &= \Big(\frac{Y_w}{L_w} D + 1-D\Big)L\\

M_c &=\Big(\frac{Y_w}{M_w} D + 1-D\Big)M\\

S_c &= \Big(\frac{Y_w}{S_w} D + 1-D\Big)S\\

\end{align}

Furthermore, if the reference white in both illuminants have the Y tristimulus value ({{nobr|1=Y_wr = Y_w}}) then:

:$\backslash begin\{align\}$

L_c &= \Big(\frac{L_{wr}}{L_w} D + 1-D\Big)L\\

M_c &=\Big(\frac{M_{wr}}{M_w} D + 1-D\Big)M\\

S_c &= \Big(\frac{S_{wr}}{S_w} D + 1-D\Big)S\\

\end{align}

After adaptation, the cone responses are converted to the Hunt–Pointer–Estévez space by going to XYZ and back:

:$$

\begin{bmatrix}

L' \\

M' \\

S'

\end{bmatrix}

=

\mathbf{M}_H

\begin{bmatrix}

X_c \\

Y_c \\

Z_c

\end{bmatrix}

=

\mathbf{M}_H

\mathbf{M}_{CAT02}^{-1}

\begin{bmatrix}

L_c \\

M_c \\

S_c

\end{bmatrix}

:$$

\mathbf{M}_H

=

\begin{bmatrix}

\;\;\,0.38971 & 0.68898 & -0.07868 \\

-0.22981 & 1.18340 & \;\;\,0.04641 \\

\;\;\,0.00000 & 0.00000 & \;\;\,1.00000

\end{bmatrix}

File:ciecat02 response compression.svg

Note that the matrix above, which was inherited from CIECAM97s,Ming Ronnier Luo & Robert William Gainer Hunt: The structure of the CIE 1997 colour appearance model has the unfortunate property that since 0.38971 + 0.68898 – 0.07868 = 1.00001, 1⃗ ≠ M_H1⃗ and that consequently gray has non-zero chroma,Chunghui Kuo, Eric Zeise & Di Lai: Robust CIECAM02 implementation and numerical experiment within an International Color Consortium workflow an issue which CAM16 aims to address.Changjun Li, Zhiqiang Li, Zhifeng Wang, Yang Xu, Ming Ronnier Luo, Guihua Cui, Manuel Melgosa, Michael Henry Brill & Michael Pointer: Comprehensive color solutions: CAM16, CAT16, and CAM16-UCS

Finally, the response is compressed based on the generalized Michaelis–Menten equation (as depicted aside):

:$$

k = \frac{1}{5 L_A + 1}

:$$

F_L = \textstyle{\frac{1}{5}} k^4 \left( 5 L_A \right) + \textstyle{\frac{1}{10}} {(1 - k^4)}^2 {\left( 5 L_A \right)}^{1/3}

F_L is the luminance level adaptation factor.

:$\backslash begin\{align\}$

L'_a &= \frac{400 {\left(F_L L'/100\right)}^{0.42}}{27.13 + {\left(F_L L'/100\right)}^{0.42}} + 0.1 \\

M'_a &= \frac{400 {\left(F_L M'/100\right)}^{0.42}}{27.13 + {\left(F_L M'/100\right)}^{0.42}} + 0.1 \\

S'_a &= \frac{400 {\left(F_L S'/100\right)}^{0.42}}{27.13 + {\left(F_L S'/100\right)}^{0.42}} + 0.1

\end{align}

As previously mentioned, if the luminance level of the background is unknown, it can be estimated from the absolute luminance of the white point as {{nobr|1=L_A = L_W / 5}} using the "medium gray" assumption. (The expression for F_L is given in terms of 5L_A for convenience.) In photopic conditions, the luminance level adaptation factor (F_L) is proportional to the cube root of the luminance of the adapting field (L_A). In scotopic conditions, it is proportional to L_A (meaning no luminance level adaptation). The photopic threshold is roughly {{nobr|1=L_W = 1}} (see F_L–L_A graph above).

CIECAM02 defines correlates for yellow-blue, red-green, brightness, and colorfulness. Let us make some preliminary definitions.

:$\backslash begin\{align\}$

C_1 &= L^\prime_a - M^\prime_a \\

C_2 &= M^\prime_a - S^\prime_a \\

C_3 &= S^\prime_a - L^\prime_a

\end{align}

The correlate for red–green (a) is the magnitude of the departure of C₁ from the criterion for unique yellow ({{nobr|1=C₁ = C₂ / 11}}), and the correlate for yellow–blue (b) is based on the mean of the magnitude of the departures of C₁ from unique red ({{nobr|1=C₁ = C₂}}) and unique green ({{nobr|1=C₁ = C₃}}).

:$\backslash begin\{align\}$

a &= C_1 - \textstyle{\frac{1}{11}}C_2

&= L^\prime_a - \textstyle{\frac{12}{11}} M^\prime_a + \textstyle{\frac{1}{11}} S^\prime_a \\

b &= \textstyle{\frac{1}{2}} \left( C_2 - C_1 + C_1 - C_3 \right) / 4.5

&= \textstyle{\frac{1}{9}} \left( L^\prime_a + M^\prime_a - 2S^\prime_a \right)

\end{align}

The 4.5 factor accounts for the fact that there are fewer cones at shorter wavelengths (the eye is less sensitive to blue). The order of the terms is such that b is positive for yellowish colors (rather than blueish).

The hue angle (h) can be found by converting the rectangular coordinate (a, b) into polar coordinates:

:$$

h = \angle (a, b) = \operatorname{atan2}(b, a),\ (0 \le h < 360^\circ)

To calculate the eccentricity (e_t) and hue composition (H), determine which quadrant the hue is in with the aid of the following table. Choose i such that {{nobr|1=h_i ≤ h′ < h_i+1}}, where {{nobr|1=h′ = h}} if {{nobr|1=h > h₁}} and {{nobr|1=h′ = h + 360°}} otherwise.

:$\backslash begin\{align\}$

H &= H_i + \frac{100 (h^\prime - h_i) / e_i}{(h^\prime - h_i) / e_i + (h_{i+1} - h^\prime) / e_{i+1}} \\

e_t &= \textstyle{\frac{1}{4}} \left[ \cos\left( \textstyle{\frac{\pi}{180}}h + 2\right) + 3.8 \right]

\end{align}

(This is not exactly the same as the eccentricity factor given in the table.)

Calculate the achromatic response A:

:$$

A = (2 L^\prime_a + M^\prime_a + \textstyle{\frac{1}{20}} S^\prime_a - 0.305) N_{bb}

where

:$\backslash begin\{align\}$

&N_{bb} = N_{cb} = 0.725 n^{-0.2} \\

&n = Y_b / Y_w

\end{align}.

The correlate of lightness is

:$$

J = 100 \left( A / A_w \right)^{c z}

where c is the impact of surround (see above), and

:$$

z = 1.48 + \sqrt{n}

.

The correlate of brightness is

:$$

Q = \left(4 / c \right) \sqrt{\textstyle{\frac{1}{100}} J} \left(A_w + 4\right) F_L^{1/4}

.

Then calculate a temporary quantity t.

:$$

t = \frac{ \textstyle{\frac{50\,000}{13}} N_c N_{cb} e_t \sqrt{a^2+b^2} }

{ L_a^\prime + M_a^\prime + \textstyle{\frac{21}{20}} S_a^\prime }

The correlate of chroma is

:$$

C = t^{0.9} \sqrt {\textstyle{\frac{1}{100}} J} (1.64 - 0.29^n)^{0.73}

.

The correlate of colorfulness is

:$$

M = C \cdot F_L^{1/4}

.

The correlate of saturation is

:$$

s = 100 \sqrt {M / Q}

.

The appearance correlates of CIECAM02, J, a, and b, form a uniform color space that can be used to calculate color differences, as long as a viewing condition is fixed. A more commonly-used derivative is the CAM02 Uniform Color Space (CAM02-UCS), an extension with tweaks to better match experimental data.{{cite journal |last1=Luo |first1=M. Ronnier |last2=Cui |first2=Guihua |last3=Li |first3=Changjun |title=Uniform colour spaces based on CIECAM02 colour appearance model |journal=Color Research & Application |date=August 2006 |volume=31 |issue=4 |pages=320–330 |doi=10.1002/col.20227|s2cid=122917960 }}

Like many color models, CIECAM02 aims to model the human perception of color. The CIECAM02 model has been shown to be a more plausible model of neural activity in the primary visual cortex, compared to the earlier CIELAB model. Specifically, both its achromatic response A and red-green correlate a can be matched to EMEG activity (entrainment), each with their own characteristic delay.{{Cite journal |last1=Thwaites |first1=Andrew |last2=Wingfield |first2=Cai |last3=Wieser |first3=Eric |last4=Soltan |first4=Andrew |last5=Marslen-Wilson |first5=William D. |last6=Nimmo-Smith |first6=Ian |title=Entrainment to the CIECAM02 and CIELAB colour appearance models in the human cortex |journal=Vision Research |volume=145 |pages=1–10 |doi=10.1016/j.visres.2018.01.011 |year=2018 |doi-access=free |id={{doi|10.17863/CAM.21754}} }}

• CIELAB color space
• CIE 1931 color space
• Color appearance model

• {{cite book|author=CIE TC 8-01|series=Publication 159|year=2004|title=A Color appearance model for color management systems|location=Vienna|publisher=CIE Central Bureau|url=http://www.cie.co.at/publ/abst/159-04.html|isbn=3-901906-29-0|access-date=2008-04-28|archive-url=https://web.archive.org/web/20110413035718/http://www.cie.co.at/publ/abst/159-04.html|archive-date=2011-04-13|url-status=dead}}
• {{Cite web|title=Color Appearance Models: CIECAM02 and Beyond|first=Mark D.|last=Fairchild|series=IS&T/SID 12th Color Imaging Conference|date=2004-11-09|url=http://rit-mcsl.org/fairchild/PDFs/AppearanceLec.pdf |archive-url=https://web.archive.org/web/20200911101920/http://rit-mcsl.org/fairchild/PDFs/AppearanceLec.pdf |archive-date=2020-09-11 |access-date=2008-02-11}}
• {{cite book |last1=Schlömer |first1=Nico |title=Algorithmic improvements for the CIECAM02 and CAM16 color appearance models |year=2018 |arxiv=1802.06067}}

• [http://isp.uv.es/code/visioncolor/colorlab.html Colorlab] MATLAB toolbox for color science computation and accurate color reproduction (by Jesus Malo and Maria Jose Luque, Universitat de Valencia). It includes CIE standard tristimulus colorimetry and transformations to a number of non-linear color appearance models (CIELAB, CIECAM, etc.).
• [http://www.cis.rit.edu/fairchild/files/CIECAM02.XLS Excel spreadsheet with forward and inverse examples] {{Webarchive|url=https://web.archive.org/web/20070109143710/http://www.cis.rit.edu/fairchild/files/CIECAM02.XLS |date=2007-01-09 }}, by Eric Walowit and Grit O'Brien
• [http://cliff.rames.googlepages.com/ciecam02plugin Experimental Implementation of the CIECAM02 Color Appearance Model in a Photoshop Compatible Plug-in] (Microsoft Windows Only), by Cliff Rames.
• [http://scanline.ca/ciecam02/ Notes on the CIECAM02 Colour Appearance Model]. Source code in C of the forward and reverse transforms, by Billy Biggs.
• [https://web.archive.org/web/20080314163606/http://www.hpl.hp.com/personal/Nathan_Moroney/ciecam02/ciecam02.html CIECAM02 Java applet], by Nathan Moroney
• :Although Java applets no longer run on any major browser, this page also offers command line executables for Windows, Mac OS X and HP-UX. Although undocumented on the page itself, the use of these executables isn't all that hard, for example on Windows:
• :>%TEMP%\cam02vc echo 95.01 100 108.82 200 18 1&&>%TEMP%\cam02xyz echo 40 20 10&&ciecam02 0 1 0 %TEMP%\cam02vc %TEMP%\cam02xyz con
• :And similarly for other platforms. The first three numbers are the white point to use, then the average surround lighting, in this case 200 cd/m², then the relative luminance of the surround on the same scale as the white point, in this case 18%, then the surround conditions, where 1 = average, 2 = dim and 3 = dark, and then XYZ coordinates of the color to check. The result will be the JCh coordinates. The bits 0 1 0 mean ‘forward, verbose, calculate D’, so change the first to 1 to convert from JCh to XYZ, the second to 0 to not print the intermediate values in the calculation, or the last to 1 to force the D parameter to 1.

{{Color space}}

Category:Color appearance models

Category:Vision