LMS (long, medium, short), is acolor space which represents the response of the three types ofcone cells of thehuman eye, named for theirresponsivity (sensitivity) peaks at long, medium, and short wavelengths.

The numerical range is generally not specified, except that the lower end is generally bounded by zero. It is common to use the LMS color space when performingchromatic adaptation (estimating the appearance of a sample under a different illuminant). It is also useful in the study ofcolor blindness, when one or more cone types are defective.

The coneresponse functions${\displaystyle \overline{l}(\lambda ),\overline{m}(\lambda ),\overline{s}(\lambda )}$ are the color matching functions (CMFs) for the LMS color space. The chromaticity coordinates (L, M, S) for a spectral distribution${\displaystyle J(\lambda )}$ are defined as:

${\displaystyle L={\int }_0^{\mathrm{\infty }}J(\lambda )\overline{l}(\lambda )d\lambda }$

${\displaystyle M={\int }_0^{\mathrm{\infty }}J(\lambda )\overline{m}(\lambda )d\lambda }$

${\displaystyle S={\int }_0^{\mathrm{\infty }}J(\lambda )\overline{s}(\lambda )d\lambda }$

The cone response functions are normalized to have their maxima equal to unity.

Typically, colors to be adapted chromatically will be specified in a color space other than LMS (e.g.sRGB). The chromatic adaptation matrix in the diagonalvon Kries transform method, however, operates ontristimulus values in the LMS color space. Since colors in most colorspaces can be transformed to theXYZ color space, only one additionaltransformation matrix is required for any color space to be adapted chromatically: to transform colors from the XYZ color space to the LMS color space.^[3]

In addition, many color adaption methods, orcolor appearance models (CAMs), run a von Kries-style diagonal matrix transform in a slightly modified, LMS-like, space instead. They may refer to it simply as LMS, as RGB, or as ργβ. The following text uses the "RGB" naming, but do note that the resulting space has nothing to do with the additive color model called RGB.^[3]

The chromatic adaptation transform (CAT) matrices for some CAMs in terms ofCIEXYZ coordinates are presented here. The matrices, in conjunction with the XYZ data defined for thestandard observer, implicitly define a "cone" response for each cell type.

Notes:

• Alltristimulus values are normally calculated using theCIE 1931 2° standard colorimetric observer.^[3]
• Unless specified otherwise, the CAT matrices are normalized (the elements in a row add up to 1) so the tristimulus values for an equal-energy illuminant (X=Y=Z), likeCIE Illuminant E, produce equal LMS values.^[3]

This articleis missing information about how the HPE matrix was derived – looks like the most "physiological" of the XYZ bunch, but where's the data?. Please expand the article to include this information. Further details may exist on thetalk page.(October 2021)

TheHunt andRLAB color appearance models use theHunt–Pointer–Estevez transformation matrix (M_HPE) for conversion fromCIE XYZ to LMS.^[4][5][6] This is the transformation matrix which was originally used in conjunction with thevon Kries transform method, and is therefore also calledvon Kries transformation matrix (M_vonKries).

• Equal-energy illuminants:
• Normalized^[7] toD65:

The originalCIECAM97s color appearance model uses theBradford transformation matrix (M_BFD) (as does theLLAB color appearance model).^[3] This is a “spectrally sharpened” transformation matrix (i.e. the L and M cone response curves are narrower and more distinct from each other). The Bradford transformation matrix was supposed to work in conjunction with a modified von Kries transform method which introduced a small non-linearity in the S (blue) channel. However, outside of CIECAM97s and LLAB this is often neglected and the Bradford transformation matrix is used in conjunction with the linear von Kries transform method, explicitly so inICC profiles.^[8]

A "spectrally sharpened" matrix is believed to improve chromatic adaptation especially for blue colors, but does not work as a real cone-describing LMS space for later human vision processing. Although the outputs are called "LMS" in the original LLAB incarnation, CIECAM97s uses a different "RGB" name to highlight that this space does not really reflect cone cells; hence the different names here.

LLAB proceeds by taking the post-adaptation XYZ values and performing a CIELAB-like treatment to get the visual correlates. On the other hand, CIECAM97s takes the post-adaptation XYZ value back into the Hunt LMS space, and works from there to model the vision system's calculation of color properties.

A revised version of CIECAM97s switches back to a linear transform method and introduces a corresponding transformation matrix (M_CAT97s):^[9]

The sharpened transformation matrix inCIECAM02 (M_CAT02) is:^[10][3]

CAM16 uses a different matrix:^[11]

As in CIECAM97s, after adaptation, the colors are converted to the traditional Hunt–Pointer–Estévez LMS for final prediction of visual results.

From a physiological point of view, the LMS color space describes a more fundamental level of human visual response, so it makes more sense to define the physiopsychological XYZ by LMS, rather than the other way around.

A set of physiologically-based LMS functions were proposed by Stockman & Sharpe in 2000. The functions have been published in a technical report by the CIE in 2006 (CIE 170).^[12][13] The functions are derived from Stiles and Burch^[1] RGB CMF data, combined with newer measurements about the contribution of each cone in the RGB functions. To adjust from the 10° data to 2°, assumptions about photopigment density difference and data about the absorption of light by pigment in thelens and themacula lutea are used.^[14]

The Stockman & Sharpe functions can then be turned into a set of three color-matching functions similar to theCIE 1931 functions.^[15]

Let${\displaystyle {\mathcal{P}}_i(\lambda )=(\overline{l}(\lambda ),\overline{m}(\lambda ),\overline{s}(\lambda ))}$ be the three cone response functions, and let${\displaystyle {\mathcal{Q}}_i(\lambda )=({\overline{x}}_{\text{F}}(\lambda ),{\overline{y}}_{\text{F}}(\lambda ),{\overline{z}}_{\text{F}}(\lambda ))}$ be the new XYZ color matching functions. Then, by definition, the new XYZ color matching functions are:

${\displaystyle {\mathcal{Q}}_i(\lambda )=\sum _{j=1}^3{T}_{ij}{\mathcal{P}}_j(\lambda )}$

where the transformation matrix${\displaystyle T_{ij}}$ is defined as:

The derivation of this transformation is relatively straightforward.^[16] The${\displaystyle {\overline{y}}_{\text{F}}(\lambda )}$ CMF is the luminous efficiency function originally proposed by Sharpe et al. (2005),^[17] but then corrected (Sharpe et al., 2011^[18][a]). The${\displaystyle {\overline{z}}_{\text{F}}(\lambda )}$ CMF is equal to the${\displaystyle \overline{s}(\lambda )}$ cone fundamental originally proposed by Stockman, Sharpe & Fach (1999)^[19] scaled to have an integral equal to the${\displaystyle {\overline{y}}_{\text{F}}(\lambda )}$ CMF. The definition of the${\displaystyle {\overline{x}}_{\text{F}}(\lambda )}$ CMF is derived from the following constraints:

1. Like the other CMFs, the values of${\displaystyle {\overline{x}}_{\text{F}}(\lambda )}$ are all positive.
2. The integral of${\displaystyle {\overline{x}}_{\text{F}}(\lambda )}$ is identical to the integrals for${\displaystyle {\overline{y}}_{\text{F}}(\lambda )}$ and${\displaystyle {\overline{z}}_{\text{F}}(\lambda )}$.
3. The coefficients of the transformation that yields${\displaystyle {\overline{x}}_{\text{F}}(\lambda )}$ are optimized to minimize the Euclidean differences between the resulting${\displaystyle {\overline{x}}_{\text{F}}(\lambda )}$,${\displaystyle {\overline{y}}_{\text{F}}(\lambda )}$ and${\displaystyle {\overline{z}}_{\text{F}}(\lambda )}$ color matching functions and the CIE 1931${\displaystyle \overline{x}(\lambda )}$,${\displaystyle \overline{y}(\lambda )}$ and${\displaystyle \overline{z}(\lambda )}$ color matching functions. — CVRL description for 'CIE (2012) 2-deg XYZ "physiologically-relevant" colour matching functions'^[15]

For any spectral distribution${\displaystyle J(\lambda )}$, let${\displaystyle P_i=(L,M,S)}$ be the LMS chromaticity coordinates for${\displaystyle J(\lambda )}$, and let${\displaystyle Q_i=(X,Y,Z{)}_{\text{F}}}$ be the corresponding new XYZ chromaticity coordinates. Then:

${\displaystyle Q_i={\int }_0^{\mathrm{\infty }}J(\lambda ){\mathcal{Q}}_i(\lambda )d\lambda ={\int }_0^{\mathrm{\infty }}J(\lambda )\sum _{j=1}^3{T}_{ij}{\mathcal{P}}_j(\lambda )d\lambda =\sum _{j=1}^3{T}_{ij}{P}_j}$

or, explicitly:

The inverse matrix is shown here for comparison with the ones for traditional XYZ:

The above development has the advantage of basing the new X_FY_FZ_F color matching functions on the physiologically-based LMS cone response functions. In addition, it offers a one-to-one relationship between the LMS chromaticity coordinates and the new X_FY_FZ_F chromaticity coordinates, which was not the case for the CIE 1931 color matching functions. The transformation for a particular color between LMS and the CIE 1931 XYZ space is not unique. It rather depends highly on the particular form of the spectral distribution${\displaystyle J(\lambda )}$ ) producing the given color. There is no fixed 3x3 matrix which will transform between the CIE 1931 XYZ coordinates and the LMS coordinates, even for a particular color, much less the entire gamut of colors. Any such transformation will be an approximation at best, generally requiring certain assumptions about the spectral distributions producing the color. For example, if the spectral distributions are constrained to be the result of mixing three monochromatic sources, (as was done in the measurement of the CIE 1931 and the Stiles and Burch^[1] color matching functions), then there will be a one-to-one relationship between the LMS and CIE 1931 XYZ coordinates of a particular color.

As of Nov 28, 2023, CIE 170-2 CMFs are proposals that have yet to be ratified by the full TC 1-36 committee or by the CIE.

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For theoretical purposes, it is often convenient to characterize radiation in terms of photons rather than energy. The energyE of a photon is given by thePlanck relation

${\displaystyle E=h\nu =hc/\lambda }$

whereE is the energy per photon,h is thePlanck constant,c is thespeed of light,ν is the frequency of the radiation andλ is the wavelength. A spectral radiative quantity in terms of energy,JE(λ), is converted to its quantal formJQ(λ) by dividing by the energy per photon:

${\displaystyle JQ(\lambda )=JE(\lambda )(\lambda /hc)}$

For example, ifJE(λ) isspectral radiance with the unit W/m²/sr/m, then the quantal equivalentJQ(λ) characterizes that radiation with the unit photons/s/m²/sr/m.

IfCE_λi(λ) (i=1,2,3) are the three energy-based color matching functions for a particular color space (LMS color space for the purposes of this article), then the tristimulus values may be expressed in terms of the quantal radiative quantity by:

${\displaystyle C{E}_i={\int }_0^{\mathrm{\infty }}JE(\lambda )C{E}_{\lambda i}(\lambda )d\lambda ={\int }_0^{\mathrm{\infty }}JQ(\lambda )(hc/\lambda )C{E}_{\lambda i}(\lambda )d\lambda }$

Define the quantal color matching functions:

${\displaystyle C{Q}_{\lambda i}(\lambda )=(C{E}_{\lambda i}(\lambda )/\lambda )/(C{E}_{\lambda i}({\lambda }_{i\text{max}})/{\lambda }_{i\text{max}})}$

whereλ_{i max} is the wavelength at whichCE_λi(λ)/λ is maximized. Define the quantal tristimulus values:

${\displaystyle C{Q}_i={\int }_0^{\mathrm{\infty }}JQ(\lambda )C{Q}_{\lambda i}(\lambda )d\lambda }$

Note that, as with the energy based functions, the peak value ofCQ_λi(λ) will be equal to unity. Using the above equation for the energy tristimulus valuesCE_i

${\displaystyle C{E}_i=(hc/{\lambda }_{imax})C{E}_{\lambda i}({\lambda }_{imax})C{Q}_i}$

For the LMS color space,${\displaystyle {\lambda }_{i\text{max}}}$ ≈ {566, 541, 441} nm and

${\displaystyle C{E}_i/C{Q}_i=\{3.49694,3.1253,0.144944\}\times {10}^{-19}}$ J/photon

The LMS color space can be used to emulate the waycolor-blind people see color. An early emulation of dichromats were produced by Brettel et al. 1997 and was rated favorably by actual patients. An example of a state-of-the-art method is Machado et al. 2009.^[20]

A related application is making color filters for color-blind people to more easily notice differences in color, a process known asdaltonization.^[21]

JPEG XL uses an XYB color space derived from LMS. Its transform matrix is shown here:

This can be interpreted as a hybrid color theory where L and M are opponents but S is handled in a trichromatic way, justified by the lower spatial density of S cones. In practical terms, this allows for using less data for storing blue signals without losing much perceived quality.^[22]

The colorspace originates fromGuetzli's butteraugli metric^[23] and was passed down to JPEG XL via Google's Pik project. Matt DesLauriers has produced aGist with the relevant parts from the reference implementation of JPEG XL translated into JavaScript.^[24]

• Color balance
• Color vision
• Luminous efficiency function
• Trichromacy

• Color space
• Color blindness

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