LMS color space

The Hunt and RLAB color appearance models use the Hunt-Pointer-Estevez transformation matrix (M_HPE) for conversion from CIE XYZ to LMS.[3][4][5] This is the transformation matrix which was originally used in conjunction with the von Kries transform method, and is therefore also called von Kries transformation matrix (M_vonKries).

• Equal-energy illuminants:
  $${\displaystyle {\begin{bmatrix}L\\M\\S\end{bmatrix}}_{\text{E}}=\left[{\begin{array}{lll}{\phantom {-}}0.38971&{\phantom {-}}0.68898&-0.07868\\-0.22981&{\phantom {-}}1.18340&{\phantom {-}}0.04641\\{\phantom {-}}0&{\phantom {-}}0&{\phantom {-}}1\end{array}}\right]{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}$$

  
• Normalized[6] to D65:
  $${\displaystyle {\begin{bmatrix}L\\M\\S\end{bmatrix}}_{\text{D65}}=\left[{\begin{array}{lll}{\phantom {-}}0.4002&{\phantom {-}}0.7076&-0.0808\\-0.2263&{\phantom {-}}1.1653&{\phantom {-}}0.0457\\{\phantom {-}}0&{\phantom {-}}0&{\phantom {-}}0.9182\end{array}}\right]{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}$$

  

The original CIECAM97s color appearance model uses the Bradford transformation matrix (M_BFD) (as does the LLAB color appearance model).[2] 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 in ICC profiles.[7]

$${\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}_{\text{BFD}}=\left[{\begin{array}{lll}{\phantom {-}}0.8951&{\phantom {-}}0.2664&-0.1614\\-0.7502&{\phantom {-}}1.7135&{\phantom {-}}0.0367\\{\phantom {-}}0.0389&-0.0685&{\phantom {-}}1.0296\end{array}}\right]{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}$$

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):[8]

$${\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}_{\text{97}}=\left[{\begin{array}{lll}{\phantom {-}}0.8562&{\phantom {-}}0.3372&-0.1934\\-0.8360&{\phantom {-}}1.8327&{\phantom {-}}0.0033\\{\phantom {-}}0.0357&-0.0469&{\phantom {-}}1.0112\end{array}}\right]{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}$$

The sharpened transformation matrix in CIECAM02 (M_CAT02) is:[9][2]

$${\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}_{\text{02}}=\left[{\begin{array}{lll}{\phantom {-}}0.7328&{\phantom {-}}0.4296&-0.1624\\-0.7036&{\phantom {-}}1.6975&{\phantom {-}}0.0061\\{\phantom {-}}0.0030&{\phantom {-}}0.0136&{\phantom {-}}0.9834\end{array}}\right]{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}$$

CAM16 uses a different matrix:[10]

$${\displaystyle {\begin{bmatrix}R\\G\\B\end{bmatrix}}_{\text{16}}=\left[{\begin{array}{lll}{\phantom {-}}0.401288&{\phantom {-}}0.650173&-0.051461\\-0.250268&{\phantom {-}}1.204414&{\phantom {-}}0.045854\\-0.002079&{\phantom {-}}0.048952&{\phantom {-}}0.953127\end{array}}\right]{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}}$$

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

A set of physiologically-based LMS functions are proposed by Stockman & Sharpe in 2000. The function has been published in a technical report by the CIE in 2006 (CIE 170).[11] The functions are derived from Stiles and Burch (1959) 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 the lens and the macula lutea are used.[12]

The Stockman & Sharpe functions can then be turned into a set of three color-matching functions similar to those in CIEXYZ:[13]

$${\displaystyle {\begin{bmatrix}X\\Y\\Z\end{bmatrix}}_{\text{F}}=\left[\,{\begin{array}{lll}1.94735469&-1.41445123&{\phantom {-}}0.36476327\\0.68990272&{\phantom {-}}0.34832189&{\phantom {-}}0\\0&{\phantom {-}}0&{\phantom {-}}1.93485343\end{array}}\right]{\begin{bmatrix}L\\M\\S\end{bmatrix}}}$$

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

$${\displaystyle {\begin{bmatrix}L\\M\\S\end{bmatrix}}=\left[{\begin{array}{lll}{\phantom {-}}0.210576&{\phantom {-}}0.855098&-0.0396983\\-0.417076&{\phantom {-}}1.177260&{\phantom {-}}0.0786283\\{\phantom {-}}0&{\phantom {-}}0&{\phantom {-}}0.5168350\\\end{array}}\right]{\begin{bmatrix}X\\Y\\Z\end{bmatrix}}_{\text{F}}}$$

Mathematical development

A spectral distribution ${\displaystyle J(\lambda )}$ composed of three monochromatic sources at wavelengths ${\displaystyle \lambda _{R},\lambda _{G},\lambda _{B}}$ may be written: ${\displaystyle J(\lambda )=a_{R}\delta (\lambda -\lambda _{R})+a_{G}\delta (\lambda -\lambda _{G})+a_{B}\delta (\lambda -\lambda _{B})=a_{i}\delta _{i}(\lambda )}$ where the index i is either R, G, or B, the ${\displaystyle a_{i}}$ are the intensities of each emitter, and index notation is being used whereby duplicate indices occur only in pairs, and summation over those indices is implied. In the above expression, the Dirac delta function vector is given by: ${\displaystyle \delta _{i}(\lambda )=(\delta (\lambda -\lambda _{R}),\delta (\lambda -\lambda _{G}),\delta (\lambda -\lambda _{B}))}$ Let ${\displaystyle P_{i}=(L,M,S)}$ be the vector of LMS coordinates for the ${\displaystyle J(\lambda )}$ spectral distribution, and let ${\displaystyle {\mathcal {P}}_{i}(\lambda )=({\mathcal {\ell }}(\lambda ),{\mathcal {m}}(\lambda ),{\mathcal {s}}(\lambda ))}$ be the vector function of wavelength whose elements are the three cone response functions (not implying any particular estimate of their values). Then the LMS coordinates of the spectral distribution are, by Grassmann's laws: ${\displaystyle P_{i}=\int _{0}^{\infty }J(\lambda ){\mathcal {P}}_{i}(\lambda )d\lambda ={\mathcal {P}}_{ij}a_{j}}$ where: ${\displaystyle {\mathcal {P}}_{ij}=\int _{0}^{\infty }\delta _{j}(\lambda ){\mathcal {P}}_{i}(\lambda )d\lambda ={\mathcal {P}}_{i}(\lambda _{j})}$ In an analogous fashion, the XYZ coordinates for the spectral distribution are ${\displaystyle Q_{i}=(X,Y,Z)}$, the XYZ color matching functions are ${\displaystyle {\mathcal {Q}}_{i}(\lambda )=({\mathcal {x}}(\lambda ),{\mathcal {y}}(\lambda ),{\mathcal {z}}(\lambda ))}$ and the XYZ coordinates may be written as: ${\displaystyle Q_{i}={\mathcal {Q}}_{ij}a_{j}}$ where ${\displaystyle {\mathcal {Q}}_{ij}={\mathcal {Q}}_{i}(\lambda _{j})}$. The XYZ coordinates may then be expressed in terms of the LMS coordinates as: ${\displaystyle Q_{i}={\mathcal {T}}_{ij}P_{j}}$ where the transformation matrix ${\displaystyle {\mathcal {T}}_{ij}}$ is given by: ${\displaystyle {\mathcal {T}}_{ij}={\mathcal {Q}}_{ik}{\mathcal {P}}_{kj}^{-1}}$ Given the improved CIE 2006 cone response functions ${\displaystyle {\mathcal {P}}_{i}(\lambda )}$[14], Stockman[15] proposed a new specification of the XYZ color matching functions ${\displaystyle {\mathcal {Q}}_{i}(\lambda )}$[16] which were more or less consistent in concept with the CIE 1931 functions, and which thereby specified the transformation matrix ${\displaystyle {\mathcal {T}}_{ij}}$ shown above.

The LMS color space can be used to emulate the way color-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.[17]

A related application is making color filters for color-blind people to more easily notice differences in color, a process known as daltonization.[18]

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

$${\displaystyle {\begin{bmatrix}X\\Y\\B\end{bmatrix}}={\begin{bmatrix}1&-1&{\phantom {-}}0\\1&{\phantom {-}}1&{\phantom {-}}0\\0&{\phantom {-}}0&{\phantom {-}}1\end{bmatrix}}{\begin{bmatrix}L\\M\\S\end{bmatrix}}}$$

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.[19]

The colorspace originates from Guetzli's butteraugli metric,[20] and was passed down to JPEG XL via Google's Pik project.