Between additive and subtractive color mixings: intermediate mixing models Lionel Simonot, Mathieu Hébert To cite this version: Lionel Simonot, Mathieu Hébert. Between additive and subtractive color mixings: intermediate mixing models. Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of America, 2013, 31 (1), pp.58-66. 10.1364/JOSAA.31.000058. hal-00962256 HAL Id: hal-00962256 https://hal.archives-ouvertes.fr/hal-00962256 Submitted on 20 Sep 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 58 J. Opt. Soc. Am. A / Vol. 31, No. 1 / January 2014 L. Simonot and M. Hébert Between additive and subtractive color mixings: intermediate mixing models Lionel Simonot1,* and Mathieu Hébert2 1Institut Pprime, Université de Poitiers, CNRS UPR 3346, Département Physique et mécanique des matériaux, SP2MI, BP 30179, 86962 Chasseneuil, Futuroscope Cedex, France 2Université de Lyon, Université Jean Monnet de Saint-Etienne, CNRS UMR5516, Laboratoire Hubert Curien, F-42000 Saint-Etienne, France *Corresponding author: lionel.simonot@univ‑poitiers.fr Received July 18, 2013; revised October 16, 2013; accepted November 11, 2013; posted November 13, 2013 (Doc. ID 194145); published December 5, 2013 The color rendering of superposed coloring components is often an issue either to predict or to simulate the ap- pearance of colored surfaces. In graphical software, for example, transparence options are available to display different layouts on top of each other. With two colored layers, tuning the transparency of the top layer enables transitioning continuously from the color of this top layer to the color of the bottom layer. However, these options based very often only on additive color mixing offer limited transitions between two base colors. It would be advantageous to introduce more advanced options, providing, for example, realistic renderings of superposed paint layers. This is the aim of the present study, where simple models are proposed to create intermediate con- figurations between additive and subtractive color mixings. These models rely on the spectral power distribution of a finite set of primaries with given proportions. They may be extended to RGB color reflective or transmissive systems if the red, green, and blue wavebands do not overlap each other. An additional parameter is introduced to tune the proportions of additive and subtractive mixings, each type of mixing being based on its set of primaries. Various simulations of color mixings are presented, illustrating the possibilities offered by this model in addition to those permitted by the purely additive and subtractive mixings. © 2013 Optical Society of America OCIS codes: (300.6550) Spectroscopy, visible; (330.1690) Color; (330.1730) Colorimetry. http://dx.doi.org/10.1364/JOSAA.31.000058 1. INTRODUCTION subtractive. The ideal additive and subtractive mixing are in- What is the result of the mixing between different colors? The cluded in these new mixing laws as limit cases. The mixing answer of course depends on the relative proportions of the laws deal with spectral data with a minimum of three spectral different primaries, but also on the type of color mixing. In bands. The models can also be satisfyingly extended to RGB additive mixing, color is obtained by mixing lights with differ- color systems in the case in which the wavebands defining the ent spectral power distributions either by projection of col- red, green, and blue channels do not overlap each other. ored light beams or by juxtaposition of small colored areas as in displays. The colors are synthesized by mixing at least 2. DEFINITION OF COLORS three primaries, generally not more: red, green, and blue. In The color P of a surface comes from the interpretation by the subtractive mixing, colors are obtained by superposing col- human visual system of the light signal reflected or transmit- ored layers behaving as spectral filters for white light, like ted by the surface, especially its spectral distribution denoted the inks in half-tone printing. Three primaries are commonly Pλ. The number N of wavebands depends on the measurement used: cyan, magenta, and yellow. The distinction between system. At least three wavebands in the visible range are additive and subtractive color mixing was first exposed by needed to define colors, but N may be larger, or much larger Helmholtz [1] in 1860 in order to clarify the previous confusion in the cases of multi and hyper-spectral systems. By normal- between light and colorant mixture. In many color ordering izing the measured spectrum by a reference spectrum consid- systems [2] and in the habits of painters even today, the ered as the maximum signal (direct incident light for primaries are often red, yellow, and blue. There is a large transmittance measurements, and light scattered by a white dispersion in the precise definition of these primaries depend- standard for reflectance measurements), the spectrum satis- ing on the available pigments. In practice, e.g., for painting, the fies the property mixing laws are complex to simulate. Nevertheless, the result- ing color in general stays between those predicted by the ideal 0 ≤ Pλ ≤ 1 ∀ λ: (1) additive and by the ideal subtractive mixing. In half-tone printing, Yule and Nielsen [3] defined such an intermediate The color P can be considered as a vector in an N- empirical mixing law. In this paper, we propose other easily dimensional space (with N ≥ 3), and Pλ are its coordinates. understandable color mixing laws governed by intuitive From the spectral values, the color can be defined more con- parameters: the component proportions and a mixing param- ventionally by using the standardization of the CIE [4,5]. The eter indicating whether the color synthesis is additive or CIEXYZ tristimulus values are obtained by multiplying the 1084-7529/14/010058-09$15.00/0 © 2014 Optical Society of America L. Simonot and M. Hébert Vol. 31, No. 1 / January 2014 / J. Opt. Soc. Am. A 59 spectral value Pλ, the standardized illuminant radiance spec- decomposed into different subareas and/or different sub- trum Sλ, and each of the matching color functions x¯ λ, y¯ λ, layers as shown in Fig. 1. The following assumptions are made and z¯λ, and then summing the resulting spectrum over the regarding the optical properties of subareas and sublayers: visible wavelength domain: — They do not scatter light and therefore satisfy Beer– 8 P – ¯ Lambert Bouguer law (i.e., the transmittance of superposed < X kPλ PλSλxλ ’ ¯ components is the product of the component s individual : Y kPλ PλSλyλ ; 2 transmittances). ¯ Z k λ PλSλzλ — The multiple reflections of light between superposed components are not taken into account. 100 where the factor k is a normalization factor giving Y — Lateral propagation of light between neighboring subar- 1 for the reference whose spectral values are Pλ for every eas can be neglected. waveband. For object colors, the CIELAB color space is classically used. The coordinates—lightness L, green/red These strong assumptions allow defining some ideal coordinate a, and blue/yellow coordinate b—can be derived mixing laws respecting the previously stated rules. from the CIEXYZ tristimulus values by nonlinear rela- Let us denote as ai the fractional areas of the subareas, and tions [4,5]. as ti the fractional thickness of the sublayers. In purely additive mixing, the color cell is composed of N juxtaposed filters of unit thickness that thus represent 3. COLOR MIXING RULES c the primaries Pi [Fig. 1(a) for Nc 3]. The fractional areas P ∈ 1 … Let us consider Nc primaries i with i f ; ;Ncg. Nc is com- of these subareas, ai, satisfying monly 2 (binary mixing) or 3 (ternary mixing). But Nc can be XNc larger to precisely model color reproduction systems; for ex- ai 1; (5) ample, for RGB displays, the black background can be num- i1 bered among the primaries (N 4), or in a CMY trichromy, c are the proportions c of the mixing. The additive mixing is a the bare paper, or the paper with one, two, or three ink layers, i linear combination of the primary spectral values: are all considered as primaries (Nc 8). Several color mixing N configurations can be defined from the spectra Pi;λ. The Xc following rules are an attempt to formalize some of these Pλ ciPi;λ (6) configurations. i1 P Let us denote as the color obtained by mixing the Nc with ci ai. primaries Pi with respective proportions ci satisfying the conditions 0 ≤ ci ≤ 1 (3) and XN ci 1: (4) i1 • Rule 1: The spectral values Pλ attached to P satisfy the property (1). • Rule 2: The proportions ci should have the same influ- ence on Pλ for every waveband λ in the visible spectrum. • Rule 3: The mixing operation is commutative: P is inde- pendent of the ordering of the components in the mixing. • Rule 4: By considering the mixing between two colors with respective proportions c and 1 − c, the relation Pλ c is continuous and monotonous with c in every waveband λ. Rule 4 induces the following logical property: the color re- sulting from the mixing of color P1 with itself is again color P1 for every proportion c. It is important to note that counterexamples can be found where these rules are not satisfied, e.g., color mixing with photoluminescence effects where light emission at one wavelength greatly depends on the absorption at a smaller wavelength.