# Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

# Clapper-Yule Model

• Mathieu Hébert
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_49

## Definition

The Clapper-Yule model is a physically based model describing the reflection of spectral light fluxes by a printed surface and enabling the prediction of halftone prints on diffusing substrates [1]. The model relies on a closed-form equation obtained by describing the multiple transfers of light between the substrate and the print-air interface through the inks. Physical parameters are attached to the inks, the diffusing support, and the surface. The model assumes that the lateral propagation distance of light within the substrate, due to scattering, is much larger than the halftone screen period. Most photons therefore cross different ink dots while traveling in the print. The reflections and transmissions of light at the surface are explicitly taken into account depending on the print’s refractive index, as well as the considered illumination and measuring geometries.

## The Clapper-Yule Equation

The Clapper-Yule equation derives from the description of multiple reflections of light between the substrate and the print-air interface, represented in Fig. 1. The substrate, strongly diffusing, has a spectral reflectance r g (λ). The print-air interface reflects on its two sides: On the top side, a fraction r S of the incident light is reflected toward the detector (this fraction may be zero according to the geometry of illumination and detection). A fraction T in enters the print. On the back side, a fraction r i is reflected and a fraction T ex exits the print toward the detector. Regarding the inks, the model assumes that they are located between the surface and the substrate. If N different inks are used, the halftone is a mosaic of 2 N colors, also called Neugebauer primaries, resulting from the partial overlap of the ink dots. In the case of typical cyan, magenta, and yellow inks, there are eight Neugebauer primaries: white (no ink), cyan, magenta, yellow, red (magenta + yellow), green (cyan + yellow), blue (cyan + magenta), and black (cyan + magenta + yellow). Each primary occupies a fractional area a i on the surface and transmits a fraction t i (λ) of the light. The global attenuation for light crossing the halftone ink layer is therefore:
$${\displaystyle \sum {a}_i{t}_i\left(\lambda \right)}.$$
By denoting, I0(λ) and I1(λ) the downward fluxes illuminating the surface, respectively, the substrate, and as J0(λ) and J1(λ) the upward fluxes exiting the surface, respectively, the substrate, the following relations are obtained:
$${J}_0\left(\lambda \right)={r}_s{I}_0\left(\lambda \right)+\left[{T}_{ex}{\displaystyle \sum {a}_i{t}_i}\left(\lambda \right)\right]\;{J}_1\left(\lambda \right),{I}_1\left(\lambda \right)=\left[{T}_{in}{\displaystyle \sum {a}_i{t}_i\left(\lambda \right)}\right]{I}_0\left(\lambda \right)+\left[{r}_i{\displaystyle \sum {a}_i{t}_i^2\left(\lambda \right)}\right] {J}_1\left(\lambda \right),{J}_1\left(\lambda \right)={r}_g\left(\lambda \right){I}_1\left(\lambda \right),$$
from which is deduced the spectral reflectance R(λ) of the halftone print:
$$R\left(\lambda \right)=\frac{J_0\left(\lambda \right)}{I_0\left(\lambda \right)}={r}_s+\frac{T_{in}{T}_{ex}{r}_g\left(\lambda \right){\left[{\displaystyle \sum {a}_i{t}_i\left(\lambda \right)}\right]}^2}{1-{r}_i{r}_g\left(\lambda \right){\displaystyle \sum {a}_i{t}_i^2\left(\lambda \right)}}.$$
This equation is the general Clapper-Yule equation. In their original paper, Clapper and Yule considered halftones of one ink, with surface coverage a and transmittance t(λ):
$$R\left(\lambda \right)={r}_s+\frac{T_{in}{T}_{ex}{r}_g\left(\lambda \right){\left[1-a+ at\left(\lambda \right)\right]}^2}{1-{r}_i{r}_g\left(\lambda \right)\left[1-a+a{t}^2\left(\lambda \right)\right]}.$$
If one Neugebauer primary covers the whole surface (“solid primary layer”), the equation becomes:
$$R\left(\lambda \right)={r}_s+\frac{T_{in}{T}_{ex}{r}_g\left(\lambda \right){t}^2\left(\lambda \right)}{1-{r}_i{r}_g{t}^2\left(\lambda \right)},$$
and in the special case where no ink is printed (white colorant), it expresses the reflectance of the printing support:
$$R\left(\lambda \right)={r}_s+\frac{T_{in}{T}_{ex}{r}_g\left(\lambda \right)}{1-{r}_i{r}_g\left(\lambda \right)}.$$

## Fresnel Terms and Measuring Geometries

The terms T in , T ex , r S , and r i are derived from the Fresnel formulae for unpolarized light. The Fresnel reflectivity of an interface between media 1 and 2, with respective indices n1 and n2, is denoted as R12(θ) when the light comes from medium 1 at the angle θ. The term T in depends on the illumination geometry. It is typically:
$${T}_{in}=1-{R}_{12}\left(\theta \right)$$
when directional light incomes at angle θ, or
$${T}_{in}=1-{\displaystyle {\int}_{\theta =0}^{\pi /2}{R}_{12}\left(\theta \right) \sin 2\theta\;d\theta }$$
when the incident light is perfectly diffuse. T ex depends on the measuring geometry. When the print is observed in one direction θ′, it is:
$${T}_{ex}={\left({n}_1/{n}_2\right)}^2{T}_{12}\left({\theta}^{\prime}\right)$$
where the factor comes from the change of solid angle containing the exiting radiance due to the refraction. When an integrating sphere captures all light reflected by the print, it is:
$${T}_{ex}=1-{\displaystyle {\int}_{\theta =0}^{\pi /2}{R}_{21}\left(\theta \right) \sin 2\theta d\theta }.$$
The term r s corresponds to the surface reflection, i.e., the gloss. It depends on the illumination and observation geometries, by it is typically 0.05 when it is captured, or 0 when it is discarded from measurement.
The reflectance r i is the fraction of diffuse light emerging from the substrate which is reflected by the surface. It corresponds to the Lambertian reflectance of flat interfaces:
$${r}_i={\displaystyle {\int}_{\theta =0}^{\pi /2}{R}_{21}\left(\theta \right) \sin 2\theta d\theta }.$$
For example, for a print made of material of refractive index 1.5 illuminated by directional light at 45° and observed at 0° (so-called 45:0° measuring geometry), one has [2]:
$$\begin{array}{c}\hfill {T}_{in}=1-{R}_{12}\left(45{}^{\circ}\right)=0.95,\hfill \\ {}\hfill {T}_{ex}={T}_{01}\left(0{}^{\circ}\right)/{n}^2=\frac{1}{n^2}\left[1-\frac{{\left(1-n\right)}^2}{{\left(1+n\right)}^2}\right]=0.42,\hfill \\ {}\hfill {r}_s=0,\hfill \\ {}\hfill {r}_i=0.6.\hfill \end{array}$$
The Clapper-Yule equation becomes:
$$R\left(\lambda \right)=\frac{0.4{r}_g\left(\lambda \right){\left[{\displaystyle \sum {a}_i{t}_i\left(\lambda \right)}\right]}^2}{1-0.6{r}_g\left(\lambda \right){\displaystyle \sum {a}_i{t}_i^2\left(\lambda \right)}}.$$

## Calibration of the Model and Prediction

The calibration of the model requires measuring the spectral reflectance of a few halftones, as the one represented in Fig. 2 in the case of CMY halftones.
By measuring the spectral reflectance R w \left(λ) of the unprinted support (patch in row A of Fig. 2), whose expression is given above, one deduces the intrinsic spectral reflectance of the substrate thanks to the following formula:
$${r}_g\left(\lambda \right)=\frac{R_w\left(\lambda \right)-{r}_s}{T_{in}{T}_{ex}+{r}_i\left({R}_w\left(\lambda \right)-{r}_s\right)}.$$
Then, by measuring the spectral reflectances R i (λ) of the solid primary patches (patches in row B of Fig. 2), whose expression is given above, one deduces the intrinsic spectral reflectance of the substrate thanks to the following formula:
$${t}_i\left(\lambda \right)=\sqrt{\frac{R_i\left(\lambda \right)-{r}_s}{r_g\left(\lambda \right)\left[{T}_{in}{T}_{ex}+{r}_i\left({R}_i\left(\lambda \right)-{r}_s\right)\right]}}.$$
All spectral parameters are now known. The spectral reflectance of a given halftone can be predicted by the Clapper-Yule general equation as soon as the surface coverages of different primaries are known. In classical clustered-dot or error diffusion prints, the primary surface coverages can be deduced from the surface coverages of the inks according to the Demichel equations. In the case of CMY halftones, the Demichel equations relating the surface coverages of the eight primaries to the surface coverages c, m, and y of the cyan, magenta, and yellow inks are:
$$\begin{array}{c}\hfill {a}_w=\left(1-c\right) \left(1-m\right) \left(1-y\right),\hfill \\ {}\hfill {a}_c=c\left(1-m\right) \left(1-y\right),\hfill \\ {}\hfill {a}_m=\left(1-c\right)m\left(1-y\right),\hfill \\ {}\hfill {a}_y=\left(1-c\right) \left(1-m\right)y,\hfill \\ {}\hfill {a}_{m+y}=\left(1-c\right) my,\hfill \\ {}\hfill {a}_{c+y}=c\left(1-m\right)y,\hfill \\ {}\hfill {a}_{c+m}= cm\left(1-y\right),\hfill \\ {}\hfill {a}_{c+m+y}=cmy.\hfill \end{array}$$
Note that the prediction accuracy of the model sensibly depends on the exactitude of the primary surface coverage values. When the inks spread on the surface, which is almost always the case, the values for c, m, and y are larger than the ones defined in prepress. This phenomenon is generally called dot gain. The amount of ink spreading cannot been estimated by advance as it depends on several parameters, such as the chemical and mechanical properties of the inks and of printing support as well as the halftone pattern. They therefore need to be estimated from measurement.
Each ink i is printed alone on paper at the nominal surface coverages q i  = 0.25, 0.5, and 0.75, which correspond to the nine color patches represented in rows C, D, and E of Fig. 2. Their respective spectral reflectances are denoted as $${R}_{q_i}^{(m)}\left(\lambda \right)$$. These halftones contain two primaries: the ink which should occupy a fractional area q i and the paper white which should occupy the fractional area 1 − q i . Applying the Clapper-Yule equation with these two primaries and these surface coverages should predict a spectral reflectance $${R}_{q_i}^{(p)}\left(\lambda \right)$$ equal to the measured one, but due to the fact that the effective ink surface coverage is different from the nominal one, these two reflectances are not the same. The effective surface coverage q i as the q i value minimizing the deviation between predicted and measured spectra, by quantifying the deviation either by the sum of square differences of the components of the two spectra, i.e.,
$${q}_i^{\prime }=\underset{0\le {q}_i\le 1}{ \arg \min }{\displaystyle \sum_{\lambda ={\lambda}_{\min}}^{\lambda_{\max }}{\left[{R}_{q_i}^{(p)}\left(\lambda \right)-{R}_{q_i}^{(m)}\left(\lambda \right)\right]}^2}$$
or by the sum of square difference of the components of their logarithm, i.e.,
$${q}_i^{\prime }=\underset{0\le {q}_i\le 1}{ \arg \min }{\displaystyle \sum_{\lambda ={\lambda}_{\min}}^{\lambda_{\max }}{\left[ \log {R}_{q_i}^{(p)}\left(\lambda \right)- \log {R}_{q_i}^{(m)}\left(\lambda \right)\right]}^2}$$
or by the corresponding color difference given, e.g., by the CIELAB metric
$${q}_i^{\prime }=\underset{0\le {q}_i\le 1}{ \arg \min}\Delta {E}_{94}\left({R}_{q_i}^{(p)}\left(\lambda \right),{R}_{q_i}^{(m)}\left(\lambda \right)\right).$$
The first method is the most classical way of determining the effective surface coverage. Taking the log of the spectra as in the second method has the advantage of providing a higher weight to lower reflectance values where the visual system is more sensitive to small spectral differences. Fitting q i from the color difference metric sometimes improves the prediction accuracy of the model but complicates the optimization. Even at the optimal surface coverage q i , the difference between the two spectra is rarely zero and provides a first indication of the prediction accuracy achievable by the model for the corresponding print setup.
Once the nine effective surface coverages are computed, assuming that the effective surface coverage is 0, respectively 1, when the nominal surface coverage is 0 (no ink), respectively 1 (full coverage), three sets of q i values are obtained which, by linear interpolation, yield the continuous ink spreading functions f i (Fig. 3). As an alternative, one can print halftones at nominal surface coverage 0.5 only and perform parabolic interpolation [3]. The number of patches needed for establishing the ink spreading curves is thus reduced to three (row C in Fig. 2).
Once the spectral parameters and the ink spreading functions are computed, the model is calibrated. The spectral reflectance of prints can be predicted for any nominal ink surface coverages c, m, and y. The ink spreading functions f i directly provide the effective surface coverages c′, m′, and y′ of the three inks:
$$\begin{array}{c}\hfill {c}^{\prime }={f}_{ c}(c),\hfill \\ {}\hfill {m}^{\prime }={f}_{ m}(m),\hfill \\ {}\hfill {y}^{\prime }={f}_{ c}(y).\hfill \end{array}$$
Plugging these effective ink surface coverages into the Demichel equations, one obtains the eight effective primary surface coverages, and the general equation of the model finally predicts of the reflectance spectrum of the considered halftone.

## Improved Ink Spreading Assessment Method

Hersch et al. observed that a given ink spreads differently according to whether it is printed alone on paper or superposed with another ink. They proposed an ink spreading assessment method taking into account the superposition conditions of the inks in the halftone [4]. This method increases noticeably the model’s prediction accuracy. It relies on the halftones represented in Fig. 2 as well as the ones represented in Fig. 4.
The nominal ink surface coverages c, m, and y are converted into effective ink surface coverages c′, m′, and y′ by accounting for the superposition-dependent ink spreading. In the case of CMY halftones, 12 in. spreading curves are obtained, similar to the ones represented in Fig. 5.
The effective surface coverage of each ink is obtained by a weighted average of the ink spreading curves, the weights being the surface coverages of the respective primaries on which the ink halftone is superposed. For example, the weight of the ink spreading curve f c (cyan halftone over the white primary) is proportional to the surface of the underlying white primary, i.e., (1 − m) (1 − y). First, c′ = c, m′ = m, and y′ = y are taken as initial values on the right side of the following equations:
$$c=\left(1-m\right) \left(1-y\right) {f}_c\left({c}_0\right)+m\left(1-y\right){f}_{c/m}\left({c}_0\right)+\left(1-m\right)\;y{f}_{c/y}\left({c}_0\right)+ my{f}_{c/m+y}\left({c}_0\right),m=\left(1-c\right) \left(1-y\right) {f}_m\left({m}_0\right)+c\left(1-y\right){f}_{c/m}\left({m}_0\right)+\left(1-c\right)\;y{f}_{m/y}\left({m}_0\right)+cy{f}_{m/c+y}\left({m}_0\right),y=\left(1-c\right) \left(1-m\right) {f}_y\left({y}_0\right)+c\left(1-m\right){f}_{y/c}\left({y}_0\right)+\left(1-c\right)\;m{f}_{y/m}\left({y}_0\right)+ cm{f}_{y/c+m}\left({y}_0\right).$$
The obtained values of c′, m′, and y′ are then inserted again into the right side of the equations, which yield new values of c′, m′, y′, and so on, until they stabilize. The final values of c′, m′ and y′ are plugged into the Demichel equations in order to obtain the effective surface coverages of the eight primaries. The spectral reflectance of the considered halftone is finally provided by the Clapper-Yule equation.

## Experimental Testing

In order to assess the prediction accuracy of the model, predicted and measured spectra may be compared on sets of printed colors. As comparison metric, one generally uses the CIELAB ΔE94, obtained by converting the predicted and measured spectra first into CIE-XYZ tristimulus values, calculated with a D65 illuminant and in respect to a 2° standard observer, and then into CIELAB color coordinates using as white reference the spectral reflectance of the unprinted paper illuminated with the D65 illuminant.

Because it assumes that the lateral propagation of light is large compared to the halftone screen period, the Clapper-Yule model is theoretically restricted to halftones with high screen frequency. For example, the model tested on two sets of 729 CMY colors printed with the same offset press on the same paper but at different frequencies, respectively, 76 and 152 lines per inch (lpi), provides better predictions for the highest frequency (average ΔE94 of 0.98 unit) than for the lowest one (average ΔE94 of 1.26 unit). Nevertheless, the experience shows that the model may also perform well for middle and low frequencies: For a set of 40 CMY colors printed in ink-jet at 90 lpi on supercalendered paper, the model achieves a fairly good prediction accuracy, denoted by the average ΔE94 of 0.47 unit. Note that the average ΔE94 is 0.70 unit when the ink superposition conditions are not taken into account in the ink spreading assessment.

## Conclusion

Despite the simplicity of its base equation, the Clapper-Yule model is one of the most accurate prediction models for halftone prints. Its main advantage compared to other models such as the Neugebauer model or the Yule-Nielsen-corrected Neugebauer model is the fact that physical parameters are attached to the different elements composing the print (inks, paper, and surface). The Fresnel terms can be adapted to the considered measuring geometry, which is particularly interesting when predictions are made for a geometry different from the one used for calibration. The model also enables controlling ink thickness at printing time by comparing the colorant transmittances in various halftones, whose log is proportional to the ink thickness [5]. Recent improvements and extensions have been proposed which enable predicting both reflectance and transmittance of halftone prints thanks to extended flux transfer model relying on similar physical concepts as the Clapper-Yule model [6, 7].

## References

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Hersch, R.D., Brichon, M., Bugnon, T., Amrhyn, P., Crété, F., Mourad, S., Janser, H., Jiang, Y., Riepenhoff, M.: Deducing ink thickness variations by a spectral prediction model. Color. Res. Appl. 34, 432–442 (2009)
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Hébert, M., Hersch, R.D.: Reflectance and transmittance model for recto-verso halftone prints: spectral predictions with multi-ink halftones. J. Opt. Soc. Am. A 26, 356–364 (2009)
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Mazauric, S., Hébert, M., Simonot, L., Fournel, T.: Two-flux transfer matrix model for predicting the reflectance and transmittance of duplex halftone prints. J. Opt. Soc. Am. A 31, 2775–2788 (2014)