Estimation of Backing Influence on Halftone Reflectance
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Lichtenauer, Matthias Scheller
Hoffstadt, Hanno
Kraushaar, Andreas
Oberhollenzer, Berthold
Zolliker, Peter
In the graphic arts industry, there is a need to convert colorimetric readings taken on one backing (usually white) to values that would have been measured with a different backing (usually black). We describe and compare different models for such a conversion. Starting from published models using linear scaling, we developed a new nonlinear model for a strongly scattering substrate. Another new model was derived from the Clapper-Yule model, which includes effects of internal reflectances. All these models are applicable in both the spectral and the tristimulus domain. For calibration, we used measurements of the bare substrate on both backings. We intentionally make only use of the measured spectral or XYZ values, and do not require knowledge of the nominal CMYK values. This is particularly useful for arbitrary patches measured with a stand-alone measurement device. The test data sets consisted of a large set of test prints, originating from digital or conventional printing processes, and covering typical ranges of mass per area. Both new models outperformed linear regression models and the spectral versions always yielded better results than their corresponding versions in tristimulus space.
 The problem we are facing is estimating the backing influence on half-ton replacements. Usually, if you use characterization targets to characterize the gamut of the device or whatever, it's measured on a white backing. But in production runs, you use a black backing. This can be because there is already a print on the backside or just because there is dust in the print-through. So the target values for a black backing are needed and it would be convenient to be able to calculate them, so you don't have to measure them. And this is really a problem. We had 35 targets as a ground-truse that we measured with targets like the one I presented. From different printing processes, printed on different subsets, all papers, but some were coated, some uncoded. And you see here the cumulative difference with the distribution from black to white backing. And if you take the median value, there are values between delta E of a unit up to a median value already of 3 delta E units. For calculations, we use the standards in printing industry for those who are familiar with these. So the models that are already available, there is one from Mr. Hans Ott, published in 2003. And one from Mr. MacBowell, I hope I pronounced his name right. And these are both linear regression models. If we plot the three stimulus values here from white backing and here from black backing, it looks as if a linear model would be appropriate. It doesn't look bad, nice regression. But if a linear model would be appropriate, then also the difference between the black and white backing values should be linear, scale linearly. And maybe we see here where we have the value on white backing plotted against the difference from black to white backing. It's not absolutely linear. So we present two new models for this calculation. The one we call the gamma regression model and the other the internal reflections model. Our word on notation, we're talking here about reflectance factors from either tints on white backing or from the substrate on white backing. And if we formulate our problem mathematically, then it is to find a function that predicts the reflectance factor of a tint on black backing, taking as input three measurements. And I'd like to draw your attention to the fact that these two measurements do not alter from patch to patch, since it's the same substrate they're printed on. Now, we want to derive these models that have been presented and published before from a radiometric approach. We say we make two flows of white. All the photons that emerge from the paper that have never seen the backing are in the part r. And the rest of the photons that have seen the backing are in the part 1 minus r. And both these parts are then attenuated by a factor that we denote t. Now, if you assume an ideal white backing, we have both these parts. If we assume a black cavity backing, there is no backing. So all the light that will see the backing is number every once. So we have this equation. And then the difference between the measurement of the substrate on a white backing or on a black backing is more or less the energy that we have that comes from the backing. So this white is then attenuated by the tint and whatever. So we can say that this difference of the substrate reflectance is scaled by the white backing measurement is the part 1 minus r. So these are the photons that see the backing. If you plug all these three equations together, we get this nice thing down here. And this is now a clear interpretation. So this is an anchor value, the measurement that we have on a white backing of a tint, plus the difference of these substrate reflectances. So this is the energy that you have more or less attenuated by a yet unspecified factor t. Now, we don't know how to set this factor t yet. We haven't specified that. Mr. Ott, he proposed to say, OK, what we measure is the attenuation of both flows together. And I just say a good approximation for this factor t would be that. And this is what we see here. Put in linear regression. Again, you see here the same plots. You have the reflectance value normalized by the substrate reflectance value on the horizontal axis. And the difference between the two reflectance values normalized by the difference of the substrate reflectance values. And the model of Mr. Ott would be a straight regression line from 1 to 0. The second model has an additional parameter. This is better minimal. It means more or less you correct this here. This fact that print has never 100% absorption. No, I've not seen that yet. So this is all it introduces. And from the model application point of view, it means that you need to know this parameter too. So this is not really a free parameter. Now, what we propose here as a new model, at first sight it looks just like, OK, let's take a nonlinear regression. But as we have shown in the paper, it can be derived from the model of Jules and Nielsen. It's really an algebraic substitution more or less. And this factor gamma is a free parameter as in Jules and Nielsen models the parameter n. Some general remarks on all these regression models. You have in all these regression models this form here. And this means that these models are from energy conservation point of view, in fact, well behaved. If we assume that this t factor denotes absorption, it should be smaller than 1 and bigger than 0 probably. So it means that the energy coming from the backing is just being reduced. And with the notable exception of the model proposed by Mr. MacDowell and his colleagues, it is of the form that we required three measurements. So two substrate measurements and one patch measurements to predict a force. And now from Mr. Jules and Nielsen to Mr. Jules and Clapper. The second model is in fact derived from the ideas of Clapper and Jules that they published in 1953. I don't go into details here because those who know will know it anyway and the other ones will be bored. So just like to point out that there are six parameters in this model with a clear physical meaning. And this leads then after a series of reflections and geometric sequences of reflectances between the substrate and the air. So this wonderful formula that we will not really discuss in detail, the problem is we want this. And it's not as obvious how we get there. We were a bit mean to them. First of all, this equation can be formulated for all four values, all these four reflectances. The measurement geometry that is used in printing industry has been designed so that direct specular reflection will not get into the sensor. So we are really, oh, let's say we're brave. This sounds nice. And we say we set this to zero. This part that represents the spectral reflections. Then we do another really bad thing that we say we assume that the colorant layer is not halftone but salt. I mean, we just assume that. And that the absorption is independent of the back. It's really a tin problem. So these simplifications allow then to outgirically substitute this T and R values in two of the three equations. Then we have two parameters left to deal with. We set the parameter S to zero, zero four. This is the value that was originally used also by Clapper and you. And then we have one free parameter. In fact, this row parameter that represents in the original model, I have to say the percentage that escapes from the paper to the air each time that light comes to the upper interface. And here we just fitted it to the data that we have. But surprisingly, the value of zero six that was already published by Mr. Yat in 1942 seemed to be fittest. So we say we probably knocked too much off. So after all this force that we applied to Clapper and Mr. Yul's ideas, we have this wonderful piece of mathematics that you don't need to understand. Further than that it is in this form. So all the things that we need can be measured. So please pardon me that all the force I did to this model. How did we evaluate these models then? We predicted with these models how it should look on black backing and we compared with the measurements that we made on black backing. So for each target printed, we have one cumulative error distribution. Now we averaged all these certifies measured targets. So this black line here represents the average of all these gray lines. So the black line represents what happens if you do not correct anything. If you take the white backing values as a prediction for what should be measured on black back. Sian and Magenta are the two linear models published previously. And the red and here is the red and the green are the two new models that we propose. And at first moment we were surprised to find that they are more or less identical. And if you remember the kind of regression function, it's not so much surprising. There is a publication by Mr. Hirsch and Mr. Mathieu Béreff that gives more or less the same form of what should be expected for regression function for Clapeyule, the original Clapeyule. So one word to how good can this model be. If we push the measurement precision of an I1 or a spectolino to the limit, we get 0.1 delta E AB for one single measurement. And there are two involved, so they can now be correlated or uncorrelated. We leave that open. Greta Macbeth, X-Rite, meanwhile I think, they specify 0.2 for the 50% fontile. We conclude that we propose two models to predict blackbacking values based on whitebacking measurements. They are applicable in both tree stimulus and spectral space. Conversions are within one delta E AB for 95% of the data. The other way around is also possible. What has not been modeled yet, what is really important in practice, is that production runs are usually also measured with polarization filter. It works from polarization measurement to polarization measurement, but not from unfiltered to polarization. Thank you for your attention. Thank you very much for this interesting presentation. Other questions about these methods? You have shown us the statistics of the arrows. Could you please comment some qualitative statements? Where does it work well? On which papers? Stuff like this. If there is a bit more insight, where it doesn't work? You have seen there are 5% more than one delta E within your best model. Where are these 5%? I am not sure whether I understand your question. If you could go perhaps to your last... Which patches are worse? Delta E is larger than one. There are patches, papers, filter positions. What point has not analyzed for smooth values or anything like this? The point is that the measurements always have some percentages beyond two delta E in the raw data. It can also be that this comes from this. But we have another idea to make money. We wanted this. This was really a design issue for practicality of the model industry. Thank you again.
 The problem we are facing is estimating the backing influence on half-ton replacements.  Usually, if you use characterization targets to characterize the gamut of the device or whatever,  it's measured on a white backing.  But in production runs, you use a black backing.  This can be because there is already a print on the backside or just because there is dust in the print-through.  So the target values for a black backing are needed and it would be convenient to be able to calculate them,  so you don't have to measure them.  And this is really a problem.  We had 35 targets as a ground-truse that we measured with targets like the one I presented.  From different printing processes, printed on different subsets, all papers,  but some were coated, some uncoded.  And you see here the cumulative difference with the distribution from black to white backing.  And if you take the median value, there are values between delta E of a unit up to a median value already of 3 delta E units.  For calculations, we use the standards in printing industry for those who are familiar with these.  So the models that are already available, there is one from Mr. Hans Ott, published in 2003.  And one from Mr. MacBowell, I hope I pronounced his name right.  And these are both linear regression models.  If we plot the three stimulus values here from white backing and here from black backing,  it looks as if a linear model would be appropriate.  It doesn't look bad, nice regression.  But if a linear model would be appropriate, then also the difference between the black and white backing values should be linear, scale linearly.  And maybe we see here where we have the value on white backing plotted against the difference from black to white backing.  It's not absolutely linear.  So we present two new models for this calculation.  The one we call the gamma regression model and the other the internal reflections model.  Our word on notation, we're talking here about reflectance factors from either tints on white backing or from the substrate on white backing.  And if we formulate our problem mathematically, then it is to find a function that predicts the reflectance factor of a tint on black backing,  taking as input three measurements.  And I'd like to draw your attention to the fact that these two measurements do not alter from patch to patch, since it's the same substrate they're printed on.  Now, we want to derive these models that have been presented and published before from a radiometric approach.  We say we make two flows of white.  All the photons that emerge from the paper that have never seen the backing are in the part r.  And the rest of the photons that have seen the backing are in the part 1 minus r.  And both these parts are then attenuated by a factor that we denote t.  Now, if you assume an ideal white backing, we have both these parts.  If we assume a black cavity backing, there is no backing.  So all the light that will see the backing is number every once.  So we have this equation.  And then the difference between the measurement of the substrate on a white backing or on a black backing is more or less the energy that we have that comes from the backing.  So this white is then attenuated by the tint and whatever.  So we can say that this difference of the substrate reflectance is scaled by the white backing measurement is the part 1 minus r.  So these are the photons that see the backing.  If you plug all these three equations together, we get this nice thing down here.  And this is now a clear interpretation.  So this is an anchor value, the measurement that we have on a white backing of a tint, plus the difference of these substrate reflectances.  So this is the energy that you have more or less attenuated by a yet unspecified factor t.  Now, we don't know how to set this factor t yet. We haven't specified that.  Mr. Ott, he proposed to say, OK, what we measure is the attenuation of both flows together.  And I just say a good approximation for this factor t would be that.  And this is what we see here.  Put in linear regression.  Again, you see here the same plots. You have the reflectance value normalized by the substrate reflectance value on the horizontal axis.  And the difference between the two reflectance values normalized by the difference of the substrate reflectance values.  And the model of Mr. Ott would be a straight regression line from 1 to 0.  The second model has an additional parameter. This is better minimal.  It means more or less you correct this here.  This fact that print has never 100% absorption.  No, I've not seen that yet.  So this is all it introduces. And from the model application point of view,  it means that you need to know this parameter too.  So this is not really a free parameter.  Now, what we propose here as a new model, at first sight it looks just like, OK, let's take a nonlinear regression.  But as we have shown in the paper, it can be derived from the model of Jules and Nielsen.  It's really an algebraic substitution more or less.  And this factor gamma is a free parameter as in Jules and Nielsen models the parameter n.  Some general remarks on all these regression models.  You have in all these regression models this form here.  And this means that these models are from energy conservation point of view, in fact, well behaved.  If we assume that this t factor denotes absorption, it should be smaller than 1 and bigger than 0 probably.  So it means that the energy coming from the backing is just being reduced.  And with the notable exception of the model proposed by Mr. MacDowell and his colleagues,  it is of the form that we required three measurements.  So two substrate measurements and one patch measurements to predict a force.  And now from Mr. Jules and Nielsen to Mr. Jules and Clapper.  The second model is in fact derived from the ideas of Clapper and Jules that they published in 1953.  I don't go into details here because those who know will know it anyway and the other ones will be bored.  So just like to point out that there are six parameters in this model with a clear physical meaning.  And this leads then after a series of reflections and geometric sequences of reflectances between the substrate and the air.  So this wonderful formula that we will not really discuss in detail, the problem is we want this.  And it's not as obvious how we get there.  We were a bit mean to them.  First of all, this equation can be formulated for all four values, all these four reflectances.  The measurement geometry that is used in printing industry has been designed so that direct specular reflection will not get into the sensor.  So we are really, oh, let's say we're brave.  This sounds nice. And we say we set this to zero.  This part that represents the spectral reflections.  Then we do another really bad thing that we say we assume that the colorant layer is not halftone but salt.  I mean, we just assume that.  And that the absorption is independent of the back.  It's really a tin problem.  So these simplifications allow then to outgirically substitute this T and R values in two of the three equations.  Then we have two parameters left to deal with.  We set the parameter S to zero, zero four.  This is the value that was originally used also by Clapper and you.  And then we have one free parameter.  In fact, this row parameter that represents in the original model, I have to say the percentage that escapes from the paper to the air each time that light comes to the upper interface.  And here we just fitted it to the data that we have. But surprisingly, the value of zero six that was already published by Mr. Yat in 1942 seemed to be fittest.  So we say we probably knocked too much off.  So after all this force that we applied to Clapper and Mr. Yul's ideas, we have this wonderful piece of mathematics that you don't need to understand.  Further than that it is in this form.  So all the things that we need can be measured.  So please pardon me that all the force I did to this model.  How did we evaluate these models then?  We predicted with these models how it should look on black backing and we compared with the measurements that we made on black backing.  So for each target printed, we have one cumulative error distribution.  Now we averaged all these certifies measured targets. So this black line here represents the average of all these gray lines.  So the black line represents what happens if you do not correct anything.  If you take the white backing values as a prediction for what should be measured on black back.  Sian and Magenta are the two linear models published previously.  And the red and here is the red and the green are the two new models that we propose.  And at first moment we were surprised to find that they are more or less identical.  And if you remember the kind of regression function, it's not so much surprising.  There is a publication by Mr. Hirsch and Mr. Mathieu Béreff that gives more or less the same form of what should be expected for regression function for Clapeyule, the original Clapeyule.  So one word to how good can this model be. If we push the measurement precision of an I1 or a spectolino to the limit, we get 0.1 delta E AB for one single measurement.  And there are two involved, so they can now be correlated or uncorrelated.  We leave that open. Greta Macbeth, X-Rite, meanwhile I think, they specify 0.2 for the 50% fontile.  We conclude that we propose two models to predict blackbacking values based on whitebacking measurements.  They are applicable in both tree stimulus and spectral space.  Conversions are within one delta E AB for 95% of the data.  The other way around is also possible. What has not been modeled yet, what is really important in practice, is that production runs are usually also measured with polarization filter.  It works from polarization measurement to polarization measurement, but not from unfiltered to polarization.  Thank you for your attention.  Thank you very much for this interesting presentation.  Other questions about these methods?  You have shown us the statistics of the arrows. Could you please comment some qualitative statements? Where does it work well?  On which papers? Stuff like this. If there is a bit more insight, where it doesn't work?  You have seen there are 5% more than one delta E within your best model. Where are these 5%?  I am not sure whether I understand your question.  If you could go perhaps to your last...  Which patches are worse?  Delta E is larger than one.  There are patches, papers, filter positions.  What point has not analyzed for smooth values or anything like this?  The point is that the measurements always have some percentages beyond two delta E in the raw data.  It can also be that this comes from this.  But we have another idea to make money. We wanted this. This was really a design issue for practicality of the model industry.  Thank you again.
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