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Dear all:

I am a beginner and I have a question regarding the determination of the ink densities to be used under the 'Ink Setup' tab when using multiple gray inks.?

O.k., Paul Roark calls it a 'judgement call'. With my very limited experience I fully agree. But regardless of that I would like to better understand the procedure which I found in several documents. Accordingly, the ink density to be entered under 'Ink Setup' should be the result of the product of the fractions of the respective next darker ink which would yield the same density (or Lab L) as the ink at 100%.

I measured the Lab L values from a print of the calibration image for Hahnemühle Photo Rag 308, the GCVT ink set of Paul Roark, resolution 2880-super and 50% global ink limit. The results for each ink are shown as circles in the image below.? All data can be well represented by exponentially decaying functions (lines).

If I follow the above mentioned standard procedure, the fractions are indicated by the labelled plusses and the densities to be entered under 'Ink Setup' are represented by the K values below the full diamonds (they do not lie on the K-curve, because the ink curves are not linear). Am I correct that these densities, i.e. products of fractions, should be used? I am asking, because the products of the fractions, although they certainly carry relevant information, have no obvious meaning which would help to visually understand the resulting ink curves.

Hendrik

PS: From the curves in my ink separation plot, is the global ink limit I selected (50%) adequate? The lighter inks still have a significant slope at 100%.

• First, I am not familar with dual K inkset described by Paul Roark. But you mention that you printed at 50% limits on HPR 308, which is a coated MK paper, and as-such should yield Lmin of approximately 17.7 to 16 (Dmax ~ 1.6 to 1.7) using single MK. So you may want to look into, perhaps increase ink limit to >50% or use "Black Boost" ~70%
• Please refer to picture below on how to set the density of lighter shade of grey inks. Mind you this is only for 3 shades of grey (K, LK, and LLK) as an example. Toner inks utilize the density numbers also, but rely on "Limit" to determine how much of the toner gets used. You that either visually, or ideally with a spectro.
• Say at the calibration ink density, the solid blue line represents density of K in 5% step file. Solid orange line represnts LK ink, and solid green line is LLK ink. What you want to do is to find Lmin of LK and LLK (dashed orange and green lines respectively), and find the patch of K which prints at the same density. In the case of LK (orange vertical dash) gives you ~25%, and LLK (green vertical dash) gives you ~15%. These are the numbers you input into "Density." If I undersand your graph, it seems you are looking at these numbers from succesive lower density inks. That would not be correct. All densities should be relative to K.

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Dear Shilesh:

Thank you very much for your reply. Very helpful, because it clarifies the procedure. What you explain is what I was thinking of, originally. But I was misguided by the descriptions in the QTR User Guide and in the Calibration Guide (from the QTR website).

In the Calibration Guide it is stated on page 4: "But it's necessary to have all the relative densities to a common value". So far so good. It is logical and in line with what you explain. But on the same page it is written: "Since the LLK will be transitioning to the LK ink in the profiles the comparison is most accurate by comparing the LLK ink to the LK ink not to the K ink." Thereafter, a calculation is made in which the relative density of LLK to LK is multiplied by the relative density of LK to K, suggesting that the result of this product would yield a more accurate relative density of LLK to K.?

1. I am not convinced that the result is more accurate, because for 3 gray inks the multiplication of fractions involves the determination of 4 data points compared to the involvement of only 2 data points for direct determination of the density ratio of LLK to K. But it may depend on the method used to measure the curves.
   
2. More importantly, the multiplication of fractions only yields the desired ratio of LLK to K if the functions L(K) for all inks would be linear, i.e. if L(K) = a + b*K with some constants a and b for all inks. But in our case the functions are exponential of the form L(K) = a + b*exp(-c*K) with a>0, b>0, c>0. For such functions the above multiplication of fractions will NOT relate the gray densities to the common K reference curve. Rather, in the example plot in my original post for HFA-PR-308 and GCVT inks the multiplication of fractions yields densities given by the abscissas of the diamonds. The densities of the gray inks (all inks are gray except for the toner) relative to the common reference ink K are given by the abscissas of the squares (confirmed by you).
   
   For an example, below I show part of my previous plot. C, LK and K denote the cartridge slots. The carts are filled as follows. C: 50%MK, LK: 100%PK, K: 100%MK. The LK cart with 100%PK prints dark gray on matte paper. From the fits (thin curves) of the data (circles) one obtains
   
   • LK to K: fraction = 0.4821 (vertical red line)
     
   • C to LK: fraction = 0.4786 (blue X)
     
   • The product of these fractions is = 0.2307 (vertical blue dotted line)
     
   • Compare this to the 'correct' value of the C to K density (vertical blue line): 0.3307.

Concerning the global ink limit, you are correct. I was thinking of using the Black Boost, because the slope of the L(K) curve for the K channel is already very small at K=100% (with global ink limit of 50%) and already L(K=100%) < 20 for the black ink.

You are correct regarding the toner. For the toner curve (open blue squares on the dashed blue line in my original plot) I have not calculated its relative density, because it requires a separate curve in the Ink Setup to neutralize the print for all gray values.

Thank you very much again!
Hendrik

Am 04.02.2025 um 04:35 schrieb shileshjani via groups.io:

• Epson P900, OEM inks
• Only used PK, LK, and LLK on Epson Premium Luster
• Calibration printed at 70% density
• Lmin of LK was 55% step of K
• Lmin of LLK was 25% step of K
• Created curve #1, K=100%, LK=55%, LLK=25%. All limits at 70
• Created curve #2, K=100%, LK=55%, LLK=0.55 * 0.25 = ~14.
• Figure of curves below shows differences in Curves 1 and 2. But not particularly instructive in-itself.
• Figure of printed L vs 5% steps much more decisive. The method of referencing LLK to LK and reflecting back to K is?much closer to linear than my previous undertanding of the method.
• I kept "Gray Curve" settings at Highlight = 5, Shadow = 5, no Overlap, and Gamma =1. The highlight and shadow values may impact results.
• I learned something.

This seems worth some comments.

First -- experimenting is the best! Nothing is obvious without trying it.
So experimenting and trying things out is always recommended.

I'm not really sure where the numbers come from for the Curve 1 & 2 -- K, LK, LLK values so its hard to be sure.
(My initial feeling is the Curve 2 has the LLK too low -- like you used the .55 factor twice; but I'm not sure.)

Looking at the ink curves you can see that LLK for Curve 2 uses a lot more ink -- you said it was a lot lighter ink (.14 vs .25).

Then looking at the density curves -- what should you care about? Curve 2 is closer to "linear" than Curve 1 but these
curves are before Linearization so that's not the end result. My take would be that curve 1 is smoother and might actually be
the better one once you do the linearization. The little wiggles in curve 2 are harder to repair.

Linearize and then do the two graphs -- things will be a lot closer but the differences may tell you more.

Roy

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I think in his example Shilesh demonstrated the difference between the two approaches we are discussing.

Curve 1 illustrates method 1: Here the printed luminosities of all lighter inks are referred to K (the common reference curve): To that end he determines the densities (for the ink setup) by those densities in the calibration plot at which the K-curve has the luminosities at which LK and LLK reach their respective minima. In his example:
??? Lmin of LK was 55% step of K
??? Lmin of LLK was 25% step of K
This method 1 corresponds to using the open square for the LLK density in my graph.

Curve 2 illustrates method 2: In this case one has to multiply the two fractions 0.55 and 0.25. This results in 0.25 * 0.55 ~ 14 for sought density of LLK. This method 2 corresponds to using the solid diamond for the LLK density in my graph.

I think in the calibration guide (p.4 of calibration.pdf) both these methods are getting mixed. Here are the sentences which are contradictory:

• Method 1: "But it's necessary to have all the relative densities to a common value".
• Method 2: "Mathematically this is very simple, just a multiply: LLK = 38% of 32.7% of 100 ... = 12.4%"? and "... the comparison is most accurate by comparing the LLK ink to the LK ink not to the K ink."

Both method yield the same result for the required density of LLK only if all ink luminosities satisfy L(K) = A - b_ink*K (with A=L(K=0) the common luminosity of the paper and different values of b_ink by for each ink). This linear relation ship holds true for very small values of K. But for larger K all L(K) curves saturate.

I hope it is clearer now.

Best,
Hendrik

Your description of method 2 makes no sense.

All the values are simple ratio -- darkness of one ink vs darkness of another ink.
So K vs LK and LK vs LLK -- then calculate K vs LLK.
It's just simple algebra ->> (a/b)*(b/c) = (a/c)

But as you can see from Shilesh's experiment: once you linearize is fixes most anything.

On Thu, Feb 6, 2025 at 11:24 AM, shileshjani wrote:

I believe Hendrik got the desciption here



I was surprised by it, so I tried it (experiment vs theory, lol)

The trouble is that he's got it wrong. He shows how he interpreted it
and it's just not correct and not what the PDF says.
Please stop saying it's an "alternate" interpretation.

Roy -- the designer and author of all this stuff

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Your description of method 2 makes no sense.   

All the values are simple ratio -- darkness of one ink vs darkness of another ink.
So K vs LK and LK vs LLK -- then calculate K vs LLK.
It's just simple algebra ->>    (a/b)*(b/c) = (a/c)

The trouble is that he's got it wrong.  He shows how he interpreted it 
and it's just not correct and not what the PDF says.  
Please stop saying it's an "alternate" interpretation.

Roy -- the designer and author of all this stuff

Dear Roy:

I am sorry to say that the math you suggest cannot be applied in the present case. The problem in your above equation is that the 'b' in the denominator of the first factor is not he same as the 'b' in the nominator of the second factor. You cannot simply replace a, b and c by K, LK and LLK in this equation. The reason is K, LK and LLK are names of ink channels and not numbers! But you need numbers to be inserted in your equation.

Here is the explanation. It is a bit longer than you may want, but I feel I have to do the explanation in very small steps for clarity, and clarity requires precision of expression. I hope to convince you.

When dealing with the calibration plot, we need to understand the printed step wedges as luminosity functions L(K) of the step variable K. Since we have 3 luminosity functions, each for one of the ink channels, I introduce three functions of K which I denote L_K, L_LK and L_LLK:

• L_K(K)
• L_LK(K)
• L_LLK(K)

Please note that the subscript K denotes the ink channel K and not the step variable K which is the argument in the parentheses. Now the independent variable K runs from 0 to 1 (100%=1). We want to find the two values K1 and K2 of the step variable K at which the luminosity function L_LK evaluated at K=1 equals the luminosity function L_K at some unknown K1, and at which the luminosity function L_LLK at K=1 equals the luminosity of L_K at some unknown K2. Mathematically, this means

• L_LK(1)?? = L_K(K1)???? (1)
  
• L_LLK(1) = L_K(K2)???? (2)
  

I have numbered the equations. These 2 equations relate the two luminosities L_LK and L_LLK at K=1 (=100%) (on the left side of the equations) to the luminosity L_K (right side of the equations), which serves as the common reference function (as intended). So we have to solve the two equations for the arguments K1 and K2. This can be done by using Newton's iteration (if one has explicit expressions for the continuous functions - I determined them by fitting an exponential), but it is easier to do this graphically. To this end one has to draw the graphs of the two functions on both sides of equation (1) and find their intersection point. At the intersection point the equation is satisfied and the abscissa of the intersection point yields the value K1. The same can be done for equations 2. Well, here we are done.

Here is the mistake

You claim that one arrives at the same value K2 when multiplying K1 with K3, where K3 is the step K at which the luminosity of L_LLK at K=1 equals the luminosity of L_LK. To find K3 we need to solve the equation

• L_LLK(1) = L_LK(K3)??? (3)
  

This equation can be solved graphically in the same way as above. However, for general nonlinear functions L_K(K), L_LK(K) and L_LLK(K) the claimed relation (multiplication of the fractions K1 and K3)

• K2 = K1*K3???? (4)
  

is not generally satisfied! This is my claim. For a proof I have given an example in my post /g/QuadToneRIP/message/19394 . Shilesh has given another proof in his post /g/QuadToneRIP/message/19398 .

It is interesting, however, that equation (4) is satisfied, if the 3 luminosity functions would be linear in K. (Note that I am talking here of the hypothetical case that one measures a linear behavior of the luminosity functions in the calibration plot. It should not be mixed up with the linearization process in QTR.) The linearity of L_K(k), L_LK(K) and L_LLK(K) is is a very special and hypothetical case. A physicist would call this a 'model'. It is unrealistic for inkjet printing (but it may possibly hold approximately for some type of matrix printers in which each pixel is made of many non-overlapping black squares which ultimately cover the whole pixel completely).?

Here I give the proof of equation (4) for this model. For luminosity functions which are linear in K equations (1) to (3) yield

• L_LK(1)??? = L_K(K1)??? = a - b1*K1???? (5)
  
• L_LLK(1)? = L_K(K2) ?? = a - b1*K2???? (6)
  
• L_LLK(1)? = L_LK(K3)? = a - b2*K3???? (7)
  

where 'a' is the luminosity of the paper white and b1 and b2 and are the constant negative slopes of the (now linear) functions L_K(K) and L_LK(K). Subtraction of the two last equations '(6) - (7)' yields

• 0 = (a - b1*K2)? - (a - b2*K3)

or

• K2 = (b2/b1) * K3?????? (8)
  

Now we evaluate (7) at K=1 to obtain

• L_LK(1) = a - b2

This is now inserted in (5) to obtain

• a - b2 = a - b1*K1

from which we get (b2/b1) = K1. This can now be inserted in (8) to find

• K2 = K1 * K3????? (9)
  

This proofs that the multiplication of fractions is equivalent to directly determining K2, if all luminosity functions are linear in K. But if the luminosity functions are exponential (as it seems to be the case)

• L_ink(K)? =? L_ink(K-> infty) + [a - L_ink(K-> infty)] * exp( -s_ink * K)

where L_ink(K-> infty) is the asymptotic luminosity for large K (saturation) and s_ink the decay rate of the luminosity (equivalent to the slope of the luminosity L_ink(K) at K=0), then (9) does not generally hold. The fact that (9) holds for linear functions can also be derived geometrically using the "Intercept theorem". I hope I did not commit any typos.

I am sorry to have found this misconception. I find it not only on page 4 of the calibration guide , but also in the user guide of Tom Moore. On page 15 one finds:?

" For QuadTone inks, this process is repeated for each lighter ink, comparing it to the next darker ink, calculating its density relative to that ink and then converting it to a density relative to black."?

Presumably, the misconception has not been discovered/discussed previously, because QTR is quite forgiving with respect to selecting the densities for multiple gray inks. So most practitioners may not care and follow their intuition. But you will understand that I had to disproof your statement "The trouble is that he's got it wrong".

Hendrik? --? a beginner in QTR but not a beginner in math ;-) :-)