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1. Lambert-Beer law and reflection density determination

One of the reasons for using the density measurement method is that the density value can easily indicate the amount of ink. In an ideal situation, as the thickness of the ink layer increases, the density value increases proportionally, that is, the thickness of the ink layer doubles, and the density value doubles accordingly. This is Lambert-Beer law. Lambert-Beer's law is suitable for transmission measurement of silver salt film, but Lambert-Beer's law is inaccurate when it is used for reflection measurement of printed matter, and it has only an approximate meaning. The following illustrates this problem through a simple transformation of Lambert-Beer law.

Lambert-Beer law can be expressed by


Φ (x) = Φ0e-ax (1-1)
In the formula: Φ-luminous flux;
Φ0——incident light flux;
α——The absorption coefficient, which is determined by the characteristics of the ink;
x-Ink layer thickness.
Taking the logarithm of both ends of (1-1), we get:
-lgΦ (x) / Φ0 = lge · α · x = α′x (1-2)
D = α′x
The formula (1-2) means that the density and the thickness of the ink layer should be proportionally proportional, which is suitable for the image observed through transmission, such as the image printed on the plastic transparent film (Figure 1-11).

(Figure 1-11)
Now multiply both ends of formula (1-2) by 2 to get 2a′x = 2D (1-3)
Equation (1-3) expresses the situation as shown in Figure 1-12. The incident light is reflected by the paper and passes through the ink layer twice, which is equal to the light absorption of the ink layer with a thickness of 2x. This means that The same ink layer thickness is printed on the transparent plastic film and paper, and the contrast obtained on the paper will be twice that of the image on the transparent film. In fact, even if the paper reflects all the incident light back, the formula (1-3) has only an approximate meaning

(Figure 1-12)
Decompose an ink layer into many thin layers (Figure 1-13), set the incident luminous flux to Φi, the transmitted luminous flux of each thin layer is Φ (xi + 1), and the absorption rate is α, then the following relationship is established;
Φ (x1) / Φ0 = α; Φ (x2) / Φ (x1) = α;
Φ (x3) / Φ (x2) = α; Φ (x4) / Φ (x3) = α.
Differentiating Φ (x) gives:

Figure 1-13
dΦ (x) / dx = -α · Φ (x) (1-4)


In the above formula, dΦ (x) / dx represents the amount of change in luminous flux Φ when the thickness of the ink layer increases from xi-1 to xi. Since the luminous flux decreases as the thickness of the ink layer increases, a negative sign is added before α.

The solution of formula (1-3) is known, namely Lambert-Beer formula: <

br> Φ (x) = Φ0 · e-ax (1-5)


To prove that this solution is correct, the following calculations can be made:

The derivative of equation (1-4) is:


dΦ (x) / dx = -α · Φ0e-ax (1-6)
Then replace the dΦ (x) / dx at the left end of the formula (1-4) with the derivative (1-6) and replace the Φ (x) at the right end of the formula (1-4) with the formula (1-5):
－Α · Φ0 · e-ax ＝ －α · Φ0 · e-ax


The two ends of the equation are the same, indicating that equation (1-5) is the solution of equation (1-4).

The additive principle of the density value is quite accurate in the case of transmission measurement. You can measure the density of several films separately and add them, and then stack them to measure the total density. You can find that the error is only the linear error of the densitometer. The reflection at the interface of each layer of film is also a factor causing errors.

However, when measuring the density of the stacked printing ink layer, people will immediately find that the density addition rate has a deviation. The ink is colored with pigments, and the photosensitive sheet is colored with dyes. The latter produces light diffusion in the film more than pigments. The light diffusion caused in the ink is much smaller, so it can better meet the density addition rate, the smaller the internal light diffusion of the ink pigment, the closer the refractive index of the pigment and the binder is closer. [next]

Second, logarithmic position error
According to the Mary-Davis formula, the calculation formula of the dot coverage can be derived as follows:
F ＝ 1－10－Dt / 1-10-Ds · 100 ％


In order to calculate the coverage of the dots, it is first necessary to convert the density value based on the measured brightness signal, which is completely unnecessary in nature, and can be directly converted according to the luminous flux received by the photodiode;

Because Dt = -lg (Φt / Φp);
Ds = -lg (Φs / Φp);

In the formula: Φp——the luminous flux first received by the diode from the white paper surface;
Φt——The luminous flux received by the photodiode from the dot surface;
Φs——The light flux received by the photodiode from the solid ground.

So F = 1-10-Dt / 1-10-Ds · 100% = Φp-Φt / Φp-ΦΦs · 100%
The result obtained by the direct conversion method is more accurate, because the logarithm calculation will produce a certain degree of error.

3. Brightness measurement error

The dot coverage rate is usually calculated by substituting the density value into the relevant formula. Since the numerical accuracy of many density meters is only two digits after the decimal point, the calculation accuracy of the dot coverage is limited. Such calculation errors are negligible in some cases and non-negligible in some cases.

First study the situation of measuring in the bright tone area. Suppose: the density of a certain dot surface is actually Dt = 0.025, and the corresponding solid density is actually Ds = 2.00. When measured with a densitometer whose indication precision is two digits behind the decimal point, the reading of the densitometer is purely random and may The case is Dt1 = 0.02 or Dt2 = 0.03, and the dot coverage derived from it is:

F1 = 8.88% or F2 = 6.74%
The measurement error of the dot coverage is:
ΔF ＝ F1－F2 ＝ 2.14%


In the plate-making process, special emphasis is placed on copying with 1% dot coverage accuracy. The accuracy of the density meter is not enough here, so in the standardized process, people do not measure the bright tone with the density meter, but set the signal bar to visually evaluate, so as to judge whether 1% of the dots are transferred correctly.

The small density deviation in the bright tone area can be clearly recognized by the eyes, which is the opposite of the situation in the dark tone area. The gray fog eyes produced in the bright tone area can be clearly recognized, but if you want to measure with a density meter, you can only use a density meter accurate to three decimal places to work.

Regardless of offset printing or flexographic printing, if negative printing is used, it will cause serious problems for printing standardization. The lone% dot block that can be accurately measured on the printed matter, the dot coverage on the negative film is only 20%, and the density meter cannot measure enough accurate data, so it must be controlled with a control strip. The dot coverage on the control bar is measured with other tools (such as a microscope). As long as the positive-negative conversion process is used, the bright tone is always replaced by the dark tone. The process of measuring and controlling with a density meter must not be marked with a question mark. The way to solve this problem is to use a density meter accurate to three decimal places.

Look again at the density measurement in the dark area. Measuring the dark tone with a density meter accurate to two decimal places does not actually produce measurement errors.

If the measured dot density and solid density in the dark area are:

Dt ＝ 1.505 Ds ＝ 2.00
The displayed value of the density meter may now be purely random, such as:
Dt1 = 1.50 Dt2 = 1.51
The resulting network coverage:
F1 = 97.816% or F2 = 97.88%
ΔF ＝ F2－F1 ＝ 0.06%
Obviously, this deviation is negligible. [next]
4. Measurement error of small contrast surface

The contrast of the measured surface has an effect on the dot coverage measurement. If the density measurement method is used to obtain the dot coverage on a surface with a small surface contrast, the result is not accurate enough, for example, a printing plate reader is used to measure the printing plate. Because the contrast on the surface of the printing plate is very small, in order to realize the standardization of the printing plate, the density of the printing plate is not controlled by the densitometer, but the quality of the printing plate is controlled by the printing control strip. For the flexographic printing plate, the control bar can currently be used to control the printing plate, but there is no proper element on the control bar to control the dot coverage. Because the change of dot coverage in printing may be more severe than that in printing, it must be observed carefully.

Now let's compare the impact of low-contrast and high-contrast surfaces on the coverage measurement of dots.

If the density change range on a low contrast surface is:

Ds = 0.60 ± 0.005 Dt = 0.20 ± 0.005

The calculation is:
Fmin = 48.5% Fmax = 50.5%

The error range ΔF = 2% This is a deviation that cannot be ignored.

The following is a comparison of the situation of high-contrast surfaces (such as printed images), which is known:


Ds ＝ 2.00 ± 0.005 Dt ＝ 0.30 ± 0.005


Then Fmin = 50.97% Fmax = 51.20% ΔF = 0.23%

The error is very small.

In the plate-making process, a simplified approximation formula can also be used to calculate the dot coverage when performing transmission measurements on high-density film. In the Mary-Davis formula:


F ＝ 1－10－Dt / 1－10－Ds · 100 ％ (1－7)


Let 10－Ds≈0

Then F = (1-10-Dt) · 100% (1-8)

The accuracy of the two formulas (1-7) and (1-8) can be compared by the following calculations:

In the case of high field density, the simplified formula (1-8) is accurate enough, but in printing, only a small field density can be obtained, which must be calculated using an accurate formula.

It is known to use (1-8) formula to calculate (1-7) formula to calculate Ds = 1.00
Dt = 0.90 87.4%
97.1%
Ds = 2.00
Dt = 1.41 96.1%
97.1%
Ds = 3.00
Dt = 1.53 97.0%
97.1%
Fifth, the difference between the indication of different density meters

It is very important that different densitometers can obtain the same measurement value, for example: the density meter used in the work preparation should get the same measurement results as the density meter used on the machine; the density meter used in the enterprise should have a good Consistency; where measured values ​​are transmitted by telephone, the densitometers on both sides should have comparable measured values.

By the way, one problem is that using multiple measuring heads to perform on-line density detection and sampling measurement shows little advantage in terms of time consumption. The time it takes to test a sample on the machine is about the same as the time it takes to manually sample a sample and test it outside the machine.

Using spectral density measurement and taking the following measures can achieve the purpose of making multiple densitometers get the same measured value.

â‘  By adjusting some parameters, different density meters can be matched with each other: density is a clearly defined value in physics. If the two densitometers show different values, then there are two possibilities. One possibility is that one of the density meters is inaccurate; another possibility is that the two density meters have different physical evaluation criteria: the geometric conditions of the optical path are different or due to differences in the light source, filter and photoelectric sensor The difference in color judgment caused. This shows that the measurement of a neutral gray silver salt film is much simpler and better comparable to the color density measurement of printed matter.

â‘¡When measuring with multiple measuring heads, the same light source and the same optical fiber should be used. For several densitometers, different light sources are also the biggest problem. Photoelectric sensors and filters can be selected.

The gray density can be used to calibrate the color density, and it is necessary to control the color of the light source based on the chromaticity diagram.

If you plan to multiply the measured value of a certain density meter by a constant to make it the same as the displayed value of another density meter, the wrong results will be obtained when calculating the dot coverage.

For example: according to the density reading = α · density measurement

Let α = 1.1
Then F ＝ 1－10－Dt1－10－Ds ≠ 1-10-Dt / 1-10-Ds
F = 1-10-0.5 / 1-10-2.0 ≠ 1-10-0.55 / 1-10-2.2
F = 69% ≠ 72%

If you intend to change the gain coefficient of the photovoltaic element to make different density meters get the same display value, it is also not feasible. At this time, the density value and the dot coverage value will not change (the working state of the amplifier should be linear).

Let Φ → aΦ ′
Then the density Dt = -logΦt / Φ0≡-loga · Φt / a · Φ0
Outlet coverage


F = Φ0-Φt / Φ0-Φs≡aΦ0-aΦt / aΦ0-aΦs

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