The problem of constructing a model for the darkness level of monochromatic halftone image printed with "black" ink based on the light reflection coefficient from the image surface using optical density in logarithmic and exponential transformation algorithms was considered. The intensity of visual perception of optical density within a specified tone reproduction range was quantitatively evaluated and analyzed. Characteristics of optical density dependence for different transformation algorithms were determined and analyzed in terms of their properties. Deviations, sensitivity to changes in parameters, and the intensity of visual perception of optical density were determined between the algorithms. For modeling and studying the transformation specifically Simulink, was applied. Based on this, a structural model scheme was developed, allowing parallel calculation and construction of optical density characteristics for different transformations, determining their deviations, sensitivity to parameter changes, and assessing the intensity of visual perception within the specified tone reproduction range. The results of simulation modeling in the form of optical density dependencies, their sensitivity, and intensity of visual perception for different transformation algorithms were presented and analyzed for their properties. It was established that the proposed exponential algorithm provides better image perception by the human visual system, which is its advantage. exponential algorithm, simulation, analysis, perception properties.
decision-making, optical density, transformation, logarithmic algorithm,
Logarithmic and exponential transformation algorithms for optical density are used in image processing to enhance contrast and image quality. Both algorithms rely on fundamental mathematical principles and utilize the logarithmic and exponential functions, respectively.
The logarithmic transformation algorithm is based on the natural logarithmic function. The main idea is to compress high optical density values, reducing them, while expanding low density values, increasing them. This allows for a more detailed image with improved contrast.
The exponential transformation algorithm utilizes the exponential function to enhance image contrast. This algorithm increases the optical density value based on the input value, highlighting details where the optical density is low. Modeling the logarithmic and exponential transformation algorithms involves steps such as loading the image with optical density, computing the logarithmic or exponential transformation based on the selected algorithm, adjusting constants to achieve the desired level of contrast, and saving the modified image with improved contrast. EMAIL: (Mikola
Lutskiv);
2023 Copyright for this paper by its authors.
By using modeling techniques for these algorithms, one can investigate their impact on image quality and find optimal constant values for different types of images and scenarios. 1.1.
In the historical evolution of photographic processes and images, the attenuation of light flux
began to be defined, and the reflection (or transmission) of light flux was quantified using the
reflection coefficient. To quantitatively evaluate the light attenuation characteristics of an object, the
decimal logarithm of opacity, known as optical density, was introduced. Later, this method was
applied in printing to determine the darkness level of images on prints based on the light reflection
from the image surface (scales), and the assessment was performed using optical density as the
decimal logarithm of the reflection coefficient [
Densitometers are devices designed to measure the intensity of reflected or transmitted light. They
are used to determine optical density, relative area of printing elements, contrast, and other
parameters. According to the international standard ISO5-2, the optical density of inked areas on a
print (scales) is determined based on the reflection coefficient and quantitatively assessed using the
optical density of reflection, which is the decimal logarithm of the inverse of the reflection coefficient
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Optical measurements are used in the field of printing for controlling originals, printing plates,
printed impressions, and more. For this purpose, densitometers (density - the ability of an object to
absorb light) are used, which are devices designed to measure the intensity of transmitted or reflected
light. They are used to determine optical density, relative area of printing elements, thickness of ink
layers, image contrast, and more [
The reflection coefficient R0 is the ratio of the reflected light flux Fp from the object under measurement to the intensity of the incident flux F0 that falls onto the object under measurement.
If the entire light flux is reflected from the image (target), its tenth, hundredth, or thousandth part, then the optical density will be 0, 1, 2, 3. Substituting the coefficient (2) into equation (1), we obtain an expression for determining the reflected optical density.
0 = 0 1
0 ≤ 0 ≤ 1, 0 = 0
, 0 = , (1) (2) (3) (4) = ( (− 0)),
0 ≤ 0 ≤ 1,
Expression (1) represents a mathematical logarithmic algorithm for determining the reflected optical density, while expression (3) represents an algorithm for the hardware implementation of a device to determine optical density and its modeling. Since the logarithmic algorithm for determining optical density has several drawbacks mentioned above, an alternative exponential algorithm for determining reflected optical density is proposed.
Dn - nominal value of optical density, b - coefficient that determines the shape of the curve and is chosen within the range (6 ≤ b ≤ 9).
Based on the above, structural schemes of optical density models have been developed using the MATLAB Simulink package, which allow for parallel computation of reflected optical density using the logarithmic algorithm (3) and the exponential algorithm (4) and are depicted in Figure 1. block Fcn. The dialog window of Fcn contains the program (expression) for calculating the reflection optical density D0 using the logarithmic algorithm. The second Ramp block generates the reflection coefficient within the range 0 ≤ R0 ≤ 1 and is fed into the second Math Function block Fcn1, which contains the program (expression 4) for determining the optical density De using the exponential algorithm. The calculation results of the optical densities are visualized using the Scope block. To compare the results obtained by different algorithms, the deviations of the optical densities are determined.
The addition and division operations, located at the bottom of the diagram, implement it.
The model parameters were set for the nominal optical density Dn = 3.0, b = 7, and other parameters are directly provided in Figure 1. The simulation results of the optical densities calculated using different algorithms are depicted in Figure 2.
0 −
= 100%, (5) curve with a steep slope. In the initial range, the optical density sharply decreases from Dm = 3.0 to D0= 0.85 at R0 = 0.15, and then gradually decreases, almost linearly, towards zero. On the other hand, the characteristic of optical density 2, determined using the potential algorithm, has a significantly smaller slope at the beginning of the range and gradually approaches zero. It should be noted that the characteristics are practically indistinguishable in the highlight tones. For comparison, the deviations of optical densities determined using different algorithms are shown in Figure 3.
The maximum deviation of optical density occurs at the beginning of the range and amounts to 27.5%. It gradually decreases and crosses zero at a coefficient of R0 = 0.12, becoming positive with a maximum deviation of +10% and gradually approaching zero. Therefore, the main deviation of optical densities occurs only at the beginning of the range with the maximum density value.
As mentioned earlier, there are problems in determining individual parameters using densitometric methods, and significant errors in their determination exist. One of the reasons for these errors is the sensitivity of the algorithms to the reflection coefficient R0, especially at low values. It is proposed to determine the sensitivity of the algorithms to changes in optical density using derivatives.
To determine the sensitivity in the model, Derivative blocks were used, which are located at the bottom of the diagram. As an example, the sensitivity of the optical density algorithms for small values of reflection coefficients was determined, as shown in Figure 4 in close-up. = 0 , (6) (7)
At nominal values of optical density D=3.0 for the logarithmic algorithm of optical density, the maximum sensitivity value is Sm=-300, gradually decreasing. At a coefficient R0=0.02, the sensitivity of optical density leads to significant errors in determining individual parameters using densitometric methods. On the other hand, the sensitivity of the exponential algorithm is practically constant and close to -25-30. Therefore, in terms of sensitivity, the proposed exponential algorithm significantly outperforms the logarithmic one.
As mentioned above, optical density does not always correspond to visual perception based on
Weber-Fechner's law, which describes the human visual system's perception of light. Since the
reproduction of images by printing methods is done using black ink, which is predominantly
described by optical density, it does not directly take into account the properties of the human visual
system. Later, it was established that Weber-Fechner's law, which is psychophysical and describes the
perception of physical quantities by sensory organs, is valid, for example, for the human perception of
sound volume, light intensity, mechanical force. In other words, sensation is proportional to the
logarithm of the stimulus intensity [
The characteristic of visual perception 2 for the logarithmic algorithm of optical density determination at the beginning of the range ≤ R0 ≤ 0.2 (0 ≤ W ≤ 14.5) follows Weber-Fechner's law of optical density perception. However, after that, the characteristic is nearly linear and converges to a final value of W = 38. Overall, the intensity characteristic of visual perception determined by the logarithmic algorithm does not comply with Weber-Fechner's law. On the other hand, the characteristic 1 of visual perception for the exponential algorithm of optical density determination follows Weber-Fechner's law throughout the entire tonal range.
Since densitometry is widely used for control and determination of various parameters in the preparation stage of image printing, plate production, and printing processes, involving various scales, understanding the peculiarities of human vision and the provided models and algorithms for transforming optical density and intensity characteristics of visual perception will contribute to enhancing the efficiency of monitoring individual stages and image quality.
A model of traditional logarithmic and exponential algorithms has been developed to determine the optical density of monochrome halftone images within a specified density range in offset printing. The algorithms are presented in mathematical expressions as hardware implementation algorithms for a measuring device. The sensitivity of the algorithms to changes in optical density is determined using derivatives. An extension of Weber-Fechner's law, which describes the perception of various physical quantities by human sensory organs and is proportional to the logarithm of stimulus intensity, is applied to perceive optical density by the human visual system. A model of visual perception intensity for optical density is proposed, enabling the determination and construction of perception characteristics for different algorithms of optical density determination.
Structural diagrams of optical density models in MATLAB: Simulink have been processed, allowing for parallel computation of reflection optical density using logarithmic and exponential algorithms. The deviations and the dependence of optical density sensitivity, leading to significant errors in determining individual parameters using densitometric methods, are determined. The results of simulation modeling indicate that the characteristic of optical density determined by the logarithmic algorithm has a steep slope in the shadows, while the characteristic determined by the exponential algorithm has a significantly gentler slope, resulting in a maximum deviation of 27.5%. The maximum sensitivity value of optical density is determined as Sm = -300, leading to significant errors in determining individual parameters using densitometric methods. However, the sensitivity of the exponential algorithm is practically constant and close to 25...30. The characteristic of intensity perception determined by the logarithmic algorithm does not comply with Weber-Fechner's law. In contrast, the characteristic of perception for the exponential algorithm complies with Weber-Fechner's law throughout the entire tonal range.