A mathematical model of optical density of digital images has been developed for print equipment management systems. The model accounts for variations in tone ranging from black to white, including intermediate shades of gray with a range of gray levels from 0 to 255. A logarithmic expression has been proposed to determine the reflection optical density, allowing for the calculation and construction of optical density characteristics for different levels of gray and various brightness gradation distributions within the tonal range. This enhances the efficiency of operators (designers) using computer publishing systems when preparing images for printing. A structural scheme has been developed for a simulator of optical density of digital images using MATLAB: Simulink. The simulator, based on the number of gray levels in an image, calculates and constructs the optical density characteristic achievable in offset printing within the range of [0≤D≤3.0]. To quantitatively evaluate technological influences on digitization results, a deviation of optical density from linearity is proposed, with a maximum allowable deviation of +30%. The results of simulation modeling are presented in the form of a gradation characteristic of the image in gray levels and the optical density characteristic of the image, depending on the number of gray levels. These results enhance the informativeness of digital images in the traditional context of the printing industry. The findings of analytical research and simulation modeling of optical density of digital images can be used by operators of computer publishing systems to select optimal gradation characteristics for reproduction at various stages of image preparation for printing. Modeling, digital image, control system, decision-making system, transformation, gray
levels, optical density, simulator, gradation characteristics, printing tools.
Optical density of digital images is a crucial parameter in the field of printing, as it indicates the concentration of ink or dye on a printing medium, such as paper or film. It is measured using a spectrophotometer or densitometer. The optical density of an image can be defined as the logarithm of the ratio between the light intensity passing through the printed material and the light intensity on a white background. It is measured on a scale of 0 to 5, where 0 represents a completely transparent material (no ink present), and 5 represents a fully opaque material (maximum ink concentration). In print management systems for graphic equipment, optical density is utilized for monitoring and calibrating the printing process. It enables the establishment of the desired printing intensity to achieve the desired levels of tone and color. Optical density can be measured separately for each color, such as process colors like cyan, magenta, yellow, and black, or as the overall density of the image. Precise measurement of optical density requires calibrated equipment and standardized
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2023 Copyright for this paper by its authors. measurement methods. It's important to note that optical density is one of several factors influencing print quality. Other factors, such as dot accuracy, dot size, halftone frequency, and the type of printing material, also play a significant role in achieving the desired outcome. 1.1.
To ensure high-quality reproduction of digital images in the field of printing, various adjustments
are often required due to the imperfect conditions under which the images obtained. This includes
imperfections in scanners, digital cameras, and non-professional equipment, as well as the presence of
various artifacts and distortions. Therefore, most digital images require different corrections,
including tonal adjustments [
Proper quality of the final product achieved when the images and textual information reproduced
in the print without significant losses, which is achieved through thorough control using step wedges,
reference scales, standards, and measuring instruments. The primary instrument used is a
densitometer, which employed to control printed reproductions, including the range of optical density,
image contrast, relative area of halftone dots, and their spread [
Hence, the determination of the optical density of digital images is a relevant task that allows for constructing the characteristics of optical density for the initial digital image after its adjustment. 1.2.
Modern prepress processes are based on the digital representation of images, which involves
converting continuous images into discrete (digital) representations using gray levels that can vary
from zero (black) to 255 (white), with 127 representing a 50% gray level [
If a digital image is reproduced on a monitor screen, laser printer, plotter, or in a printed reproduction, its tone will appear differently. This is because the image on the monitor is displayed using emitted light, while in printing it is reproduced using toner or ink, which needs to be taken into account when controlling the image at different stages of preparation for printing. To enhance the efficiency of digital image processing, it is necessary to have information about the optical density that can be used to predict image contrast, the area of halftone dots, and their spread. This contributes to improving the quality of image preparation for printing. The goal of the article is to develop a mathematical model of optical density for digital images, determine and construct reflection optical density characteristics for different levels of grayscale, and analyze their properties.
To construct a mathematical model of optical density for digital images, we assume that the digital
images are obtained through scanning originals, which is based on the phenomenon of light reflection
from the image. The main physical parameter involved is the reflectance coefficient R, which is
determined by the relationship [
The equation you mentioned describes the relationship between the intensity of the incident light
(Ф0) and the intensity of the reflected light (Ф1). If the scanned original reflects 10% of the light, then
the reflectance coefficient of the analyzed image would be 0.1. The main parameter for scanning is
optical density, which is measured in logarithmic units [
After substituting the reflection coefficient in (1) optical density.
In scanners, the intensity of reflected light is measured by photosensitive elements (e.g., CCD), and their output electrical signal corresponds to the brightness L of the image. Through analog-todigital conversion (ADC), it is transformed into a corresponding number of gray levels within the range of 0 to 255 (an 8-bit representation of black pixels). The optical density of scanned digitized originals (digital image) is then determined by the number of gray levels that correspond to light intensity. where L0 represents the nominal number of gray levels of the ADC and L corresponds to the brightness level of the image
The optical density of a digital image is determined by the expression: = ,
0 ≤ ≤ 255, 0 = 255 + 1
, = Ф0 Ф1
, = 1
, = Ф1 Ф0
, = (
) ∗ 0, 0 1 = 255 + 1 , (1) (2) (3) (4) (5) (6) (7) output of this block is connected to the input of the Mathematical Functions (Fcn) block.
2. Mathematical Functions (Fcn) Block: This block contains the program (equation 5) for calculating the optical density. It takes the gray levels from the Ramp block as input and performs the necessary calculations.
The unit is introduced to account for the first level of black.
If we linearly change the amount of gray in the range of 0≤L≤255 in equation (5), we can calculate
and construct the optical density characteristic of the digital image on a linear scale. In most cases,
digital images have low quality and require adjustments, which can be done using computer graphics
software such as Photoshop [
Where Lout – is the output brightness value, L0 = 255 – is the number of gray levels, and r is the exponent that determines the desired correction of the digital image. Then, using equation (5), we can determine the optical density of the corrected image.
To simplify the task of determining the optical density of a digital image, an imitation modeling approach was applied using the MATLAB Simulink package. Based on the available blocks from the Simulink library, a structural diagram of the simulator model for optical density of digital images was constructed, as shown in Figure 1. following blocks:
The structural diagram of the simulator model for the optical density of digital images includes the 3. Gain Block (Gajn): This block receives the gray levels in parallel and applies scaling to them. The scaled gray levels are then passed to the next Mathematical Functions (Fcn4) block.
4. Mathematical Functions (Fcn4) Block: This block contains the program (equation 6) for determining the corrected number of levels based on the scaled gray levels. The result is then fed into the Mathematical Functions (Fcn1) block.
5. Mathematical Functions (Fcn1) Block: This block contains the program (equation 7) for calculating the optical density of the corrected digital image. It takes the corrected number of levels as input and performs the necessary calculations.
6. Scope Block: This block is used for visualization and displays the intermediate results or variables during the simulation.
7. Display Block: This block is used to visualize and display the final results of the simulation.
The results of the simulation, depicting the gradient characteristics of the digital images, are presented in Figure 2. The second characteristic is a straight line and corresponds to a linear digital scale with 256 levels of gray. The image exhibits more or less uniform information content and has a constant contrast value of 1. The first gradient characteristic of the corrected image is a convex curve. Shifting the characteristic curve upwards increases the brightness of the image, enhances contrast, and improves the visibility of details in darker areas of the image, but may lead to the loss of details in brighter areas. Shifting the characteristic curve downwards results in image darkening, increased color saturation, better visibility of details in brighter areas of the image, but at the expense of losing details in the shadows. The results of the simulated modeling of the optical density of digital images are presented in Figure 3.
Maximum optical density value, Dm = 2.5, corresponds to the achievable level in offset printing. The optical density of the linear digital scale changes according to the logarithmic law (equation 5), gradually approaching zero in the gray tones. The characteristic curve of the optical density of the corrected digital image (equation 2) is positioned below the previous one. Shifting the curve downwards results in a decrease in optical density and lightening of the image.
Comparing Figures 2 and 3, we can conclude that the optical density of digital images is inversely related to the grayscale characteristic provided by the gray levels, which is due to the logarithmic algorithm for calculating the optical density (equation 7). Therefore, optical density and brightness of digital images describe different aspects of the image. It should be noted that the characteristics of optical density more fully describe the black tone of digital images compared to the grayscale characteristic. To assess the impact of image correction, it is proposed to determine the deviation of the optical density in percentage.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. where Dm is the maximum value of optical density of the digital image.
The results of the simulated modeling of the optical density deviation are presented in Figure 4. = 0– , (8)
The maximum deviation of optical density occurs at low gray levels, with a maximum value of 30%. It gradually decreases and approaches zero at the end of the interval.
Let's examine the influence of the number of gray levels on the optical density. As an example, we set the number of gray levels to 128. The simulator was adjusted accordingly to have 128 levels (see equation 7). The results of the simulated modeling of optical density for different numbers of gray levels are shown in Figure 5.
The initial value of the first characteristic of optical density is 2.5 and decreases logarithmically, approaching zero at 255 gray levels. On the other hand, the second characteristic has an initial value of D = 2.4, positioned below the previous one, and approaches zero at 128 gray levels. Decreasing the number of gray levels compresses the optical density characteristic, reduces tonality, and brightens the image. Adjusting the image in this way will result in fewer details in the dark areas of the image, which should be taken into account when preparing the image for printing.
A mathematical model of optical density for digital images has been developed, expressed logarithmically as the ratio of the nominal number of gray levels (255) to the given digital image. A structural scheme of the accumulator model for optical density of digital images in the MATLAB package Simulink has been constructed, which allows for computing and generating characteristics of optical density for specified digital images and analyzing their properties.
The presented results of simulation modeling are in the form of optical density characteristics for typical digital images with linear scales and convex gradation characteristics. It has been established that the optical density characteristic for a linear digital scale has a maximum value of Dm=2.5 and gradually approaches zero for gray tones. The characteristic for the adjusted image is positioned below the previous one, resulting in a decrease in optical density and image brightening. The deviation of optical densities has been determined, with the maximum value occurring at low gray levels and reaching 30%. The influence of the number of gray levels on optical density has been examined. It has been found that with 128 gray levels, there is compression of the optical density characteristic, a reduction in tonality, and image brightening. Consequently, the adjusted image will have fewer details in the dark areas.
It has been established that the optical density characteristic of a digital image is inversely related to the gradation characteristic provided by gray levels, which is due to the logarithmic algorithm for calculating optical density. Optical density and brightness describe images in different planes. The optical density characteristics more fully describe the black tone of digital images compared to gradation characteristics. The results of this work enhance the informativeness of digital images and can be used in computer publishing systems to select optimal gradation characteristics for reproduction at different stages of image preparation for printing.