An analytical method of formation of a linear characteristic of normalized raster transformation for rhombic elements with a normalized relative area that changes linearly within [0,1] that is a carrier of information has been developed, the parameters of the adjustment unit that forms the linearity of the transformation have been determined. The structural scheme of the model that gives the chance to calculate and visualize the characteristics of raster transformation has been constructed. The results of simulation modelling of gradation characteristics, deviations from the linear one have been presented, the parameters of the adjustment unit that provides the formation of the linear characteristic of the normalized raster transformation for rhombic elements have been determined. The advantage of the method is that it allows one to provide the linearization of the raster for a given lineature without changing the parameters of the adjustment unit relatively simply by scaling method.
Rasterization is practically the most important basis for reproducing images in printing industry.
Rasterization is used to reproduce images of different shades of tone on the imprint, i.e. the
decomposition of the image into small elements, the transformation into a raster plate, manufacturing
a raster plate consisting of space and printing elements necessary for printing itself. The information
carrier is the area of raster elements that corresponds to the tone of the original. Modern applications
for image processing include a rasterization unit, rasterization algorithms, procedures and sequence of
area output in the form of bitmaps, which are reduced to the formation of raster dots of a given shape,
lineature and the area corresponding to the tone of the original [
In many areas of science and technology, the basic information about real objects or processes is
presented in the form of images. When obtaining and reproducing images by printing techniques, it is
necessary to ensure high quality, which is reduced in the preparation of images for printing, plate
manufacturing and printing itself, which necessitates the development of simple and effective
methods and technologies to improve their quality. In digital image processing, grey gradations in the
range [0.255] are represented by arrays of numbers. Instead, in the raster transformation one can
operate by changing the geometric dimensions and areas of raster elements, and in plate
manufacturing and printing one can operate by changing the printing elements that are the carrier of
information. Therefore, the existing methods of digital image processing [
The monographs [
The authors [
Rasterization is the basis for reproducing images in printing industry. The information carrier is
the area of raster elements that corresponds to the optical density of the original, or the number of
levels of brightness of the digital image. To generalize the analysis, a mathematical model of
normalized raster transformation in the form of a double function with a domain as a closed single
square and a set of relative areas with a limited interval [
= ( , , ) , where XH is a normalized size of a raster element, which is located in a single square, LP is a number of grey levels, D is the optical density of the original image.
The geometric representation of the suggested normalized raster transformation for the rhombic element, located in the centre of the square of single dimensions is presented in Figure 1.
The raster square (cell) ABCD has constant single dimensions. The rhombic element is located in the centre of square O. In the process of raster transformation, the size of the raster element changes, which is represented by a quarter of the diagonal of the square, which varies within [0,L0]. According to Figure.1, one can write the function of raster transformation for the first interval: 1 = 4 2, if 0 ≤
≤ 0 = 0,3538, where XH is an argument (space variable), L0 is a quarter of the diagonal of the raster square АВСD. (1) (2) 0 4 2 = 1
+ 2 , = [ − 0] ∗ 100%, (3) (4) (5) (6) 2 = 8
( − ) , if 0 ≤ ≤ М Based on the diagram of Figure.2, we determine necessary parameters for calculating the area: = √1 + 1 = 1,414, = = = 0,7071; 0 = = 0,3538 Then, the function of the normalized raster transformation is
If in expressions (3) and (4) the spatial variable is linearly changed, then they can be used to calculate and construct the characteristic of the normalized raster transformation. Since the characteristic of raster transformation is nonlinear, it is suggested to determine the deviation of the characteristic from the linear one to assess the nonlinearity where S0 – is a linear characteristic
To simplify the solution of problems and construction of the raster transformation characteristic
and its linearization, we use simulation modeling. Based on the above information and the paradigm
of object-oriented programming, a block diagram of the model of the normalized raster transformation
is designed in the Matlab:Simulink package [
In the upper part there is a diagram of the model to calculate the area by the first interval operation according to expression (3). [0≤ ≤ 0,3538]. The Ramp block generates a linear normalized size which is limited by the Saturation block to the level of 0,3536. Mathematical functions block Fcn calculates the area 1 according to the expression (2). There is a diagram in each part for calculating the area of the raster element on the second range [0,3536≤ ≤0,7071]. The TransferFcn unit performs the integration
The Step unit switches the specified ranges. At the output of the summation unit, the result of calculating the normalized raster transformation is obtained. Scope and Display units are used to visualize the simulation results. At the bottom there is a diagram that determines the deviation of the characteristics of the normalized raster transformation from the linear. Let's adjust the model to certain parameters. Figure 3 presents the results of simulation modeling of the graphic characteristics of the normalized raster transformation for raster elements of rhombic shape. approaches to zero. The nonlinearity of the normalized raster transformation characteristic causes function for linearization of the normalized raster transformation is offered for this purpose л = 0,599 0,5, 0 ≤ ≤ 0,7071 which precedes the raster transformation function (2). (7)
On the basis of the above information and Figure 2, a block diagram of the model of linear characteristic forming of the normalized raster transformation for rhombic elements with sequential inclusion of the adjustment unit (Figure 5) implemented using a block of mathematical functions.
The Ramp block generates a linearly increasing normalized size of the raster element, which is provided at the input of the Fcn1 mathematical functions block in the dialogue box of which the equation (7) is recorded for linearization, the output of which is transferred to the next block of mathematical functions Fcn1 where the expression (2) is recorded of the normalized raster transformation, the calculated area is obtained at its output. The Scope unit is used to visualize the simulation results.
At the bottom there is a diagram that determines the deviation of the characteristic from the linear according to the expression (6), which is visualized by the second unit Scope1. Figure 6 shows the results of linearization of the normalized raster transformation characteristic.
For comparison, the figure shows linear characteristic that is quite close to the normalized raster transformation characteristic, and the difference between them is barely noticeable. Figure 7 shows the results of simulation of the deviation of the linearized characteristic from the linear one.
The deviation characteristic is almost linear and varies from zero to 0.141%. Therefore, the suggested sequential adjustment is simple and provides sufficient accuracy of linearization of the normalized raster transformation characteristic for raster elements of rhombic shape, which is an advantage.
Analytical method of linear characteristic forming of the normalized raster transformation for
rhombic raster elements with a normalized relative area that changes linearly within [
The results of simulation modeling of gradation characteristics, deviation from the linear one are presented, the parameters of the adjustment unit are determined, that provides the formation of the linear characteristic of the normalized raster transformation for rhombic elements. It is defined, that the suggested method of linearization of the natural characteristic of the normalized raster transformation provides a linearization error of not more than 0.141%. The results of this work can be used for further research with the aim to compensate the impact of dot gain of raster elements while making a plate, and in the process of printing.