An RGB Pseudo-Colorization Method for Filtering of Multi-Source Graphical Data

Abstract

Artificial colorization (pseudo-colorization) is a commonly used method to improve the readability of images obtained from sources (sensors) that do not reflect the original color of the object of observation (e.g., X-ray). It is designed to draw the observer’s attention to the important details of the analyzed image (e.g., disease changes in medical imaging). Analogous needs occur in the process of assessing the emission security (EMSEC) of imaging devices used to process classified information, which is made on the basis of the analysis of images reproduced from compromising emanations related to the operation of these devices. The presence of many graphic elements in an image may reduce the level of perception of the information contained in it. Such images may be very noisy or contain overlapping graphic symbols, the source of which is devices processing graphic information operating in close proximity to each other. The use of various types of measures enabling data filtration at various stages of their processing, e.g., the use of a directional antenna, frequency filtering, point filtering or contextual contrast modification, does not always prove effective. The solution to the filtration problem is the pseudo-colorization of the image. However, the image colorization used based on the typical “Hot”, “Radar” or “Cold” color palettes does not meet the requirements for filtering graphic data from many sources. It is necessary to use a filter that will allow the sharp cut-off of graphic data at the border between the background and the graphic symbol. For the pseudo-colorization process itself, the exponential function as a function of transforming the amplitudes of image pixels from the gray color space to the RGB color space is sufficient. However, the smooth transition of the function shape from zero values to values greater than zero results in a low efficiency of filtering graphic data from noise. In this article, a method of filtering an image based on the pseudo-colorization of its content, i.e., reproduction of a compromising emanation signal level in the RGB value of image pixel color components, was proposed. A quadratic function was proposed as the transformation function. The higher effectiveness of the method based on the use of a square function (compared to the exponential function) was shown by conducting tests on many images, some of which are presented in this article. The proposed solution is a universal approach and can be used in various fields related to image analysis and the need for their filtration. Its universality is related to the possibility of changing function parameters affecting its position on the value axis from 0 to 255, its width, its minimum and its maximum value for each RGB channel.

1. Introduction

The issue of electromagnetic protection, i.e., minimizing threats to the confidentiality of information related to the possibility of using the so-called disclosing emissions, has been present in the awareness of a wide group of people from the IT (Information Technology) industry since at least half of the 1980s [1]. Considering that classified research programs related to this issue date back to the 1940s, one might assume that all threats have been recognized and eliminated [2]. Indeed, in the case of specialized devices produced in short series (e.g., encryption devices), their electromagnetic safety can be ensured by using filtration and shielding techniques [3]. However, modern information processing systems are based on the widespread use of computer hardware. Both economics and ergonomics mean that this equipment cannot significantly differ from commercial equipment [4].

It is true that technical progress related to the development of digital signal processing and transmission techniques (extensive modulation techniques, considerable processing speed) make it practically impossible to use radiated electromagnetic emissions related to the operation of ultra-fast interfaces currently used in computer systems in the classical sense of electromagnetic eavesdropping (SATA (Serial Advanced Technology Attachment), USB (Universal Serial Bus), Ethernet) [5]. However, it is still possible to explore these types of system elements related to the display of information in a form understandable to the user (monitors, displays, printers) [6]. These devices still use raster imaging techniques, and the information entered into them is processed serially, at a relatively slow pace. Such technologies still allow the reconstruction of information in the form of images, which can become an easy source of non-invasive data acquisition. As the experimental results show, laser printers are not safe in this respect, and digital DVI/HDMI (Digital Visual Interface/High-Definition Multimedia Interface) interfaces are characterized by their even greater susceptibility to electromagnetic eavesdropping than the analog VGA (Video Graphics Array) interface [7]. Data reproduced from emission signals leaking from analog sources have in their graphic structure only vertical and diagonal edges, which are very susceptible to disturbances inside the propagation channels. Compatibility requirements with older-generation hardware mean that the use of the DisplayPort interface is also not completely safe [8]. The reason for this phenomenon is the parallel existence of signals from analog VGA or digital HDMI standards. Each graphic imaging device may become a source of undesirable emissions [9]. This was demonstrated in [10,11], where the source of signals correlating with the displayed information was an LCD panel. A similar conclusion was reached by Guri in [12], who conducted research and showed the existence of a risk of electromagnetic infiltration in the case of LED monitors. The requirements related to ensuring the security of confidential information mean that devices intended for processing classified information, both in special and commercial versions, must be subject to tests to determine their ability to produce usable electromagnetic compromising emanations.

This type of research is conducted in specialized research laboratories operating under the auspices of national security services. It is obvious that they must have tools to identify compromising emanation signals. Especially in the case of signals that are video signals and therefore can be used using relatively unsophisticated reception and imaging methods, one cannot afford to make the mistake of missing the target. At the final stage of analysis, the reproduced images are assessed by the operator [13]. The verdict regarding the level of security of the tested device may depend on operator experience and predispositions, regardless of the colors of the processed images [14], the printing technology used for laser printers [15] or the quality of prints for LED and laser printers [16]. Therefore, it is important to equip it with tools enabling an objective assessment of the degree of similarity of the reproduced image to the source image.

It should be borne in mind that electromagnetic compromising emanation signals arise naturally and are the result of the laws of physics, so their nature has not been designed to ensure the best possible propagation, free from losses and distortions [17]. In the case of video signals, they usually reflect only part of the imaged information (edges of shapes) [18] and do not contain information about the colors used to present it [19]. Therefore, they are residual emissions, often distorted and disturbed by other signals accompanying the device’s operation. These may be emissions correlated with vertical and horizontal deflection signals of computer monitors [20], or signals of cooling systems, heaters and stepper motors as in the case of multifunctional devices [21]. Due to such a special form of these signals, it becomes difficult to use classical methods of assessing their quality based on comparing the replica with a standard to evaluate the images reproduced from them. In this case, a significant problem is the fact that even a distorted, noisy image reconstructed from the compromising emanation signal has information content that may contribute to the loss of confidentiality of the data associated with it [22]. Classical methods of assessing image quality would treat these types of images as illegible and unacceptable. It is therefore necessary to subject the reconstructed image to transformations that will minimize the impact of those elements of the reconstructed image that are distortive and highlight those that may be important for its content [23]. Filtering operations in the frequency domain, edge detection [24], edge detection using histogram equalization [25] and edge detection for images in the YUV color space [26] are most often used for this purpose. This method is effective when the bands of the useful signal and the interference overlap only slightly. In the case of analyzing such specific signals as compromising emanation signals, however, we quite often deal with situations in which the interference is a signal coming from another element of the graphic circuit or another imaging device. In the reproduced image, at least two unsynchronized replicas of the same or different images are observed. It is necessary to use a spatial filtering technique based on strengthening, weakening and manipulating the color [27] of specific elements of the reproduced image in order to obtain a higher level of perception [28] or increase its similarity to one of the source images. Color manipulation [28] is also used to reduce the sensitivity of processed data to the effectiveness of electromagnetic infiltration [29].

Color imaging devices use a color creation technique involving mixing the so-called primary colors. For example, in typical monitors, the color of each pixel on the screen is determined through appropriate control of each of the three elements responsible for producing colors, red ($R$), green ($G$) and blue ($B$), which is referred to as the $RGB$ color space. The source of the compromising emanation signals are therefore all the elements processing these three signals [30]. This processing is carried out in parallel; the resulting compromising emanation signal does not allow for a simple reconstruction of the analyzed image in its original color [31]. At the same time, the human sense of sight, or rather the way we perceive the world around us, has been adapted to receive color information. The very structure of the human eye means that we can see some shades of color clearly but cannot distinguish others. Receptors called cones located in the retina of the eye are responsible for color vision. There are approximately 6 million of them and they are concentrated mainly in the so-called macula lutea. There are three types of cones in the retina of the human eye, each of which has a different spectral characteristic, i.e., different sensitivity to each wavelength of light. These three types of cones are marked as S (the maximum response for short waves), M (the maximum response for medium waves) and L (the maximum response for long waves). The individual colors perceived by humans are created as a result of the interactions of nerve impulses produced by all three types of cones. The first one has maximum sensitivity for a wavelength of approximately 445 nm, which corresponds to the blue color. The second one is “tuned” to waves with a length of 535 nm (this corresponds to the green color), and the third one is 575 nm, which in turn is usually considered as red. It is estimated that the ratio of the number of different types of suppositories is $R:G:B=40:20:1$. The resultant sensitivity of the human eye is therefore the lowest for blue.

Reproducing the color of an image reconstructed from electromagnetic compromising emanation signals is a difficult, but not impossible, task [32]. However, in typical applications of assessing the electromagnetic security of devices intended for processing classified information, the original color of the images being the “source” of compromising emanation signals is of secondary importance. The most important thing is to assess their information content [33]. Therefore, the images reconstructed from them are subjected to operations aimed at eliminating noise and interference, improving contrast [34] and edge detection. Assessment of the accuracy of reproducing the original information can be supported by changing the color space in which the image is reconstructed. Typically, these images are reproduced in shades of gray [8,12,16,29]. As mentioned, cones located in the retina of the eye are responsible for color vision. In addition to them, the human eye is equipped with other types of receptors, the so-called stamens. They are primarily responsible for night vision and shape and movement detection. Moreover, there are many more of them than cones (100 million). It would seem, therefore, that images reproduced using shades of gray are suitable for analysis in terms of identifying the details they contain. It turns out, however, that the detection of details is made thanks to cones, which are also responsible for detecting colors. Additionally, it should be noted that compromising emanation signals are usually characterized by a negligible difference between the useful signal level and the environmental noise background level and are significantly lower than the interference signals coming from the components of the tested device. In such conditions, the linear transformation of the signal level value into the gray level of the pixel does not ensure adequate contrast of the monochromatic image or extraction of its details from the noise background. For this purpose, various approaches are proposed for the analytical assessment of the quality of color images. In [35], the Root Mean Enhancement method is proposed, which explores the three-dimensional contrast relationships of the RGB color channels. In turn, in [36], contrast measurement of a color image is carried out based on a stimulus in the human visual system (HVS). Grayscale images are also assessed for quality as an initial stage of image qualification [37]. A more effective solution is to use a modified color map, which can be created by reflecting a non-linear mapping in the LUT (Look-up Table), enabling compression or expansion of the dynamics of the input signal (electromagnetic compromising emanation) [38]. At the same time, the human eye is adapted to perceive the surrounding reality in color. However, the appropriate selection of image colors can increase or decrease its clarity and readability. Modifying the color of an image can be an effective compositional tool that can diversify more of its details than when using a color palette limited to shades of gray. Color is often one of the first things that catches the eye. We are naturally drawn to brighter colors. Colors have long been associated with emotions, green with safety and red with threat. The color of an object contrasting with the background attracts the observer’s eye. Appropriate use of complementary (contrasting) colors allows the observer’s eye to capture the image as a whole. At the same time, the improper use of colors may result in a deterioration of image readability, e.g., the simultaneous use of intensely saturated colors whose wavelengths significantly differ should be avoided. Due to differences in eye accommodation depending on the wavelength of light, colors such as red, orange, yellow and green can be observed simultaneously without the need for further accommodation. However, this is not possible in the case of the red, blue and purple combination. For this reason, pairs of colors that are far apart in the visible light spectrum should not be combined. Similarly, using the blue color to represent text, thin lines or small objects should also be avoided. The human visual system is not adapted to the perception of stimuli in the form of high-frequency waves that represent small objects. It should be mentioned that the number of “blue” cones is the smallest, and “short” waves are absorbed significantly more in the eye. For the same reasons, shades of blue are indicated as background colors, so grouping colors that differ only in the blue component is not recommended. Since the pigment responsible for the perception of blue waves is not sensitive to changes in brightness, the color border areas are not perceived as distinct edges.

The above-mentioned conditions indicate that the method of the colorization of images reproduced from electromagnetic compromising emanation signals and filtering important graphic data, proposed in this article, becomes an effective tool for assessing their information content (the degree of correlation with the original images). Therefore, it is an effective tool for assessing the electromagnetic security of devices used to display classified information. It uses both the effects related to the application of knowledge provided by color psychology and, above all, the filtering of image content by means of the attenuation/expansion of signal dynamics.

The proposed method is called the pseudo-colorization of images. This is directly related to the final effect obtained. The colors of the restored image do not match the colors of the original image. Colorization is intended to improve the readability of the data contained therein. The aim of the method is therefore not to attempt to faithfully reproduce the colors used during image processing on the original side.

Images created using shades of gray are present in many applications. Very often, their quality corresponds to the applications, and their undeniable advantage is the economical use of resources related to their storage or analysis [39]. However, a significant disadvantage of such images is the difficulty in perceiving many details, especially in cases that may be important from the point of view of assessing health [40], life [41], the body’s bone structure [42] or information security. In the case of images acquired in the process of electromagnetic infiltration, the information contained in such images is distorted by other graphic elements due to very “weak” electromagnetic compromising emanation signals [20,21,43], which significantly impede data recognition and perception [44]. For better readability of the image, they should be removed (filtered). One of the methods is pseudo-colorization, which, on the one hand, introduces color to the image and, on the other hand, allows the removal of unnecessary elements from it.

The possibility of practical application of the pseudo-colorization of images appearing in shades of gray is of great interest in the field of scientific research [45]. This applies not only to the analysis of X-ray images (e.g., medicine, border control or monitoring systems) [46] but also to image perception in difficult climatic conditions [47] and to the evaluation of images of the Earth’s surface [48]. An important field of application of the results of work on image colorization methods, related to increasing the possibilities of visual perception, is the protection of information against electromagnetic penetration. In many cases, the use of methods to improve the brightness, contrast or binarization of image pixel amplitude values does not bring expected results [49].

Colorization techniques are commonly used in each of the above-mentioned areas. The authors propose various solutions enabling the perception of the smallest details important in a given activity, while at the same time carrying out considerations that take into account the psychological and physiological processes taking place in human color perception, as well as the effects of using various models and color transformations. In each case, the statistical analyses conducted show the advantage of using a color image over one using shades of gray. The most frequently used methods are the use of typical color palettes such as “Hot”, “Jet”, “Warm” or “Spring”, linearly modified for X-ray scanning applications, to assist, for example, airport control workers in identifying and detecting dangerous objects [50].

Problems with analyzing grayscale images have also been noticed in the case of analyzing Earth’s areas using SAR (Synthetic Aperture Radar). In [51], a pseudo-colorization method based on the CIE L*a*b* uniform color space was proposed. As the authors of the article point out, the main advantage of this method is that it does not generate SAR image distortions. Moreover, such an image contains all the information contained in the SAR image in grayscale, while improving the visual perception and cognitive abilities of the SAR image. Therefore, this method has no filtering properties.

An important area of using image colorization is medicine. The use of various methods of transforming images from the gray color palette into the $RGB$ color space makes areas that are important for health assessment, visible. In [39], a colorization method based on Gabor filtering and Welsh colorization was presented, which is an effective method for the specificity of medical X-ray images. To assess the effectiveness of the method, the authors used a measure in the form of $SSIM$ (Structural Similarity Index Measure), which takes into account image brightness, contrast and graphic structure. Another approach to the problem of colorization is presented in [41,42]. Coloring images has also found its interest in the case of X-ray images, which are very important in the detection and diagnosis of diseases [52]. The key idea of this method is to apply sine and cosine transformations to the pixel amplitudes of the input image, which facilitates the detection of the edges of important graphic areas contained in the image [53]. In the authors’ opinion, this approach facilitates and improves the process of detecting and diagnosing diseases for medical radiologists and specialists in this field [54].

Attempts at the colorization of images aimed at emphasizing the details contained in them are intensively carried out in the mentioned areas. Limitations on their use in other applications are caused by the specificity of the analyzed images. Different image analysis methods are used in the field of medicine and others in the field of Earth analysis. In each of these areas, we usually deal with good-quality images. Their only weakness is the color palette of shades of gray. In the case of research on information protection against electromagnetic penetration, the reproduced images are characterized by a large content of redundant data that must be removed.

At this point, it is worth mentioning attempts to use image colorization to recreate its original color [31], which may also contribute to increasing the readability of the reconstructed image. The method proposed by the authors uses the relationship between the color of the video pixel (pixel amplitude value) and the value of the compromising emanation signal level. However, the presented results show that the method does not provide faithful color reproduction and still requires improvement. Another problem is the possibility of using the method when data filtration is necessary.

As mentioned, compromising emanation signals resulting from the operation of imaging systems are generated by many mutually interfering sources. These signals, which are both video and control signals, radiate at similar frequencies and cover similar bands. For this reason, the use of classical signal filtering methods in the frequency domain may significantly distort the information content of the images reproduced from them. Therefore, it seems that it is more useful to use spatial filtration methods that enable the elimination of irrelevant or unnecessary elements of the reconstructed image that disturb the perception of the information contained in it. These methods are based on the analysis of the statistical properties of the image pixel set. However, classic contextual methods leave the final image in a color space corresponding to shades of gray, and therefore have a limited impact on the human sense of sight. It seems beneficial to transfer them to a color space built taking into account the properties of the human eye and thus highlight their important content.

This article presents a pseudo-colorization method that not only highlights important data by using the $RGB$ color palette. Appropriate shaping of the color and brightness of image pixels is achieved by manipulating the signal amplitudes of the $R$, $G$ and $B$ channels by using a quadratic transformation function. In this way, the proposed method also makes it possible to remove unnecessary image elements that interfere with its readability.

In addition to the introduction, this article contains several other sections. Section 2, Materials and Methods, presents a form of graphic filter: it is a mathematical description, an algorithm for determining the value of the width of the quadratic function of each RGB component for input data specified for the exponential function and also test conditions. Section 3, Results and Discussions, contains results of the tests carried out for different images. The discussions apply to a comparison of SSIM values for exponential and quadratic functions. Finally, Section 4 concludes the main outcomes of this article.

2. Materials and Methods

2.1. Form of Graphic Filter

Images reconstructed from electromagnetic compromising emanation signals obtained during the electromagnetic eavesdropping process may be subjected to pseudo-colorization. However, this is not colorization, the effect of which is to reproduce the original colors of the image, i.e., the image displayed on graphic displays of ICT devices. The purpose of such colorization is to increase the level of readability of the reproduced image and increase the level of visual perception. Pseudo-colorization involves generating signals of three $RGB$ color channels based on a transformation using the values of a one-dimensional signal and an adopted mathematical function. An important element of the entire pseudo-colorization process is the ability to selectively change the pixel amplitudes of the transformed image. This means that a typical linear transformation will not meet the expectations in terms of filtration, exposure and colorization of the image into a human-friendly form, due to the structure of the visual system. In one article, an exponential function was adopted as the function for calculating the $RGB$ value [38]. The nature of the SCA (Side-Channel Attack) of the radiated type corresponds to the properties of a high-pass filter, which preserves the vertical edges of the original image and practically eliminates the horizontal ones. At the same time, the technique of creating characters (fonts) used in text editors means that the video signal (and the compromising emanation signal) does not abruptly change but increases/decreases quite gently. This is the result of MS Windows using an algorithm to smooth the edges of graphic characters. Its nature is best reflected by an exponential function. In older versions of MS Windows, the smoothing option of edges could be turned off. However, the sharpness of the edges then had a negative impact on the readability of the processed data.

Exponential function with mathematical notation (1) corresponding to the $R$ channel:

$$E_R\left(x\right)=\left\{\begin{array}{c}{F}_{E_R}+\left(C_{E_R}-F_{E_R}\right)·e^{\left(\frac{-22.5}{A_{E_R}^2}·{\left(x-B_{E_R}\right)}^2\right)} \mathrm{for} x\in \left[x_{1E_R},x_{2E_R}\right]\\ \\ F_{E_R} \mathrm{for} \mathrm{other} x\in \left[0,\left.x_{1E_R}\right)\right.\cup \left(x_{2E_R}\right.,\left.255\right]\end{array}\right.$$

(1)

where:

$$x_{1E_R}=B_{E_R}-\frac{A_{E_R}}{2},$$

(2)

$$x_{2E_R}=B_{E_R}+\frac{A_{E_R}}{2},$$

(3)

• $A_{E_R}$—width of the shape of the exponential function defined as

  $$A_{E_R}=x_{2E_R}-x_{1E_R},$$

  (4)
• $B_{E_R}$—the maximum value of the exponential function in the range of x $\left[x_{1E_R},x_{2E_R}\right]$;
• $x_{1E_R}$ and $x_{2E_R}$—zeros of the exponential function $E_R\left(x\right)$ for $x\in \left[x_{{1E}_R},x_{{2E}_R}\right]$;
• $C_{E_R}$—maximum value (if $C_{E_R}>F_{E_R}$) or minimum value (if $C_{E_R}<F_{E_R}$) of the exponential function $E_R\left(x\right)$ in the range of x $\left[x_{1E_R},x_{2E_R}\right]$;
• $F_{E_R}$—value of the exponential function $E_R\left(x\right)$ for $x\in \left[0,\left.x_{1E_R}\right)\right.\cup \left(x_{2E_R}\right.,\left.255\right]$.

And a shape as in Figure 1 was used to determine the values of the $RGB$ color components for each pixel of the analyzed images, while maintaining the properties of human vision related to the perception of appropriate colors. The effectiveness of the adopted methodology for determining $RGB$ values has been proven both by the results of visual analyses of the obtained images and the values of the adopted image contrast assessment measure for the $Y$ channel of the $YUV$ color model [25].

It should be noted that the pseudo-colorization of images should support tasks related to image analysis in the following areas (the authors focused on the first mentioned area in the next stage of the analyses):

• Data acquisition in the electromagnetic infiltration process;
• Analysis of medical images allowing for highlighting disease lesions;
• Identification of dangerous elements during border control;
• Research on the structures of the Earth’s surface;
• Other.

In each of these areas, colorization is related to the transformation of the image from the gray color scale into a color $RGB$ image. The first task of colorization is for this transformation to significantly increase the level of perception of the data contained in the image [31,39,41,55,56]. The second task is to filter the image being analyzed. An appropriately adopted transformation function and thus appropriately determined $RGB$ values allow for the removal of graphic elements from the image that disturb the readability of the image, while maintaining its colorful characteristic.

The exponential function is a solution suitable for the pseudo-colorization of images. However, it is too mild a function for image filtration. Its gentle transition at the limits of the image pixel amplitude range from the value of “0” to the value of “255” causes a barely noticeable border between the background and the sign. In many cases, it is necessary to use a sharper transition at the boundary of two areas than provided by the exponential function. The authors propose the use of a quadratic function, the form of which is described by Formula (5):

$$K_R\left(x\right)=\left\{\begin{array}{c}{C}_{K_R}-\frac{4·\left(C_{K_R}-F_{K_R}\right)}{A_{K_R}^2}·{\left(x-B_{K_R}\right)}^2 \mathrm{for} x\in \left[x_{1K_R},x_{2K_R}\right] \\ \\ F_{K_R} \mathrm{for} \mathrm{other} x\in \left[0,\left.x_{1K_R}\right)\right.\cup \left(x_{2K_R}\right.,\left.255\right]\end{array}\right.,$$

(5)

where:

$$x_{1K_R}=B_{K_R}-\frac{A_{K_R}}{2},$$

(6)

$$x_{2K_R}=B_{K_R}+\frac{A_{K_R}}{2},$$

(7)

• $A_{K_R}$—width of the shape of the quadratic function defined as

  $$A_{K_R}=x_{2K_R}-x_{1K_R},$$

  (8)
• $B_{K_R}$—the maximum value of the quadratic function in the range of x $\left[x_{1K_R},x_{2K_R}\right]$;
• $x_{1K_R}$ and $x_{2K_R}$—zeros of the quadratic function $K_R\left(x\right)$ for $x\in \left[x_{{1K}_R},x_{{2K}_R}\right]$;
• $C_{K_R}$—maximum value (if $C_{K_R}>F_{K_R}$) or minimum value (if $C_{K_R}<F_{K_R}$) of the quadratic function $K_R\left(x\right)$ in the range of x $\left[x_{1K_R},x_{2K_R}\right]$;
• $F_{K_R}$—value of the quadratic function $K_R\left(x\right)$ for $x\in \left[0,\left.x_{1K_R}\right)\right.\cup \left(x_{2K_R}\right.,\left.255\right]$.

The general shape of the function is shown in Figure 2.

As previously mentioned, the shape of the analyzed exponential function for the pseudo-colorization of images is not accidental. It is related to the form of signs contained in the reproduced images from compromising emanation signals. SCA, through which the radiated emission signal from the source to the receiver passes, has the characteristics of a high-pass filter. On the receiving side, the vertical and diagonal edges of graphic elements are identified. The value of the observed changes in image pixel amplitudes depends on the sharpness of the original image. The Windows system uses an anti-aliasing algorithm to display graphics. This affects the blurring effect of the reproduced edges of graphic characters, hence the proposed form of the function, the course of which is to adapt to the properties of the graphic symbols displayed on the receiving side. The proposed function, by creating the LUT values, performs a multi-level thresholding operation, which is important in cases of analyses of images containing graphics in the presence of other elements that make it difficult to recognize the information sought. However, to increase the effect of image filtration, the authors based their method on a quadratic function. The course of such a function allows for a sharper filtering of data at the border between the background and the graphic symbol, as opposed to the gentle transition that occurs for the exponential function.

The analyses performed showed a greater efficiency of filtration based on a quadratic function as opposed to an exponential function. Therefore, this article presents a filtering method for images reconstructed from compromising emanation signals, using pseudo-colorization based on a quadratic function.

In order to compare the effectiveness of the proposed methods, it was necessary to make an assumption regarding the width of the shape of the quadratic function in relation to the width of the shape of the exponential function while maintaining the remaining parameters of both functions, such as the maximum of the function, the minimum of the function and the shift in the function in the range $\left[0;255\right]$. This is intended to enable the automation of the process of pseudo-colorization and the filtering of images at a later stage. The pseudo-colorization method based on the exponential function was the starting point as the basis for transforming the image from grayscale colors to $RGB$ colors. This function meets the appropriate criterion that allows the acceptance of the method of pseudo-colorization of images in the electromagnetic infiltration process.

Moving on to defining the course of the quadratic function, in the first stage, we can assume a constant width $A$ ($A_E=A_K$) of the exponential and quadratic functions, which means that $x_{1E}=x_{1K}$ and $x_{2E}=x_{2K}$, with $C_E=C_K$, $F_E=F_K$ and $B_E=B_K$ (Figure 3).

This assumption causes the following differences:

$$D_{EK}\left(A_K\left(m\right),x_{1E}+n\right)=E\left(A_E,x_{1E}+n\right)-K\left(A_K\left(m\right),x_{1E}+n\right),$$

(9)

where

$$A_K\left(m\right)=x_{2K}\left(m\right)-x_{1K}\left(m\right),$$

(10)

$$x_{2K}\left(m\right)=x_{2E}-m,$$

(11a)

$$x_{1K}\left(m\right)=x_{1E}+m,$$

(11b)

that is,

$$A_K\left(m\right)=A_E-2·m,$$

(12)

for $n=0,\dots ,x_{2E}-x_{1E}$ and $m=0$, i.e., $A_K=A_E$, achieves significant values and pseudo-colorization with a quadratic function, taking into account only image colorization, which does not bring acceptable results. In such a case, the quadratic function amplifies the pixels on the background-character boundary too quickly and does not effectively filter the character from the background.

Therefore, it was proposed to connect the width of the quadratic function with the width of the exponential function by using the average deviation parameter and minimizing its value in the range from $x_{1E}$ to $x_{2E}$.

The above assumption allows for a given parameter $A_E$, i.e., the values $x_{1E}$ and $x_{2E}$, of the exponential function to determine the value $A_K$ of a quadratic function that will best approximate the shape of the function $K\left(x\right)$ to the function $E\left(x\right)$. The average deviation described by Formula (13) was adopted as a measure of the approximation of the quadratic function $K\left(x\right)$ to the exponential function $E\left(x\right)$:

$$d_{EK}\left(A_K\left(m\right)\right)=\frac{{\sum }_{n=0}^{N=A_E-1}\left(D_{EK}\left({A_K\left(m\right),x}_{1E}+n\right)-\overline{D_{EK}\left(A_K\left(m\right)\right)}\right)}{N},$$

(13)

where

$$\overline{D_{EK}\left(A_K\left(m\right)\right)}=\frac{{\sum }_{n=0}^{N=A_E-1}{D}_{EK}\left({A_K\left(m\right),x}_{1E}+n\right)}{N},$$

(14)

and $m=\mathrm{0,1},\dots ,M,M+1$, where $M$ is the value for which

$$d_{EK}\left(A_K\left(M\right)\right)=min\left\{d_{EK}\left(A_K\left(m\right)\right)\right\}.$$

(15)

The $min\left\{d_{EK}\left(A_K\left(m\right)\right)\right\}$ values calculated according to the algorithm presented in Figure 4 for each $RGB$ channel determine the widths of the $K\left(x\right)$ function by determining the values of the zeros $x_{1K}$ and $x_{2K}$, and thus the width of the $A_K$ function.

This allows for obtaining waveforms (Figure 5) for which the differences $D_{EK}$ provide both acceptable pseudo-colorization and image filtering efficiency.

The algorithm presented in Figure 4 begins with determining the parameters of the exponential function, the effectiveness of which in the pseudo-colorization process was demonstrated in [38]. Parameters $x_{1E}$, $x_{2E}$, $C_E$, $F_E$ and $B_E$ for each RGB channel place three runs of the $E\left(x\right)$ function on the grayscale image pixel amplitude value axis (horizontal axis), which allow pseudo-colorization of the analyzed image by transforming the pixel amplitude values’ input image to a color image. Since the problem considered in this article concerns the connection of pseudo-colorization with the filtration of graphic data contained in the image, a quadratic function was proposed as a function of transforming the amplitude values of image pixels in grayscale into the RGB color space. For this function, parameters must also be determined as for the exponential function, determining its shape and position on the axis of the image pixel amplitude values on the grayscale. Since the characteristic of the image after pseudo-colorization using the exponential function should be preserved (image coloring), the appropriate parameters of the quadratic function ($x_{1\mathrm{K}}$ and $x_{2\mathrm{K}}$) should be related to the corresponding parameters of the exponential function ($x_{1E}$ and $x_{2E}$,). Taking into account the shape of the quadratic function, the values of these parameters should not be equal (Figure 4), because the colors of the image will change significantly. The remaining parameters $C_K$, $F_K$ and $B_K$ of the quadratic transformation function should be equal to the parameter values as for the exponential function, i.e., $C_K=C_E$, $F_K=F_E$ and $B_K=B_E$. For the conditions adopted in this way, the next stage of the algorithm determines the value of the average deviation $d_{EK}\left(A_K\left(m\right)\right)$ for $m=0$ in accordance with relationship (13), i.e., for the case when $x_{1K}{=x}_{1E}$ and $x_{2K}{=x}_{2E}$ (Figure 4). This is the initial value to which the value of $d_{EK}\left(A_K\left(m\right)\right)$ calculated for ($m+1$) is compared. Increasing the m parameter by “1” increment reduces the range of the function, i.e., $x_{1\mathrm{K}}\uparrow$ and $x_{2\mathrm{K}}\downarrow$. Then, the formula

$$d_{EK}\left(A_K\left(m\right)\right)<d_{EK}\left(A_K\left(m+1\right)\right),$$

(16)

is checked. If Formula (16) is not met, the average deviation is iteratively calculated until condition (16) is met. The value of m for which Formula (16) is met clearly determines $x_{1K}$ and $x_{2K}$ and thus the width of the interval (the range of the quadratic function) $A_K$. It should be noted that the process of determining the $A_K$ value must be carried out three times, i.e., for each R, G and B channel separately.

The selection of the values of the parameters of the exponential function, and therefore of the quadratic function, must be preceded by the analysis of the pixel amplitudes of the pseudo-colored image. The maxima of the function should correspond to the pixel amplitude values of graphic data (grayscale image), which are to be exposed in the pseudo-colorization process, i.e., filtered from the disturbance background. Since graphic data in its original form are not built from pixels with the same amplitude values (the influence of the information transfer channel and intra-channel disturbances), these data are filtered out by different positions of quadratic functions for individual RGB channels on the axis of pixel amplitude values of the pseudo-colored image. In this way, image filtering and coloring are achieved, which increases the readability of the reproduced graphic data.

2.2. Experimental Settings

2.2.1. Test Images and Test System Configuration

The electromagnetic compromising emanation signals were recorded in an anechoic chamber in the arrangement shown in Figure 6. The environment inside the chamber allowed for the separation from possible sources of disturbances not coming from the tested computer sets.

Measurements of the level of emission signals and their recordings were carried out with the measurement antenna positioned at a distance of approximately 2 m and 3 m from the first and second workstations, respectively. The measurement antenna and the mentioned computer sets were placed in one line, which made it impossible to point the antenna at one selected set (Figure 7).

The research system consisted of two computer sets whose displays worked in a digital or analog standard, depending on the measurement scenario. Both commercial desktop computers and laptops were used to conduct the experiments. Various hardware configurations were used, selecting pairs of displays equipped with digital and analog video interfaces. The interaction of signals coming from images representing both text and graphic data was studied. Finally, the combinations that best illustrated the problem of overlapping images were selected to be presented in this article.

The tests were carried out for six scenarios:

• Scenario I: monitor operating mode 1280 × 1024, HDMI graphic standard (Figure 8a), monitor operating mode 1280 × 1024, HDMI graphic standard (Figure 8b), frequency of the measured compromising emanation signal $f_o=1391 \mathrm{M}\mathrm{H}\mathrm{z}$, reception band $\mathrm{B}\mathrm{W}=50 \mathrm{M}\mathrm{H}\mathrm{z}$;
• Scenario II: monitor operating mode 1280 × 1024, HDMI graphic standard (Figure 8c), monitor operating mode 1280 × 1024, VGA graphic standard (Figure 8d), frequency of the measured compromising emanation signal $f_o=235 \mathrm{M}\mathrm{H}\mathrm{z}$, reception band $\mathrm{B}\mathrm{W}=50 \mathrm{M}\mathrm{H}\mathrm{z}$;
• Scenario III: monitor operating mode 1280 × 1024, HDMI graphic standard (Figure 8e), monitor operating mode 1280 × 1024, HDMI graphic standard (Figure 8f), frequency of the measured compromising emanation signal $f_o=1647 \mathrm{M}\mathrm{H}\mathrm{z}$, reception band $\mathrm{B}\mathrm{W}=50 \mathrm{M}\mathrm{H}\mathrm{z}$;
• Scenario IV: monitor operating mode 1280 × 1024, VGA graphic standard (Figure 8g), monitor operating mode 1280 × 1024, VGA graphic standard (Figure 8h), frequency of the measured compromising emanation signal $f_o=415 \mathrm{M}\mathrm{H}\mathrm{z}$, reception band $\mathrm{B}\mathrm{W}=50 \mathrm{M}\mathrm{H}\mathrm{z}$;
• Scenario V: monitor operating mode 1280 × 1024, VGA graphic standard (Figure 8i), monitor operating mode 1280 × 1024, VGA graphic standard (Figure 8j), frequency of the measured compromising emanation signal $f_o=286 \mathrm{M}\mathrm{H}\mathrm{z}$, reception band $\mathrm{B}\mathrm{W}=50 \mathrm{M}\mathrm{H}\mathrm{z}$;
• Scenario VI: monitor operating mode 1280 × 1024, HDMI graphic standard (Figure 8k), monitor operating mode 1280 × 1024, HDMI graphic standard (Figure 8l), frequency of the measured compromising emanation signal $f_o=1647 \mathrm{M}\mathrm{H}\mathrm{z}$, reception band $\mathrm{B}\mathrm{W}=50 \mathrm{M}\mathrm{H}\mathrm{z}$.

The reception of electromagnetic compromising emanation signals was made at frequencies at which compromising emanation signals from two computer sets occurred simultaneously, which resulted in the mutual overlap of images reflecting the data processed by these computers. In each case, due to different configurations of the computer sets, the compromising emanation signal from one of the sets was more dominant. This phenomenon caused the reproduced image to contain graphic data processed on this set with clearer shapes compared to the data processed on the other computer set (Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14). Each figure contains two images. The image on the left is an image subject to filter analysis based on the pseudo-colorization method. This is an image reconstructed from the leaking emission signal generated during the simultaneous operation of two computer sets. The image on the right is an image reconstructed from the emission signal leaking where only one of the computer sets is operating. The purpose of the filtration process is to obtain reference images.

The process of the pseudo-colorization of images reproduced in the electromagnetic infiltration process on a gray color scale is carried out in accordance with the algorithm presented in Figure 4. The output data are the values of the color components for the three $RGB$ color channels calculated on the basis of Formula (5). The evaluation of the image filtration efficiency is carried out using the Y channel of the $YUV$ color space model according to the following formula [25]:

$$\left[Y_{x\left(m,n\right)}\right]=\left[\begin{array}{ccc}0.299& 0.587& 0.114\end{array}\right]\left[\begin{array}{c}{R}_{x\left(m,n\right)}\\ G_{x\left(m,n\right)}\\ B_{x\left(m,n\right)}\end{array}\right].$$

(17)

The luminance values $Y_{x\left(m,n\right)}$ of a pixel are created by mixing the primary colors of an image pixel stored in the $RGB$ color space.

2.2.2. Measure and Criterion for Assessing Image Filtration

To assess the effectiveness of the pseudo-colorization method based on a quadratic function in the process of filtering graphic images, its effects were compared with other digital image processing methods. These methods were:

• Histogram extension: Histogram equalization is a non-linear operation that changes the brightness of the image so that the histogram presents an even distribution. Each pixel of the image is modified depending on the layout of the cumulative histogram, not in the same way. Equalizing the histogram gives better results than expanding. More details can be seen; the “peaks” are expanded and the “valleys” are compressed through equalization.
• Histogram equalization: This method increases the global contrast of images, especially when the signal used to describe the image is represented by values from a small range. By extending these values to a wider range, the intensities can be better distributed on the histogram. This allows for increased contrast in fragments with a low level of contrast. The method allows for a more accurate presentation of details in overexposed or underexposed images.
• Median filtration 3 × 3: As mentioned in the introduction, the aim of this article is to present a method of filtering low-quality images, which results from the field of operation of electromagnetic infiltration. Reproduced images from leaking emission signals contain many interferences that make it difficult to read the sought data. In the case of interference in the form of single pixels, an effective method of improving quality may be median filtering with a 3 × 3 window. Using a larger window may result in too much smoothing of the edges of graphic symbols, leading to their blurring.
• FFT transformation and high-pass filtering with window sizes of 20 × 20, 40 × 40, 80 × 80 and 100 × 100: FFT transformation and high-pass filtering allow for the simple filtering of low-frequency components that occupy the center of the image spectrum after FFT transformation. By adjusting the window size, appropriate frequency components can be eliminated. This approach allows for filtering image noise, leaving the components that determine the information content. For the image as a whole, the effect is usually to increase contrast by emphasizing the sharp edges of objects.
• DCT transformation and high-pass filtering with window sizes of 10 × 10, 20 × 20, 40 × 40, 80 × 80 and 100 × 100: DCT transformation is based on the cosine function. Similarly to the FFT transformation, high-pass filtering can be performed by eliminating very low-frequency components. Five windows of different sizes are used in the filtration.
• Image colorization with the exponential function: Pseudo-colorization of the image based on the exponential transformation function from the grayscale space to the RGB color space allows for image colorization, emphasizing certain areas of the analyzed image. The method also allows for a certain level of filtration of such an image. However, due to the shape of the function, filtration during its use is not fully effective, hence the proposal to use a quadratic function as a transformation function.

The above methods are used in the electromagnetic infiltration process to assess and classify electromagnetic compromising emanation sources. The reconstructed image was subjected each time to the above-mentioned transformations. The effect of processing was assessed using the measure proposed by the authors of this article, which was the modified $SSIM$. This measure is described by Formula (18):

$$SSIM={\mu }_x^{\alpha }·{\sigma }_x^{\beta }·{corr}_{xy}^{\gamma },$$

(18)

$${\mu }_x^{\alpha }={\left(\frac{1}{M·N}\sum _{m,n=0}^{M-1,N-1}{x}_{m,n}\right)}^{\alpha },$$

(19)

$${\sigma }_x^{\beta }={\left(\sqrt{\frac{{\sum }_{m,n=0}^{M-1,N-1}{\left(x_{m,n}-{\mu }_x\right)}^2}{M·N}}\right)}^{\beta },$$

(20)

$${corr}_x^{\gamma }={\left(\frac{{\sum }_{m,n=0}^{M-1,N-1}\left(x_{m,n}-{\mu }_x\right)·\left(y_{m,n}-{\mu }_y\right)}{\sqrt{{\sum }_{m,n=0}^{M-1,N-1}{\left(x_{m,n}-{\mu }_x\right)}^2·{\sum }_{m,n=0}^{M-1,N-1}{\left(y_{m,n}-{\mu }_y\right)}^2}}\right)}^{\gamma },$$

(21)

$${\mu }_y=\frac{1}{M·N}\sum _{m,n=0}^{M-1,N-1}{y}_{m,n},$$

(22)

• $M$—horizontal size of the analyzed (pattern) image;
• $N$—vertical size of the analyzed (pattern) image;
• $x_{m,n}$—amplitude of the analyzed image pixel with coordinates $\left(n,m\right)$;
• $y_{m,n}$—amplitude of the pattern image pixel with coordinates $\left(n,m\right)$.

The weights of the individual components of the proposed measure, described as α, β and γ, determine the contribution of luminance ${\mu }_x^{\alpha }$, contrast ${\sigma }_x^{\beta }$ and cross-correlation ${corr}_x^{\gamma }$ in the calculated $SSIM$ value.

The specificity of the analysis of images reproduced in the process of electromagnetic infiltration, particularly related to the poor quality of these images, requires an appropriate balance of the participation of the three above-mentioned factors of $SSIM$. The authors proposed the following weight values:

$$\begin{gathered} \alpha =0.1, \\ \beta =1.1, \\ \gamma =3.0, \end{gathered}$$

(23)

which were used in further image filtration analyses using the pseudo-colorization method. It should be noted that a higher modified $SSIM$ value indicates a greater similarity of the filtered image to the image used as a pattern image in the analysis.

The pseudo-colorization method proposed by the authors is primarily used in the area of obtaining graphic data reconstructed on the basis of recorded electromagnetic compromising emanations. This is an area in which the reproduced data in the form of images are accompanied by various types of interference, causing difficulties in reading or searching for important data. Nevertheless, the method can be used in other areas of image analysis where it is necessary to sharpen graphical information from many data samples. This includes, among others, medical images, images of the Earth and others, which in their original form appear in shades of gray. The presented pseudo-colorization algorithm allows the full manipulation of RGB color values. This means that, depending on the needs, certain pixel amplitude values can be exposed and others suppressed, and the image is transformed into the RGB color space, increasing the perception of the data contained in it. Available image colorization methods are based on pre-defined RGB color palettes (e.g., Hot, Radar, Spring [38]), for which it is only possible to change the width of the range of individual colors in a given palette. The analysis results presented in [38] showed that this solution is not effective when trying to increase the level of perceptibility of graphic data or filtering it.

3. Results and Discussion

3.1. Input Data of Analyses

In the analyses of the effectiveness of the pseudo-colorization method in filtering graphic data, the images presented in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 were used. The values of the $RGB$ color channels were determined based on exponential and quadratic functions. According to the processing algorithm (Figure 4), in the pseudo-colorization process, the input data were the parameters of the exponential function for individual images. Based on this, also in accordance with the algorithm described in Section 2.1, the parameters of the quadratic function were determined. The modified $SSIM$ was used as a measure for assessing filtration efficiency, and maximizing the value of this measure was adopted as a criterion.

Images reproduced from electromagnetic compromising emanation signals very often contain elements that are not useful in the process of their analysis (reading the content). These are most often accompanying elements related to the source software used on the computer and may cover a significant part of the playback area. These elements are:

• Graphics program window (text editor);
• Areas not used in the graphics (text editing) program that are outside the editing sheet;
• Lines related to the synchronization signals of the eavesdropped monitor;
• Other graphics affecting the size of the area occupied by useful data.

The above elements may significantly negatively affect the components of luminance ${\mu }_x^{\alpha }$, contrast ${\sigma }_x^{\beta }$ and cross-correlation ${corr}_x^{\gamma }$ of the measure determined by Formula (17). Hence, further analyses were performed for two cases:

• Whole reproduced images and pattern images (Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14);
• Images that are fragments of the reconstructed images, and the pattern containing the filtered data (Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20).

According to the adopted algorithm (Figure 4), an important element of image pseudo-colorization is determining the parameters of the exponential function $E\left(x\right)$. Then, for the shape and position of the function determined in this way, the width $A_K$ of the quadratic function $K\left(x\right)$ should be determined in accordance with Formula (10) and the criterion $min\left\{{\sigma }_{EK}\left(A_K\left(m\right)\right)\right\}$. The determined values of the parameters $A_E$ (i.e., $x_{1E}$ and $x_{2E}$), $C_E$, $F_E$ and $B_E$ of the exponential function $E\left(x\right)$ and the quadratic function $K\left(x\right)$ are presented in

Table 1. Parameters of the exponential function $E\left(x\right)$ and the quadratic function $K\left(x\right)$ for the $R$ channel, used to transform the pixel amplitude values of the grayscale color space into the $RGB$ color space for the images in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

Function Parameters	Figure 9		Figure 10		Figure 11		Figure 12		Figure 13		Figure 14
	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$
$x_{1E}$ ($x_{1K}$)	80	100	114	136	110	150	80	120	70	110	100	122
$x_{2E}$ ($x_{2K}$)	170	150	214	192	290	250	260	220	250	210	200	172
$A_E$ ($A_K$)	90	50	100	56	180	100	180	100	180	100	100	56
$C_E$ ($C_K$)	255	255	255	255	255	255	255	255	255	255	255	255
$F_E$ ($F_K$)	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *
$B_E$ ($B_K$)	125	125	164	164	200	200	170	170	160	160	150	150

*—values $F_E$ and $F_K$ of functions $E\left(x\right)$ and $K\left(x\right)$ in ranges $x\in \left[0,x_{1E}\right]\cup \left[x_{2E},255\right]$ or $x\in \left[0,x_{1K}\right]\cup \left[x_{2K},255\right]$.

,

Table 2. Parameters of the exponential function $E\left(x\right)$ and the quadratic function $K\left(x\right)$ for the $G$ channel, used to transform the pixel amplitude values of the grayscale color space into the $RGB$ color space for the images in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

Function Parameters	Figure 9		Figure 10		Figure 11		Figure 12		Figure 13		Figure 14
	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$
$x_{1E}$ ($x_{1K}$)	27	60	59	70	15	70	27	82	23	78	15	73
$x_{2E}$ ($x_{2K}$)	177	144	109	98	265	210	277	222	273	218	275	277
$A_E$ ($A_K$)	150	84	50	28	250	140	250	140	250	140	260	144
$C_E$ ($C_K$)	255	255	255	255	255	255	255	255	255	255	255	255
$F_E$ ($F_K$)	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *	0 *
$B_E$ ($B_K$)	102	102	84	84	140	140	152	152	148	148	145	145

*—values $F_E$ and $F_K$ of functions $E\left(x\right)$ and $K\left(x\right)$ in ranges $x\in \left[0,x_{1E}\right]\cup \left[x_{2E},255\right]$ or $x\in \left[0,x_{1K}\right]\cup \left[x_{2K},255\right]$.

and

Table 3. Parameters of the exponential function $E\left(x\right)$ and the quadratic function $K\left(x\right)$ for the $B$ channel, used to transform the pixel amplitude values of the grayscale color space into the $RGB$ color space for the images in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

Function Parameters	Figure 9		Figure 10		Figure 11		Figure 12		Figure 13		Figure 14
	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$
$x_{1E}$ ($x_{1K}$)	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A
$x_{2E}$ ($x_{2K}$)	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A
$A_E$ ($A_K$)	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A
$C_E$ ($C_K$)	70 *	70 *	70 *	70 *	10 *	10 *	80 *	80 *	50 *	50 *	90 *	90 *
$F_E$ ($F_K$)	70 *	70 *	70 *	70 *	10 *	10 *	80 *	80 *	50 *	50 *	90 *	90 *
$B_E$ ($B_K$)	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A

*—values $F_E$ and $F_K$ of functions $E\left(x\right)$ and $K\left(x\right)$ in ranges $x\in \left[0,x_{1E}\right]\cup \left[x_{2E},255\right]$ or $x\in \left[0,x_{1K}\right]\cup \left[x_{2K},255\right].$ N/A—for $C_E=F_E$ and $C_K=F_K$. Functions $E\left(x\right)$ and $K\left(x\right)$ are constant and their values do not depend on parameters $x_{1E},x_{1K},x_{2E},x_{2K},A_E,A_K,A_{E}i{C}_K$.

.

The parameters $A_K$ and therefore $x_{1K}$ and $x_{2K}$ were determined based on Formula (13) for the condition $min\left\{d_{EK}\left(A_K\left(m\right)\right)\right\}$. The variability in the value of the $d_{EK}\left(A_K\left(m\right)\right)$ parameter is presented in

Table 4. Values of the average deviation $d_{EK}\left(A_K\left(m\right)\right)$ and width $A_K$ of quadratic function $K$ for a given input width $A_E=50$ of the exponential function $E\left(x\right)$.

$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$
0	50	16.5380	5	40	9.4357	10	30	2.8783
1	48	15.1250	6	38	8.0297	11	28	1.8706
2	46	13.7030	7	36	6.6459	12	26	2.5316
3	44	12.2770	8	34	5.2961	-	-	-
4	42	10.8540	9	32	4.0395	-	-	-

,

Table 5. Values of the average deviation $d_{EK}\left(A_K\left(m\right)\right)$ and width $A_K$ of quadratic function $K$ for a given input width $A_E=90$ of the exponential function $E\left(x\right)$.

$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$
0	90	26.8470	8	74	16.9630	16	58	7.1681
1	88	25.6460	9	72	15.7020	17	56	6.0688
2	86	24.4310	10	70	14.4420	18	54	5.0105
3	84	23.2040	11	68	13.1870	19	52	4.0446
4	82	21.9670	12	66	11.9460	20	50	3.4284
5	80	20.7250	13	64	10.7150	21	48	3.9717
6	78	19.4770	14	62	9.5000	-	-	-
7	76	18.2220	15	60	8.3062	-	-	-

,

Table 6. Values of the average deviation $d_{EK}\left(A_K\left(m\right)\right)$ and width $A_K$ of quadratic function $K$ for a given input width $A_E=100$ of the exponential function $E\left(x\right)$.

$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$
0	100	29.0370	8	84	19.5490	16	68	9.8781
1	98	27.8880	9	82	18.3270	17	66	8.7406
2	96	26.7250	10	80	17.1020	18	64	7.6286
3	94	25.5510	11	78	15.8800	19	62	6.5485
4	92	24.3660	12	76	14.6620	20	60	5.5352
5	90	23.1750	13	74	13.4480	21	58	4.5753
6	88	21.9740	14	72	12.2400	22	56	3.7076
7	86	20.7650	15	70	11.0430	23	54	4.1892

,

Table 7. Values of the average deviation $d_{EK}\left(A_K\left(m\right)\right)$ and width $A_K$ of quadratic function $K$ for a given input width $A_E=150$ of the exponential function $E\left(x\right)$.

$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$
0	150	37.926	12	126	26.263	24	102	13.774
1	148	37.012	13	124	25.235	25	100	12.760
2	146	36.088	14	122	24.200	26	98	11.759
3	144	35.15	15	120	23.158	27	96	10.781
4	142	34.202	16	118	22.111	28	94	9.8285
5	140	33.246	17	116	21.062	29	92	8.8944
6	138	32.277	18	114	20.015	30	90	7.9915
7	136	31.299	19	112	18.966	31	88	7.1317
8	134	30.313	20	110	17.918	32	86	6.3051
9	132	29.316	21	108	16.871	33	84	5.5498
10	130	28.307	22	106	15.834	34	82	5.8988
11	128	27.290	23	104	14.801	-	-	-

,

Table 8. Values of the average deviation $d_{EK}\left(A_K\left(m\right)\right)$ and width $A_K$ of quadratic function $K$ for a given input width $A_E=180$ of the exponential function $E\left(x\right)$.

$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$
0	180	41.654	14	152	29.788	28	124	16.552
1	178	40.876	15	150	28.869	29	122	15.611
2	176	40.085	16	148	27.943	30	120	14.674
3	174	39.282	17	146	27.012	31	118	13.750
4	172	38.472	18	144	26.074	32	116	12.840
5	170	37.648	19	142	25.129	33	114	11.948
6	168	36.814	20	140	24.179	34	112	11.070
7	166	35.970	21	138	23.229	35	110	10.209
8	164	35.114	22	136	22.275	36	108	9.3766
9	162	34.246	23	134	21.318	37	106	8.5729
10	160	33.372	24	132	20.360	38	104	7.7945
11	158	32.489	25	130	19.406	39	102	7.0644
12	156	31.599	26	128	18.451	40	100	6.7606
13	154	30.699	27	126	17.497	41	98	7.0615

and

Table 9. Values of the average deviation $d_{EK}\left(A_K\left(m\right)\right)$ and width $A_K$ of quadratic function $K$ for a given input width $A_E=250$ of the exponential function $E\left(x\right)$.

$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$	$\mathit{m}$	${\mathit{A}}_{\mathit{K}}\left(\mathit{m}\right)$	${\mathit{d}}_{\mathit{E}\mathit{K}}$
0	250	46.144	19	212	35.078	38	174	21.185
1	248	45.654	20	210	34.400	39	172	20.420
2	246	45.150	21	208	33.716	40	170	19.660
3	244	44.633	22	206	33.025	41	168	18.900
4	242	44.11	23	204	32.328	42	166	18.144
5	240	43.578	24	202	31.621	43	164	17.390
6	238	43.033	25	200	30.911	44	162	16.642
7	236	42.477	26	198	30.192	45	160	15.899
8	234	41.913	27	196	29.464	46	158	15.165
9	232	41.340	28	194	28.732	47	156	14.442
10	230	40.756	29	192	27.994	48	154	13.731
11	228	40.160	30	190	27.250	49	152	13.030
12	226	39.558	31	188	26.502	50	150	12.341
13	224	38.946	32	186	25.751	51	148	11.674
14	222	38.325	33	184	24.994	52	146	11.025
15	220	37.691	34	182	24.235	53	144	10.394
16	218	37.048	35	180	23.475	54	142	9.7947
17	216	36.401	36	178	22.711	55	140	9.2366
18	214	35.745	37	176	21.948	56	138	9.3501

for the input data $A_E=50,90,100,150,180,250$ included in Table 1, Table 2 and Table 3. The minimum values of $d_{EK}$ indicated the $A_K$ values of the quadratic function, which was used to determine the pixel amplitudes of the pseudo-colored image.

3.2. Output Data of Analyses

The pseudo-colorization process and its effects primarily involve the analysis of images and the assessment of the effectiveness of the process in terms of the level (in these specific cases, increasing this level) of the perceptibility of the data contained in the analyzed images. In the area of digital image processing, the first assessment of the effectiveness of the methods used by humans is visual assessment. The next stage is to confirm or not the initial assessment with a quantitative assessment. This assessment is the value of the $SSIM$ parameter.

Detailed analysis results regarding the form of the obtained images after transformations (Figure 21) were limited to the images from Figure 9, while the numerical data of the modified $SSIM$ values for each analyzed image are presented in

Table 10. The modified $SSIM$ values for the image subjected to quality improvement (filtering) transformations and the pattern image for the cases presented in Figure 9 and Figure 15.

A Type of Image Transformation			Analysis of Entire Images	Analysis of Image Fragments
1	Image without transformation		62.21430876	44.41559036
2	Histogram extension		61.96496451	44.39428523
3	Histogram equalization		110.5927163 *	57.87137273 *
4	Median filtering 3 × 3		66.42757065	50.8331846
5	FFT transformation and high-pass filtering	20 × 20 window size	1.374061373	3.22144137
6		40 × 40 window size	0.439452908	1.137323642
7		80 × 80 window size	0.223030551	0.687738941
8		100 × 100 window size	0.184376931	0.589140298
9	DCT transformation and high-pass filtering	10 × 10 window size	5.586333807	16.12400992
10		20 × 20 window size	1.235724263	3.564110134
11		40 × 40 window size	0.435227304	1.217007112
12		80 × 80 window size	0.238154443	0.690460499
13		100 × 100 window size	0.193186189	0.594737456
14	Pseudo-colorization—exponential function		110.8905649 **	74.16620629
15	Pseudo-colorization—quadratic function		108.9343769	90.70664067 **

*—maximum value of modified $SSIM$ for image transformations from 1 to 13 items. **—maximum value of modified $SSIM$ for pseudo-colorization method (transformations for 14 and 15 items).

,

Table 11. The modified $SSIM$ values for the image subjected to quality improvement (filtering) transformations and the pattern image for the cases presented in Figure 10 and Figure 16.

A Type of Image Transformation			Analysis of Entire Images	Analysis of Image Fragments
1	Image without transformation		2.564842593	19.8162198
2	Histogram extension		2.566870083	19.83121156
3	Histogram equalization		10.65714581 *	24.07995041 *
4	Median filtering 3 × 3		1.507557708	15.3068329
5	FFT transformation and high-pass filtering	20 × 20 window size	6.533470602	17.70837157
6		40 × 40 window size	7.173832531	14.92272688
7		80 × 80 window size	5.332066726	9.864773506
8		100 × 100 window size	4.717695106	8.862156332
9	DCT transformation and high-pass filtering	10 × 10 window size	5.80164346	18.20773913
10		20 × 20 window size	7.040676565	21.43672255
11		40 × 40 window size	10.20590809	22.35596197
12		80 × 80 window size	6.191275415	14.01123133
13		100 × 100 window size	5.261621853	10.84721382
14	Pseudo-colorization—exponential function		25.48283713	53.6008577
15	Pseudo-colorization—quadratic function		25.8821422 **	53.60838096 **

*—maximum value of modified $SSIM$ for image transformations from 1 to 13 items. **—maximum value of modified $SSIM$ for pseudo-colorization method (transformations for 14 and 15 items).

,

Table 12. The modified $SSIM$ values for the image subjected to quality improvement (filtering) transformations and the pattern image for the cases presented in Figure 11 and Figure 17.

A Type of Image Transformation			Analysis of Entire Images	Analysis of Image Fragments
1	Image without transformation		59.99718996 *	91.08796756
2	Histogram extension		59.9154782	91.14891281 *
3	Histogram equalization		37.59217146	89.79987081
4	Median filtering 3 × 3		55.93392534	66.34285569
5	FFT transformation and high-pass filtering	20 × 20 window size	14.19911427	15.31094889
6		40 × 40 window size	8.582338104	10.8830412
7		80 × 80 window size	5.215632501	8.898050701
8		100 × 100 window size	3.952754123	8.314354137
9	DCT transformation and high-pass filtering	10 × 10 window size	26.35678086	41.36257195
10		20 × 20 window size	14.97982916	15.45372877
11		40 × 40 window size	8.271485532	10.948628
12		80 × 80 window size	5.148795934	8.760441325
13		100 × 100 window size	3.924452481	8.481787531
14	Pseudo-colorization—exponential function		80.5659707	107.3398241
15	Pseudo-colorization—quadratic function		82.8305449 **	110.7675894 **

*—maximum value of modified $SSIM$ for image transformations from 1 to 13 items. **—maximum value of modified $SSIM$ for pseudo-colorization method (transformations for 14 and 15 items).

,

Table 13. The modified $SSIM$ values for the image subjected to quality improvement (filtering) transformations and the pattern image for the cases presented in Figure 12 and Figure 18.

A Type of Image Transformation			Analysis of Entire Images	Analysis of Image Fragments
1	Image without transformation		42.6255306	59.28368261 *
2	Histogram extension		41.59900154 *	56.48035173
3	Histogram equalization		24.54900839	33.88031203
4	Median filtering 3 × 3		36.83538324	50.4188848
5	FFT transformation and high-pass filtering	20 × 20 window size	30.13573216	54.07756317
6		40 × 40 window size	15.89508915	24.52390787
7		80 × 80 window size	10.77893639	16.84266551
8		100 × 100 window size	9.368389438	14.71888291
9	DCT transformation and high-pass filtering	10 × 10 window size	31.30773499	55.87887428
10		20 × 20 window size	30.41408686	55.24359722
11		40 × 40 window size	16.10181922	25.44827526
12		80 × 80 window size	10.87035357	16.74417214
13		100 × 100 window size	9.337752323	14.5609634
14	Pseudo-colorization—exponential function		60.8928515 **	90.68509161
15	Pseudo-colorization—quadratic function		60.82328748	93.47786244 **

*—maximum value of modified $SSIM$ for image transformations from 1 to 13 items. **—maximum value of modified $SSIM$ for pseudo-colorization method (transformations for 14 and 15 items).

,

Table 14. The modified $SSIM$ values for the image subjected to quality improvement (filtering) transformations and the pattern image for the cases presented in Figure 13 and Figure 19.

A Type of Image Transformation			Analysis of Entire Images	Analysis of Image Fragments
1	Image without transformation		43.18645589	48.64494326 *
2	Histogram extension		42.74807808	48.24242467
3	Histogram equalization		46.84996288 *	42.68665069
4	Median filtering 3 × 3		41.90743254	46.12603689
5	FFT transformation and high-pass filtering	20 × 20 window size	−0.000354213	−0.001611553
6		40 × 40 window size	−0.001686477	−0.003272034
7		80 × 80 window size	-0.000545819	−0.001420342
8		100 × 100 window size	−0.002727702	−0.006738712
9	DCT transformation and high-pass filtering	10 × 00 window size	35.19744353	43.76048327
10		20 × 20 window size	34.05713659	41.59246773
11		40 × 40 window size	29.26227155	36.6702639
12		80 × 80 window size	23.09714201	29.8464573
13		100 × 100 window size	21.30228893	26.30868997
14	Pseudo-colorization—exponential function		48.45822	57.56721151
15	Pseudo-colorization—quadratic function		52.49378323 **	63.07583852 **

*—maximum value of modified $SSIM$ for image transformations from 1 to 13 items. **—maximum value of modified $SSIM$ for pseudo-colorization method (transformations for 14 and 15 items).

and

Table 15. The modified $SSIM$ values for the image subjected to quality improvement (filtering) transformations and the pattern image for the cases presented in Figure 14 and Figure 20.

A Type of Image Transformation			Analysis of Entire Images	Analysis of Image Fragments
1	Image without transformation		40.00073349	18.26016129 *
2	Histogram extension		39.94281826	18.22403402
3	Histogram equalization		35.80573409	17.84075797
4	Median filtering 3×3		50.25126503 *	15.71348382
5	FFT transformation and high-pass filtering	20 × 20 window size	8.386409096	12.16370983
6		40 × 40 window size	4.990868079	7.247821637
7		80 × 80 window size	3.201859588	6.480875639
8		100 × 100 window size	2.65609977	6.455038649
9	DCT transformation and high-pass filtering	10 × 10 window size	17.81137484	13.35577024
10		20 × 20 window size	8.332374039	11.94655579
11		40 × 40 window size	4.692065006	7.407443368
12		80 × 80 window size	3.012139433	6.53649631
13		100 × 100 window size	2.619354802	6.455038649
14	Pseudo-colorization—exponential function		78.37291257	31.22178904
15	Pseudo-colorization—quadratic function		80.38576308 **	32.12163831 **

*—maximum value of modified $SSIM$ for image transformations from 1 to 13 items. **—maximum value of modified $SSIM$ for pseudo-colorization method (transformations for 14 and 15 items).

.

Figure 21 shows the effect of using digital image processing methods aimed at filtering them and increasing the level of perception. Additionally, examples of the DCT and FFT spectrum of the image and the DCT and FFT spectrum with a 40 × 40 filter window are presented. The methods used in the analyses are listed in Table 10, Table 11, Table 12, Table 13, Table 14 and Table 15.

Figure 22 shows images corresponding to the image in Figure 9a after pseudo-colorization based on exponential and quadratic functions. The values of the $SSIM$ parameter for this image are included in Table 10 and on Figure 23. When performing visual analysis, it is difficult to assess which of the pseudo-coloring functions is more effective. Certainly, the use of each function results in obtaining an image with a higher level of perception and filtering out interference. The analyzed image is an example of an image in which the presence of graphics unrelated to information useful from the point of view of electromagnetic infiltration has a significant impact on the values of the $SSIM$ parameter, determining the superiority of the quadratic function over the exponential one. However, the appropriate limitation of the area of the analyzed image (Figure 15) shows that the choice of the quadratic function is correct and the difference in the value of the $SSIM$ parameter is greater than 16.

The next analyzed image was the one shown in Figure 10a. Similarly to the image in Figure 9a, it was subjected to the transformations listed in Table 11 and on Figure 24. As a result of pseudo-colorization, the images shown in Figure 25 were obtained. $SSIM$ values calculated for both the entire image and its fragment are higher for the quadratic function than for the exponential function.

It should also be noted that for the image fragment, the values obtained (for both the quadratic and exponential functions) are more than twice as large. This proves the higher structural similarity of the image to the pattern.

Subsequent images transformed in accordance with the algorithm presented in Figure 4 confirm the approach proposed by the authors to the issue of image filtration in connection with their pseudo-coloring. In each case (the images differ in the graphic structure of the data contained), the SSIM values calculated for the quadratic function in relation to image fragments are higher than the values of this parameter calculated for the exponential function (Table 12, Table 13, Table 14 and Table 15, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31, Figure 32 and Figure 33). Analysis of image fragments makes it possible to become independent from unwanted interference that does not carry any useful information (Figure 26, Figure 28, Figure 30 and Figure 32) and makes it difficult to perceive the data you are looking for. This is often caused by a much stronger electromagnetic disturbance, which, in the reproduced image, is presented in the form of high-brightness pixels (very often the amplitudes of such pixels reach values of 255). At the same time, $SSIM$ values for image fragments refer to image areas that contain data fully correlated with the processed information on the monitored graphic display.

In the case of images that are obtained as a result of other human activity, the analysis of image fragments (the ability to focus on an important part increases the level of perception) and its filtration using the pseudo-colorization method can increase the level of detection of any threats related to human health and safety.

The analysis concerned the determination of the modified $SSIM$ value for the image filtration methods listed in Table 10, Table 11, Table 12, Table 13, Table 14 and Table 15 including the pseudo-colorization method using the exponential function and the quadratic function. The obtained results indicate that the quadratic function has better filtering properties in terms of graphic data contained in images. At the same time, the values of the modified $SSIM$ draw attention to a very important aspect of processing images containing interference in the form of various graphic forms unrelated to the processed data. Removing them by limiting the size of the analyzed image area significantly improves the numerical values of the modified $SSIM$ measure. This, in turn, allows us to clearly indicate the proposed method based on a quadratic function as effective in filtering graphic data.

Table 16. Determining cases for which a quadratic or exponential function achieves better filtering properties by using pseudo-colorization method.

Area of Analyzed Image	Figure 9		Figure 10		Figure 11		Figure 12		Figure 13		Figure 14
	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$	$\mathit{E}\left(\mathit{x}\right)$	$\mathit{K}\left(\mathit{x}\right)$
Entire image	+			+		+	+			+		+
Fragment of image		+		+		+		+		+		+

shows the cases for which a quadratic function or an exponential function achieves better filtering properties. The “+” sign indicates the highest modified $SSIM$ value obtained, calculated for the entire image and its fragment for the exponential function and quadratic function.

4. Conclusions

This article presents the possibility of using the pseudo-colorization method to improve the quality (level of visual perception) of images acquired in the electromagnetic infiltration process, both by using color and filtering important graphic data. The pseudo-colorization method involves the use of an exponential function in the transformation of the gray color space into the $RGB$ color space. The effectiveness of this approach to the pseudo-colorization problem was demonstrated in [38]. The authors of this article proposed a modified solution based on the use of a quadratic function as a transformation function of the pixel amplitude values of the image reproduced in the $RGB$ color space. Each of the functions, the exponential function $E\left(x\right)$ and the quadratic function $K\left(x\right)$, is characterized by four basic parameters: the width of the shapes $A_E$ and $A_K$, the maximum value, the minimum value and offset on the pixel amplitude value axis for a grayscale image. The determination of the shape width parameter of the quadratic function $K\left(x\right)$ in relation to the width of the exponential function $E\left(x\right)$, while maintaining the remaining parameters, was based on the parameter of average deviation. The choice of a quadratic function as a better alternative to the exponential function was related to the properties of the shape of the function on the edges of the transition between the background and the graphic symbol contained in the image. The increase in the value of the $K\left(x\right)$ function with an increase in the parameter n is faster than for the exponential function $E\left(x\right)$, so its filtering properties may be better. This means that a filter based on a quadratic function is a “sharper” filter. As a result, the pseudo-colorization process for the quadratic transformation function more clearly marks the boundaries between the image background and the graphic data contained in it. The exponential function introduces smooth transitions and the filtered graphic shapes become blurred at the above-mentioned background-graphic symbol boundaries.

Comparative analyses, in terms of the filtration efficiency of graphic data contained in images reproduced for two shapes of the transformation function of the gray color space into the $RGB$ color space, were carried out for six images. The selected images differ in their data structure and information content. In addition, appropriate calculations were performed for entire images and their fragments containing the most important data from the point of view of obtaining information. In this way, image areas strongly disturbed by elements whose information content is very low were eliminated from the analysis. The modified $SSIM$ was adopted as a measure for assessing the effectiveness of the proposed method, the maximum value of which for a given analyzed case (image) is a criterion for the acceptance of the method’s superiority over other data filtration methods. The modification of $SSIM$ consisted in adopting different weights of α, β and γ for the product of the luminance, contrast and correlation coefficients between two images, respectively.

The weight values of the luminance, contrast and cross-correlation parameters were selected in a way that allows for a greater share of contrast and cross-correlation in the value of the $SSIM$ parameter. The contrast is primarily related to the quadratic transformation function used. Cross-correlation allows you to avoid the significant impact of disturbing elements in the evaluation of images, which is primarily visible in the analysis of image fragments subject to the filtering process using pseudo-colorization.

The obtained values of the modified $SSIM$ show that the proposed pseudo-colorization method based on the quadratic function and the values of the weights α, β and γ give better results than the exponential function. This applies in particular to analysis limited to image fragments.

As mentioned, the analyses carried out were based on six images with different graphic structures. Modified $SSIM$ values were calculated for each analyzed image (the entire image and its fragment) subjected to the pseudo-colorization process based on exponential and quadratic functions. The obtained $SSIM$ values clearly indicate the superiority of pseudo-colorization with a quadratic function over an exponential function (and thus the effectiveness of image filtration) for image fragments containing important data from the point of view of electromagnetic infiltration. The $SSIM$ values are 74.1662 and 90.7066 (Figure 9a), 53.6008 and 53.6008 (Figure 10a), 107.3398 and 110.7675 (Figure 11a), 90.6850 and 93.4778 (Figure 12a), 57.5672 and 63.0758 (Figure 13a) and 31.2217 and 32.1216 (Figure 14a), for the exponential and quadratic functions, respectively. The results of the analysis of whole images depends on the graphic content, which does not constitute information about the processed data. Most often, these are interferences related to the operation of graphic channels, i.e., they are signals correlated with signals related to data display parameters on graphic displays.

Further work in the area presented in this article will focus on the development of an algorithm for automating the process of pseudo-colorization and filtering images. This will allow the development of an additional application module of the Software Raster Generator supporting the analysis of images reproduced from compromising emanation signals. Additionally, analyses of the effectiveness of using other functions of transforming the gray color space into the $RGB$ space will be carried out. These functions can be, e.g., a triangular or rectangular function or the sum of these functions and functions already analyzed (exponential and quadratic functions). In such a case, for each $RGB$ channel, it will be possible to expose several areas of the pixel amplitude values of the filtered image above the background.