At the beginning of the article, we name the definition of Ateb-functions and prove some properties. Afterwards, a digital watermark constructing method based on Ateb-functions is considered. The proposed method is tested for many cases of attack. The results of experiments are shown in tables and figures. We discussed the obtained results, and finally, some conclusions are presented.
2.1. Definition of Ateb-Functions
The task of constructing a DWM based on the generalization of the Fourier transform in the form of Ateb-transform in order to protect and identify electronic data in the Internet is considered. To define the Ateb-transform, let us consider the definition of Ateb-functions. In [
3], the properties of these functions are given in detail; therefore, only the most necessary definitions for the understanding of the further presentation are given.
Since the statement relates to special functions, it is known [
13] that the incomplete beta function is determined by equation
where
p and
q are real numbers. In a partial case, if
x = 1, Equation (1) takes the form of the Euler integral of the first kind
i.e., the full Beta function.
All
x values from interval [0, 1] of functions
Bx (
p, q) and
B1 (
p, q) given by Equations (1) and (2), are positive and satisfy the conditions.
Consider the possible values for the parameters
p > 0,
q > 0, namely
where
m and
n are determined by the formulas
If p > 0, q > 0, than the Beta-function is definite and continuous, and for other valid values p and q, the function goes to infinity with t → 0 or t → 1. The mathematical definition of the Ateb-function is defined as an inversion to the Beta-function. Therefore, the name of the functions Ateb was proposed as the inversion of the word Beta.
Let us consider the expression
where
m,
n are defined by Equations (5) and (4). Replacing the variables of the type
Equation (5) turns into a look
In Equation (7),
ω is a function from
ν, and also from
m and
n. To build Ateb-functions, let us consider the inverse dependence
ν from
ω, which is a function
m,
n, is called Ateb-sine introduced in [
14], and looks like
Similarly, the replacement of variables
from Equation (1) get the ratio
Dependency
u from
ω for Equation (9) is a function of
m and
n and is called the Ateb-cosine and looks like
The basic relation for periodic Ateb-functions is
The Equation (11) is a generalization of the main trigonometric identity. Ateb-functions constructed for the values of the Equation (4) are periodic Ateb-functions. On the basis of these functions, analytic solutions of the system of ordinary differential equations are constructed
where
α,
β are some real constants.
If
m,
n satisfies the Equation (4), then Equation (12) describes oscillatory motion with one degree of freedom. The Ateb-functions are used successfully for modeling a vibration motion in [
15,
16]. However, in this investigation, we propose to use them for constructing a digital watermark.
From Equations (7) and (9), it is obvious that if n = m = 1, then there are received u = cos ω, v = sin ω. This property is the basis for the development of this study. Since Ateb-functions are a generalization of ordinary trigonometric functions, one can construct a generalization of the Fourier transform on the basis of these functions.
In the simulation and development of systems for the transformation and protection of information, methods based on the mathematical apparatus of orthogonal trigonometric transforms are widely used (OTT) [
2]. A method of orthogonal transforms is proposed, which is based on periodic Ateb-functions. In the future, it will be called as an orthogonal Ateb-transform (OAT). The ability to build an OAT is based on such provisions. First, in [
3,
4], Ateb-functions are shown as a generalized case of ordinary trigonometric functions. Secondly, in [
2], the orthonormality of a system of periodic Ateb-functions is proved. In the works [
3,
4], the methods and algorithms for calculating Ateb-functions depending on the parameters have been developed, which allows successfully using the proposed method of OAT.
Let us introduce a generalization of the Fourier transform based on periodic Ateb-functions.
2.2. Construction of Orthogonal Ateb Transforms
In this section, continuous one-dimensional direct and inverse Ateb-transforms will be constructed first with one parameter, and then with two parameters.
Let us introduce the function of the Ateb-sine and cosine in the form
sa (n, 1, t) and
ca (1, n, t). Let
x (t) be a real function, then Ateb-transform will be the next
where
Given the parity and the oddity of the Ateb-functions [
3], let us write the inverse Ateb-transform in the form
where П is the half period of Ateb-functions. The right side of Equation (16) depends on the parameter
n. For every value of
n, the decomposition of function
x (t) will be different.
Character—that is, the rate of growth or decline of the period of the Ateb-functions ca (1, n, ωt) and sa (n, 1, ωt)—will vary depending on n. Dependence of the Ateb-function on the parameter n makes it possible to pick up the corresponding to x (t) appearance of ca (1, n, ωt) and sa (n, 1, ωt), which corresponds to the task of constructing of security elements.
Let’s introduce the Hartley function
csa (1, n, t) as
Let’s introduce straight and inverse Hartley Ateb-transform, using formulas
In case of n = 1 introduced by Equations (13)–(16), (18), and (19), Ateb transformations will be known as Fourier and Hartley orthogonal transforms. For the existence of the Ateb-transform function x (t), it is sufficient to fulfill the same conditions that are sufficient for the existence of an orthogonal Fourier transform.
Let us consider that
x (t) is a real function; then, the analog of the known Fourier transform Ateb-transform will be constructed in the form
where
where
is an Ateb-cosine function, and
is an Ateb-sine function. Given the basic identity for the Ateb-functions in Equation (11), we obtain an expression for the inverse transform
where П (
m, n) is a half period of Ateb-functions. Let us take into consideration
Then, direct and inverse Hartley Ateb transforms will be written by formulas
In case n = 1, m = 1 introduced by Equations (21)–(23), (25), and (26), Ateb-transforms will be known as orthogonal Fourier and Hartley transforms. Let us prove the validity of such properties for the introduced transforms: linearity, symmetry, similarity, displacement, modulation, and convolution, which are similar to the properties of ordinary Fourier and Hartley transforms.
2.5. Construction of the Digital Watermark
The need to increase the level of security of information transmission is connected with the new methods of creating, storing, and distributing information on paper carriers and with the change of the material carriers themselves, namely the introduction of plastic carriers of information, new types of paper, and other factors.
Therefore, raising the level of safety of documents on tangible media in the conditions of informatization of social processes is an urgent task. The original approaches for creating protection elements based on fractals are proposed in [
9,
10]. The efficient pre-processing procedure for images is developed in [
17].
The development of methods of protection and identification in order to increase the level of security of documents and thus prevent violation of the integrity of information on tangible media to ensure the appropriate level of security of information transmission was also described.
Along with the development of new methods of protection, it is necessary to create new methods for identifying documents. In order to increase the level of security of printed documents by methods of graphical protection and identification in this work, the apparatus of the theory of Ateb-functions, in particular Ateb-transforms, is used. The work also describes the method of identifying a document based on the embedding of hidden information. A new method for identifying a document based on values of parameters m, n of Ateb-functions f (m, n, x), and the introduced analogue of discrete orthogonal transformations are constructed.
To embed a hidden image, an additive algorithm was used using the discrete Ateb-transform (DAT), given by Equation (20) with different values of parameters m and n. An image conversion Equation (23) was used to read the image. The attached image or message is invisible because changes are made in a small number of elements, so the proposed method refers to the methods of hiding data in the frequency domain.
With DAT, let us convert the image and then use the following four ways to embed a hidden image [
1]. In the first way, the
r largest values are changed by the formula for embedding the hidden image in the form
where
zwp is the converted image,
zp is the original image,
w is the hidden image size
r, and
α is the coefficient for adjusting the value of embedding.
In the second way, instead of Equation (28), let us use the next formula:
The third way is to apply the formulas of the form
To implement the fourth method, we use the formula
To verify the efficiency and stability of the proposed method of identification, the following experiments were carried out for parameter values m = 3, n = 1/7. Testing was performed for 10 standard images from the test base USC-SIPI, in particular for test images “LENA”, “BABUIN”, and others. The following series of attacks was carried out:
File resize 10%, 25%, 50%, 75%, 150%, and 200%;
Turn 1°, 5°, 10°, 45°, 90°, and 180°;
Compress the image to 10%, 25%, or 50%;
Change the color depth of an image 256 → 128, 256 → 64, 256 → 32, 256 → 16, 256 → 8, 256 → 4, 256 → 2.
According to [
1,
7], these four types of attacks are the most common for images, when an image is not changing visually.
The criterion for the presence of a hidden image is correlation, which is calculated by the formula:
where
K is the correlation criterion,
r is the hidden image size,
is the
i-th image element,
is the average value of image elements,
is the
i-th hidden image element,
is the average value of the hidden image elements,
is the standard deviation of image elements with a hidden image,
is the standard deviation of hidden image elements.