# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 1 Introduction
As is known, the transition from real variables to complex variables can be considered as one of the examples of using the method of algebraic extensions [30], in which the root of an irreducible algebraic equation (in this case, the imaginary unit) is added to the initial mathematical object (in this case, the set of real numbers). It is essential that this method can be used also when the initial set is discrete. In particular, Galois fields \(GF(p^{n})\) can be constructed by this method. In this case, Galois fields \(GF(p)\), which is a ring of subtraction classes of integers modulo prime number \(p\), are considered as initial sets. It is appropriate to emphasize that the fields \(GF(p^{n})\) are currently used to solve a wide variety of applied problems [31; 32; 33]. In the case when the irreducible equation has degree two, the method of algebraic expansions allows one to pass from the field \(GF(p)\) to the field \(GF\big{(}p^{2}\big{)}\). An arbitrary element of such a field can be represented as  
\[x=x_{1}+\theta x_{2} \tag{1}\]where \(\theta\) is the root of some equation, which is irreducible in the field \(GF(p)\), and the variables \(x_{1}\) and \(x_{2}\) take values in the field \(GF(p)\). The analogy with the representation of a complex variable in the next conventional form is obvious. \[z=x+iy \tag{2}\]  
The difference is that in Formula (1), the variables \(x_{1}\) and \(x_{2}\) refer to a discrete (and, moreover, finite) set, while the variables \(x\) and \(y\) in Formula (2) change continuously. This analogy allows one to assert that discrete Cartesian coordinates admit mapping through expressions of the form (1), and for clarity, the root of the irreducible equation \(\theta\) can be treated as a logical imaginary unit of the first kind [34]. Note that the tasks in which it is acceptable to use a discrete coordinate grid are not only related to digital processing of such images as human faces, as well as with the information protection issues mentioned above. Another example is tasks related to remote sensing of the Earth (RSE) based on satellite images [35; 36; 37], monitoring of agricultural lands by means of overflight by unmanned aerial vehicles equipped with video recorders [38; 39; 40]. In the latter case, as well as in any mapping tasks, there arises a necessity to combine images acquired at different spatial locations of the recorder, which, from the mathematical point of view, is reduced to a coordinate transformation problem. Of course, such tasks can be solved by traditional methods [41; 42]; however, the primary processing of video information using onboard computing systems, which are often based on convolutional neural networks, is of growing interest. As applied to remote sensing tasks, such primary processing is associated with the need to exclude images related to the case when the Earth's surface is hidden by cloud cover [43; 44], as applied to agricultural land monitoring, to exclude the same type of video information [45; 46; 47], etc. The solution to such tasks obviously requires use of onboard computing systems that perform primary data processing. There, reports [48; 49], in which computing systems suitable for such purposes are proposed to be realized on the basis of chips with tunable logic structure (PLD), are known. However, in any case, the solution to this problem requires an appropriate algorithmization that allows to obtain a general picture while minimizing the amount of data coming from video recorders located on certain technical means to the data processing center, which makes it urgent to develop appropriate discrete geometry tools, which in the future will allow to simplify the logical structure of the used computational means. Tools allowing to simplify onboard computing systems are also relevant for realization of physical methods of information protection mentioned above.