# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 3 Background: Theoretical Aspect  
The main idea of the proposed method is based on the next fact. Some function-taking values in Galois fields [81] (or other finite algebraic structures, e.g., finite algebraic rings [82]) can serve as a model for any digital signal varying in a finite range of amplitudes. Indeed, functions of a real or complex variable, traditionally used in practice, are purely mathematical objects; hence, they represent nothing more than a model of a signal. The choice of such a mathematical object is nothing more than a matter of convention and convenience. The number of levels of a digital signal varying in a finite range of amplitudes is also known to be finite; hence, functions taking values in Galois fields (finite commutative algebraic bodies) can be used as a signal model. Consequently, a function-taking value in Galois fields whose argument also takes values in the same field may be used for representation of any digital image. In particular, one can use the mapping  
\[y=F(x);\ x,y\in GF\Big{(}p^{2}\Big{)} \tag{3}\]  
With such mapping, the argument \(x\) actually corresponds to a two-dimensional data array since the representation (1) has two components, i.e., one element of the Galois field uniquely characterizes, for example, a specific pixel. Eligibility of such a mapping is determined by the fact that the set of elements of the field \(GF(p)\) can be considered as a subset of the field elements \(GF\big{(}p^{2}\big{)}\). This clarification is essential since the solutions of the algebraic problems discussed above assume that the values of the function have only one discrete dimension (the values of this function can be assigned a scale corresponding to a finite set of real numbers), while the coordinates of a pixel (if we consider a typical television screen) are two-dimensional. Accordingly, a separate element of the Galois field, represented in the form (1), defining, for example, the coordinates of a pixel, is also two-dimensional in the following sense. Each element of the Galois field \(GF(p)\) can be assigned a natural number, but in order to uniquely characterize the element x, represented in the form (1), two such numbers are needed. However, this does not cancel the possibility of using Formula (3), in which both the values of the function and its argument formally belong to the same set, specifically the field \(GF\big{(}p^{2}\big{)}\). Indeed, all elements of the field \(GF(p)\) are a subset of the field \(GF\big{(}p^{2}\big{)}\), which justifies the use of Formula (3). However, representation (1) can only be considered as a special case corresponding to the use of vectors with discrete coordinates. Along with Galois fields, algebraic rings can also be used for this purpose. As shown in this paper, this approach has quite definite advantages. To demonstrate this, consider the matrix representation of the elements of Galois fields, taking into account that the most common way of representing coordinate transformations is based on the use of matrices. Let us consider the product of two elements, represented in the form (1), on each other. \[C=AB=(A_{1}+\theta A_{2})(B_{1}+\theta B_{2})=A_{1}B_{1}-gA_{2}B_{2}+\theta(A_ {2}B_{1}+A_{1}B_{2}) \tag{4}\]  
where, for definiteness, it is assumed that \(\theta\) is the root of the field irreducible \(GF(p)\) equation  
\[x^{2}+g=0 \tag{5}\]  
Finding elements \(g\) that generate irreducible equations is, in general, an independent problem, the consideration of which is beyond the scope of this paper. From the point of view of practical applications, it is sufficient to point out that such elements can be found directly (by squaring the elements of the basic field). We emphasize that there exists a number of Galois fields \(GF(p)\) in which Equation (5) at \(g=1\) has solutions. In particular, such a field is the field \(GF(13)\) since there is  
\[5^{2}\equiv-1(13) \tag{6}\]  
Consequently, it is reasonable to consider the irreducible equation in a more general form (5).