# Electronic Fourier-Galois Spectrum Analyzer for the Field \(Gf(31)\)
## 1 Introduction  
Nowadays, Fourier-Galois transforms find more and more applications [1, 2, 3]. They are of considerable interest in terms of cryptography [4, 5] as well as digital signal processing [6, 7]. Such transformations are of no less interest from the point of view of digital image processing, especially those for which orthogonal bases can be constructed in a natural (for using the Fourier-Galois transform) way. Namely, the vast majority of works devoted to the problem of digital image processing, as a rule, are oriented to the processing of images of arbitrary character, which is obviously dictated by the needs of practice (user face identification [8, 9], object recognition [10], predictive control model of a vehicle [11], etc.). Along with this, however, there is quite a definite range of tasks where digital image processing is required, with quite certain specificity. Numerous examples of this kind are related, in particular, to the study of objects of interest for physical chemistry, which was emphasized in [12]. The most obvious example in this respect is the registration of the kinetics of swelling [13] or deswelling [14] of polymer hydrogels. Polymer hydrogels are cross-linked polymer networks (which makes them analogous to rubber), capable, however, of swelling in water and aqueous solutions, increasing the volume by 500 times or more. This property determines their numerous practical applications [15, 16], including those of interest in medicine [17, 18], ecology [19, 20], green energy [21], etc. The most convenient way to study the swelling kinetics of hydrogels is through video recording of this process, which already makes us pay attention to the digital processing of images that have quite certain specificity. It should be noted that in many studies in which the swelling kinetics of hydrogels were investigated, catheterometers were used as a measuring device [22; 23], i.e., in the end, it was also a question of image processing, but only with direct human participation. Another example from the field of physical chemistry is related to surface tension measurements, which are also of considerable interest [24; 25]. By properly processing the image of a liquid drop on a surface that is deformed due to surface tension effects, the necessary quantitative information can be obtained. These examples show that images whose processing is of interest from the point of view of experimental studies in physical chemistry often either have circular symmetry or are close to it. Moreover, such images (from the point of view of registering the characteristics of the object under study) can be partitioned into a relatively small number of pixels. A method based on the numerical convolution theorem, a visual proof of which was given in [26] based on the results of [27], can be considered a promising means of processing such images. This theorem is an analog of the classical (analog) convolution theorem, on the basis of which, in particular, the whole apparatus of analysis of linear electronic circuits is built. This classical theorem allows us to pass to the description of the circuit in terms of the frequency range, since the Fourier image of the convolution is the product of the Fourier images of the functions under its sign. In particular, this theorem allows us to correctly justify the use of such terms as amplitude-frequency response, frequency range of a radio signal, etc. [28; 29]. Accordingly, if the image processing procedure is described by a digital convolution operation (which is the case in practice when using convolutional neural networks [30; 31]), one can raise the question of analogs of the amplitude-frequency response (transfer functions) and the transition to the description in terms of digital spectra. The orthogonal bases allowing for the construction of such spectra are currently known to be quite numerous (the Walsh basis and its modifications [32], etc.).