Комплексные программные методы обработки интерференционных картин/Сomplex Software Methods for Processing Interference Patterns тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Хирьянов Тимофей Федорович

• Хирьянов Тимофей Федорович
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Хирьянов Тимофей Федорович

Contents

Pages

Introduction

Chapter 1. Comprehensive Review of Interferogram Processing Methods

1.1 Statement of the problem

1.2 Transform methods

1.3 Deep learning methods

1.4 Regularized Phase-Tracking methods

1.5 Tracing methods

1.6 Methods overview

Chapter 2. Parabola method

2.1 Preprocessing

2.2 Search of extremum points

2.3 Tracing

2.4 Phase unwrapping

2.5 Overview

Chapter 3. Extended Approach Parabola method

3.1 Fringes with negative curvature

3.2 Preprocessing

3.2.1 One-dimensional filtering

3.2.2 Multi-dimensional filtering

3.2.3 Determination of the local direction of the fringe

3.3 Tracing

3.4 Phase unwrapping

3.5 Overview

Chapter 4. WISP: Workframe for Interferogram Signal Phase-unwrapping

4.1 Workframe overview

4.1.1 Assumptions

4.1.2 Workframe structure

Pages

4.2 Workframe algorithms

4.2.1 Pre-tracing

4.2.2 Tuning the line position

4.2.3 Geometrical isophases construction

4.2.4 Local fringe directions field

4.2.5 Smoothing intensity along isophases

4.2.6 Gradient line formation

4.2.7 Phase unwrapping

4.2.8 Tuning dark fringes and isophase positions

4.2.9 Stopping criterion

4.3 Algorithmic pseudo code

4.4 Noise robustness

4.5 Comparison with alternative methods

4.6 Overview

Conclusion

List of Abbreviations and Symbols

Glossary of Terms

References

List of Figures

List of Tables

Введение диссертации (часть автореферата) на тему «Комплексные программные методы обработки интерференционных картин/Сomplex Software Methods for Processing Interference Patterns»

Introduction

Optical methods for studying phase objects and algorithms for processing the corresponding data are of significant interest to a wide range of fundamental and applied research fields. These include, in particular, microbiology [1], quality control of optical components [2], measurement of gas flows and flame dynamics [3], investigation of plasma formation evolution generated in gas discharges [4] or under intense electromagnetic radiation exposure to the environment [5], and many others. Of particular interest in this regard are interferometric techniques for measuring the parameters of the studied phase object when illuminated by coherent laser radiation.

Laser interferometry serves as a crucial approach for determining the density of transparent objects [6]. The interference patterns captured in interferograms contain detailed information about variations in the refractive index within the sample. The accuracy of extracting these details is largely influenced by the quality of the initial measurements and the methodology applied to interpret the interferogram. Obtaining an accurate map of optical density requires precise phase retrieval from the interference pattern. This step can be quite challenging due to the presence of noise, intricate fringe patterns, and the inherent restrictions of current phase extraction algorithms.

There are many different optical interferometry schemes, such as holographic, moiré, speckle, shift, multi-pass, resonant interferometry systems, etc. A shift interferogram, primarily considered in this work, is an interference pattern from a beam that has passed through an object with an unperturbed beam. The task is to reconstruct a two-dimensional pattern of the difference in phase incursion in the unperturbed and perturbed probing beams from a frame with their interference pattern.

The focus of this work is primarily on interferograms of phase objects that are of significant interest in the physics of gas discharges. More specifically, we consider highly ionized plasma channels generated during the development of spark discharges in an air environment. Previous studies [4] have highlighted the considerable challenges in optically investigating this type of phase object due to its rapid evolution (nanosecond timescale), small transverse dimensions, and complex internal structure. The intricate structure of spark channels can lead to small-scale perturbations in interference fringes against the background of natural noise, as well as the appearance of fringes with negative curvature. Moreover, preprocessing (noise removal) of such frames is a

challenging task due to the object's parameters being comparable to various artifacts such as diffraction patterns from airborne dust, fringe blurring, and the nonlinear response function of the detector matrix to illumination intensity, among others. Processing such frames requires analytical validation of the achieved accuracy and the optimality of the obtained solution. Accounting for these factors necessitates the use of specialized interferogram processing algorithms capable of reliably extracting phase shift information of the probing radiation with high precision.

A well-known and reliable method for high-precision phase retrieval is temporal phase-shifting [6—15]. In this case, controlled phase shifts are introduced into the interferometer, implementing a different optical path of the probing beam - for example, by moving a mirror. Several interference patterns with different phase shifts are captured. The distribution of the wave front phase is calculated from these frames [6]. However, this type of diagnostics is only feasible in the case of stationary objects under study. In applications such as plasma interferometry, where the lifetime of an object is nanoseconds, this method is not feasible [16]. A similar method, parallel phase-shifting, is based on the use of interference signals in quadrature (shifted by a predictable phase difference between each other, usually n/2k) which allows the phase shift to be determined with high accuracy. It is subject to the condition of absolute synchronization of paired signals registration, which is also inapplicable in the case of fast-flowing processes [17; 18].

Thus, in scenarios involving ultrafast phenomena, where capturing just one fringe pattern image is possible, traditional phase-shifting techniques mentioned above cannot be employed straightforwardly. This is due to the fact that temporal phase scanning depends on acquiring multiple frames with predetermined phase shifts to accurately retrieve phase data. Meanwhile, parallel phase shifting relies on recording several images simultaneously with a single camera, which demands exact control of optical path lengths during the experiment as well as meticulous alignment, potentially leading to a reduction in the effective image size.

For such processes, alternative methods exist for extracting phase information from a single fringe pattern image. One of the most popular methods is windowed Fourier transform [19; 20]. In these works, the windowed Fourier transform serves firstly as a method for preliminary cleaning of images from small-scale noise. The two-dimensional Fourier transform applied to the interference frame is subjected to a window filter, which nullifies the amplitudes of high frequencies, with which white noise is most often associated. Then the inverse Fourier transform is performed, the

result of which is a spatial interference pattern cleared of small-scale noise. Secondly, the WFT method provides the opportunity to get phase of the local spectrum at the peak of the frequency distribution at each point. The outcome of the WFT technique is highly influenced by the selection of the window size. This involves an inherent compromise between spatial and frequency resolution: using smaller windows improves spatial precision but degrades frequency accuracy. Because the Fourier transform must be computed at every window position, the WFT approach tends to be relatively demanding in terms of computational resources [19]. Moreover, it is quite susceptible to noise and irregularities, suffers from boundary-related artifacts, and often requires meticulous, sometimes manual, tuning of the window parameters [19].

An alternative approach for phase extraction from interferograms is the wavelet transform, which offers a multiresolution analysis framework. The wavelet transform decomposes a signal into low-frequency components, known as approximations, and high-frequency components, referred to as details. These coefficients create a multilevel, pyramid-shaped representation of the original signal. Unlike the WFT, the wavelet transform offers variable resolution—achieving finer spatial detail at higher frequencies and better frequency precision at lower frequencies. This characteristic makes it especially effective for examining interferograms that contain regions with different fringe densities and localized patterns [21; 22]. When dealing with interferograms that exhibit curvilinear fringes or abrupt changes, the curvelet transform outperforms traditional wavelet and Fourier techniques [23]. This is because curvelets effectively capture edges and curved structures by integrating directional selectivity with anisotropic scaling, which makes them particularly well-suited for analyzing fringe patterns with intricate shapes. However, the curvelet method has extremely high computational cost, is parameter-sensitive and requires careful tuning near image edges.

The S-transform [24] provides a unique time-frequency representation that combines advantages of both the windowed Fourier transform and wavelet transform. Unlike WFT which uses fixed window sizes or WT which employs scale-based analysis, the ST adapts its window width according to frequency, offering progressive resolution similar to wavelets while maintaining a direct relationship with Fourier spectrum. Parameter selection for this method requires empirical tuning, and computational cost remains higher than WFT (though lower than curvelets).

The Hilbert-Huang transform applied to 2D images is called Bidirectional Empirical Mode Decomposition. Unlike WFT and CWT, it does not use the

presumed decomposition basis; which is derived from the signal itself being called characteristic intrinsic mode functions. The method shows better accuracy results than WFT [25], but the practical usage of BEMD is limited by the calculation time. Fast Adaptive Bidirectional Empirical Mode Decomposition [26], which is declared to give better results and not be so algorithmically complex, still remains a relatively resource-intensive method. Since the algorithm does not have a strict mathematical justification, the selection of parameters for the correct restoration of phase information requires additional analysis and an expert approach [26].

In recent years, deep learning methods have been actively used to solve the phase reconstruction problem, offering an alternative to traditional approaches. Neural network methods have several advantages, including high processing speed after training, resistance to noise, and the ability to learn complex nonlinear dependencies that are difficult to describe analytically. Since we are dealing with images, the preferential use of convolutional neural networks, which are optimally designed for image processing, is natural. In [27], a CNN with a U-Net-like architecture is employed to process interferograms. The network is trained to directly predict the unwrapped phase from noisy or wrapped phase images exhibiting 2n discontinuities. Work [28] proposes a comprehensive approach combining interferogram pre-denoising and phase extraction. Specifically, a cascaded CNN architecture is utilized: initially, a DnCNN module denoises the wrapped phase, followed by a U-Net that performs phase unwrapping. In [29], a hybrid network combining CNN and recurrent neural network components is applied. The CNN extracts spatial features, while recurrent layers capture temporal dependencies across sequences of phase images. In conventional CNNs, increasing network depth often leads to degraded accuracy due to training difficulties. ResNet addresses this issue by introducing residual connections that allow signals to bypass multiple layers unchanged. Work [30] employs a ResNet architecture with residual blocks, enabling the training of very deep networks without gradient vanishing problems. Method described in work [31] utilizes a generative adversarial network framework. Here, the generator reconstructs the unwrapped phase, while the discriminator evaluates the authenticity of the output. This approach facilitates more realistic reconstructions under severe distortions.

Although spectral methods and empirical decompositions are effective in certain settings, they have limitations: sensitivity to noise (especially HHT and WFT), the need for manual parameter selection (e.g., windows in WFT or scales in CWT), extremely high computational cost when processing large data, and problems with local distortions

(gaps, sharp edges). Neural networks automatically extract features, adapt to different types of data, and work faster on inference, which makes them attractive for real-time and industrial applications. However, their main drawback is the need for large labeled data for training. The lack of correctly processed experimental data of the type of frames under study is a well-known problem among researchers of medical biological objects, plasma objects, etc. Moreover, in phase reconstruction, deep learning methodologies aim to automatically determine relationships between data through optimization of neural network parameters on empirical datasets. Neural networks, with their multilayered architecture and millions of adjustable parameters, demonstrate the capacity to capture complex dependencies. However, unlike physics-based algorithms, generalized neural network structures frequently exhibit opacity; it remains challenging to discern precisely what the network has learned and what function a specific parameter performs. This opacity presents significant challenges when neural network failures occur: it becomes impossible to either analyze the underlying cause of the failure or implement targeted improvements to prevent similar errors in subsequent applications. This is a critical disadvantage when it is necessary to extract numerical data with the highest possible predictable accuracy.

Here we first touch upon the final and central part of the problem of reconstructing a phase object from an interference frame: phase unwrapping as such [32]. A feature of shear interferometry is the fact that the phase is reconstructed with an accuracy of 2n. Thus, the standard taking of the argument from the wave in transform methods will lead to the appearance of jumps of the order of 2 n, i.e., the phase will be restored in the frame not continuously, but stepwise. Such a phase is usually called wrapped. The procedure for obtaining a continuous phase shift pattern (assuming that the density of the object does not experience discontinuities), or a minimally discontinuous phase shift pattern, is called phase unwrapping actually.

We see that deep learning implies automatic reproduction of the smoothest possible picture, i.e., includes phase unwrapping. As for transform methods, they require a second stage after phase extraction — the unwrapping method itself. In this aspect, we will have to highlight a number of methods dedicated to phase unwrapping after its extraction from the interferogram.

As well as deep learning approaches, Regularized Phase-Tracking methods remain extensively employed in both research and industrial settings [33—35]. This is especially true when high reliability, interpretability, and expert oversight are essential. While modern methods often emphasize automation, RPT techniques provide

mathematically rigorous and noise-resilient solutions that continue to be favored for mission-critical applications. Empirical studies demonstrate the relevance and competitiveness of RPT methods in space interferometry [36]: Flynn's method gave lower error than ResNet, and in industrial measurements [33]: RPT was chosen for nanoscale problems due to its stability.

RPT refers to a family of algorithms designed to recover phase information from interferograms by minimizing energy functional that incorporates regularization. These approaches frame the problem as an optimization task that balances fidelity to the observed data with smoothness constraints. Moreover, they iteratively refine the phase estimate, effectively reducing phase discontinuities and noise. In [37], Goldstein's branch-cut method establishes the classical approach by identifying 2n discontinuities through phase gradient analysis, creating minimal-length branch cuts to isolate residues, and performing path-independent integration around these barriers. This method excels in optical interferometry but shows limitations with high noise. Work [36] introduces Flynn's minimum discontinuity approach, which reformulates the problem. The regularized phase-tracking technique by Servin [37] employs local quadratic approximation: it operates within sliding 5x5 to 7x7 pixel windows, uses Tikhonov regularization to prevent noise amplification, and becomes particularly effective for speckle noise in coherent imaging systems. Recent advances in [37] incorporate robust statistics: L1-norm regularization for impulse noise resilience, total variation constraints for edge preservation, and uses Huber loss functions for balanced noise suppression.

Here we describe methods that solve the phase scan problem in the inverse way to the previous section. While RPT methods split the area into continuous phase zones and stitch them together, tracing methods focus on finding discontinuity lines and processing them correctly. This includes various variations of the active contour method, as well as phase interpolation methods between reference lines. Most often, lines with maximum visibility — black ones — are used as reference lines, and their search is called tracing [16]. The phase values in the k-th black line are assigned in shear interferometry to be n + 2nk, the remaining phase values are interpolated or algorithmically supplemented. The method proposed in [38] identifies black lines by fitting to local intensity minima. Phase between adjacent lines is approximated using cubic splines, constrained by the known 2n phase jump across each fringe. This method achieves ±0.1n accuracy for smoothly varying fringes but struggles with bifurcations. As introduced in [39], Active Contours method dynamically adapts to

black-line trajectories by minimizing an energy functional. Phase interpolation between contours employs Laplace's equation V2^ = 0 with Dirichlet boundary conditions (phase values fixed at the black lines). This excels in tracking complex topologies but requires manual initialization for closed loops. Steger's Line Detection with Graph Optimization in [40] combines ridge detection with graph cuts to trace black lines. First, Hessian-based filtering locates line centers via eigenvalue analysis. Then, a graph prioritizes connectivity: edges represent pixel linkages, and weights encode intensity coherence. Phase between lines is reconstructed using a weighted Poisson solver. This method is robust to noise but computationally intensive.

Of course, the most modern and progressive is phase restoration through convolutional networks, but to train the network, a base of ready, fairly accurately processed frames similar to the one being studied is required. In the case where the restored phase must be analytically substantiated, and a quantitative assessment of the error is also required, it is the tracing methods that come to the fore. However, in their current form, methods of this type not only work slower than others, but also work too poorly in the case of highly noisy images.

Actuality of this work is justified, first of all, by the urgent need in the field of plasma micron interferometry for a reliable and predictable tool for processing experimental interferograms, namely, a tool for cleaning noise, improving the image without introducing structural changes into it and increasing the error of the result. In addition, correct phase extraction and phase unwrapping are required using techniques that take into account the physics of the process, the object and the occurrence of noise, and that also give a predictably reliable result.

Deep learning is not applicable in the existing published versions, since this method lacks predictability of error and mathematical transparency of the algorithm for obtaining the result, and there is no large reliable training sample for a specific type of interferogram.

The combination of transform and RPT methods or transform and tracing methods is assumed to be optimal, but it requires tuning for the specifics of the task: small-scale nature of the object (which increases the weight of artifacts and noise, band widening, nonlinearity of the recording equipment matrix, graininess and overexposure), rapid flow of the object (narrowing the diagnostic capabilities), the need for high accuracy and numerical optimality of the result in relation to the experimental data.

Aim of this work is the creation of transparent software code implementing a full cycle of phase restoration based on an interference shift frame with minimal manual adjustment, implementing analytically sound optimal algorithms taking into account the physical characteristics of the process with precise error estimates.

To achieve this aim, it was necessary to solve the following tasks:

1. Development and implementation of an algorithm for preprocessing an interference frame.

2. Development and implementation of an algorithm for extracting phase from an interference frame.

3. Development and implementation of an algorithm for obtaining a phase shift and estimating the error of the result.

Scientific novelty of this work is represented by:

1. Robust Parabola method of fringes tracing that overcomes traditional methods (i.e. WFT and RPT) by RMSD metrics being an algorithm with linear complexity to the total amount of image pixels.

2. Improvement of the Parabola method with nonlinear locally adaptive processing technique which allowes to use it for frames with complex patterms.

3. Smoothed Kernel-based Intensity Measuring (SKIM) function as a quality criterion of fringes tracing.

4. Application of anizotrophic smoothing to the processing of interferograms.

5. Proposal of SKIM-based iterative approach of phase unwrapping that overcomes for at least 60% modern approaches including DCNN (by RMSD metrics).

Practical significance. The algorithms developed in this work were used to analyze interferograms of phase objects in the physics of gas discharges. Specifically, highly ionized plasma channels formed during spark discharges in air. These phase objects exhibit rapid temporal changes on the nanosecond scale, very small transverse sizes, and intricate internal structures, making their optical investigation highly demanding.

Optical methods for examining phase objects, along with advanced data processing algorithms like the one proposed here, have broad applicability across numerous fundamental and applied scientific disciplines. These include microbiology, where phase imaging aids in studying transparent biological specimens; quality

assurance in optical component manufacturing; diagnostics of gas flow and flame dynamics; and investigations of plasma formation processes induced by gas discharges or exposure to intense electromagnetic radiation. Interferometric techniques, particularly those employing coherent laser illumination, are invaluable for precisely measuring the parameters of such phase objects by capturing detailed refractive index variations.

Thus, the developed algorithms can be effectively utilized in fields such as plasma physics, combustion diagnostics, optical metrology, and biological imaging, where accurate interpretation of interferometric data is critical for advancing both scientific understanding and technological applications.

The implemented code is open source, which confirms the reproducibility of the results and allows the scientific community to apply the result to their experimental data.

Methodology and research methods. To enhance the quality of raw interference patterns, a noise suppression and distortion correction algorithm has been developed. The method employs contrast- and structure-preserving filtering techniques, improving the extraction of interference fringes.

An efficient algorithm for automatic fringe (phase isoline) detection has been implemented using a parabolic fitting approach. The algorithm analyzes local extrema and connects them into continuous lines using seed points on an initial slice, followed by iterative neighbor search.

To improve noise robustness, an iterative refinement method has been applied. It progressively updates the fringe map by reconstructing the local fringe orientation field while preserving contrast and shape.

An advanced tracing algorithm version has been developed to correctly handle interferograms featuring negative-curvature fringes, typical of complex phase objects.

Phase distribution reconstruction employs a multi-step approach:

1. Dark fringe tracing via parabolic fitting;

2. Iso-phase curve construction using an original fitting method;

3. Local phase gradient field computation;

4. Anisotropic diffusion-based smoothing;

5. Phase unwrapping via an original gradient-field-based technique;

6. Convergence and error assessment by comparing forward-modeled interference with experimental frames.

Main statements submitted for defense:

1. A novel approach to pre-processing interference patterns, which effectively removes noise and distortions from the recorded images, enhancing the quality of subsequent analysis.

2. A simple and computationally efficient algorithm developed for tracing interferograms of phase objects, enabling rapid and reliable extraction of fringe information.

3. An iterative noise and distortion removal method that preserves fringe contrast, thereby maintaining the integrity of the interferometric data during preprocessing.

4. An extended algorithm designed for the analysis of interferograms from complex-structured phase objects, capable of accurately processing fringes exhibiting negative curvature.

5. An iterative phase reconstruction approach for complex phase objects, involving sequential steps of dark fringe tracing, isophase pattern mapping, local gradient estimation, anisotropic diffusion smoothing, phase unwrapping, and convergence evaluation. This method incorporates a novel loss function specifically tailored for isophase fitting, which substantially improves curve approximation quality and enhances the accuracy of phase reconstruction.

Reliability. All the algorithms were tested in the context of analyzing the interference patterns of complex-structured spark channels developing from the surface of a point cathode in ambient air. The experimental findings demonstrate the algorithm's reliability and precision. When compared with existing methods, the framework technique consistently delivers better performance in phase unwrapping tasks, especially in scenarios with significant noise interference. Based on RMSD evaluations, WISP attains the smallest reconstruction errors, achieving a 39.7% reduction relative to the second-best approach (Deep Convolutional Neural Network), underscoring its enhanced robustness, accuracy, and efficiency in computation.

Work approbation. The main results of the work were reported at:

1. Relationship between high-energy radiation parameters and the stage of development of an atmospheric discharge. Fast Electron-Explosive, Electronic, and Electromagnetic Processes in Pulsed Electronics and Optoelectronics Conference, Moscow, Russia, November 15-17, 2022 [41].

2. Laser probing of laboratory sparks and leaders in application to the study of lightning discharges. Trovant Conference: 21st All-Russian Youth Samara

Competition-Conference on Optics, Laser Physics, and Plasma Physics, dedicated to the 300th anniversary of the Russian Academy of Sciences, Samara, Russia, November 14-18, 2023 [42]. The materials developed within the framework of the dissertation work were published in peer-reviewed journals in the following publications:

1. Development of the Interfan Client-Server Software for Efficient Analysis of Interference Patterns of Plasma Objects. Bulletin of the Lebedev Physics Institute. - 2020. - T. 47. - P. 376-380 [43].

2. Algorithm of Interferogram Tracing. I. The Parabola Method: Pros and Cons. Journal of Russian Laser Research. -2021. -V. 42. - P. 25-31 [16].

3. Algorithm of Interferogram Tracing. II. Fringes with Negative Curvature and Extended Approach to Their Processing. Journal of Russian Laser Research. -2021. -V. 42. - P. 161-170 [44].

4. WISP: Workframe for Interferogram Signal Phase-unwrapping. IEEE Access. - 2025. [45]

Based on the dissertation materials, a patent for the software was registered: 1. INTERFAN SERVER. Federal Service for Intellectual Property (Rospatent), Russia. Certificate of state registration of computer program 2021662053, 2021. [46].

Personal contribution. The author's contribution is the development of the described algorithms, as well as their software implementation.

Publications. The main results on the topic of the dissertation are presented in 6 printed publications, 4 of which were published in journals recommended by the Higher Attestation Commission, 2 — in abstracts of reports. 1 patent was published based on the results. Volume and structure of work.

The full volume of the dissertation is 96 pages, including 20 figures and 9 tables. The list of references contains 63 titles.

Заключение диссертации по теме «Другие cпециальности», Хирьянов Тимофей Федорович

Conclusion

The main results of the work are as follows.

This research has developed and validated a comprehensive methodological framework for interferometric analysis of rapidly evolving phase objects, addressing critical challenges in studying phenomena such as spark discharge plasma channels that evolve on nanosecond timescales and exhibit complex features like negative curvature fringes under significant noise conditions.

The principal outcome is a suite of novel algorithms—the Parabola Method (PM), Extended Approach Parabola Method (EAPM), and the complete WISP framework—forming an integrated pipeline from preprocessing and fringe tracing to robust phase unwrapping and error estimation. The PM provides computationally efficient tracing of sharp oscillatory fringes, while EAPM extends this capability to negative curvature cases. The WISP framework integrates these with adaptive SKIM-based tuning and anisotropic diffusion, achieving iterative phase reconstruction with proven convergence.

The methods were rigorously validated using experimental data from spark discharges in atmospheric air. Quantitative evaluation against established baselines, including Deep Convolutional Neural Networks, demonstrated that WISP achieves the lowest reconstruction errors (RMSD), outperforming the next best method by 39.7%, confirming its robustness in high-noise environments critical for plasma diagnostics.

Key original contributions include: a novel preprocessing technique preserving fringe contrast while eliminating noise; a computationally lightweight tracing algorithm for complex interferograms; enhanced capability for analyzing negative curvature fringes; and the iterative WISP framework with its specialized loss function for isophase fitting. These advances are accompanied by an open-source software package with comprehensive documentation.

Future research directions include real-time implementation for dynamic process monitoring, extension to other phase objects in fluid dynamics and biophysics, and integration with machine learning techniques for enhanced automation. This work provides both immediate practical tools for high-precision plasma diagnostics and a versatile foundation for advancing quantitative interferometric analysis across scientific and engineering disciplines.

In conclusion, the author expresses sincere gratitude and deep appreciation to the scientific supervisor, I.A. Makarov, for support, assistance, discussion of results, and scientific guidance. The author also thanks E.V. Parkevich for the experimental part of the work, as well as for the joint creative process on the algorithms. Additionally, the author thanks A.I. Khiryanova for the collaborative creative work on the algorithms and the dissertation text.

Список литературы диссертационного исследования кандидат наук Хирьянов Тимофей Федорович, 2025 год

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