Псевдобулевский полиномиальный подход к решению задач компьютерного зрения / Pseudo- Boolean Polynomial approach to solving Computer Vision tasks тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Чикаке Тендай Мапунгвана

• Чикаке Тендай Мапунгвана
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Чикаке Тендай Мапунгвана

Contents

Page

Introduction

Chapter 1. Pseudo-Boolean polynomials

1.1 Introduction

1.2 Background

1.2.1 Historical Development

1.2.2 Mathematical Foundations

1.2.3 Representation Methods

1.2.4 Applications in Optimization

1.3 Method

1.3.1 Mathematical Formulation

1.3.2 Comparative Analysis of Pseudo-Boolean Polynomial Representation Methods

1.3.3 Generalized Framework for Data Analysis

1.3.4 Illustrative Example

1.3.5 Computational Advantage and Properties

1.3.6 Integer Encoding Computational Mechanism

1.3.7 Variable Embedding and Efficient Implementation

1.4 Computational formulation of pseudo-Boolean polynomials

1.4.1 Aggregation and Analysis of Polynomial Terms

1.4.2 Computational Complexity Analysis

1.5 Comparison of Methods

1.5.1 Direct Enumeration

1.5.2 Fourier Transform Method

1.5.3 Algebraic Normal Form (ANF)

1.5.4 Graph Cut Methods

1.5.5 Integer Encoding Method

1.5.6 Qualitative and Quantitative Comparison

1.6 Chapter Summary

1.6.1 Transition to Dimensionality Reduction and Clustering

Chapter 2. Dimensionality Reduction and Cluster analysis

2.1 Introduction

2.2 Related work

2.2.1 Classical Dimensionality Reduction Methods

2.2.2 Non-linear Approaches

2.2.3 Challenges in Contemporary Approaches

2.2.4 Pseudo-Boolean Approaches

2.3 Methods

2.3.1 Theoretical Foundations

2.3.2 Mathematical Formulation

2.3.3 Reduction and Equivalence Properties

2.3.4 Application to Dimensionality Reduction

2.3.5 Distance Metrics and Clustering

2.3.6 Advanced Aggregation Strategies

2.3.7 Intelligent Sorting Methods

2.3.8 Supervised Probes Framework

2.3.9 Configuration Optimization Protocol

2.3.10 Dimensionality Reduction Characteristics

2.3.11 Visualization and Classification

2.3.12 Theoretical Basis for Feature Reduction

2.3.13 Application to Language Assessment Systems

2.4 Experimental Setup

2.4.1 The Iris Flower dataset

2.4.2 The Wisconsin Diagnostic Breast Cancer (WDBC) dataset

2.4.3 Real World Implementation

2.5 Results and Discussion

2.5.1 Comprehensive Dataset Evaluation Protocol

2.5.2 Overall Performance Comparison

2.5.3 Dataset-Specific Performance Analysis and Best Configurations

2.5.4 Performance Comparison Against Baseline Methods

2.5.5 Computational Efficiency Analysis

2.5.6 Visual Analysis of Clustering Performance

2.5.7 Performance on Diverse Data Types

2.5.8 The Iris Flower Dataset: Detailed Analysis

2.5.9 The Wisconsin Diagnostic Breast Cancer (WDBC) Dataset

2.6 Chapter Summary

Chapter 3. Image Edge detection

3.1 Introduction

3.2 Related work

3.2.1 Classical Gradient-Based Methods

3.2.2 Statistical and Variational Methods

3.2.3 Machine Learning and Deep Learning Approaches

3.2.4 Polynomial and Mathematical Morphology Approaches

3.2.5 Research Gap and Our Contribution

3.3 Pseudo-Boolean Polynomials for image edge detection

3.4 Experimental Setup

3.5 Results and Discussion

3.5.1 Experimental Setup

3.5.2 Visual Comparison with Classical Methods

3.5.3 Comparison with Advanced Methods

3.5.4 Application to Natural Scene Images

3.5.5 Application to Medical Images

3.5.6 Limitations and Future Directions

3.5.7 Summary of Contributions

3.6 Chapter Summary

Conclusion

List of Abbreviations and Notations

Glossary of Terms

Bibliography

List of Figures

List of Tables

Appendix A. Application document listings

Appendix B. Application code listings

Введение диссертации (часть автореферата) на тему «Псевдобулевский полиномиальный подход к решению задач компьютерного зрения / Pseudo- Boolean Polynomial approach to solving Computer Vision tasks»

Introduction

This work investigates the application of pseudo-Boolean polynomials (pBp) in processing image data for various computer vision (CV) tasks. While the utilization of pBp in CV remains relatively underexplored, this work introduces novel interpretations and algorithms that leverage their inherent properties to address challenges in areas such as dimensionality reduction, clustering, image edge detection, and segmentation.

Pseudo-Boolean polynomials, mathematical constructs involving binary variables, offer a powerful framework for representing complex optimization problems. Penalty-based pseudo-Boolean polynomials are effectively employed to streamline the computational demands of optimization problems, including the p-Median Problem and the Simple Plant Location Problem [1—3].

In this work, we demonstrate the reduction in processing complexity achieved through the non-linear compression of matrices into penalty-based pseudo-Boolean polynomials for pattern analysis. The concept of equivalence, as discussed in [1; 2], illustrates that a single pBp can encapsulate multiple matrix representations, establishing a many-to-one relationship between the original matrix and the polynomial. This property facilitates data embedding, which is valuable in clustering and classification tasks.

Furthermore, we exploit the compacting and embedding properties of pBp for dimensionality reduction and leverage polynomial degree properties for image edge and blob detection.

The methods presented in this thesis prioritize algorithmic solutions for computer vision problems, contrasting with the more prevalent data-driven approaches.

Relevance of the topic. The field of computer vision has witnessed remarkable advancements in recent years, largely propelled by the development of deep learning techniques. These methods, particularly deep neural networks, have achieved considerable success in various tasks, including image classification, object detection, and image segmentation. Deep neural networks have demonstrated improved performance with increased data and computational resources, owing to the deepening of their layers [4]. However, the reliance on substantial volumes of labelled data and significant computational resources presents challenges for

their widespread adoption, especially in resource-constrained environments [5; 6]. Contemporary deep neural networks often possess billions of trainable parameters and necessitate operation on multiple or high-performance Graphics Processing Units (GPUs), resulting in considerable electrical power consumption [7].

While deep neural networks have become a mainstay in machine learning, issues of interpretability and performance remain pertinent [8]. For instance, when a neural network misclassifies an image of a cat as a rabbit, elucidating the reasoning behind this erroneous prediction proves challenging. The decision-making process becomes more transparent when features are human-interpretable. Therefore, the question of interpretability in computer vision and general data science persists, motivating the algorithmic focus of this work.

Interpretability is paramount in domains such as banking, medical diagnosis, and critical financial decision-making at the corporate and economic levels. The high accuracy of a neural network alone does not provide sufficient justification for adopting a decision. Interpretability is crucial for defending suggested decisions and substantiating legal cases. The "black-box" nature of these models hinders trust and validation of their predictions, potentially leading to risks and ethical dilemmas.

Deep neural networks are also susceptible to over-parameterization, which can engender unfair decision-making in critical domains [9]. In contrast, while other algorithms may offer interpretability, they can suffer from accuracy saturation, where additional data yields no significant improvement [10].

The primary challenges in contemporary computer vision encompass interpretability, reproducibility, and performance. A central question is how to effectively balance these considerations.

In light of these challenges, there is a growing interest in exploring alternative approaches that minimize dependence on extensive data and computational resources. These challenges, among others, motivate the exploration of alternative algorithms and paradigms that do not necessarily require data-driven learning to address computer vision tasks.

Pseudo-Boolean polynomials offer a deterministic and interpretable framework for processing image data, rendering them an attractive complement to prevalent deep learning methods. This work proposes approaches that leverage the mathematical properties of pseudo-Boolean polynomials to perform tasks such as dimensionality reduction, clustering, and image edge detection. These approaches

mitigate the need for large volumes of labelled data or extensive computational resources, making them suitable for a broad spectrum of applications.

Furthermore, the deterministic nature of pseudo-Boolean polynomials ensures reproducible and interpretable results. This transparency is particularly valuable in domains where understanding the underlying decision-making process is paramount. By providing a transparent and explainable framework, this work offers potential avenues for addressing the interpretability concerns associated with deep learning-based methods.

The goal of this work is to discover and showcase the utility of properties within discrete optimization paradigms for application in image processing, computer vision, and general data processing. This work highlights the application of pseudo-Boolean polynomial properties in dimensionality reduction, data clustering, image processing, and optical character recognition. Practical examples of dimensionality reduction, image edge detection, and image region equivalence are presented.

To achieve this aim, the following tasks were undertaken:

1. Identify the exploitable properties of pseudo-Boolean polynomials for addressing computer vision tasks.

2. Determine data representations suitable for processing through pseudo-Boolean polynomial formulation.

3. Provide rigorous justifications for the utilization of the selected properties and/or paradigms.

4. Benchmark the output results against existing paradigms.

5. Substantiate the potential for adoption of the proposed paradigms in real-world applications.

Scientific novelty: This thesis introduces novel approaches to dimensionality reduction, data clustering, and image edge detection through the application of pseudo-Boolean polynomials, addressing specific limitations of contemporary deep learning techniques. The proposed methods offer deterministic, interpretable, and resource-efficient solutions for various computer vision tasks, positioning the pseudo-Boolean polynomial concept as a valuable contribution to the field. The findings of this research hold the potential to impact a wide range of applications,

from medical imaging to industrial inspection, where reliable and explainable image analysis is crucial for informed decision-making. This novelty is manifested through:

1. Development of efficient data encoding software for representing and processing pseudo-Boolean polynomials.

2. Projection of matrix data into multidimensional hypersphere vectors, enabling the measurement of Euclidean distance to assess matrix similarity.

3. Identification of gradient transitions in image data based on the degrees of calculated pseudo-Boolean polynomials.

4. Successful application of the dimensionality reduction methodology in language proficiency assessment, demonstrating its effectiveness in real-world feature selection tasks.

Practical significance This work presents a straightforward, transparent, and reproducible algorithm for generating pseudo-Boolean polynomials with properties suitable for data analysis tasks, specifically image edge detection and clustering. The practical impact of this work is demonstrated through its successful implementation in the Интеллектуальная Система Тестирования Общеязыковых Компетенций (ИСТОК) [11; 12], where it significantly improved feature selection and classification accuracy in language proficiency assessment.

Methodology and research methods.

This work proposes, tests, evaluates, and compares theoretical arguments concerning the properties of selected discrete optimization methods for image and data processing tasks against conventional paradigms. For each shortlisted property or characteristic, typical use cases are reviewed, experimented with, and the results are compared.

Main results. First, the dissertation develops a comprehensive formulation of pseudo-Boolean polynomials together with an efficient integer-encoding scheme enabling compact representation and fast manipulation for computational tasks. Second, it introduces a deterministic dimensionality-reduction method grounded in PBP properties that preserves structural relationships in data and serves as an interpretable alternative to stochastic embeddings, achieving competitive performance on standard datasets. Third, it proposes a digital image edge-detection approach that exploits polynomial degree characteristics, yielding notable resilience to noise and improved edge continuity across varied imagery.

For edge detection on 512x512 images, the PBP method runs in 32 ms on CPU (Intel i5, 8 GB RAM), comparable to Canny (25 ms) and substantially faster

than Fisher-information-based (187 ms) and HED (312 ms CPU / 45 ms GPU). Qualitatively, it produces superior edge continuity relative to Sobel and continuity comparable to Canny with fewer preprocessing steps, while exhibiting higher noise robustness owing to stability of polynomial degree under perturbations.

For dimensionality reduction (PBP embedding) across 11 datasets, the method attains strong linear separability (LinSep) with low boundary complexity (BC): Iris 0.9800 (BC 0.0333, V 0.9488, ARI 0.9603, Silhouette 0.6561), HTRU2 0.9740 (BC 0.0377), Seeds 0.9524 (BC 0.0714), WDBC 0.9508 (BC 0.0738); with near-separable embeddings on Banknote 0.9738 (BC 0.0211) and Penguins 0.9211 (V 0.7203, ARI 0.7027). Against the strongest baseline per dataset, PBP leads LinSep on 5/11 (e.g., Iris 0.9733 vs 0.9467; Seeds 0.9524 vs 0.9286) and yields the smoothest boundaries on 2/11 (Iris BC 0.0333; Seeds BC 0.0714).

Basic provisions submitted for defense:

1. Pseudo-Boolean polynomials can be effectively employed for dimensionality reduction in image data, offering a compact, interpretable, and analysable representation.

2. The equivalence property of pseudo-Boolean polynomials facilitates the clustering of structurally equivalent matrices in data analysis tasks.

3. The degree properties of pseudo-Boolean polynomials can be leveraged for accurate image edge and blob detection.

4. The proposed methods present a deterministic and resource-efficient alternative to deep learning techniques for specific computer vision tasks.

5. The interpretability and reproducibility of the results render the methods suitable for critical applications where transparency is paramount.

Credibility The credibility of this thesis is substantiated through rigorous theoretical analysis and comprehensive experimental validation. The proposed methods have been evaluated on diverse datasets, including synthetic images with basic shapes and real-world images, to demonstrate their effectiveness and robustness. The results have been benchmarked against existing state-of-the-art methods to highlight the advantages and limitations of the proposed approaches. Furthermore, the reproducibility of the results has been verified through independent replication of the experiments. The theoretical underpinnings of the methods are rooted in well-established principles of discrete optimization and pseudo-Boolean polynomial theory, further bolstering the credibility of this work.

Approbation of work. The research presented in this thesis has been subjected to rigorous scrutiny and validation across diverse scholarly and professional venues. Key findings were disseminated at the following conferences:

1. Sixteenth International Conference on Machine Vision (ICMV), 2023, Yerevan, Armenia.

2. 65th All-Russian Scientific Conference of MIPT, 2024, Moscow, Russia.

3. International Conference on Data Analysis, Optimization, and their Applications, 2023, Moscow, Russia.

Peer-reviewed articles detailing the methods and results have been published in reputable journals such as the 24th International BIOMAT Consortium, International Institute for Interdisciplinary Sciences and Journal of Computer Optics . Moreover, the methods have been implemented in open-source software tools, facilitating independent verification and application by the broader research community and also applied in the development of the Интеллектуальная Система Тестирования Общеязыковых Компетенций (ИСТОК) for feature selection using the dimensionality reduction methods.

Personal contribution. The primary author made substantial contributions to the identification of exploitable properties of pseudo-Boolean polynomials in computer vision tasks, the conceptualization and implementation of program code for the identified properties, and the rigorous comparison of the proposed methods with existing paradigms. All results were obtained by the applicant personally or in collaboration with direct participation, under the advising of Prof. B. I. Goldengorin. In co-authored works, the author's contributions include: identification, implementation, and reporting of edge detection properties; implementation for registered software; identification, implementation, and reporting of unsupervised clustering properties of pseudo-Boolean polynomials; software for Gramian optimization in optimal control; domain-specific medical imaging edge detection software and analysis.

Volume and structure of work. The dissertation consists of an introduction, 3 chapters, conclusion and 2 attachments The full volume of the dissertation is 145 pages, including 17 figures and 7 tables. The list of references contains 133 titles.

Chapter 1. Pseudo-Boolean polynomials

This research examines the theoretical foundations and computational applications of penalty-based pseudo-Boolean polynomials, extending the seminal work of [1] and [2]. We systematically identify and formalize key properties of these polynomials, demonstrating their utility across diverse computer vision and data processing tasks.

The chapter begins by rigorously formulating these polynomials within the context of p-Median problems, establishing a mathematical framework that serves as the foundation for subsequent applications. We then analyze their intrinsic properties—compression, equivalence, matrix-to-vector projection, and polynomial degree characteristics—and develop novel methodologies for their exploitation in image edge detection, dimensionality reduction, and clustering tasks.

Furthermore, we present an efficient computational algorithm for generating these polynomials and analyzing their properties, implementing optimized data structures and bitwise operations that significantly reduce computational complexity. This algorithmic approach achieves 0(m • n log(m • n)) time complexity for matrices of size m x n, enabling practical applications to large-scale data analysis problems. Our contribution advances both the theoretical understanding and practical implementation of pseudo-Boolean polynomial techniques, establishing a bridge between discrete optimization theory and computer vision applications.

1.1 Introduction

The intersection of discrete optimization and computer vision represents a fertile ground for innovative algorithmic approaches that can transcend traditional domain boundaries. This dissertation investigates penalty-based pseudo-Boolean polynomials—mathematical constructs traditionally employed in combinatorial optimization—and establishes their novel utility in solving complex computer vision and data processing challenges.

Pseudo-Boolean polynomials are formally defined as mappings f : Bn ^ R, where B = {0,1} represents the Boolean domain and n denotes a natural number

corresponding to the dimension of the Boolean space [3]. These polynomials provide a mathematically rigorous framework for representing discrete optimization problems, capturing the inherent combinatorial nature of many computational challenges in both theoretical and applied contexts [21; 22].

Specifically, penalty-based pseudo-Boolean polynomials, also known as Hammer-Beresnev polynomials, possess distinctive algebraic properties that render them particularly valuable for data-intensive applications [23]. Previous research by [1] and [2] has elucidated the reduction and equivalence properties of these polynomials, demonstrating their efficacy in compacting both dense and sparse matrices through strategic aggregation. This compaction capability has proven instrumental in reducing the computational complexity of classical combinatorial optimization problems, including the p-Median and Simple Plant Location problems [2; 24].

The primary contribution of this research lies in the systematic identification and formalization of four key properties of penalty-based pseudo-Boolean polynomials that render them exceptionally valuable for computer vision and data processing applications:

Заключение диссертации по теме «Другие cпециальности», Чикаке Тендай Мапунгвана

Conclusion

This thesis presented a novel approach to formulating and representing penalty-based pseudo-Boolean polynomials and their applications to dimensionality reduction and edge detection in image analysis. By extending the theoretical foundations established by [1] and [2], we demonstrated the efficacy of these mathematical structures across multiple computational domains. Summary of Contributions.

Our research made contributions in three interconnected areas.

1. Pseudo-Boolean

Polynomial Formulation. Established a comprehensive mathematical foundation for penalty-based pseudo-Boolean polynomials in computer vision; developed an efficient integer-encoding representation; achieved complexity 0(mn log(mn)) versus classical 0(2n).

2. Dimensionality Reduction. Introduced a PBP-based reduction technique that preserves structural relationships; provided a deterministic, mathematically grounded alternative to stochastic embeddings; demonstrated competitive performance against PCA, t-SNE, and UMAP on suitable benchmarks.

3. Digital Image Edge Detection. Proposed an edge detector leveraging polynomial degree properties; showed improved noise resilience and edge continuity; validated across natural and medical images with clear interpretability (degree corresponds to edge characteristics).

Advantages and Limitations.

Advantages. (1) Unsupervised operation. works without labelled data where annotations are scarce; (2) Interpretability. deterministic, mathematically justified outputs with explicit guarantees; (3) Efficiency. CPU-friendly implementations without specialized hardware; (4) Versatility. one framework adaptable across applications while preserving theoretical consistency.

Limitations. (1) Parameter selection. performance depends on informed choices and lacks full automation; (2) Extreme scale. very large data may still pose computational burdens; (3) Integration. limited exploration of hybridization with deep learning. Recommendations.

For practical use and reporting, we recommend: (1) selecting parameters via cross-validated grids or image/statistical heuristics aligned with task objectives; (2) documenting complexity, runtime, and hardware (CPU/GPU) to support reproducibility and fair comparison; (3) providing public code/data artifacts where possible; (4) using deterministic seeds and versioning (libraries, datasets) to ensure replicable results across platforms. Future Research Directions.

Future work includes:

1. Theoretical Extensions: Extend PBP theory to higher-dimensional data structures; exploit connections to submodular optimization [29] and extended submodularity [40] for algorithmic gains.

2. Dimensionality Reduction:

Develop adaptive neighbourhood preservation, by using local polynomial cues; integrate with manifold learning for strongly non-linear data; extend to temporal data and dynamic embeddings.

3. Image Analysis: Automate parameter selection from image statistics; extend to volumetric (3D) modalities in medical imaging; characterize semantic regions via degree distributions; expand benchmarking on BSDS500 [133] and related datasets.

4. Hybrid Approaches: Combine pBp features with deep models; design architectures that incorporate PBP-based extraction; study pBp-guided attention for interpretability.

In conclusion, this research establishes a novel paradigm spanning discrete optimization theory, dimensionality reduction, and computer vision through the unifying framework of pseudo-Boolean polynomials. By bridging these traditionally separate domains, we have demonstrated the value of cross-disciplinary approaches in computational science.

Список литературы диссертационного исследования кандидат наук Чикаке Тендай Мапунгвана, 2025 год

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