Komütatı̇f elı̇ptı̇k oktonyon matrı̇s denklem çözümlerı̇ ve görüntü ı̇yı̇leştı̇rme uygulamaları = Commutatı̇ve ellı̇ptı̇c octonı̇on matrı̇x equatı̇on solutı̇ons and ımage enhancement applı̇catı̇ons

Abstract

Birinci bölümde, ilk alt başlık literatür incelemelerine odaklanmaktadır. Tezin temelini oluşturarak mevcut bilgileri derinlemesine incelemeyi amaçlamaktadır. Literatürdeki çalışmaların kapsamlı bir analizi yapılarak, konunun önceki araştırmalardaki yeri ve önemi vurgulanmaktadır. İkinci alt başlıkta ise bu tezin amacı belirtilmektedir. İkinci bölümde, eliptik kuaterniyon cebri ve temel özellikleri incelenmektedir. Eliptik kuaterniyon matris yapısı, denklem çözümleri için kullanılmakta ve bu alanda etkili algoritmalar sunulmaktadır. Ayrıca, renkli görüntü iyileştirme çalışmaları da eliptik kuaterniyonlar temelinde gerçekleştirilmektedir. Üçüncü bölüm tezin orijinal kısmının ilk bölümü oluşturmaktadır ve bu bölümde komütatif eliptik oktonyon cebri tanımlanmaktadır. Bu tanımlamalar, komütatif eliptik oktonyonların eliptik ve eliptik kuaterniyon temel matrislerini ifade etmek için matris cebri ile izomorfizmalar aracılığıyla ilişkilendirilir. Ayrıca, komütatif eliptik oktonyon matrisleri kapsamlı bir şekilde ele alınır. Bu matris yapısı, Kalman Yakubovich s- eşlenik, Sylvester s-eşlenik ve AX = B denklemlerinin çözümlerini elde etmek için kullanılır. Dördüncü bölüm tezin orijinal kısmının ikinci bölümünü oluşturmaktadır. Bu bölümde, bir görüntünün kırmızı (R), yeşil (G) ve mavi (B) kanallarında nasıl temsil edildiği ayrıntılı bir şekilde açıklanır ve görüntü iyileştirme süreci detaylarıyla ele alınır. Bu aşamada, görüntülerin komütatif eliptik oktonyon temsil matrisiyle nasıl ilişkilendirildiği incelenir ve görüntülerin komütatif eliptik oktonyonik olarak nasıl ifade edildiği ortaya konur. Bu yöntemle, görüntülerin daha karmaşık ve zengin bir temsilini elde etmek mümkün olur. Son olarak, komütatif eliptik oktonyonik görüntü üzerinde çeşitli görüntü iyileştirme teknikleri uygulanır ve bu tekniklerin sonuçları görsel olarak değerlendirilir. Bu bölüm, komütatif eliptik oktonyonların görüntü işleme alanınd aki pot ansiyelini vurgular ve d aha et kili görünt ü iyileşt irm e yöntemlerinin geliştirilmesine katkı sağlar. Beşinci bölümde, tezde elde edilen sonuçlar kapsamlı bir şekilde değerlendirilmiştir ve gelecekte yapılacak araştırmalar için öneriler sunulmuştur. Bu değerlendirme, komütatif eliptik oktonyon cebri alanında sağlanan ilerlemeleri vurgular ve bu alanda daha fazla keşif yapılması gerektiğini belirtir. Ayrıca, görüntü işleme uygulamalarında komütatif eliptik oktonyonların potansiyelini daha da araştırmak için öneriler sunulur. Bu bölüm, tezin önemli katkılarını özetler ve gelecekteki çalışmalara ilham verir.

The thesis consists of five comprehensive chapters, each covering important topics that form the fundamental basis of the research inquiry. The first chapter serves as the introduction, providing an initial overview of the study. Its first subsection focuses on the crucial aspect of the literature review. By conducting an extensive analysis of existing knowledge, the literature review aims to establish a solid foundation for the thesis. This section critically examines previous research related to the topic, highlighting the significance and relevance of the subject in earlier studies. Through synthesizing and integrating the existing scientific works, the literature review explains the rationale and importance of the research. Additionally, this section identifies unresolved or contradictory points in the literature and discusses the reasons for focusing the research on these aspects. The literature review also demonstrates how the research integrates with the existing knowledge and emphasizes the originality of the study. Overall, this subsection elaborates on the purpose of the thesis and provides a detailed explanation of the general objective and specific research goals, enabling read ers t o bet t er und erst and t he signif icance and pot ent ial contributions of the study. The second subsection comprehensively explores the topic of elliptic quaternion algebra, providing a detailed examination of its matrix structure and equation solutions. In this section, important tools in the field of elliptic quaternion algebra, such as the Kalman Yakubovich's conjugate and Sylvester equation solutions, are also discussed. The focus on matrix structure and equation solutions highlights the fundamental properties of the elliptic quaternion matrix structure and emphasizes its effective utilization in the field of image processing. Furthermore, this section delves into the applications of elliptic quaternion algebra and matrix structure in color image enhancement. The combination of elliptic quaternion algebra and matrix structure with Kalman Yakubovich's conjugate and Sylvester equation solutions further underscores their significance in both mathematical and practical applications. This section aims to highlight advancements in elliptic quaternion algebra, serving as a foundation for future research and encouraging further exploration, innovation, and progress in the field. The third chapter constitutes the original part of the thesis. It defines commutative elliptic octonion algebra, which represents elliptic and elliptic quaternion fundamental matrices through isomorphisms associated with matrix algebra. Furthermore, it extensively covers commutative elliptic octonion matrix structures. This matrix structure serves as an effective tool to obtain solutions for Kalman-Yakubovich's equation, Sylvester's equation, and other equations. This chapter highlights the original contributions of the thesis, showcasing how commutative elliptic octonion algebra and matrix structures provide advantages in solving equations. By establishing a connection between commutative elliptic octonion algebra and general matrix algebra xxiii principles, this approach demonstrates the utilization of matrices in various equation- solving problems. In this context, important equations such as Kalman-Yakubovich's equation and Sylvester's equation can be solved using commutative elliptic octonion matrix structures. This new approach enables faster and more accurate results in the equation-solving process. The fourth chapter represents the second original part of the thesis and signifies a significant step in the field of image processing. In this chapter, a detailed explanation is provided on how an image is represented in the red (R), green (G), and blue (B) channels. The process of image enhancement is systematically addressed, examining various image enhancement techniques and algorithms and demonstrating their application. Moreover, this chapter showcases the utilization of the commutative elliptic octonion matrix structure, enabling the attainment of more complex and rich representations of images. Commutative elliptic octonions are employed as a tool to reproduce images in higher quality and realism. This matrix structure facilitates the application of diverse image enhancement techniques to images. The effectiveness of these enhancements is evaluated visually through analysis. By emphasizing the potential of commutative elliptic octonions in the field of image processing, this chapter contributes to the development of more efficient image enhancement methods. Consequently, utilizing advanced and robust tools in the domain of image processing enables the attainment of higher quality results. The fifth chapter is an important stage where the results of the thesis are thoroughly evaluated, and recommendations for future research are presented. In this stage, the results obtained by the thesis are examined comprehensively, and the significance of these findings is emphasized. The analysis includes a detailed evaluation of the effectiveness of the commutative elliptic octonion algebra and matrix structures in solving equations and enhancing images. The implications of the research findings are discussed, highlighting their contributions to the field of commutative elliptic octonion algebra and image processing. Additionally, recommendations are provided for further research, emphasizing the importance of conducting more exploration in the field of commutative elliptic octonion algebra. Specifically, suggestions are made to further investigate the potential of commutative elliptic octonions in image processing applications. These recommendations underscore the belief that commutative elliptic octonions could offer a broader range of utility in the field of image processing and contribute to the development of more effective image enhancement techniques. The conclusions of this thesis demonstrate the necessity of advancing further research in the field of commutative elliptic octonion algebra and image processing, introducing a new perspective for future studies. The research conducted in this thesis, focusing on the utilization of commutative elliptic octonions and matrix structures, highlights the progress made in these areas. It is believed that in the future, commutative elliptic octonions could find a wider range of applications, and more advanced image enhancement techniques could be developed. This thesis aims to serve as an inspiration for future research by highlighting the advancements and contributions made in this field. Furthermore, this thesis contributes to the development of more efficient solutions for equation-solving problems and image enhancement tasks. The innovative use of commutative elliptic octonions and matrix structures provides advantages in terms of accuracy, speed, and quality of results. By establishing a connection between commutative elliptic octonion algebra and general matrix algebra principles, this xxiv research expands the applicability of matrices in various mathematical and practical contexts. Moreover, the integration of commutative elliptic octonions into image processing techniques opens up new possibilities for enhancing the quality, realism, and visual appeal of images. The commutative elliptic octonion matrix structure serves as a powerful tool for representing and manipulating color images, enabling the application of advanced enhancement algorithms. This thesis showcases the effectiveness of these techniques and highlights their potential in producing visually appealing and high- quality images. The evaluation of the research findings in the fifth chapter emphasizes the significance and contributions of the thesis. The comprehensive analysis of the results demonstrates the effectiveness and practicality of the proposed approaches in commutative elliptic octonion algebra and image processing. The successful application of commutative elliptic octonion matrix structures in solving equations and enhancing images highlights their relevance and potential in these domains. Based on the outcomes of the thesis, recommendations for future research are presented. The identified areas of further exploration in commutative elliptic octonion algebra and image processing encourage researchers to delve deeper into these fields. The suggested directions include investigating additional applications of commutative elliptic octonions in image processing, exploring novel enhancement techniques, and further developing the theoretical foundations of commutative elliptic octonion algebra. In conclusion, this thesis makes a significant contribution to the fields of commutative elliptic octonion algebra and image processing. By introducing innovative approaches and techniques, it expands our understanding of these domains and offers more efficient solutions. The findings highlight the necessity for continued research and exploration in commutative elliptic octonion algebra and image processing, with the potential for broader applications and advancements in the future. This thesis aims to inspire and guide future studies in these areas, encouraging researchers to build upon the progress made and explore new frontiers in commutative elliptic octonion algebra and image processing.