Eliptik kuaterniyon matrislerinin tekil değer ayrışımı ve onların görüntü işlemedeki uygulamaları = Singular value decomposition of elliptic quaternion matrices and their applications in image processing

Abstract

Dijital ortamlardaki görüntüler m satır, n sütuna sahip sayısal matrislerle ifade edilirler. Görüntü sıkıştırma, iyileştirme, ayrıştırma gibi görüntü işleme teknikleri görüntüyü ifade eden bu matrislere uygulanır. Görüntü işleme süreçlerinde renkli görüntüleri temsil için kullanılan geleneksel renkli görüntü seyrek matris modelleri üç ayrı renk kanalı (Red, Green ve Blue kanalları) arasındaki ilişkiyi göz ardı eder. Dolayısıyla birçok görüntü işleme süreçlerinde renkli bir görüntünün kanalları tek tek işlenmekte veya görüntü gri tonlamalı hale dönüştürülmektedir. Bu durum görüntü işleyen cihazların hem bellek yükünü hem de işlemci yükünü artırmakta ve performans değerlerini düşürmektedir. Bu tez kapsamında yukarıda bahsedilen problemin üstesinden gelmek amacıyla renkli görüntülerin bağımsız renk kanallarını bir bütün olarak ele alabilen, eliptik kuaterniyon matris teorisini kullanan yeni bir renkli görüntü seyrek matris modeli önerilerek elde edilen teoriler yardımıyla eliptik kuaterniyon tabanlı renkli görüntü sıkıştırma, iyileştirme ve ayrıştırma gibi metotlar elde edilmiştir. Oluşturulan bu tez 6 bölümden oluşmaktadır ve planı aşağıdaki gibidir: Tezin birinci bölümünde tezdeki problemin tanıtıldığı giriş bölümüne yer verilmiştir. Bu bölümde tezin kapsamı ve amacından bahsedilip literatürde konuyla ilgili yer alan çalışmalar hakkında gerekli bilgiler verilmiştir. İkinci bölümde çalışmamız boyunca kullanacağımız eliptik kompleks sayılar ve onların matrislerinin, eliptik kuaterniyonlar ve onların matrislerinin temel cebirsel özellikleri verilmiştir. Daha sonra ise renkli görüntüler ve renkli görüntü işleme konularının temel kavram ve uygulamalarına yer verilmiştir Üçüncü bölümde çalışmanın orjinal teorik kısmı yer almaktadır. Bu bölümde öncelikle eliptik sayı katsayılı ve değerli N'inci dereceden monik polinomların köklerine, eliptik kompleks matrislerin özdeğer-özvektörlerine ve tekil değer ayrışımı kavramlarına dair tanımlar verilerek ilgili teoremler ifade ve ispat edilmiştir. Eliptik kompleks matrislerin tekil değer ayrışımının bir sonucu olarak eliptik kompleks matrislerin pseudo tersleri ve en küçük kareler çözümü elde edilmiştir. Son olarak elde edilen veriler ışığında çeşitli algoritmalar geliştirilmiştir. Çalışmanın dördüncü bölümünde eliptik kompleks matrisler için elde edilen tüm tanım ve teoremler, kurulan yapı koruyan dönüşümler yardımıyla elemanları eliptik kuaterniyon olan matrislere genelleştirilmiştir. Ayrıca çalışmanın ilerleyen kısımlarında elde edilmiş olan teoremler ışığında eliptik kompleks ve eliptik kuaterniyon matrislerin özdeğer-özvektör, tekil değer ayrışımı, en küçük kareler metodu, pseudo tersleri vb. cebirsel özelliklerinin elde edilmesine dair algoritmalar verilmiştir. Beşinci bölüm projenin uygulamaya dair olan kısmını oluşturmaktadır. Bu bölümde öncelikle renkli görüntülerin eliptik kuaterniyon temsilleri verilmiştir. Daha sonra eliptik kuaterniyon matrisler için geliştirdiğimiz tekil değer ayrışımı yardımı ile renkli görüntülerin yeniden yapılandırması ve sıkıştırılması yöntemi ele alınmıştır. Bu amaçla geliştirdiğimiz sıkıştırma yöntemini kullanarak yeniden yapılandırılan renkli görüntülerin görsel kalitesi literatürde mevcut olan diğer hiperkompleks tabanlı sıkıştırma algoritmaları ile karşılaştırılarak elde ettiğimiz metodun performans değerlendirmesi yapılmıştır. Son bölümde çalışmada elde edilmiş olan sonuçlara ve önerilere yer verilerek tez tamamlanmıştır.

Images in digital media are represented by numeric matrices with m rows and n columns. Image processing techniques such as image compression, enhancement, and decomposition are applied to these matrices that express the image. Traditional color image sparse matrix models used to represent color images in image processing processes ignore the relationship between the three distinct color channels (the red, green, and blue channels). Therefore, in many image processing processes, the channels of a color image are processed one by one or the image is converted to grayscale. This situation increases both the memory load and the processor load of the image processing devices and decreases their performance values. In this thesis, in order to overcome the aforementioned problem, a new color image sparse matrix model using elliptic quaternion matrix theory is proposed that can handle the independent color channels of color images as a whole, and elliptic quaternion-based color image compression, enhancement, and decomposition methods are obtained with the help of the obtained theories. The thesis consists of six chapters, and the plan is as follows: In the first part of the thesis, the problem to be discussed in the thesis is introduced. In this section, the scope and purpose of the thesis are stated in detail, and the topics covered by the study and the purposes for which it was prepared are explained. In addition, information is given about the studies in the literature on the subject. In this way, the innovations and contributions that the thesis brought to the current subject were also emphasized. In the second part, the mathematical tools that we will use in our study—elliptic complex numbers, elliptic quaternions, and the basic algebraic properties of the matrices of these objects—are discussed in detail. Definitions and important features of these concepts are presented with the necessary details for understanding the subject. Elliptic quaternions, which are the focus of our study, are of great importance because of their properties related to mathematical transformations. Similar properties of these properties are also valid for elliptic complex numbers, and therefore, explaining the link between elliptic quaternions and elliptic complex numbers plays a critical role in our study. In this context, in our study, structure-preserving transformations from elliptic quaternions to elliptic complex numbers and their properties are explained in detail. In addition, the basic concepts and applications of color spaces, color images, and color image processing, which have an important place in the processing of color images, are also discussed. Color spaces are a mathematical concept used to describe colors in image processing, and different color spaces allow colors to be represented in different ways. In the processing of color images, different color channels (for example, RGB or YUV) in color spaces can be handled separately. In this section, basic techniques such as filtering, segmentation, and segmentation used in color image processing techniques are also mentioned. The third part of the study is the main part, where the original theoretical part of the study is included. In this section, the main research questions and hypotheses of the study are examined in detail to develop the theoretical framework. The original contribution and creative ideas of the work are presented in this section and are therefore critical to the originality and scientific value of the work. The theoretical part presented in this section provides a basis for the rest of the study and explains the methods used to arrive at the main conclusions and recommendations of the research. In this section, firstly, the roots of Nth-order monic polynomials with elliptic coefficients and valued elliptic numbers are obtained with the help of structure-preserving transformations from elliptic complex numbers to complex numbers. Then, by giving definitions about the eigenvalue-eigenvectors and singular value decomposition concepts of these elliptic complex matrices, the related theorems are stated and proved. As a result of singular value decomposition of elliptic complex matrices, pseudo-inverses of elliptic complex matrices and the least squares solution of the elliptic complex matrix equation are obtained. In the light of the obtained data, algorithms that find eigenvalue-eigenvectors, singular value decompositions, pseudo-inverses, and least squares solutions of elliptic complex matrices have been developed, and various problems have been solved with the help of these algorithms. In the fourth part of the study, the roots of valued Nth-order monic polynomials with elliptic quaternion coefficients are obtained with the help of structure-preserving transformations from the elliptic quaternions sentence to the elliptic complex numbers set. After this theory, eigenvalue-eigenvector, singular value decomposition of elliptic quaternion matrices, least squares method, pseudo-inverses, etc. Definitions, theorems, and algorithms for algebraic properties have been obtained. In addition, examples are given to support our results. The fifth section is the main section that includes the implementation part of the project. In this section, previously developed theoretical frameworks and methods are applied to solve real-world problems. This section details how the theoretical framework can be used for real-world applications, using examples and scenarios. In the chapter, different implementation scenarios are presented depending on the purpose and objectives of the project, and the methods used to realize these scenarios are explained in detail. This chapter shows how project work can offer solutions to real-world problems and explains the practical value of project work. Elliptic quaternions have one real and three imaginary parts. Each pixel of a color image expressed in RGB (red-green-blue) space contains three basic color components, namely red, green, and blue. Based on this information, each pixel of a color image is expressed as a pure imaginary (zero real component) elliptical quaternion. According to this representation, the red, green, and blue components of each pixel of the color images correspond to the components of the pure imaginary elliptic quaternions. Therefore, it is expressed as an elliptical quaternion matrix in the form of a color image with pixel resolution. Then, the method of reconstruction and compression of color images with the help of singular value decomposition that we developed for elliptic quaternion matrices is discussed. For this purpose, the visual quality, time, and memory complexity of the reconstructed color images using the compression method we developed were compared with those of other hypercomplex-based compression algorithms available in the literature, and a performance evaluation of the method we obtained was made. In the last chapter, the theoretical concepts presented in the previous chapters and the results of their applications are interpreted in detail. The results were obtained using the algorithms and methods described in the previous sections, and various problems were solved in line with them. In addition, based on the results obtained from the study, recommendations were presented, and it was emphasized that the study could be a potential resource for future research.