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All Journal Desimal: Jurnal Matematika Tensor: Pure and Applied Mathematics Journal Journal of Computing Theories and Applications
Shiraj, Md. Morshed Bin
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Computer Science & IT Decision Sciences, Operations Research & Management Education Mathematics Social Sciences

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A Cubical Persistent Homology-Based Technique for Image Denoising with Topological Feature Preservation Al-Imran, Md.; Liza, Mst Zinia Afroz; Shiraj, Md. Morshed Bin; Murshed, Md. Masum; Akhter, Nasima
Journal of Computing Theories and Applications Vol. 2 No. 2 (2024): JCTA 2(2) 2024
Publisher : Universitas Dian Nuswantoro

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.62411/jcta.11488

Abstract

Image denoising is a fundamental challenge in image processing, where the objective is to remove noise while preserving critical image features. Traditional denoising methods, such as Wavelet, Total Variation (TV) minimization, and Non-Local Means (NLM), often struggle to maintain the topological integrity of image features, leading to the loss of essential structures. This study proposes a Cubical Persistent Homology-Based Technique (CPHBT) that leverages persistence barcodes to identify significant topological features and reduce noise. The method selects filtration levels that preserve important features like loops and connected components. Applied to digit images, our method demonstrates superior performance, achieving a Peak Signal-to-Noise Ratio (PSNR) of 46.88 and a Structural Similarity Index Measure (SSIM) of 0.99, outperforming TV (PSNR: 21.52, SSIM: 0.9812) and NLM (PSNR: 22.09, SSIM: 0.9822). These results confirm that cubical persistent homology offers an effective solution for image denoising by balancing noise reduction and preserving critical topological features, thus enhancing overall image quality.
Exploring the Lazy Witness Complex for Efficient Persistent Homology in Large-Scale Data Liza, Mst Zinia Afroz; Al-Imran, Md.; Shiraj, Md. Morshed Bin; Hossain, Tozam; Murshed, Md. Masum; Akhter, Nasima
Tensor: Pure and Applied Mathematics Journal Vol 5 No 2 (2024): Tensor: Pure and Applied Mathematics Journal
Publisher : Department of Mathematics, Faculty of Mathematics and Natural Sciences, Pattimura University, Ambon, Indonesia

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/tensorvol5iss2pp79-92

Abstract

In this paper, topological data analysis (TDA) techniques have been explored, with a focus on the selection of the Witness Complex and Persistent Homology of some nested families of Lazy Witness Complex as approximations for analyzing complex datasets. The Witness Complex was chosen for its efficiency and scalability, as it constructs a simplicial complex using landmark points, reducing computational load compared to methods like the Vietoris-Rips and Čech complexes. This makes it suitable for large, high-dimensional datasets, accurately representing the dataset's intrinsic geometry even with varying data densities. Persistent Homology was then reviewed with the aim of calculating it on some nested families of the Witness Complex. Subsequently, the nested families of the Lazy Witness Complex were introduced mathematically, with an example of the entire construction process for a well-known point cloud dataset. For this purpose, 50 points were generated randomly from a circle, and persistent diagrams of the point cloud data were analyzed to understand and compare the behavior among the approximations of the Witness Complex after choosing 10 landmarks using the Maxmin method. Since the families are nested, the filtration process became faster for each successive family, thus reducing computational complexity. For all three cases , the persistent barcodes indicated the same shape of the dataset. This study may help in choosing the suitable family of the Witness Complex over Persistent Homology to balance computational feasibility with topological accuracy, enabling efficient handling of large datasets while preserving important topological features. This approach allows for extracting meaningful insights from complex data while effectively managing computational resources.
A review of some properties of persistent homology Ullah, Md. Safik; Mou, Mst. Sima Akhter; Shiraj, Md. Morshed Bin; Rahman, Md. Mizanur; Murshed, Md. Masum; Akhter, Nasima
Desimal: Jurnal Matematika Vol. 6 No. 2 (2023): Desimal: Jurnal Matematika
Publisher : Universitas Islam Negeri Raden Intan Lampung

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.24042/djm.v6i2.17812

Abstract

Every day, enormous complex geometric data are accumulating rapidly, and qualitative analysis is needed, which cannot be done properly without studying the shapes of those data. Persistent homology describes the homology of data sets of arbitrary size, producing state-of-the art results in data analysis across a significant number of fields and sparking a rigorous study of persistence in homology theory. In this study, persistent homology has been demonstrated as a homology theory by satisfying the Eilenberg-Steenrod axioms. A brief background on persistent homology groups has been written to understand their construction. Then other definitions of persistent homology based on functors and graded modules have also been reviewed. The Mayer-Vietoris-Vietorisfor persistent homology has been derived as a property of persistent homology. Subsequently, a long, exact sequence for persistent homology has been constructed. Furthermore, the stability of persistent homology has been examined carefully. Finally, the Diamond principle of persistent homology has been explained briefly. This study can be used to investigate new properties of persistent homology, among other benefits.
Co-Authors Akhter, Nasima Al-Imran, Md. Hossain, Tozam Liza, Mst Zinia Afroz Md. Mizanur Rahman Mou, Mst. Sima Akhter Murshed, Md. Masum Ullah, Md. Safik