<p>This paper introduces the Biquaternion Windowed Linear Canonical Transform (BiQWLCT), a powerful new mathematical tool that brings the windowed linear canonical transform into the realm of biquaternions. Designed to handle the complexities of multidimensional data, BiQWLCT offers a robust framework for analyzing signals and images with intricate phase and polarization characteristics. The paper explores the essential properties of BiQWLCT, such as linearity, shift, parity, orthogonality, inversion, the Plancherel theorem, and the Heisenberg uncertainty principle, which highlights the inherent trade-offs in simultaneously localizing signals in both spatial and frequency domains. To connect theory with real-world applications, the paper includes an example that illustrates how BiQWLCT can be used to analyze signals or images, making its practical value clear. Beyond this, the paper discusses the potential applications of BiQWLCT across various fields, including time-frequency analysis, feature extraction, image enhancement, texture analysis, and compression. By blending theoretical insights with practical examples, the paper demonstrates how BiQWLCT has the potential to transform signal processing and image analysis, particularly when dealing with complex, multidimensional data in innovative and efficient ways.</p>

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Biquaternion Windowed Linear Canonical Transform and Associated Uncertainty Inequalities

  • Aijaz Ahmad Dar,
  • Owais Ahmad

摘要

This paper introduces the Biquaternion Windowed Linear Canonical Transform (BiQWLCT), a powerful new mathematical tool that brings the windowed linear canonical transform into the realm of biquaternions. Designed to handle the complexities of multidimensional data, BiQWLCT offers a robust framework for analyzing signals and images with intricate phase and polarization characteristics. The paper explores the essential properties of BiQWLCT, such as linearity, shift, parity, orthogonality, inversion, the Plancherel theorem, and the Heisenberg uncertainty principle, which highlights the inherent trade-offs in simultaneously localizing signals in both spatial and frequency domains. To connect theory with real-world applications, the paper includes an example that illustrates how BiQWLCT can be used to analyze signals or images, making its practical value clear. Beyond this, the paper discusses the potential applications of BiQWLCT across various fields, including time-frequency analysis, feature extraction, image enhancement, texture analysis, and compression. By blending theoretical insights with practical examples, the paper demonstrates how BiQWLCT has the potential to transform signal processing and image analysis, particularly when dealing with complex, multidimensional data in innovative and efficient ways.