对偶四元数傅里叶变换
Dual-Quaternion Fourier Transform
AI总结：
本研究提出对偶四元数傅里叶变换（DQFT）及其逆变换（IDQFT），将傅里叶变换扩展到对偶四元数域，为多分量数据的分析、处理等提供新的数学工具，可用于求解相关偏微分或函数方程。
摘要：
傅里叶变换（FT）在从增强、复原、分析到安全、压缩和处理的广泛应用中发挥着关键作用。傅里叶变换（FT）是将函数转换为描述频率的形式的过程，该过程已被扩展到许多域和数值表示（包括四元数）。然而，在本文中，我们提出了一种使用对偶四元数的新方法，因为对偶四元数提供了具有独特数学性质的高效紧凑的符号形式。尽管对偶四元数作为一种高效的数学模型，在科学和计算的许多领域中已确立了自身地位，可提供明确、简便、计算高效的多分量数据表示方式，但将其与傅里叶过程相结合的研究还很少。对偶四元数简单来说就是对偶数理论与超复数的统一；这一数学概念允许多变量数据集以统一的非线性方式进行变换、组合、处理和插值。我们定义了适用于对偶四元数域上对偶四元数值数据的对偶四元数傅里叶变换（DQFT），这为新型分析、处理和滤波技术打开了大门。我们还提出了逆对偶四元数傅里叶变换（IDQFT）。DQFT释放了高维超复数代数所提供的潜力，可用于求解对偶四元数偏微分方程或函数方程（例如用于多分量数据分析）。
英文摘要：
Fourier transform (FT) plays a crucial role in a broad range of applications, from enhancement, restoration and analysis through to security, compression and manipulation. The Fourier transform (FT) is a process that converts a function into a form that describes the frequencies. This process has been extended to many domains and numerical representations (including quaternions). However, in this article, we present a new approach using dual-quaternions. As dual-quaternions offer an efficient and compact symbolic form with unique mathematical properties. While dual-quaternions have established themselves in many fields of science and computing as an efficient mathematical model for providing an unambiguous, un-cumbersome, computationally effective means of representing multi-component data, not much research has been done to combine them with Fourier processes. Dual-quaternions are simply the unification of dual-number theory with hypercomplex numbers; a mathematical concept that allows multi-variable data sets to be transformed, combined, manipulated and interpolated in a unified non-linear manner. We define a Dual-Quaternion Fourier transform (DQFT) for dual-quaternion valued data over dual-quaternion domains. This opens the door to new types of analysis, manipulation and filtering techniques. We also present the Inverse Dual-Quaternion Fourier Transform (IDQFT). The DQFT unlocks the potential provided by hypercomplex algebra in higher dimensions useful for solving dual-quaternion partial differential equations or functional equations (e.g., for multicomponent data analysis)
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