The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose three different forms of uncertainty principles for the two-sided QLCT, which include Hardy’s uncertainty principle, Beurling’s uncertainty principle and Donoho–Stark’s uncertainty principle. These consequences actually describe the quantitative relationships of the quaternion-valued signal in arbitrary two different QLCT domains, which have many applications in signal recovery and color image analysis. In addition, in order to analyze the non-stationary signal and time-varying system, we present Lieb’s uncertainty principle for the two-sided short-time quaternion linear canonical transform (SQLCT) based on the Hausdorff–Young inequality. By adding the nonzero quaternion-valued window function, the two-sided SQLCT has a great significant application in the study of signal local frequency spectrum. Finally, we also give a lower bound for the essential support of the two-sided SQLCT.

中文翻译：

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两侧四元数线性正则变换的不确定性原理

四元数线性正则变换 (QLCT) 作为四元数傅立叶变换的一种广义形式，是图像和信号处理中的强大分析工具。在本文中，我们提出了三种不同形式的双面 QLCT 不确定性原理，包括 Hardy 不确定性原理、Beurling 不确定性原理和 Donoho-Stark 不确定性原理。这些结果实际上描述了任意两个不同 QLCT 域中四元数值信号的定量关系，这在信号恢复和彩色图像分析中有许多应用。此外，为了分析非平稳信号和时变系统，我们提出了基于 Hausdorff-Young 不等式的双边短时四元数线性正则变换 (SQLCT) 的 Lieb 不确定原理。通过加入非零四元数窗函数，双边SQLCT在信号局部频谱的研究中具有重要意义。最后，我们还给出了双边 SQLCT 的基本支持的下限。