Directional Uncertainty Principle for Quaternion Fourier Transform

Quaternion Fourier Transform on Quaternion Fields and Generalizations

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear ($GL$) transformation behavior of the QFT with matrices, Clifford geometric algebra and with examples. We finally arrive at wide-ranging non-commutative multivector FT generalizations of the QFT. Examples given are new volume-time and spacetime algebra Fourier transformations.

OPS-QFTs: A new type of quaternion Fourier transforms based on the orthogonal planes split with one or two general pure quaternions

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions $f,g$. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by $f,g$, or by only one pure unit quaternion $f$, comment on their geometric meaning, and establish inverse transformations. Keywords: Clifford geometric algebra, quaternion geometry, quaternion Fourier transform, inverse Fourier transform, orthogonal planes split

四元数傅里叶变换与四元数卷积的矩阵形式

On the Matrix Form of the Quaternion Fourier Transform and Quaternion Convolution

本研究探讨四元数傅里叶变换与卷积的矩阵形式，阐明其与复数域的关系，并利用理论结果提出限制四元数卷积神经网络Lipschitz常数的方法。

Quaternionic Fourier-Mellin Transform

In this contribution we generalize the classical Fourier Mellin transform [S. Dorrode and F. Ghorbel, Robust and efficient Fourier-Mellin transform approximations for gray-level image reconstruction and complete invariant description, Computer Vision and Image Understanding, 83(1) (2001), 57-78, DOI 10.1006/cviu.2001.0922.], which transforms functions $f$ representing, e.g., a gray level image defined over a compact set of $\mathbb{R}^2$. The quaternionic Fourier Mellin transform (QFMT) applies to functions $f: \mathbb{R}^2 \rightarrow \mathbb{H}$, for which $|f|$ is summable over $\mathbb{R}_…

The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

The two-sided quaternionic Fourier transformation (QFT) was introduced in \cite{Ell:1993} for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical investigations \cite{10.1007/s00006-007-0037-8, EH:DirUP_QFT} a special split of quaternions was introduced, then called $\pm$split. In the current \change{chapter} we analyze this split further, interpret it geometrically as \change{an} \emph{orthogonal 2D planes split} (OPS), and generalize it to a freely steerable split of $\H$ into two orthogonal 2D analysis planes. The new general form of the OPS split a…

Two-sided Clifford Fourier transform with two square roots of -1 in Cl(p,q)

We generalize quaternion and Clifford Fourier transforms to general two-sided Clifford Fourier transforms (CFT), and study their properties (from linearity to convolution). Two general \textit{multivector square roots} $\in \cl{p,q}$ \textit{of} -1 are used to split multivector signals, and to construct the left and right CFT kernel factors. Keywords: Clifford Fourier transform, Clifford algebra, signal processing, square roots of -1 .