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Quaternion Fourier Transform on Quaternion Fields and Generalizations
Eckhard Hitzer
TL;DR
The paper addresses limited understanding of quaternionic Fourier transforms for quaternion fields and their application-relevant properties. It develops componentwise, Plancherel, geometric-algebra, and automorphism-based analyses, culminating in volume-time and spacetime multivector transforms. The main outcome is a systematic bridge between real-signal QFT results and quaternion-field results, together with broad Clifford-algebra generalizations.
Problem
The paper seeks deeper understanding of QFTs applied to quaternion fields, including useful properties, Plancherel theorems, and transformation behavior beyond real signals.
Method
The paper rewrites quaternion functions into real components, studies alternative QFT forms and GL(R2) automorphisms, and uses Clifford-algebra isomorphisms and coordinate-free formulations for generalization.
Results
The paper systematically reduces quaternion-field QFT computation to real-signal computations, generalizes real-signal results, and establishes volume-time and spacetime multivector Fourier transforms.
Takeaways & Limitations
The resulting non-commutative multivector transforms are presented as applicable to spatial data recorded with time, including fluid and gas flows, seismic analysis, and electromagnetic phenomena.
Takeaways & Limitations
For rotations, the expression for the transformed component f̂+ is generally not valid in higher dimensions.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper develops quaternionic Fourier-transform methods for quaternion fields, extending beyond real signals toward application-relevant properties, automorphisms, and multivector generalizations.
- The paper studies the QFT for quaternion fields f: R2 → H, rather than only real signals, with applications to PDEs, image processing, and numerical implementations.
- It examines how different QFT forms yield scalar- and quaternion-valued Plancherel theorems.
- The quaternion-field computation is systematically reduced to four real-component QFT computations, while real-signal results can be generalized back to quaternion fields.
- The QFT’s behavior under GL(R2) automorphisms is studied using matrix formulations, Clifford geometric algebra, and examples involving stretches, reflections, and rotations.
- The paper introduces non-commutative multivector Fourier-transform generalizations on Rm,n-valued Clifford-algebra functions, including volume-time and spacetime algebra examples.
- Quaternions are conveniently rewritten using k = ij, keeping i on the left and j on the right of terms.
2. The quaternion Fourier transform
The paper defines the QFT for quaternion-valued functions, establishes its principal structural and Plancherel properties, and shows how quaternion computations connect systematically to real-component transforms.
- The QFT is defined for f ∈ L2(R2; H) using left and right quaternion exponential factors, with an inverse transform also provided.
- Quaternion-valued functions are decomposed into four real component functions, enabling QFT calculations to be expressed through real-signal computations.
- Theorem 2.2 establishes a Plancherel identity for the scalar product of quaternion module functions and their QFTs.
- The QFT Parseval theorem relates the L2(R2; H)-norm of a quaternion function to the norm of its QFT, multiplied by 1/(2π).
- Theorem 2.5 reduces arbitrary quaternion-function transforms to four real-function QFTs and transfers real-signal theorems componentwise to quaternion functions.
- The QFT of quaternion-valued functions obeys the generalized GL(R2) transformation law established for real signals.
3. The right side quaternion Fourier transform (QFTr)
The QFTr modifies the placement of quaternionic exponential factors so fully quaternion-valued functions admit a quaternion-valued Plancherel theorem based on the original inner product.
- Motivation: The original quaternion-valued inner product lacks the cyclic symmetry needed for a general QFT Plancherel theorem.The paper addresses this either by modifying the inner-product symmetry or by modifying the transform.
- Definition: The right side QFT (QFTr) shifts the j-exponential factor to the left of the quaternion function.Its inverse uses the exponential factors in reversed order relative to the transform definition.
- Properties: For general quaternion-valued functions, QFTr retains left linearity and dilation properties, while x-shift, derivative, and monomial-power rules require modification.The left-linearity coefficients may be fully quaternionic constants.
- Limitations: A modulation property analogous to the standard table does not hold because non-commuting exponential factors obstruct it.Some power identities also require shifting both factors to the left, depending on the function and transform form.
- Plancherel theorem: The QFTr establishes a quaternion-valued Plancherel theorem for the original inner product of arbitrary quaternion module functions.The corresponding Parseval theorem follows by setting the two functions equal, yielding an L2 norm identity.
4. Understanding the GL(R2) transformation properties of the QFT
The paper gives the QFT’s GL(R2) behavior a geometric-algebra interpretation by splitting the transform into two components and analyzing stretches, reflections, and rotations.
- Geometric formulation: The matrix transformation law for quaternion-valued signals is recast geometrically using invariant Clifford geometric algebra techniques.The geometric formulation clarifies the four terms and matrices appearing in the matrix law.
- Component split: The quaternion QFT is split into f− and f+ components whose transforms have complex forms, with the f+ component involving a reflection.The split operation commutes with the QFT operation.
- Automorphisms: Every GL(R2) automorphism is decomposed into rotations and symmetric transformations, with positive eigenvalues giving stretches and negative eigenvalues combining reflections with stretches.Stretches and reflections generate the elementary transformations, while rotations correspond to two reflections.
- Coordinate-free form: The coordinate-free formulation rewrites the exponential arguments as scalar products involving the identity or a reflection operator.This explains why the f+ transform carries the reflected frequency argument.
- Transformation theorem: The GL(R2) transformation theorem applies an automorphism to the spatial argument while transforming frequency arguments through the inverse adjoint and an absolute determinant factor.The determinant factor arises from exchanging the order of integration.
- Generalizations: The geometric interpretation motivates non-commutative Fourier transforms on Rm,n-valued domains and demonstrates extensions to volume-time and spacetime algebra.The spacetime example targets functions from R3,1 to the Clifford algebra Cl3,1, with a volume-time subalgebra as an intermediate step.
5. Generalization of the QFT to a new spacetime algebra Fourier transform
Quaternion–Clifford isomorphisms and geometric splitting extend the QFT from quaternion fields to volume-time and full spacetime-algebra Fourier transforms, with corresponding GL transformation laws.
- Generalization framework: Quaternion-to-Clifford subalgebra isomorphisms provide the route from quaternion-valued QFTs to higher-dimensional multivector Fourier transforms.The construction uses H ∼= Cl(0,2) and related Clifford subalgebras.
- Full spacetime algebra functions: The resulting spacetime Fourier transform extends the QFT to L2(R3,1; Cl3,1), while analogous transformations remain possible in other dimensions and Clifford algebras.The paper presents this as one example of a broader family of generally non-commutative multivector Fourier transforms.
- Volume-time Fourier transform: The volume-time Fourier transform acts on functions from R3,1 to the volume-time subalgebra Vt and has an inverse transform.Its integration uses the spacetime volume d4x = dtdxdydz.
- Volume-time Fourier transform: The f± split yields a volume-time transform corresponding to the geometric QFT decomposition, with kernels reflecting the physical spacetime split.The f+ component uses the flat Minkowski metric ts − x⃗·u⃗ in its exponent.
- GL transformation properties: The volume-time transform preserves the QFT’s GL transformation structure, including a determinant factor that can be omitted for proper Lorentz transformations.For proper Lorentz transformations, |det A| = 1.
- Full spacetime algebra functions: Right linearity decomposes general Cl3,1-valued functions into four Vt-valued functions, enabling the spacetime Fourier transform and its invertibility.The decomposition represents all 16 coefficient functions through four volume-time module functions.
6. Conclusions
The paper uses quaternion splitting, coordinate-free geometric methods, and Clifford isomorphisms to analyze QFT properties and construct volume-time and spacetime Fourier transforms.
- Conclusions: Quaternion rewriting and splitting support analysis of QFT properties, including behavior under general linear automorphisms.The split is closely related to the choice of time direction in spacetime applications.
- Conclusions: Coordinate-free formulations combined with quaternion-to-Clifford isomorphisms produce non-commutative multivector Fourier transforms on Rm,n and Clifford algebras Clm,n.The paper demonstrates this approach with volume-time and spacetime Fourier transforms.
- Conclusions: The proposed volume-time and spacetime transforms are presented as awaiting applications including dynamic flows, seismic analysis, and electromagnetic phenomena.The stated application scope is wherever spatial data are recorded with time.