# Quaternion Fourier Transform on Quaternion Fields and Generalizations

arXiv: https://arxiv.org/abs/1306.1023 · HTML: https://mlsanity.com/paper/1306.1023/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2013.

## Built on by (most cited first, 3 of 3)

- [Novel computational workflows for natural and biomedical image processing based on hypercomplex algebras](https://mlsanity.com/paper/2502.07758/) (2025)
- [Quaternion Optimized Model with Sparse Regularization for Color Image Recovery](https://mlsanity.com/paper/2204.08629/) (2022)
- [A Robust Lane Detection Associated with Quaternion Hardy Filter](https://mlsanity.com/paper/2108.04356/) (2021)

## Abstract (arXiv)

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.