Quaternionic Fourier-Mellin Transform

TL;DR

This paper introduces the quaternionic Fourier-Mellin transform (QFMT), extending the classical Fourier-Mellin transform to quaternion-valued functions, and explores its properties similar to the quaternionic Fourier transform.

Contribution

It generalizes the Fourier-Mellin transform to quaternion-valued functions and investigates its properties, expanding the mathematical tools for image analysis and signal processing.

Findings

QFMT applies to quaternion-valued functions with summable magnitude.

Properties of QFMT are analogous to those of the quaternionic Fourier transform.

The paper provides foundational analysis for potential applications in image processing.

Abstract

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 $\mathbb{R}_+^* \times \mathbb{S}^1$ under the measure $d\theta \frac{dr}{r}$. $\mathbb{R}_+^*$ is the multiplicative group of positive and non-zero real numbers. We investigate the properties of the QFMT similar to the investigation of the quaternionic Fourier Transform (QFT) in [E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Advances in Applied Clifford Algebras, 17(3) (2007), 497-517.; E. Hitzer, Directional Uncertainty Principle for Quaternion Fourier Transforms, Advances in Applied Clifford Algebras, 20(2) (2010), 271-284, online since 08 July 2009.].