  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.].
