Physical interpretation of the mellin transform variable?

Question

I shall keep this to the point: Given a time domain signal say microphone recording of a conversation:

1. Laplace tranfrom of x is some function X(s) say defined in the complex plane. I like to think of this variable s as a measure of frequency and damping inherrent in the signal x(t) (If anyone has a better interpretation of this I'd love to hear it).

2. Fourier transform of x is some function x(f) defined in a real domain f that splits the function x into it's constituent frequency components, like only listeneing to the drums in a jazz band.

3. Mellin transform? I cannot find any such explanation for the melling transform of a function?

Also it seems the Mellin tranform has limited applications in engineering, physics has always been ahead of engineers in adopting new techniques. Is there any reason for this or any new research?

Answer 1

• Physical interpretation: To develop a physical intuition, this article might be informative:

The power spectrum of the Mellin transformation with applications to scaling of physical quantities

The Mellin transform is used to diagonalize the dilation operator in a manner analogous to the use of the Fourier transform to diagonalize the translation operator. A power spectrum is also introduced for the Mellin transform which is analogous to that used for the Fourier transform. Unlike the case for the power spectrum of the Fourier transform where sharp peaks correspond to periodicities in translation, the peaks in the power spectrum of the Mellin transform correspond to periodicities in magnification.

For this reason the Mellin transform is also referred to as the "scale transform" (just as the Fourier transform is called a frequency transform). Just as the Fourier transform is insensitive (in absolute value) to a translation, the Mellin transform is insensitive (in absolute value) to a magnification.

• Applications in radar engineering: A classic application of the Mellin transform is in radar technology. Recall that the resolution in time of a signal is the reciprocal of the spread of its Fourier transform. In radar (or sonar) a signal is reflected from a moving target and you would like to accurately determine its velocity. The velocity resolution of the signal is the reciprocal of the spread of its Mellin transform. This application is worked out in:

Cramer Rao bound computation for velocity estimation in the broad-band case using the Mellin transform

• Applications in pattern recognition: The invariance properties of the Mellin and Fourier transforms can be combined in the socalled Fourier-Mellin transform to detect for rotation and scale invariant patterns. This combination is realized by a logarithmic mapping followed by a Fourier transformation, and it is believed that the visual cortex operates in this way.

A biologically plausible transform for visual recognition that is invariant to translation, scale, and rotation