The Mellin Transform


(C) C. Valens, 1999-2003

 

 

Mathematics

Note

This text was ripped from Eric W. Weisstein's Treasure Troves

 

Definition

$\displaystyle \phi(z)$ = $\displaystyle \int^\infty_0 t^{z-1}f(t)\,dt$  
f(t) = $\displaystyle {1\over 2\pi i} \int_{c-i\infty}^{c+i\infty} t^{-z}\phi(z)\,dz.$  

The transform $\phi(z)$ exists if the integral

\begin{displaymath} \int_0^\infty \vert f(x)\vert x^{k-1}\,dx \end{displaymath}

is bounded for some k>0, in which case the inverse f(t) exists with c>k. The functions $\phi(z)$ and f(t) are called a Mellin transform pair, and either can be computed if the other is known. See also Fourier Transform, Integral Transform, Strassen Formulas

 

References

Arfken, G.
Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, p. 795, 1985.

Bracewell, R.
The Fourier Transform and Its Applications.
New York: McGraw-Hill, pp. 254-257, 1965.

Gradshteyn, I. S. and Ryzhik, I. M.
Mellin Transform.
§17.41 in Tables of Integrals, Series, and Products, 5th ed.
San Diego, CA: Academic Press, pp. 1193-1197, 1979.

Morse, P. M. and Feshbach, H.
Methods of Theoretical Physics, Part I.
New York: McGraw-Hill, pp. 469-471, 1953.

Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I.
Evaluation of Integrals and the Mellin Transform.
Itogi Nauki i Tekhniki, Seriya Matemat. Analiz 27, 3-146, 1989.

Zwillinger, D. (Ed.).
CRC Standard Mathematical Tables and Formulae
Boca Raton, FL: CRC Press, p. 567, 1995.

End of ripped text

 

 

From Usenet

Original posters are unknown.

 

joseph7971@my-deja.com writes:

> Hi,
> Does anyone know a good reference to see some examples of how the
> scale/mellin transform works or is implemented. I'm looking at it at
> the moment but I am finding that side of it confusing. all advice is
> appreciated

Bracewell's book ("The Fourier Transform and its Applications") has a brief section on the Mellin transform. For a causal signal, the MT of a signal and its scaled counterpart differ by just a phase factor, analogous to the "shift <-> phase" property of the Fourier transform. The connection is closer when one realizes that the MT of a signal can be computed by sampling it at exponentially spaced intervals and then computing its Fourier transform. In MATLAB, plot a function, its scaled version, and an exponentially sampled version of the scaled version. The last one should look like just like the first one except for a time shift.

Some references that I had collected in the last millenium:

Robbins, G.M. and Huang, T.S.
Inverse Filtering for Linear Shift-Variant Imaging Systems,
Proc. IEEE, Jul. 1972, pp. 862-872.

Brill, M. and West, G.
A Fast Mellin Transform,
J. of Information Processing and Cybernetics, Vol. 20, 1984, pp. 229-233.

Zwicke, P.E. and Kiss Jr., I.
A New Implementation of the Mellin Transform and its Application to Radar Classification of Ships,
IEEE Trans. PAMI, Vol. 5, Mar. 1983, pp. 191-199.

Moses, H.E. and Quesada, A.F.
The power spectrum of the Mellin Transform with Applications to scaling of Physical Quantities,
J. Math. Phys., Vol 15, No. 6, Jun. 1974, pp. 748-752.

Altes, R.A.
The Fourier-Mellin transform and mammalian hearing,
J. Acoust. Soc. Am., Vol. 63, Jan. 1978, pp. 174-183.

Then there is

Fritz Oberhettinger
Tables of Mellin transforms
(unfortunately out-of-print).

Sheng, Y.
Fourier-Mellin spatial filters for invariant pattern recognition
Optical Engineering, Vol 28, No 5, May 1989, pages 494-500.

Sheng, Y. and Duvernoy, J.
Circular-Fourier-radial-Mellin transform descriptors for pattern recognition
Journal Optical Society America A, Vol 3, No 6, June 1986, pages 885-888.

Sheng, Y. and L.Shen, L.
Orthogonal Fourier-Mellin moments for invariant pattern recognition
JOSA, Vol 11, No 6, June 1994, pages 1748-1757.

If you do a technical journals literature search on authors: David Casasent and/or Dmitri Psaltis, a number of papers should pop up on theory and use of Mellin transforms.

 

 

Links

Eric W. Weisstein's Treasure Troves
Philippe Flajolet's Research Papers