A Fast Mellin and Scale Transform
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Research Article A Fast Mellin and Scale Transform Antonio De Sena1 and Davide Rocchesso2 1 Dipartimento 2 Dipartimento
di Informatica, Universit`a di Verona, Strada Le Grazie, 15-37134 Verona, Italy di Arti e Disegno Industriale, Universit`a Iuav di Venezia, Dorsoduro 2206, 30123 Venezia, Italy
Received 24 August 2006; Revised 30 December 2006; Accepted 5 March 2007 Recommended by Jar-Ferr Kevin Yang A fast algorithm for the discrete-scale (and β-Mellin) transform is proposed. It performs a discrete-time discrete-scale approximation of the continuous-time transform, with subquadratic asymptotic complexity. The algorithm is based on a well-known relation between the Mellin and Fourier transforms, and it is practical and accurate. The paper gives some theoretical background on the Mellin, β-Mellin, and scale transforms. Then the algorithm is presented and analyzed in terms of computational complexity and precision. The effects of different interpolation procedures used in the algorithm are discussed. Copyright © 2007 A. De Sena and D. Rocchesso. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1.
INTRODUCTION
The Mellin transform, and the particular version called scale transform, can represent a signal in terms of scale. The scale can be interpreted, similarly to frequency, as a physical attribute of signals. The proposed fast (subquadratic) implementation allows this transform to be used in practical applications. The algorithm can compute the Mellin transform M f (p) =
∞ 0
f (t)t p−1 dt,
(1)
in the complex variable p = − jc + β, with β ∈ R fixed parameter and c ∈ R independent variable. We call this family of transforms the β-Mellin transform. It is indeed a restriction of the Mellin transform, as the real part of the complex variable p is parameterized. The β parameter allows to select among: (i) a scale-invariant transform (β = 1/2, scale transform); (ii) a compression/expansion invariant transform (β = 0); (iii) a shape-invariant transform (β = −1, the ratio between the maximum of the function and its extension is a constant). The proposed algorithm is based on the well-known relation between the Mellin and Fourier transforms. While methods that exploit such relation have been proposed long ago [1, 2], efficiency and practicality are still remarkable objectives to be achieved. Mellin and scale transforms are important in vision and image processing. In particular, a so-called Fourier-Mellin transform can be used for pattern recognition for its in-
variance to shift, scale, and rotation. In [3], various techniques have been presented for the implementation of the Fourier-Mellin transform, including a polar-log coordinates remapping. In [4], the problem of estimation of scale and orientation differences between objects in images has been approached using the analytical Fourier-Mellin transform [3]. Other approaches to the Mellin transform impl
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