Báo cáo hóa học: Research Article A Fast Mellin and Scale Transform Antonio De Sena1 and Davide Rocchesso2

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Báo cáo hóa học: " Research Article A Fast Mellin and Scale Transform Antonio De Sena1 and Davide Rocchesso2"Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2007, Article ID 89170, 9 pagesdoi:10.1155/2007/89170Research ArticleA Fast Mellin and Scale Transform Antonio De Sena1 and Davide Rocchesso2 1 Dipartimento di Informatica, Universit` di Verona, Strada Le Grazie, 15-37134 Verona, Italy a 2 Dipartimento di Arti e Disegno Industriale, Universit` Iuav di Venezia, Dorsoduro 2206, 30123 Venezia, Italy a 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 approx- imation 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 variance to shift, scale, and rotation. In [3], various tech- niques have been presented for the implementation of theThe Mellin transform, and the particular version called scale Fourier-Mellin transform, including a polar-log coordinatestransform, can represent a signal in terms of scale. The scale remapping. In [4], the problem of estimation of scale andcan be interpreted, similarly to frequency, as a physical at- orientation differences between objects in images has beentribute of signals. The proposed fast (subquadratic) imple- approached using the analytical Fourier-Mellin transformmentation allows this transform to be used in practical ap- [3].plications. The algorithm can compute the Mellin transform Other approaches to the Mellin transform implementa- tion have been taken by J. Bertrand et al. [5–7]. In their ∞ f (t )t p−1 dt , M f ( p) = (1) studies, the authors tackled the transform in the frequency 0 domain by considering analytic signals. An implementationin the complex variable p = − jc + β, with β ∈ R fixed pa- based on exponential resampling in the time domain shouldrameter and c ∈ R independent variable. We call this family give a solution to a few practical problems. Namely, a startingof transforms the β-Mellin transform. It is indeed a restric- point near 0 implies an impossible exponential resampling, and if the signal support in time is very small compared totion of the Mellin transform, as the real part of the complexvariable p is parameterized. The β parameter allows to se- the starting point of the signal, the exponential samplinglect among: (i) a scale-invariant transform (β = 1/ 2, scale becomes a quasiuniform sampling. An implementation that follows this idea has been made by Goncalv´ s and Lemoine ¸etransform); (ii) a compression/expansion invariant trans-form (β = 0); (iii) a shape-invariant transform (β = −1, the (http://gdr-isis.org/tftb/refguide/node32.html), but the algo- rithm appears to be quadratic in complexity. The authorsratio between the maximum of the function and its extension have searched for other implementatio ...