Mellin transform

Summary

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a function f is :\left{\mathcal{M}f\right}(s) = \varphi(s)=\int_0^\infty x^{s-1} f(x) , dx. The inverse transform is :\left{\mathcal{M}^{-1}\varphi\right}(x) = f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} \varphi(s), ds. The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Me

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Nonperturbative Mellin Amplitudes

João Pedro Alves Da Silva

Conformal field theory lies at the heart of two central topics in theoretical high energy physics: the study of quantum gravity and the mapping of quantum field theories through the renormalization group. In this thesis we explore a technique to study conformal field theories called Mellin amplitudes, which are essentially the Mellin transforms of conformal correlation functions. The thesis is divided into two parts. In the first part we study the fundamental properties of Mellin amplitudes. We clarify the conditions for the existence of Mellin amplitudes nonperturbatively. We formulate a conjecture for the analytic properties of Mellin amplitudes and partially prove it. Finally, we discuss Polyakov conditions, which are nonperturbative zeros of Mellin amplitudes. In the second part of the thesis we consider applications of Mellin amplitudes. We apply Mellin amplitudes to: the Wilson-Fisher model in 4-epsilon dimensions, three dimensional conformal field theories with slightly broken higher spin symmetry, two dimensional minimal models and loop diagrams in Anti-de-Sitter space. Our main result is the derivation of nonperturbative sum rules that constrain effective field theories in Anti-de-Sitter space.

EPFL2021

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In this paper we propose a low complexity method for Rotation, Scale and Translation (RST) invariant content-based image retrieval, suitable for a handheld image recognition device. The RST compensation method is based on Fourier-Mellin Transform (FMT) which we implement efficiently using log-polar grid interpolation. This RST compensation method is used in conjunction with an image recognition algorithm based on Discrete Cosine Transform (DCT) phase matching. A pre-selection algorithm is also added for decreasing the complexity. This algorithm is based on color proportions within concentric circular zones encompassing the edge pixels. The resulting RST invariant image recognition system was tested on 1500 pictograms and 1000 pictures with different RST conditions, showing an average recognition accuracy of 95.2% for pictograms and 96.9% for pictures.

EURASIP2011