The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the Lévy stable distributions, and more generally the probability distributions governed by generalized diffusion equations of fractional order in space and/or in time. In these cases the related stochast...

Combined use of the X-ray (Radon) transform and the wavelet transform has proved to be useful in application areas such as diagnostic medicine and seismology. In the present paper, the wavelet X-ray transform is introduced. This transform performs one-dimensional wavelet transforms along lines in R n , which are parameterized in the same fashion as for the X-ray transform. It is shown that the ...

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-M...

We study a new Radon-like transform that averages projected pforms in Rn over affine (n − k)-spaces. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in the work by Gelfand, Graev and Shapiro (1969). Moreover, if it can be extended to a somewhat larger space of p-forms, our inversion formula will al...

We employ the Mellin transform to facilitate the bit error ratio (BER) analysis of a fast frequency hopping (FFH)-assisted, M -ary frequency-shift keying (MFSK) using product combining (PC) when the transmitted signal is subjected to both Rayleigh fading and partial-band noise jamming. Exploiting the fact that the Mellin transform of the product of independent random variables is the product of...

In two dimensions, the exponential X-ray transform has been wellstudied due to its applications of correcting attenuation effects in Single Photon Emission Computed Tomography (SPECT). Explicit inversion formulas have been known for over 15 years. The threedimensional (3D) case has not been as thoroughly examined, and inversion formulas are available for only a few of the wide range of possible...

In this paper, we develop some important Fourier analysis tools in the context of time scales. In particular, we present a generalized Fourier transform in this context as well as a critical inversion result. This leads directly to a convolution for signals on two (possibly distinct) time scales as well as several natural classes of time scales which arise in this setting: dilated, closed under...

A pattern descriptor invariant to rotation, scaling, translation (RST), and robust to additive noise is proposed by using the Radon, Fourier, and Mellin transforms. The Radon transform converts the RST transformations applied on a pattern into transformations in the radial and angular slices of the Radon transform data. These beneficial properties of the Radon transform make it an useful interm...

We explore the possibility of using the method of classical integral transforms to solve a class of q-difference-differential equations. The Laplace and the Mellin transform of q-derivatives are derived. The results show that the Mellin transform of the qderivative resembles most closely the corresponding expression in classical analysis, and it could therefore be useful in solving certain q-di...

In this article, an image feature extraction method based on two-dimensional (2D) Mellin cepstrum is introduced. The concept of one-dimensional (1D) mel-cepstrum that is widely used in speech recognition is extended to two-dimensions using both the ordinary 2D Fourier transform and the Mellin transform. The resultant feature matrices are applied to two different classifiers such as common matri...