The “twisted Mellin transform” is a slightly modified version of the usual classical Mellin transform on L([0,∞)). In this short note we investigate some of its basic properties. From the point of view of combinatorics one of its most interesting properties is that it intertwines the differential operator, df/dx, with its finite difference analogue, ∇f = f(x)−f(x−1). From the point of view of a...

Many digital audio effects rely on transformations performed in the Fourier-transformed (frequency) domain. However, other transforms and domains exist and could be exploited. We propose to use the Mellin transform for a class of sound transformations. We present a fast implementation of the Mellin transform (more precisely a Fast Scale Transform), and we provide some examples on how it could b...

In this contribution we generalize the classical Fourier Mellin transform [3], which transforms functions f representing, e.g., a gray level image defined over a compact set of R. The quaternionic Fourier Mellin transform (QFMT) applies to functions f : R → H, for which |f | is summable over R+ × S under the measure dθ dr r . R ∗ + is the multiplicative group of positive and non-zero real numbe...

We introduce a forward model for the cone-beam X-ray Computed Tomography projection data measured in native geometries as a Fourier Integral Operator and present a corresponding filtered-backprojection type inversion formula. Our model and inversion formula can accommodate arbitrary source trajectories, arbitrary detector plane orientation, detector surface geometries, and other system related ...

Given a rather general weight function n0, we derive a new cone beam transform inversion formula. The derivation is explicitly based on Grangeat’s formula (1990) and the classical 3D Radon transform inversion. The new formula is theoretically exact and is represented by a 2D integral. We show that if the source trajectory C is complete in the sense of Tuy (1983) (and satisfies two other very mi...

Recently, a 'gammachirp' function has been derived as an optimal auditory filter function in terms of minimal uncertainty in a joint time and modified-scale representation if the scale transform defined by Cohen is used in the auditory system. The gammatone function, which is widely used as the impulse response of a linear auditory filter, is a first-order approximation of the 'gammachirp' func...

We develop the technique of inverse Mellin transform for processes occurring in a background magnetic field. We show by analyticity that the energy (momentum) derivatives of a field theory amplitude at the zero energy (momentum) is equal to the Mellin transform of the absorptive part of the amplitude. By inverting the transform, the absorptive part of the amplitude can be easily calculated. We ...