Tomograms and other transforms: A unified view (Q2774586)

The authors touch a burning topic in integral transforms viz. unification of treatment of Fourier transform, wavelet transform (both linear transforms), and Wigner-Ville quasidistributions and tomograms. These transforms are used for signal processing in physics, engineering, medicine etc.NEWLINENEWLINENEWLINEExplicit formulae relating the three types of transforms (viz. wavelet-like, quasidistribution and tomographic transforms) are tried to be obtained. Special emphasis is given to the properties of the scale-time and scale-frequency tomograms so important in applications.NEWLINENEWLINENEWLINETomograms are stated to be the class of positive bilinear transforms referring to noncommutative tomography, which in addition to the time-frequency domain also applies to other noncommutative pairs like time-scale, frequency-scale etc.NEWLINENEWLINENEWLINEFor generalization of Fourier transform in yet another dimension the reader is referred to \textit{J. M. C. Joshi} [J. Nat. Phys. Sci. 11, 65-78 (1997; Zbl 0976.44001)], where S. M. Joshi's generalized Fourier, Laplace, Stieltjes and Hankel transforms and their extensions in the space of distributions are introduced.