Local Radon transform and variable scale resolution with application to time-dependent wave radiation

Windowing techniques are applied to develop local projection transforms of two-dimensional functions (images). The two schemes considered here are termed local Radon transform (LRT) and multiscale projection transform. Emphasis is on the LRT, which is viewed as a windowed version of the Radon transform (RT); directional information is resolved via local projections in the former and via global line projections in the latter. The inverse LRT therefore synthesizes the image function in terms of its distribution in a phase space, comprised of the observation point coordinates and of the local spectral (directional) coordinate. The choice of the window in the LRT implies implicit variable-scale resolution of the image. The properties of the LRT are therefore contrasted with those of the multiscale projection transform, which is based on the recently developed wavelet (or multiscale) transform. Both the LRT and the multiscale projection transform have similar formal structure, but, in the latter, the local measures of directions are replaced by local measures of scale. Finally, the applications of these transforms to time-dependent wave propagation is demonstrated. Two phase-space representation schemes that are based on these transforms are considered. In both cases, the field is expressed in terms of space-time wave packets (pulsed beams (PBs)) that emanate from an initial plane where the time-dependent field is given. In the LRT-based representation, these wave packet propagators emerge in all directions, having essentially the same space-time scale. In the multiscale projection transform-based representation, however, they emerge in the same direction, but differ in the space-time confinement. General considerations are discussed regarding the applications of these new alternative representations.