Windowed Radon transform frames

Two windowed Radon transform (WRT) frame formulations for the decomposition of band limited functions f(x)∈L2(Rℓ), ℓ=2,3, are presented. The “basic” formulation consists of two dual frame sets of shifted and rotated windows, one is used to synthesize f and the other to calculate the expansion coefficients as projections of f onto this set. The latter operation is a WRT that samples f at the discrete phase-space lattice of locations and directions. Explicit expressions are derived for a class of isodiffracting (ID) windows, which are matched to the lattice to yield snug frames. The basic formulation is then generalized to multiscales-WRT frames, where the large scales elements are associated with wider windows and sparser (rotation-direction) phase-space lattices that are decimated subsets of the lattice at the smallest scale. The analysis is presented for 3D, with a summary of the modifications for 2D. Finally, we discuss applications to time-dependent wave theory, whereby the source distribution is expanded using a WRT frame. The WRT extracts the local radiation properties of the source, thus describing the radiated field as a sum of collimated isodiffracting pulsed beams (ID-PB) that emerge from the source along the preferred radiation directions.