Region-of-interest tomography using the wavelet transform and angular harmonics

Some authors have shown that wavelets can be used to localize the Radon transform, i.e., to reconstruct a small region of interest (ROI) from projections densely sampled in the ROI and sparsely sampled outside it. A different approach using Fourier expansions of the image and its projections shows that this localization is a property of the Radon transform itself, not just the wavelet representation of it, provided that a wavelet-reminiscent exponential sampling scheme is used. This suggests that the Radon transform naturally lends itself to a multiresolution representation. We present a new procedure for reconstructing the wavelet transforms of the angular harmonics of an image from the wavelet transforms of the angular harmonics of its projections. The relation is a convolution over scale, rather than translation, and preservation of compact support of the wavelet basis functions since a derivative-Hilbert-transform is not required. It generalizes a procedure using exponential radial sampling, and decouples over angular harmonics