Linear and Non-linear Image Processing Operations on Digital Projections

This paper outlines methods to perform basic spatial operations on 2D digital images that have been mapped into an equivalent set of digital projections. The digital projections are generated using the Discrete Radon Transform (DRT). Each digital projection is similar to an oriented straight-line integral of the Radon transform and the set of digital projections resembles the sinogram from classical tomography. Mapping discrete 2D data to a set of discrete 1D projections provides representational advantages and significant computational efficiency for some operations. The 2D distribution of image pixels viewed along each direction is expressed through one 1D digital projection. The integration of image values to form a directed projection smooths and hence compresses the variance of the original data. The DRT may, however, complicate the comparison of individual pixel values across a local area of the image, because it stores values that are global sums of local intensities. Pixel locations across a region of interest can be linked to those digital projections that contain contributions from those local values. Groups of pixels can be selected by particular translations of digital rays to form periodic lines. Algorithms for generalized linear convolution operations in the DRT domain are presented, as is an initial review of the effect of non-linear, rank-based operators applied to digital projections in the DRT space.