Algorithms for an Adaptive Discrete Radon Transform

From the discrete radon transform (DRT), R(t, m) of a p x p image I(x, y), intensity at (t, m) results from the sum of intensities, I of elements, (x, y) along a projection of x = my + t (mod p), the resultant transform of the image presented has an averaging effect. If information is required for a subset of the image, it cannot be extracted from transform of the complete image. It is therefore useful to perform DRT operations on a p′ x p′ subset, I ′ of the image to preserve the information of interest. If it is then required to shift the transform of this subset around the image to locate a specific projection of interest, it is inefficient to recalculate the entire DRT for any translation of less than p′ as much of the information would already be represented in the current transform. An algorithm to allow the transform to be updated adaptively and an investigation of the efficiency of this method is presented. An expanded representation of Radon Space, R(k, θ) based on the work by Svalbe in [4] is presented. This space expands the (t, m) transform by removing the modulo arithmetic thus isolating all projections separated by a distance γp where γ is an integer. Thus the (k, θ) transform has no periodic nature as in R(t, m). Whilst reducing this transforms advantage in adaptive translation, it does cause it to be conducive to scaling the transform of a p′ subset up or down to the transform of a subset of size p′′ adaptively. Algorithms are presented to perform translation and both scaling up and down adaptively and the efficiency of each method is investigated.