Image Representation Via a Finite Radon Transform

{ This paper presents a model of nite Radon transforms composed of Radon projections. The model generalizes to nite groups projections in the classical Radon transform theory. The Radon projector averages a function on a group over cosets of a subgroup. Reconstruction formulae formally similar to the convolved backprojection ones are derived and an iterative reconstruction technique is found to converge after nite number of steps. Applying these results to the group Z 2 p , new computationally favourable image representations have been obtained. A numerical study of the transform coding aspects is attached. I. INTRODUCTION Development of various representations of image data continues to be an area of active research. It has had substantial impact on many image processing and image analysis problems including manipulation, compression, pattern recognition, coding, computer vision , etc. The most common manner of image representation is to use an energy preserving transform. Classical and thoroughly investigated examples include Fourier, cosine, sine, Hadamard, Haar and other unitary transforms 15], 5]. The Radon transform entered the center of interest with its rst radioastronomic and tomographic applications 8] and spread quickly into many elds. Recall that the Radon transform (this notion is from integral geometry, cf. 7], 11], 18]) of a real function f deened on Euclidean plane is the function ^ f, deened on the family of all lines in the plane, having the value ^ f(p) equal to the line integral along the line p. It is often advantageous to view the Radon transform as the family (^ f q) of projections where the projection ^ f q is the restriction of ^ f on the set of all lines parallel to the line q containing the origin.