Pixel-driven computation of parallel and fan-beam projections of a digital image based on pixel-representation using a new formula. (April 2022) - Galigekere, Ramesh R.
Highlights: A new formula for the Radon transform of a square pixel (simpler than that given by SR Deans) has been derived. The formula allows pixel-driven computation of Riemann-sum approximation to the Radon transform i.e., the weighted sum of pixel-values, the weights being the exact lengths of intersection of the ray with the pixels along the ray-path, respectively. This is unlike the current pixel-driven methods, which involvethe use of arbitrary kernels in the projection-domain. The method has been extended to compute divergent- (fan-) beam projections. The parallel-beam projections can replace the Hough transform (in the sense that it forms a generalized version of the Hough transform) – in many computer vision applications. Pixel driven methods have the advantages of the ability to exploit region-of-interest computing, thus reducing the computation significantly in such situations. The method is explained with suitable diagrams in such a way, that anyone with some programming experience can implement it – reducing one's dependence on commercial software. Abstract: Background: The computation of the projections of a digital image – modelled as a superposition of square pixels – is essential in several algorithms in computed tomography, and also in many machine vision applications. Projections of digital images are computed through the ray-driven approach. Current pixel-driven methods, though simpler, involve interpolation kernels in the projection-domain – not based on the exact Radon transform (RT) of a square. Methods: A new analytical formula – for the line-integral of the unit pixel – simpler than that published previously, is derived. The formula allows easy, pixel-driven computation of the RT of a digital image based on the pixel model i.e., Riemann-sum approximation to the line integral. The method naturally allows pixel-driven backprojection, based on the same (pixel) model. The approach is extended to computing projections over divergent (fan-) beams, and its application as a generalized version of the traditional Hough transform, is discussed. Results: The Radon transform of the unit-pixel match that of a digital square image. The RT, of a mathematical phantom consisting of a superposition of elliptical disks, compares well with that based on analytical formula. A comparative study with the pixel driven approach with interpolation in the projection-domain, and its important variant, is included. The fan-beam projections of the square image and the phantom are presented. The applicability of the RT in estimating the Hough transform over precise lines, is shown. Conclusion: The new formula, a simplified version of that of Deans, is useful in pixel-driven computation of parallel and fan-beam projections based on Riemann-sum approximation, which is exact in the case of the common pixel-based image model. Pixel-driven approach is amenable to parallel, and also region-of-interest computation. The method is useful in CT as well as machine vision applications.