TY - JOUR

T1 - Pixel-driven computation of parallel and fan-beam projections of a digital image based on pixel-representation using a new formula

AU - Galigekere, Ramesh R.

N1 - Funding Information:
The author thanks the anonymous reviewers for their comments and suggestions, which have resulted in significant improvements in the manuscript. He also thanks his good old friend Dr. Ganesh Murthy, Mercedes-Benz R & D India Pvt. Ltd. Bangalore, India, for reading, and suggesting to write the algorithmic steps explicitly.
Publisher Copyright:
© 2022

PY - 2022/4

Y1 - 2022/4

N2 - 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.

AB - 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.

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U2 - 10.1016/j.cmpb.2022.106672

DO - 10.1016/j.cmpb.2022.106672

M3 - Article

AN - SCOPUS:85124791160

VL - 216

JO - Computer Methods and Programs in Biomedicine

JF - Computer Methods and Programs in Biomedicine

SN - 0169-2607

M1 - 106672

ER -