Abstract
A rigorous method for band-limited images reconstruction from sampled projections is presented. The problem is formulated as the minimization of a quadratic criterion expressed in the frequency domain; consequences of spectral support limitation and sampling are taken into account. Then, the optimal reconstruction is obtained in two steps: a classical 1D-convolution/backprojection followed by the computation of the solution of a 2D convolution equation. The proposed solution for this second step is based on a conjugate gradient method. In both steps, the computations are performed in the frequency domain. This reduces the number of computations and allows the execution in a reasonable amount of time. A particular choice of the weights establishes a link between this approach and Radon's classical backprojection method: the classical method is optimal when the number of projections is infinite. In practice, the method proposed here improves the results obtained when computing an estimate of the solution by backprojections from a finite number of samples and projections when the projections are noisefree.
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References
R. H. Chan and G. Strang, “Toeplitz Equations by Conjugate Gradients with Circulant Preconditioner,”SIAM J. Sci. Statist. Comput, no. 10, 1989, pp. 104–119.
D. E. Dudgeon and R. M. Mersereau,Multidimensional Digital Signal Processing, Prentice Hall, 1984.
P. Edholm, G. T. Herman, and D. A. Roberts, “Image Reconstruction from Linograms: Implementation and Evaluation,”IEEE Trans. on Medical Imaging, vol. 7, no. 3, 1988, pp. 239–246.
R. Gordon, “A Tutorial on ART,”IEEE Transactions on Nuclear Sciences, vol. NS-21, 1974, pp. 78–93.
G. T. Herman (Editor), “Image Reconstruction from Projections,”Topics in Applied Physics, vol. 32, Springer Verlag, 1979.
G. T. Herman,Image Reconstruction from Projections, the Fundamentals of Computerized Tomography, New York: Academic Press, 1980.
G. T. Herman and A. Lent, “A Computer Implementation of a Bayesian Analysis of Image Reconstruction,”Inform. Contr., vol. 31, 1976, pp. 364–384.
G. T. Herman, A. Lent, and S. W. Rowland, “ART, Mathematics and Applications,”J. Theor. Biol., vol. 42, 1973, pp. 1–32.
G. T. Herman and D. Odhner, “Performance Evaluation of an Iterative Image Reconstruction Algorithm for Positrion Emission Tomography,”IEEE Trans. on Medical Imaging, vol. 10, no. 3, 1991, pp. 336–345.
A. K. Jain, “Fundamentals of Digital Image Processing,”Image Reconstruction from Projections, Prentice Hall, 1989.
A. K. Jain and S. Ansari, “Radon Transform Theory for Random Fields and Image Reconstruction from Noisy Projections,”Proc. ICASSP, San Diego, 1984.
A. Kak and M. Slaney,Principles of Computerized Tomographic Imaging, IEEE press, 1988.
G. A. Korn and T. M. Korn,Mathematical Handbook for Scientists and Engineers, McGraw Hill, 1968.
S. S. Kuo and R. Mamone, “Resolution Enhancement of Tomographic Images Using the Row Action Projection Method,”IEEE Trans. on Medical Imaging, vol. 10, no. 4, 1991, pp. 593–601.
J. Le Roux, P. E. Lise, and E. Zerbib, “Least Squares Reconstruction of Band Limited Images from Projections,”Proceedings of the IEEE Workshop on IMDSP, Cannes, France, Sept. 8–10, 1993, pp. 22–23.
S. Matej and I. Bajla, “A High Speed Reconstruction from Projections Using Direct Fourier Method with Optimized Parameters—An Experimental Analysis,”IEEE Trans. on Medical Imaging, vol. 9, no. 4, 1990, pp. 421–429.
R. M. Mersereau and A. V. Oppenheim, “Digital Reconstruction of Multidimensional Signals from their Projections,”Proceedings of the IEEE, vol. 62, no. 10, 1974, pp. 1319–1338.
J. M. Ollinger, “Iterative Reconstruction-Reprojection and Expectation-Maximization Algorithms,”IEEE Trans. on Medical Imaging, vol. 9, no. 1, 1990, pp. 94–98.
J. D. O'Sullivan, “A Fast Sinc Function Gridding Algorithm for Fourier Inversion in Computer Tomography,”IEEE Trans. on Medical Imaging, vol. MI-4, no. 4, 1985, pp. 200–207.
T. S. Pan and A. Yagle, “Numerical Study of Multigrid Implementations of Some Iterative Image Reconstruction Algorithms,”IEEE Trans. on Medical Imaging, vol. 10, no. 4, 1991, pp. 572–588.
W. H. Press, et al.,Numerical Recipies, the Art of Scientific Computing, Cambridge University Press, 1988.
J. Radon, “Uber die Bestimmung von Functionnen durch ihre Integralwerte langs gewisser Mannifaligkeiten,”Berchte Saechsicdhe Akademie des Wissenschafter, vol. 69, 1917, pp. 262–279.
S. W. Rowland, “Computer Inplementation of Image Reconstruction Formulas” in “Image Reconstruction from Projections,”Topics in Applied Physics, vol. 32, Springer Verlag, 1979.
K. Sauer and C. Bouman, “A Local Update Strategy for Iterative Reconstruction from Projections,”IEEE Trans. on Signal Processing, vol. 41, no. 2, 1993, pp. 534–548.
H. Stark, J. W. Woods, I. Paul, and R. Hingorani, “An Investigation of Computerized Tomography by Direct Fourier Inversion and Optimum Interpolation,”IEEE Trans. on Biomed. Eng., vol. BME-28, no. 7, 1981, pp. 496–505.
M. Tabei and M. Ueda, “Backprojection by Upsampled Fourier Series Expansion and Interpolated FFT,”IEEE Trans. on Image Processing, vol. 1, no. 1, 1992, pp. 77–87.
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Le Roux, J., Lise, P., Zerbib, E. et al. A formulation in concordance with the sampling theorem for band-limited images reconstruction from projections. Multidim Syst Sign Process 7, 27–52 (1996). https://doi.org/10.1007/BF02106105
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DOI: https://doi.org/10.1007/BF02106105