Abstract
In image reconstruction from projections, one attempts to estimate values of an unknown function f of two variables at points of a finite grid (usually uniformly spaced in two orthogonal directions) from a finite number of (measured) line integrals of f. Lines can be characterized by two variables, typically a distance l and an angle θ. Usually the lines along which integrals of f are measured correspond to a grid of sample points in the (l,θ) space. Thus, both the input (the line integrals) and the output (values of f) can be thought of as samples of a function on a grid. The problem then translates into solving a system of linear equations, the size of which depends on the sizes of the grids associated with the input and the output.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
G. T. Herman: Image Reconstruction from Projections, the Fundamentals of Computerized Tomography (Academic Press, New York, 1980)
P. P. B. Eggermont, G. T. Herman, A. Lent: Iterative algorithms for large partitioned linear systems, with applications to image reconstruction, Linear Algebra and its Applications 40 37–67 (1981)
S. Kaczmarz: Angenäherte Auflosung von Systemen linearer Gleichungen, Bull. Acado Polon. Sci. Lett. A35 355–357 (1937)
L. F. Richardson: The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam, Phil. Trans. Royal Society (London) A21Q, 307–357 (1910)
A. Ben-Israel, T. N. E. Greville: Generalized Inverses: Theory and Applications (Wiley-Interscience, New York, 1974)
G. Cimmino: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, La Ricerca Scientifica (Roma) XVI, Serie II, Anno IX, Vol. 1, 326–333 (1938)
G. T. Herman, R. A. Robb, J. E. Gray, R. M. Lewitt, R. A. Reynolds, B. Smith, H. Tuy, D. P. Hanson, C. M. Kratz: “Reconstruction algorithms for dose reduction in x-ray computed tomography”, in Proc, MEDCOMP ’82, Philadelphia, PA, 1982, pp. 448–455
G. T. Herman, H. Hurwitz, A. Lent: A storage efficient algorithm for finding the regularized solution of a large inconsistent system of equations, J. Inst. Maths. Applies. 25, 361–366 (1980)
Y. Censor, P. P. B. Eggermont, D. Gordon: Strong underrelaxation in Kaczmarz’s method for inconsistent systems. Numerische Mathematik to appear
A. Brandt: Myltilevel adaptive solutions to boundary value problems. Math. Comp. 31, 333–390 (1977)
K. Kouris, H. Tuy, A. Lent, Go T. Herman, R. M. Lewitt: Reconstruction from sparsely sampled data by ART with interpolated rays, IEEE Trans. Medical Imaging, to appear
G. T. Herman, A. Lent, S. W. Rowland: ART: mathematics and applications, a report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques, J. Theor. Biol. 42 1–32 (1973)
G. T. Herman, A. Lent: Quadratic optimization for image reconstruction I, Computer Graphics Image Processing 5 319–332 (1976)
J. E. Gray, D. P. Hanson, G. T. Herman, R. M. Lewitt, R. A. Reynolds, R. A. Robb, B. Smith, H. Tuy: “Reconstruction Algorithms for Dose Reduction in X-Ray Computed Tomography”, Technical Report MIPG63, Medical linage Processing Group, Dept. of Radiology, University of Pennsylvania (1982)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Herman, G.T., Levkowitz, H., Tuy, H.K., McCormick, S. (1984). Multilevel Image Reconstruction. In: Rosenfeld, A. (eds) Multiresolution Image Processing and Analysis. Springer Series in Information Sciences, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-51590-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-51590-3_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-51592-7
Online ISBN: 978-3-642-51590-3
eBook Packages: Springer Book Archive