Abstract
Using constraints to better define regions of known intensity can improve the signal-to-noise ration in computed tomography (CT). This is accomplished in positron emission tomography (PET) by reducing the region of unknown activity with time-coincidence circuitry. Mathematically, constraints can be implemented into reconstruction algorithms using a priori information, such as the use of an x-ray CT image to define regions of radionuclide uptake in single photon emission computed tomography (SPECT) or PET imaging. In addition to regional constraints, intensity constraints can also be included into the model equations. This may be especially useful where the data has a high degree of contrast as in DSA limited-angle tomography.
The maximum entropy reconstruction problem with equality constraints can be formulated either as a primal optimization program whose optimum is determined from a solution to a system of nonlinear equations or reduced to a dual optimization program using duality principles. The dual program is an unconstrained optimization problem with an objective function that is nonlinear and nonquadratic, but is an essentially smooth function as feasible solutions approach boundary points. Therefore, a solution can be determined using gradient type of algorithms. Simulations using a Gauss-Seidel-Newton iteration to solve the primal program are compared with a conjugate gradient algorithm that solves the dual optimization program. Simulations show that the two approaches produce reconstructions with very different texture.
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References
Budinger, T. F. (1971). “Transfer Function Theory and Image Evaluation in Biology: Applications in Electron Microscopy and Nuclear Medicine,” Ph.D. dissertation, University of California, Berkeley.
Budinger, T. F., Gullberg, G. T. and Huesman, R. H. (1979). Emission computed tomography, in: “Image Reconstruction From Projections: Implementation and Applications,” G. T. Herman, ed., Springer-Verlag, New York, 147–246.
D’Addario, L. R. (1975). Maximum a posteriori probability and maximum entropy reconstruction, in: “Image Processing for 2-D and 3-D Reconstruction from Projections,” Technical Digest of Papers Presented at Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from projections, Stanford Univ., Stanford, WAS-1-WAS-4.
Frieden, B. R. (1972). Restoring with maximum likelihood and maximum entropy, J Opt Soc Amer., 62, 511–518.
Frieden, B. R. (1975). Image enhancement and restoration, in: “Topics in Applied Physics-Picture Processing and Digital Filtering,” T. S. Huang, ed., Springer-Verlag, New York, 177–248.
Frieden, B. R. (1977). Estimation—A new role for maximum entropy, in: “1976 SPSE Conference Proceedings,” R. Shaw, ed., Society of Photographic Scientists and Engineers, Washington D.C., 261–265.
Frieden, B. R. (1980). Statistical models for the image restoration problem, Comput Graph and Image Process, 12, 40–59.
Gordon, R., Bender, R. and Herman, G. T. (1970). Algebraic reconstruction techniques ( ART) for three-dimensional electron microscopy and x-ray photography, J Theor Biol., 29, 471–481.
Gullberg, G. T. (1975). Entropy and transverse section reconstruction, in: “Information Processing in Scintigraphy,” C. Raynaud and A. ToddPokropek, eds., “Proceedings of the IVth International Conference,” Orsay, France, 249–257.
Gullberg, G.T. and Malko, J.A. (1984). Attenuation correction for quantitative tomography-different methods, Europ J Nucl Med, 9, A24.
Gullberg, G.T. and Ghosh Roy, D.N. (1986). Maximum entropy reconstruction with constraints: Reducing the problem using duality principles, in: “Proceedings of the International Workshop on Physics and Engineering of Computerized Multidimensional Imaging and Process,” SPIE 671, 25–33.
Huesman, R. H., Gullberg, G.T., Greenberg, W.L. and Budinger, T.F. (1977). “Users Manual Donner Algorithms for Reconstruction Tomography,”
Jaynes, E. T. (1957). Information theory and statistical mechanics, Phys Rev, 106, 620.
Lange, K. and Carson, R. (1984). EM reconstruction algorithms for emission and transmission tomography, J Comput Assist Tomogr., 8, 306–316.
Lent, A. (1977). A convergent algorithm for maximum entropy image restoration, with a medical x-ray application, in: “1976 SPSE Conference Proceedings,” R. Shaw, ed., Society of Photographic Scientists and Engineers, Washington, D.C., 249–257.
Lewis, M. H., Willerson, J. T., Lewis, S. E., Bonte, F. J., Parker, R. W. and Stokely, E. M. (1982). Attenuation compensation in single-photon emission tomography: A comparative evaluation, J Nucl Med, 23, 1121–1127.
Minerbo, G. (1979). MENT: A maximum entropy algorithm for reconstructing a source from projection data,“ Comput Graph and Image Process, 10, 48–68.
Ortega, J. M., and Rheinboldt, W.C. (1970). “Iterative Solutions of Nonlinear Equations in Several Variables,” Academic Press, New York.
Rockefeller, R. T. (1970). “Convex Analysis,” Princeton University Press, Princeton.
Shepp, L. A. and Vardi, V. (1982). Maximum likelihood reconstruction for emission tomography, IEEE Trans Med Imaging, MI-1, 113–122.
Walters, T. E., Simon, W., Chesler, D. A. and Correia, J. A. (1981). Attenuation correction in gamma emission computed Tomography, J Comput Assist Tomogr, 5, 89–94.
Wernecke, S. J. (1975). Maximum entropy image reconstruction, in: “Image Processing for 2-D and 3-D Reconstruction from Projections,” Technical Digest of Papers Presented at Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections, Stanford Univ., Stanford, WAG-1-WAG-4.
Wernecke, S. J. (1977). Maximum entropy techniques for digital image reconstruction, in: “1976 SPSE Conference Proceedings,” R. Shaw, ed., Society of Photographic Scientists and Engineers, Washington, D.C., 238–243.
Wernecke, S. J. and D’Addario, L. R. (1977). Maximum entropy image reconstruction,IEEE Transactions on Computers, C-26, 351–364.
Zwick, M. and Zeitler, E. (1973). Image reconstruction from projections, Optik, 38, 550–565.
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Gullberg, G.T., Tsui, B.M.W. (1988). Maximum Entropy Reconstruction with Constraints: Iterative Algorithms for Solving the Primal and Dual Programs. In: de Graaf, C.N., Viergever, M.A. (eds) Information Processing in Medical Imaging. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-7263-3_11
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DOI: https://doi.org/10.1007/978-1-4615-7263-3_11
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