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Maximum Entropy Reconstruction with Constraints: Iterative Algorithms for Solving the Primal and Dual Programs

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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.

Keywords

  • Single Photon Emission Compute Tomography
  • Maximum Entropy
  • Model Constraint
  • Conjugate Gradient Algorithm
  • Duality Principle

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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