Computational Optimization and Applications

, Volume 71, Issue 1, pp 171–191 | Cite as

Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm

  • E. Loli PiccolominiEmail author
  • V. L. Coli
  • E. Morotti
  • L. Zanni


We propose a scaled gradient projection algorithm for the reconstruction of 3D X-ray tomographic images from limited data. The problem arises from the discretization of an ill-posed integral problem and, due to the incompleteness of the data, has infinite possible solutions. Hence, by following a regularization approach, we formulate the reconstruction problem as the nonnegatively constrained minimization of an objective function given by the sum of a fit-to-data term and a smoothed differentiable Total Variation function. The problem is challenging for its very large size and because a good reconstruction is required in a very short time. For these reasons, we propose to use a gradient projection method, accelerated by exploiting a scaling strategy for defining gradient-based descent directions and generalized Barzilai–Borwein rules for the choice of the step-lengths. The numerical results on a 3D phantom are very promising since they show the ability of the scaling strategy to accelerate the convergence in the first iterations.


3D Computed tomography Image reconstruction Total variation regularization Nonnegatively constrained minimization Scaled gradient projection methods 



This work has been partially supported by the Italian Institute GNCS - INdAM and by the FAR2015 project of the University of Modena and Reggio Emilia, Italy.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BolognaBolognaItaly
  2. 2.Department of Physics, Informatics and MathematicsUniversity of Modena and Reggio EmiliaModenaItaly
  3. 3.Department of MathematicsUniversity of PadovaPadovaItaly

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