Reconstruction of 3D X-ray CT images from reduced sampling by a scaled gradient projection algorithm
- 172 Downloads
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.
Keywords3D 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.
- 4.Bertero, M., Lantéri, H., Zanni, L.: Iterative image reconstruction: a point of view. In: Censor, Y., et al. (eds.) Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), pp. 37–63. Birkhauser-Verlag, Basel (2008)Google Scholar
- 5.Bertsekas, D.: Convex Optimization Theory. Supplementary Chapter 6 on Convex Optimization Algorithms. Athena Scientific, Belmont (2009)Google Scholar
- 13.Coli, V.L., Ruggiero, V., Zanni, L.: Scaled first-order methods for a class of large-scale constrained least square problems. In: Sergeyev, Y.D., Kvasov, D.E., Dell’Accio, F., Mukhametzhanov, M.S. (eds.) Numerical Computations: Theory and Algorithms (NUMTA-2016), pp. 040002-1–040002-4. AIP Publishing, Melville (2016)Google Scholar
- 16.di Serafino, D., Ruggiero, V., Toraldo, G., Zanni, L.: On the steplength selection in gradient methods for unconstrained optimization. Appl. Math. Comput. 318, 176–195 (2018)Google Scholar
- 22.Jørgensen, J.H., Jensen, T.L., Hansen, P.C., Jensen, S.H., Sidky, E.Y., Pan, X.: Accelerated gradient methods for total-variation-based CT image reconstruction. In: 11th Fully 3D Image Reconstruction in Radiology and Nuclear Medicins, pp. 435–438 (2011)Google Scholar
- 36.Sidky, E.Y., Kao, C.M., Pan, X.: Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT. J. X-ray Sci. Technol. 14(2), 119–139 (2006)Google Scholar