Far East Journal of Dynamical Systems
Volume 14, Issue 1, Pages 51 - 70
(September 2010)
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BLOCK CONTINUOUS-TIME IMAGE RECONSTRUCTION FOR COMPUTED TOMOGRAPHY
Omar M. Abou Al-Ola, Ken’ichi Fujimoto and Tetsuya Yoshinaga
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Abstract:
We have presented a novel approach for reconstructing tomographic images based on the idea of continuous dynamical methods; the method consists of a continuous-time image reconstruction (CIR) system described by differential equations for solving ill-posed inverse problems. We have extended our CIR method, by introducing subsets of projections, to what be called block CIR system. According to many simulations and numerical discussions, we see that a switched system with a piecewise smooth vector field, which describes a block CIR system, can yield frankly good-quality image reconstructions. The goodness of our results about block CIR approach tempts us to show that this approach works well theoretically, based on dynamical systems theory, but the road is not smooth. Since the main challenge in stability analysis based on Lyapunov theory is always to find a suitable Lyapunov function, in this paper, we introduce an illustrative idea to search for a common Lyapunov function for all subsystems in our block CIR system. This idea is efficient and does succeed in solving many problems, but it may be interesting to develop other ideas for other problems. |
| Keywords and phrases: tomographic image, nonlinear differential equation, block CIR, switched systems, stability, common Lyapunov function. |
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