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Performance analysis of regularization algorithms used for image reconstruction in computed tomography

Wysoczański D., Mroczka J., Polak A. G.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2013

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Vol. 61, nr 2

467--474

EN

Computed tomography is one of the most significant diagnostic techniques in medicine. This work is focused on hard-field imaging, where signals take a form of straight rays and the reconstructed image can be presented as a matrix with unknown pixels. Algebraic methods for direct computation of the image have not been used in practice because of the scale of the problem and numerical errors appearing in the solution. The aim of this work was to analyse the performance of direct algebraic algorithms for tomographic image reconstruction including regularisation mechanism such as: generalised regularisation, Tikhonov regularisation, Twomey regularisation and ridge regression (RR), as well as comparing the results with the filtered backprojection (FBP) as the reference method. The performed analyses demonstrated that the regularised algebraic methods are more accurate than the commonly used FBP, and RR appeared the most precise among them. Additionally it was shown that the invariant system matrix (inverted during calculations) can be easily determined by solving the forward problem. Finally, potential directions of further research have been pointed out.

2

Redundancy relations for fault diagnosis in nonlinear uncertain systems

Shumsky A.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 4

477-489

EN

The problem of fault detection and isolation in nonlinear uncertain systems is studied within the scope of the analytical redundancy concept. The problem solution involves checking the redundancy relations existing among measured system inputs and outputs. A novel method is proposed for constructing redundancy relations based on system models described by differential equations whose right-hand sides are polynomials. The method involves a nonlinear transformation of the initial system model into a strict feedback form. Algebraic and geometric tools are used for this transformation. The features of the method are made particular for uncertain systems with a linear structure.

3

Nonlinear diagnostic filter design: Algebraic and geometric points of view

Shumsky A., Zhirabok A.

International Journal of Applied Mathematics and Computer Science

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2006

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Vol. 16, no 1

115-127

EN

The problem of diagnostic filter design is studied. Algebraic and geometric approaches to solving this problem are investigated. Some relations between these approaches are established. New definitions of fault detectability and isolability are formulated. On the basis of these definitions, a procedure for diagnostic filter design is given in both algebraic and geometric terms.

4

Analysis of some dual properties in discrete dynamic systems

Zhirabok A.

International Journal of Applied Mathematics and Computer Science

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2006

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Vol. 16, no 3

297-308

EN

The problem of duality in nonlinear and linear systems is considered. In addition to the known duality between controllability and observability, new dual notions and their properties are investigated. A way to refine these properties through an isomorphic transformation of the original systems is suggested.