×

zbMATH — the first resource for mathematics

On quasi-local inversion of spiral CT data. (English) Zbl 0939.92021
Efficient mathematical methods are of immense use in applied sciences. Here is an explicit example for how one can deal with a medical imaging situation involving spiral computed tomography (CT) and the selected handling of data. The author has presented a quasi-local algorithm for computing an approximation \(f_0\) of a well-defined function \(f\) which retains all the sharp properties of \(f\). An in-depth study of the algorithm is discussed together with results of numerical testing on simulated data.
After constructing the function of \(f_0\), a proof is provided asserting that the map \(f\to f_0\) is a pseudo-differential operator (PDO). Then stability simplification and computation of \(f_0\) are discussed. Some refinements on the choice of components, sampling of spirals and selection of parameters of the computational scheme adopted are also described at length. A numerical experiment is performed in order to test the proposed algorithm and the emerging data have been displayed in 6 figures. There are altogether 9 sections, 113 equations, 8 theorems including one lemma and 25 references.
MSC:
92C55 Biomedical imaging and signal processing
65R10 Numerical methods for integral transforms
44A12 Radon transform
PDF BibTeX Cite
Full Text: DOI
References:
[1] et al., ?Large area 97-{\(\mu\)}m pitch indirect-detection active-matrix flat-panel imager (AMFPI)?, Proc. Medical Imaging 1998: Physics of Medical Imaging (J. T. Dobbins III and J. M. Boone, eds), Proceedings SPIE-3336, pp. 2-13, 1998.
[2] and Asymptotic Expansions of Integrals, Dover, Mineola, New York, 1986.
[3] Boman, Trans. AMS 335 pp 877– (1993)
[4] Defrise, IEEE Trans. Med. Imaging 13 pp 186– (1994)
[5] Defrise, IEEE Trans. Med. Imaging 12 pp 622– (1993)
[6] ?Results, old and new, in computed tomography?, Inverse problems in wave propagation ( et al., eds.), The IMA Volumes in Mathematics and its Applications, Vol. 90, Springer Verlag, New York, 1997, pp. 167-193.
[7] Finch, SIAM J. Appl. Math. 45 pp 665– (1985) · Zbl 0579.65136
[8] and (eds), Spiral CT. Principles, Techniques, and Clinical Applications, Raven Press, New York, NY, 1995.
[9] ?Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform?, In: Mathematical Methods in Tomography. Lecture Notes in Math., Vol. 1497 ( and eds), 1991.
[10] Greenleaf, Duke Math. J. 58 pp 205– (1989) · Zbl 0668.44004
[11] The Analysis of Linear Partial Differential Operators, Vol III, Springer, New York, 1985.
[12] The Analysis of Linear Partial Differential Operators, Vol IV, Springer, New York, 1985.
[13] Kirillov, Soviet Math. Dokl. 2 pp 268– (1961)
[14] Kudo, Phys. Med. Biol. 43 pp 2885– (1998)
[15] et al., ?Selenium flat panel detector-based volume tomographic imaging: phantom studies?, Proc. Medical Imaging 1998: Physics of Medical Imaging (J. T. Dobbins III and J. M. Boone, eds), Proceedings SPIE-3336, pp. 316-324, 1998.
[16] Quinto, SIAM J. Math. Anal. 24 pp 1215– (1993) · Zbl 0784.44002
[17] and The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996. · Zbl 0863.44001
[18] Ramm, Appl. Math. Lett. 5 pp 91– (1992)
[19] Ramm, J. Math. Anal. Appl. 183 pp 528– (1994) · Zbl 0813.44005
[20] Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1980.
[21] Smith, IEEE Trans. Med. Imaging 4 pp 14– (1985)
[22] Smith, Opt. Eng. 29 pp 524– (1990)
[23] ?Method and apparatus for converting cone beam x-ray projection data to planar integral and reconstructing a three-dimensional computerized tomography (CT) image of an object?, US Patent 5257183, October 1995.
[24] Tuy, SIAM J. Appl. Math. 43 pp 546– (1983)
[25] et al., Helical/Spiral CT. A Practical Approach, McGraw-Hill, Inc., New York, NY, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.