Application of optimal quadrature formulas for reconstruction of CT images. (English) Zbl 1460.65025
Summary: In the present paper, the construction process of the optimal quadrature formulas for weighted integrals is presented in the Sobolev space \(\widetilde{L}_2^{(m)}(0,1]\) of complex-valued periodic functions which are square integrable with \(m\)th order derivative. In particular, optimal quadrature formulas are given for Fourier coefficients. Here, using these optimal quadrature formulas the approximation formulas for Fourier integrals \(\int_a^b\operatorname{e}^{2\pi\operatorname{i}\omega x}f(x) \operatorname{d}x\) with \(\omega\in\mathbb{R}\) are obtained. In the cases \(m=1,2\) and 3, the obtained approximation formulas are applied for reconstruction of Computed Tomography (CT) images coming from the filtered back-projection method. Compared with the optimal quadrature formulas in non-periodic case, the approximation formula for the periodic case is much simpler, therefore it is easy to implement and costs less computation.
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 41A05 | Interpolation in approximation theory |
| 41A15 | Spline approximation |
| 41A55 | Approximate quadratures |
| 65T40 | Numerical methods for trigonometric approximation and interpolation |
Keywords:
optimal quadrature formula; Sobolev space; error functional; Fourier transform; Radon transform; filtered back-projection method; CT image reconstructionReferences:
| [1] | Buzug, T. M., Computed Tomography from Photon Statistics to Modern Cone-Beam CT (2008), Springer: Springer Berlin |
| [2] | Feeman, T. G., The Mathematics of Medical Imaging, A Beginner’s Guide (2015), Springer · Zbl 1351.92002 |
| [3] | Kak, A. C.; Slaney, M., Principles of Computerized Tomographic Imaging (1988), IEEE Press: IEEE Press New York · Zbl 0721.92011 |
| [4] | Filon, L. N.G., On a quadrature formula for trigonometric integrals, Proc. Roy. Soc. Edinburgh, 49, 38-47 (1928) · JFM 55.0946.02 |
| [5] | Avdeenko, V. A.; Malyukov, A. A., A quadrature formula for the fourier integral based on the use of a cubic spline, USSR Comput. Math. Math. Phys., 29, 783-786 (1989), (Russian) · Zbl 0686.65098 |
| [6] | Babuška, I., Optimal quadrature formulas, Dokl. Akad. Nauk. SSSR, 149, 227-229 (1963), (Russian) |
| [7] | Babuška, I.; Vitasek, E.; Prager, M., Numerical Processes in Differential Equations (1966), Wiley: Wiley New York · Zbl 0156.16003 |
| [8] | Bakhvalov, N. S.; Vasil’eva, L. G., Evaluation of the integrals of oscillating functions by interpolation at nodes of Gaussian quadratures, USSR Comput. Math. Math. Phys., 8, 241-249 (1968), (Russian) · Zbl 0196.17502 |
| [9] | Iserles, A.; Nørsett, S. P., Efficient quadrature of highly oscillatory integrals using derivatives, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 1383-1399 (2005) · Zbl 1145.65309 |
| [10] | Milovanović, G. V., Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures, Comput. Math. Appl., 36, 19-39 (1998) · Zbl 0932.65023 |
| [11] | Novak, E.; Ullrich, M.; Woźniakowski, H., Complexity of oscillatory integration for univariate Sobolev space, J. Complexity, 31, 15-41 (2015) · Zbl 1318.65088 |
| [12] | Xu, Z.; Milovanović, G. V.; Xiang, S., Efficient computation of highly oscillatory integrals with Henkel kernel, Appl. Math. Comput., 261, 312-322 (2015) · Zbl 1410.65074 |
| [13] | Zhang, S.; Novak, E., Optimal quadrature formulas for the Sobolev space \(H^1\), J. Sci. Comput., 78, 274-289 (2019) · Zbl 1455.65041 |
| [14] | Deaño, A.; Huybrechs, D.; Iserles, A., Computing Highly Oscillatory Integrals (2018), SIAM: SIAM Philadelphia · Zbl 1400.65004 |
| [15] | Milovanović, G. V.; Stanić, M. P., Numerical integration of highly oscillating functions, (Milovanović, G. V.; Rassias, M. Th., Analytic Number Theory, Approximation Theory, and Special Functions/ (2014), Springer: Springer New York), 613-649 · Zbl 1325.65040 |
| [16] | Olver, S., Numerical Approximation of Highly Oscillatory Integrals (2008), University of Cambridge, (PhD dissertation) · Zbl 1131.65028 |
| [17] | Sobolev, S. L., Introduction To the Theory of Cubature Formulas (1974), Nauka: Nauka Moscow, (Russian) |
| [18] | Sobolev, S. L.; Vaskevich, V. L., The Theory of Cubature Formulas (1997), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0877.65009 |
| [19] | Ahlberg, J. H.; Nilson, E. N.; Walsh, J. L., The Theory of Splines and their Applications, Mathematics in Science and Engineering (1967), Academic Press: Academic Press New York · Zbl 0158.15901 |
| [20] | Boltaev, N. D.; Hayotov, A. R.; Shadimetov, Kh. M., Construction of optimal quadrature formula for numerical calculation of Fourier coefficients in Sobolev space \(L_2^{( 1 )}\), Am. J. Numer. Anal., 4, 1-7 (2016) |
| [21] | Boltaev, N. D.; Hayotov, A. R.; Shadimetov, Kh. M., Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space \(L_2^{( m )} ( 01 )\), Numer. Algorithms, 74, 307-336 (2017) · Zbl 1364.65302 |
| [22] | Boltaev, N. D.; Hayotov, A. R.; Milovanović, G. V.; Shadimetov, Kh. M., Optimal quadrature formulas for Fourier coefficients in \(W_2^{( m m - 1 )}\) space, J. Appl. Anal. Comput., 7, 1233-1266 (2017) · Zbl 07247213 |
| [23] | Hayotov, A. R.; Jeon, S. M.; Lee, C.-O., On an optimal quadrature formula for approximation of Fourier integrals in the space \(L_2^{( 1 )}\), J. Comput. Appl. Math., 372, Article 112713 pp. (2020) · Zbl 1437.65011 |
| [24] | Hayotov, A. R.; Jeon, S. M.; Lee, C.-O.; Shadimetov, Kh. M., Optimal quadrature formulas for non-periodic functions in Sobolev space and its application to CT image reconstruction (2020), arXiv:2001.02636v2 [math.NA] |
| [25] | Shadimetov, Kh. M., On an optimal quadrature formula, Uzbek Math. Zh., 3, 90-98 (1998) |
| [26] | Shadimetov, Kh. M., Weight optimal cubature formulas in Sobolev’s periodic space, Siberian J. Numer. Math. -Novosibirsk, 2, 185-196 (1999), (Russian) · Zbl 0936.46025 |
| [27] | Kh.M. Shadimetov, The discrete analogue of the differential operator \(\operatorname{d}^{2 m} / \operatorname{d} x^{2 m}\) and its construction, in: Questions of Computations and Applied Mathematics. Tashkent, no. 79, 1985, pp. 22-35. ArXiv:1001.0556v1 [math.NA], 2010. · Zbl 0592.65008 |
| [28] | F.G. Frobenius, Über die Bernoullischen Zahlen und Eulerischen Polynome. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin (1910) 809-847. · Zbl 1264.11013 |
| [29] | Sobolev, S. L., The coefficients of optimal quadrature formulas, (Selected Works of S.L. Sobolev (2006), Springer: Springer US), 561-566 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
