×

Sampling for the V-line transform with vertex on a circle. (English) Zbl 1478.44004

Summary: In this paper, we consider a special V-line transform in the two dimensional space. It integrates a given function \(f\) over the V-lines whose vertices are on a circle centered at the origin and whose symmetric axes pass through the origin. We study the sampling problem of this V-line transform. Namely, we consider the problem of recovering the continuous data from its discrete samples. Under suitable conditions, we prove an error estimate of the recovery. The error estimate is explicitly expressed in terms of \(f\). We then elaborate the required conditions for two sampling schemes: standard and interlaced ones. Finally, we analyze the number of sampling points needed for each case.

MSC:

44A12 Radon transform
65R10 Numerical methods for integral transforms
92C55 Biomedical imaging and signal processing
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol 55 (1964), Washington, DC: US Government Printing Office, Washington, DC · Zbl 0171.38503
[2] Allmaras, M.; Darrow, D.; Hristova, Y.; Kanschat, G.; Kuchment, P., Detecting small low emission radiating sources, Inverse Probl. Imag., 7, 47-79 (2013) · Zbl 1266.82083 · doi:10.3934/ipi.2013.7.47
[3] Ambartsoumian, G.; Moon, S., Inversion of the V-line radon transform in a disc and its applications in imaging, Comput. Math. Appl., 64, 260-265 (2012) · Zbl 1252.44003 · doi:10.1016/j.camwa.2012.01.059
[4] Basko, R.; Zeng, G. L.; Gullberg, G. T., Analytical reconstruction formula for one-dimensional Compton camera, IEEE Trans. Nucl. Sci., 44, 1342-1346 (1997) · doi:10.1109/23.597011
[5] Budinger, T. F.; Gullberg, G. T.; Huesman, R. H., Emission computed tomography, Image Reconstruction from Projections (1979), New York: Springer, New York
[6] Cree, M. J.; Bones, P. J., Towards direct reconstruction from a gamma camera based on Compton scattering, IEEE Trans. Med. Imag., 13, 398-407 (1994) · doi:10.1109/42.293932
[7] Cormac, A. M., Representation of a function by its line integrals, with some radiological applications, J. Appl. Phys., 34, 2722-2727 (1963) · Zbl 0117.32303
[8] Cormack, A. M., Sampling the Radon transform with beams of finite width, Phys. Med. Biol., 23, 1141 (1978) · doi:10.1088/0031-9155/23/6/010
[9] Desbat, L., Efficient sampling on coarse grids in tomography, Inverse Problems, 9, 251 (1993) · Zbl 0773.65088 · doi:10.1088/0266-5611/9/2/007
[10] Desbat, L., Interpolation of lacking data in tomography, 123-128 (2001)
[11] Desbat, L.; Roux, S.; Grangeat, P.; Koenig, A., Sampling conditions of 3D parallel and fan-beam x-ray CT with application to helical tomography, Phys. Med. Biol., 49, 2377 (2004) · doi:10.1088/0031-9155/49/11/018
[12] Everett, D. B.; Fleming, J. S.; Todd, R. W.; Nightingale, J. M., Gamma-radiation imaging system based on the Compton effect, Proc. Inst. Electr. Eng., 124, 995-1000 (1977) · doi:10.1049/piee.1977.0203
[13] Faridani, A., An application of a multidimensional sampling theorem to computed tomography, Contemp. Math., 113, 65-80 (1990) · Zbl 0729.65107 · doi:10.1090/conm/113/1108645
[14] Faridani, A., Reconstructing from efficiently sampled data in parallel-beam computed tomography, Inverse Probl. Imag., 245, 68-102 (1991) · Zbl 0748.65088
[15] Faridani, A., Results, old and new, in computed tomography, Inverse Problems in Wave Propagation, 167-193 (1997), New York: Springer, New York · Zbl 0870.44001
[16] Faridani, A., Sampling in parallel-beam tomography, Inverse Problems, Tomography, and Image Processing, 33-53 (1998), Berlin: Springer, Berlin · Zbl 0904.65129
[17] Faridani, A., Sampling theory and parallel-beam tomography, Sampling, Wavelets, and Tomography, 225-254 (2004), Boston, MA: Birkhäuser, Boston, MA · Zbl 1062.94537
[18] Faridani, A., Fan-beam tomography and sampling theory, vol 63, 43-66 (2006) · Zbl 1095.65114
[19] Florescu, L.; Schotland, J. C.; Markel, V. A., Single-scattering optical tomography, Phys. Rev. E, 79 (2009) · doi:10.1103/physreve.79.069903
[20] Gouia-Zarrad, R.; Ambartsoumian, G., Exact inversion of the conical radon transform with a fixed opening angle, Inverse Problems, 30 (2014) · Zbl 1306.44002 · doi:10.1088/0266-5611/30/4/045007
[21] Gelfand, I. M.; Gindikin, S. G.; Graev, M. I., Selected Topics in Integral Geometry. Providence, RI, vol 220 (2003), American Mathematical Society · Zbl 1055.53059
[22] Haltmeier, M., Exact reconstruction formulas for a radon transform over cones, Inverse Problems, 30 (2014) · Zbl 1291.44005 · doi:10.1088/0266-5611/30/3/035001
[23] Haltmeier, M., Sampling conditions for the circular radon transform, IEEE Trans. Image Process., 25, 2910-2919 (2016) · Zbl 1408.94225 · doi:10.1109/tip.2016.2551364
[24] Hristova, Y., Inversion of a V-line transform arising in emission tomography, Journal of Coupled Systems and Multiscale Dynamics, 3, 272-277 (2015) · doi:10.1166/jcsmd.2015.1086
[25] Helgason, S., The Radon Transform, vol 2 (1999), Berlin: Springer, Berlin · Zbl 0932.43011
[26] Jung, C-Y; Moon, S., Inversion formulas for cone transforms arising in application of Compton cameras, Inverse Problems, 31 (2015) · Zbl 1315.44001 · doi:10.1088/0266-5611/31/1/015006
[27] Jung, C-Y; Moon, S., Exact inversion of the cone transform arising in an application of a Compton camera consisting of line detectors, SIAM J. Imag. Sci., 9, 520-536 (2016) · Zbl 1346.44001 · doi:10.1137/15m1033617
[28] Katsevich, A., Resolution analysis of inverting the generalized Radon transform from discrete data in R^3, SIAM J. Math. Anal., 52, 3990-4021 (2020) · Zbl 1469.44005 · doi:10.1137/19m1295039
[29] Kruse, H., Resolution of reconstruction methods in computerized tomography, SIAM J. Sci. Stat. Comput., 10, 447-474 (1989) · Zbl 0691.65095 · doi:10.1137/0910030
[30] Kuchment, P., The Radon Transform and Medical Imaging (2013), Philadelphia, PA: SIAM, Philadelphia, PA
[31] Kuchment, P.; Terzioglu, F., Three-dimensional image reconstruction from Compton camera data, SIAM J. Imag. Sci., 9, 1708-1725 (2016) · Zbl 1354.44002 · doi:10.1137/16m107476x
[32] Kuchment, P.; Terzioglu, F., Inversion of weighted divergent beam and cone transforms, Inverse Probl. Imag., 11, 6 (2017) · Zbl 1372.44004 · doi:10.3934/ipi.2017049
[33] Kwon, K., An inversion of the conical Radon transform arising in the Compton camera with helical movement, Biomed. Eng. Lett., 9, 233-243 (2019) · doi:10.1007/s13534-019-00106-y
[34] Moon, S., On the determination of a function from its conical Radon transform with a fixed central axis, SIAM J. Math. Anal., 48, 1833-1847 (2016) · Zbl 1381.44009 · doi:10.1137/15m1021945
[35] Moon, S.; Haltmeier, M., Analytic inversion of a conical radon transform arising in application of Compton cameras on the cylinder, SIAM J. Imag. Sci., 10, 535-557 (2017) · Zbl 1365.65275 · doi:10.1137/16m1083116
[36] Mathison, C., Sampling in thermoacoustic tomography, J. Inverse Ill-Posed Probl., 28, 881-897 (2020) · Zbl 1460.35405 · doi:10.1515/jiip-2020-0001
[37] Morvidone, M.; Nguyen, M. K.; Truong, T. T.; Zaidi, H., On the V-Line Radon transform and Its imaging applications, Int. J. Biomed. Imag., 2010 (2010) · doi:10.1155/2010/208179
[38] Natterer, F., Recent developments in x-ray tomography, Tomography, Impedance Imaging, and Integral Geometry, 177-198 (1991), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0823.92012
[39] Natterer, F., Sampling in fan beam tomography, SIAM J. Appl. Math., 53, 358-380 (1993) · Zbl 0773.65089 · doi:10.1137/0153021
[40] Natterer, F., The Mathematics of Computerized Tomography (2001), Philadelphia, PA: SIAM, Philadelphia, PA · Zbl 0973.92020
[41] Nguyen, M. K.; Truong, T. T.; Grangeat, P., Radon transforms on a class of cones with fixed axis direction, J. Phys. A: Math. Gen., 38, 8003 (2005) · Zbl 1086.44002 · doi:10.1088/0305-4470/38/37/006
[42] Palamodov, V. P., Localization of harmonic decomposition of the Radon transform, Inverse Problems, 11, 1025 (1995) · Zbl 0836.44001 · doi:10.1088/0266-5611/11/5/006
[43] Rattey, P.; Lindgren, A., Sampling the 2D Radon transform, IEEE Trans. Acoust. Speech Signal Process., 29, 994-1002 (1981) · doi:10.1109/tassp.1981.1163686
[44] Schiefeneder, D.; Haltmeier, M., The Radon transform over cones with vertices on the sphere and orthogonal axes, SIAM J. Appl. Math., 77, 1335-1351 (2017) · Zbl 1371.44001 · doi:10.1137/16m1079476
[45] Siegel, K. M., An inequality involving Bessel functions of argument nearly equal to their order, Proc. Am. Math. Soc., 4, 858 (1953) · Zbl 0052.06503 · doi:10.1090/s0002-9939-1953-0058775-0
[46] Singh, M., An electronically collimated gamma camera for single photon emission computed tomography. Part I: theoretical considerations and design criteria, Med. Phys., 10, 421-427 (1983) · doi:10.1118/1.595313
[47] Stefanov, P., Semiclassical sampling and discretization of certain linear inverse problems (2018)
[48] Terzioglu, F., Some inversion formulas for the cone transform, Inverse Problems, 31 (2015) · Zbl 1329.35356 · doi:10.1088/0266-5611/31/11/115010
[49] Terzioglu, F.; Kuchment, P.; Kunyansky, L., Compton camera imaging and the cone transform: a brief overview, Inverse Problems, 34 (2018) · Zbl 1478.68410 · doi:10.1088/1361-6420/aab0ab
[50] Terzioglu, F., Some analytic properties of the cone transform, Inverse Problems, 35 (2019) · Zbl 1461.44003 · doi:10.1088/1361-6420/aafccf
[51] Truong, T. T.; Nguyen, M. K., On new V-line Radon transforms in \(####\) and their inversion, J. Phys. A: Math. Theor., 44 (2011) · Zbl 1210.44001 · doi:10.1088/1751-8113/44/7/075206
[52] Webber, J. W.; Quinto, E. T., Microlocal analysis of generalized Radon transforms from scattering tomography (2020)
[53] Zhang, Y., Recovery of singularities for the weighted cone transform appearing in Compton camera imaging, Inverse Problems, 36 (2020) · Zbl 1442.44003 · doi:10.1088/1361-6420/ab3cc8
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.