Computing the fibre orientation from Radon data using local Radon transform

Previous Article
Cumulative wavefront reconstructor for the Shack-Hartmann sensor
IPI Home
This Issue
Next Article
Reconstructions from boundary measurements on admissible manifolds

November 2011, 5(4): 879-891. doi: 10.3934/ipi.2011.5.879

Computing the fibre orientation from Radon data using local Radon transform

Michael Krause ^1, , Jan Marcel Hausherr ^1, and Walter Krenkel ^1,

1.	Ceramic Materials Engineering, University of Bayreuth, 95440 Bayreuth, Germany, Germany, Germany

Received March 2010 Revised June 2011 Published November 2011

Abstract
Related Papers

Computed tomography (CT) has become a common analysis method in the materials sciences. It allows the internal visualisation of the complete volume of an object, providing 3D information about the internal structures. One field where CT is applied is the examination of fibre-reinforced composite structures. Fibre-reinforced composites are typically composed of two types of material, mainly of high strength fibres embedded in a surrounding matrix. In this material class, the fibres typically determine the strength of the composite materials, which is largely dependent on the orientation of the fibres. Knowledge of the fibre orientation is therefore essential for the evaluation of maximal loading or for the prediction of failure. The easiest way to determine the fibre orientation is to compute it from the reconstructions received from the tomograph. A different method to determine fibre orientation is to compute it directly from Radon data using the combination of reconstruction and image analysis introduced by Louis [A. K. Louis, Combining Image Reconstruction and Image Analysis with an Application to 2D - Tomography, SIAM J. Imaging Sciences 1 (2008), 188--208]. This can be achieved by adapting the reconstruction process in computed tomography by the use of anisotropic, elongated convolution filters, leading to a set of reconstruction kernels that are dependent on the angle of the projections, thereby reflecting the anisotropy of the filters. In this paper, the two-dimensional case of computing fibre orientation directly from simulated Radon data is presented.

Keywords: non-destructive testing, Radon transform., Computed tomography.

Mathematics Subject Classification: Primary: 65R32; Secondary: 68U1.

Citation: Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems and Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

References:

[1]	E. J. Candès, "Ridgelets: Theory and Applications," Ph.D. Thesis, Department of Statistics, Stanford University, 1998.
[2]	E. J. Candès and D. L. Donoho, Curvelets-a surprisingly effective nonadaptive representation for objects with edges, in "Curves and Surfaces," Vanderbilt University Press, Nashville, TN, 1999.
[3]	E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise $C^2$ singularities, Comm. on Pure and Appl. Math., 57 (2004), 219-266.
[4]	E. J. Candès and J. Romberg, Practical signal recovery from random projections, Wavelet Applications in Signal and Image Processing XI, Proc. SPIE Conf. 5914, 2004.
[5]	A. Faridani, E. Ritman, K. Smith and T. Kennan, Local tomography, SIAM, J. Appl. Math., 52 (1992), 459-484. doi: 10.1137/0152026.
[6]	A. Faridani, D. Finch, E. Ritman, K. Smith and T. Kennan, Local tomography II, SIAM, J. Appl. Math., 57 (1997), 1095-1127. doi: 10.1137/S0036139995286357.
[7]	L. Gang, O. Chutape and M. Krishnan, Detection and measurement of retinal vessels in fundus images using amplitude modified second-order gaussian filter, IEEE Transactions on Biomedical Engineering, 49 (2002), 168-172. doi: 10.1109/10.979356.
[8]	A. Hoover, V. Kouznetsova and M. Goldbaum, Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response, IEEE Transactions on Medical Imaging, 19 (2000), 203-210.
[9]	M. Krause, R. M. Alles, B. Burgeth and J. Weickert, Retinal vessel detection via second derivative of local Radon transform, Technical Report No. 212, Universität des Saarlandes, 2009.
[10]	M. Krause, J. M. Hausherr, B. Burgeth, C. Herrmann and W. Krenkel, Determination of the fibre orientation in composites using the structure tensor and local X-ray transform, Journal of Materials Science, 45 (2010), 888-896. doi: 10.1007/s10853-009-4016-4.
[11]	A. K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), 175-190.
[12]	A. K. Louis, Combining image reconstruction and image analysis with an application to two-dimensional tomography, SIAM J. Imaging Sciences, 1 (2008), 188-208. doi: 10.1137/070700863.
[13]	A. K. Louis, Diffusion reconstruction from very noisy tomographic data, Inverse Problems and Imaging, 4 (2010), 675-683. doi: 10.3934/ipi.2010.4.675.
[14]	F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.
[15]	A. Rieder, R. Dietz and T. Schuster, Approximate inverse meets local tomography, Math. Meth. Appl. Sci., 23 (2000), 1373-1387. doi: 10.1002/1099-1476(200010)23:15<1373::AID-MMA170>3.0.CO;2-A.
[16]	M. Sofka and C. Stewart, Retinal vessel centerline extraction using multiscale matched filters, confidence and edge measures, IEEE Transactions on Medical Imaging, 25 (2006), 1531-1546.
[17]	M. Van Ginkel, "Image Analysis Using Orientation Space Based on Steerable Filters," Ph.D thesis, Delft University of Technology, 2002.
[18]	K. Vermeer, F. Vos, H. Lemij and A. Vossepoel, A model based method for retinal blood vessel detection, Computers in Biology and Medecine, 34 (2004), 209-219. doi: 10.1016/S0010-4825(03)00055-6.

show all references

References:

[1]	E. J. Candès, "Ridgelets: Theory and Applications," Ph.D. Thesis, Department of Statistics, Stanford University, 1998.
[2]	E. J. Candès and D. L. Donoho, Curvelets-a surprisingly effective nonadaptive representation for objects with edges, in "Curves and Surfaces," Vanderbilt University Press, Nashville, TN, 1999.
[3]	E. J. Candès and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise $C^2$ singularities, Comm. on Pure and Appl. Math., 57 (2004), 219-266.
[4]	E. J. Candès and J. Romberg, Practical signal recovery from random projections, Wavelet Applications in Signal and Image Processing XI, Proc. SPIE Conf. 5914, 2004.
[5]	A. Faridani, E. Ritman, K. Smith and T. Kennan, Local tomography, SIAM, J. Appl. Math., 52 (1992), 459-484. doi: 10.1137/0152026.
[6]	A. Faridani, D. Finch, E. Ritman, K. Smith and T. Kennan, Local tomography II, SIAM, J. Appl. Math., 57 (1997), 1095-1127. doi: 10.1137/S0036139995286357.
[7]	L. Gang, O. Chutape and M. Krishnan, Detection and measurement of retinal vessels in fundus images using amplitude modified second-order gaussian filter, IEEE Transactions on Biomedical Engineering, 49 (2002), 168-172. doi: 10.1109/10.979356.
[8]	A. Hoover, V. Kouznetsova and M. Goldbaum, Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response, IEEE Transactions on Medical Imaging, 19 (2000), 203-210.
[9]	M. Krause, R. M. Alles, B. Burgeth and J. Weickert, Retinal vessel detection via second derivative of local Radon transform, Technical Report No. 212, Universität des Saarlandes, 2009.
[10]	M. Krause, J. M. Hausherr, B. Burgeth, C. Herrmann and W. Krenkel, Determination of the fibre orientation in composites using the structure tensor and local X-ray transform, Journal of Materials Science, 45 (2010), 888-896. doi: 10.1007/s10853-009-4016-4.
[11]	A. K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), 175-190.
[12]	A. K. Louis, Combining image reconstruction and image analysis with an application to two-dimensional tomography, SIAM J. Imaging Sciences, 1 (2008), 188-208. doi: 10.1137/070700863.
[13]	A. K. Louis, Diffusion reconstruction from very noisy tomographic data, Inverse Problems and Imaging, 4 (2010), 675-683. doi: 10.3934/ipi.2010.4.675.
[14]	F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.
[15]	A. Rieder, R. Dietz and T. Schuster, Approximate inverse meets local tomography, Math. Meth. Appl. Sci., 23 (2000), 1373-1387. doi: 10.1002/1099-1476(200010)23:15<1373::AID-MMA170>3.0.CO;2-A.
[16]	M. Sofka and C. Stewart, Retinal vessel centerline extraction using multiscale matched filters, confidence and edge measures, IEEE Transactions on Medical Imaging, 25 (2006), 1531-1546.
[17]	M. Van Ginkel, "Image Analysis Using Orientation Space Based on Steerable Filters," Ph.D thesis, Delft University of Technology, 2002.
[18]	K. Vermeer, F. Vos, H. Lemij and A. Vossepoel, A model based method for retinal blood vessel detection, Computers in Biology and Medecine, 34 (2004), 209-219. doi: 10.1016/S0010-4825(03)00055-6.

[1]	Simon Hubmer, Alexander Ploier, Ronny Ramlau, Peter Fosodeder, Sandrine van Frank. A mathematical approach towards THz tomography for non-destructive imaging. Inverse Problems and Imaging, 2022, 16 (1) : 68-88. doi: 10.3934/ipi.2021041
[2]	Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021
[3]	Philippe Destuynder, Caroline Fabre. Few remarks on the use of Love waves in non destructive testing. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 427-444. doi: 10.3934/dcdss.2016005
[4]	Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems and Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649
[5]	Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029
[6]	Shousheng Luo, Tie Zhou. Superiorization of EM algorithm and its application in Single-Photon Emission Computed Tomography(SPECT). Inverse Problems and Imaging, 2014, 8 (1) : 223-246. doi: 10.3934/ipi.2014.8.223
[7]	Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023
[8]	Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems and Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
[9]	Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems and Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693
[10]	C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems and Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457
[11]	Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems and Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341
[12]	Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
[13]	C. Xiong, J.P. Miller, F. Gao, Y. Yan, J.C. Morris. Testing increasing hazard rate for the progression time of dementia. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 813-821. doi: 10.3934/dcdsb.2004.4.813
[14]	Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
[15]	Zhenglin Wang. Fast non-uniform Fourier transform based regularization for sparse-view large-size CT reconstruction. STEM Education, 2022, 2 (2) : 121-139. doi: 10.3934/steme.2022009
[16]	Helmut Harbrecht, Thorsten Hohage. A Newton method for reconstructing non star-shaped domains in electrical impedance tomography. Inverse Problems and Imaging, 2009, 3 (2) : 353-371. doi: 10.3934/ipi.2009.3.353
[17]	Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227
[18]	Tak Kuen Siu, Yang Shen. Risk-minimizing pricing and Esscher transform in a general non-Markovian regime-switching jump-diffusion model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2595-2626. doi: 10.3934/dcdsb.2017100
[19]	Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems and Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631
[20]	Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006

2021 Impact Factor: 1.483

Tools

Metrics

Other articles
by authors

on AIMS
Michael Krause Jan Marcel Hausherr Walter Krenkel
on Google Scholar
Michael Krause Jan Marcel Hausherr Walter Krenkel

[Back to Top]