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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

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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

A. Faridani, E. Ritman, K. Smith and T. Kennan, Local tomography, SIAM, J. Appl. Math., 52 (1992), 459-484. doi: 10.1137/0152026.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[11]

A. K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), 175-190.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[17]

M. Van Ginkel, "Image Analysis Using Orientation Space Based on Steerable Filters," Ph.D thesis, Delft University of Technology, 2002. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

E. J. Candès, "Ridgelets: Theory and Applications," Ph.D. Thesis, Department of Statistics, Stanford University, 1998.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[5]

A. Faridani, E. Ritman, K. Smith and T. Kennan, Local tomography, SIAM, J. Appl. Math., 52 (1992), 459-484. doi: 10.1137/0152026.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[11]

A. K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), 175-190.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[17]

M. Van Ginkel, "Image Analysis Using Orientation Space Based on Steerable Filters," Ph.D thesis, Delft University of Technology, 2002. Google Scholar

[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.  Google Scholar

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