# American Institute of Mathematical Sciences

November  2010, 4(4): 675-683. doi: 10.3934/ipi.2010.4.675

## Diffusion reconstruction from very noisy tomographic data

 1 Department of Mathematics, Saarland University, D-66041 Saarbrucken, Germany

Received  January 2009 Published  September 2010

As a consequence of very noisy tomographic data the reconstructed images are contaminated by severely amplified noise. Typically two remedies are considered. Firstly, the data are smoothed, this is called pre-whitening in the engineering literature. The disadvantage here is that the individually treated data sets could become inconsistent. Secondly, the image, reconstructed from the original data sets, is treated by methods of image smoothing. As example diffusion filters are mentioned. In this paper we present a method where the reconstruction of the smoothed image is performed in one step; i.e., we develop special reconstruction kernels, which directly compute the image smoothed by a diffusion filter. Examples from synthetic data are presented.
Citation: Alfred K. Louis. Diffusion reconstruction from very noisy tomographic data. Inverse Problems and Imaging, 2010, 4 (4) : 675-683. doi: 10.3934/ipi.2010.4.675
##### References:
 [1] M. Abramowith and I. Stegun, "Handbook of Mathematical Functions," Dover, New York, 1972. [2] S. Bonnet, F. Peyrin, F. Turjman and R. Prost, Multiresolution reconstruction in Fan-Beam tomography, IEEE Transactions on Image Processing, 11 (2002), 169-176. doi: doi:10.1109/83.988951. [3] J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: doi:10.1215/S0012-7094-87-05547-5. [4] J. F. Canny, A computational approach to edge detection, IEEE TPAMI, 8 (1986), 679-698. [5] D. Fanelli and O. Öktem, Electron tomography: a short overview with an emphasis on the absorption potential of the forward problem, Inverse Problems, 24 (2008), 013001(51pp). [6] A. Katsevich, Improved cone beam local tomography, Inverse Problems, 22 (2006), 627-643. doi: doi:10.1088/0266-5611/22/2/015. [7] M. K. Likht, On the calculation of functionals in the solution of linear equations of the first kind, Comput. Math. Math. Phys. (USSR), 7 (1967), 271-278. doi: doi:10.1016/0041-5553(67)90046-8. [8] A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind, Inverse Problems, 6 (1990), 427-440. doi: doi:10.1088/0266-5611/6/3/011. [9] A. K. Louis and P. Maass, Contour reconstruction in 3-D X-ray CT, IEEE Transactions on Medical Imaging, 12 (1993), 764-769. doi: doi:10.1109/42.251129. [10] A. K. Louis, P. Maass and A. Rieder, "Wavelets," Teubner, Stuttgart, 1994 (English translation: Wiley, Chichester, 1997). [11] A. K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), 175-190. doi: doi:10.1088/0266-5611/12/2/005. [12] A. K. Louis, A unified approach to regularization methods for linear ill-posed problems, Inverse Problems, 15 (1999), 489-498. [13] A. K. Louis, Combining image reconstruction and image analysis with an application to 2D tomography, SIAM J. Imaging Sciences, 1 (2008), 188-208. doi: doi:10.1137/070700863. [14] A. K. Louis and E. T. Quinto, Local tomographic methods in SONAR, in "Surveys on Solution Methods for Inverse Problems" (D. Colton, H. W. Engl, A. K. Louis and J. McLaughlin, editors), 147-155, Springer, Wien/New York, 2000. [15] F. Natterer, "The Mathematics of Computerized Tomography," Classics in Applied Mathematics, Philadelphia, SIAM, 2001. [16] S. Oeckl, T. Schön, A. Knauf and A. K. Louis, "Multiresolution 3D-computerized Tomography and its Application to NDT," Proc. ECNDT, 9, 2006. [17] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: doi:10.1109/34.56205. [18] E. T. Quinto and O. Öktem, Local tomography in electron microscopy, SIAM J. Appl. Math., 68 (2008), 1282-1303. doi: doi:10.1137/07068326X. [19] K. T. Smith, Inversion of the x-ray transform, SIAM-AMS Proc., 14 (1984), 41-52. [20] E. Vainberg and I. A. Kazak and V. P. Kurozaev, Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections, Soviet J Nondestructive Testing, 17 (1981), 415-423. [21] J. Weickert, "Anisotropic Diffusion in Image Processing," Teubner: Stuttgart, 1998.

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##### References:
 [1] M. Abramowith and I. Stegun, "Handbook of Mathematical Functions," Dover, New York, 1972. [2] S. Bonnet, F. Peyrin, F. Turjman and R. Prost, Multiresolution reconstruction in Fan-Beam tomography, IEEE Transactions on Image Processing, 11 (2002), 169-176. doi: doi:10.1109/83.988951. [3] J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: doi:10.1215/S0012-7094-87-05547-5. [4] J. F. Canny, A computational approach to edge detection, IEEE TPAMI, 8 (1986), 679-698. [5] D. Fanelli and O. Öktem, Electron tomography: a short overview with an emphasis on the absorption potential of the forward problem, Inverse Problems, 24 (2008), 013001(51pp). [6] A. Katsevich, Improved cone beam local tomography, Inverse Problems, 22 (2006), 627-643. doi: doi:10.1088/0266-5611/22/2/015. [7] M. K. Likht, On the calculation of functionals in the solution of linear equations of the first kind, Comput. Math. Math. Phys. (USSR), 7 (1967), 271-278. doi: doi:10.1016/0041-5553(67)90046-8. [8] A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind, Inverse Problems, 6 (1990), 427-440. doi: doi:10.1088/0266-5611/6/3/011. [9] A. K. Louis and P. Maass, Contour reconstruction in 3-D X-ray CT, IEEE Transactions on Medical Imaging, 12 (1993), 764-769. doi: doi:10.1109/42.251129. [10] A. K. Louis, P. Maass and A. Rieder, "Wavelets," Teubner, Stuttgart, 1994 (English translation: Wiley, Chichester, 1997). [11] A. K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), 175-190. doi: doi:10.1088/0266-5611/12/2/005. [12] A. K. Louis, A unified approach to regularization methods for linear ill-posed problems, Inverse Problems, 15 (1999), 489-498. [13] A. K. Louis, Combining image reconstruction and image analysis with an application to 2D tomography, SIAM J. Imaging Sciences, 1 (2008), 188-208. doi: doi:10.1137/070700863. [14] A. K. Louis and E. T. Quinto, Local tomographic methods in SONAR, in "Surveys on Solution Methods for Inverse Problems" (D. Colton, H. W. Engl, A. K. Louis and J. McLaughlin, editors), 147-155, Springer, Wien/New York, 2000. [15] F. Natterer, "The Mathematics of Computerized Tomography," Classics in Applied Mathematics, Philadelphia, SIAM, 2001. [16] S. Oeckl, T. Schön, A. Knauf and A. K. Louis, "Multiresolution 3D-computerized Tomography and its Application to NDT," Proc. ECNDT, 9, 2006. [17] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. on Pattern Analysis and Machine Intelligence, 12 (1990), 629-639. doi: doi:10.1109/34.56205. [18] E. T. Quinto and O. Öktem, Local tomography in electron microscopy, SIAM J. Appl. Math., 68 (2008), 1282-1303. doi: doi:10.1137/07068326X. [19] K. T. Smith, Inversion of the x-ray transform, SIAM-AMS Proc., 14 (1984), 41-52. [20] E. Vainberg and I. A. Kazak and V. P. Kurozaev, Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections, Soviet J Nondestructive Testing, 17 (1981), 415-423. [21] J. Weickert, "Anisotropic Diffusion in Image Processing," Teubner: Stuttgart, 1998.
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