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ROI reconstruction from truncated conebeam projections
1.  Department of Mathematics, University of Houston, Houston, TX 772043008, USA 
2.  Department of Biomedical Informatics, Columbia University, New York, NY 10032, USA 
3.  Matematicas, Instituto Tecnologico Autonomo de México, 01080 Ciudad de México, CDMX, México 
RegionofInterest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only xray projections intersecting $C$ and it is useful to reduce overall radiation exposure when only a small specific region of a body needs to be examined. We consider xray acquisition from sources located on a smooth curve $Γ$ in $\mathbb R^3$ verifying the classical Tuy condition. In this generic situation, the nontrucated conebeam transform of smooth density functions $f$ admits an explicit inverse $Z$ as originally shown by Grangeat. However $Z$ cannot directly reconstruct $f$ from ROItruncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities $f$ in $L^{∞}(B)$ where $B$ is a bounded ball in $\mathbb R^3$, our method iterates an operator $U$ combining ROItruncated projections, inversion by the operator $Z$ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI $C \subset B$, given $ε >0$, we prove that if $C$ is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an $ε$accurate approximation of $f$ in $L^{∞}$. The accuracy depends on the regularity of $f$ quantified by its Sobolev norm in $W^5(B)$. Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an $ε$accurate approximation of $f$. We have numerically verified these theoretical results using simulated acquisition of ROItruncated conebeam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region $B$.
References:
[1] 
R. Alaifari, M. Defrise and A. Katsevich, Stability estimates for the regularized inversion of the truncated Hilbert transform, Inverse Problems, 32 (2016), 065005, 17 pp, arXiv: 1507.01141. 
[2] 
T. Aubin, Nonlinear Analysis on Manifolds: MongeAmpére Equations, Grundlehren der mathematischen Wissenschaften, Springer, New York, 1982. 
[3] 
R. Clackdoyle, F. Noo, A large class of inversion formulae for the 2d Radon transform of functions of compact support, Inverse Problems, 20 (2004), 12811291. 
[4] 
M. Defrise, R. Clack, A conebeam reconstruction algorithm using shiftvariant filtering and conebeam backprojection, Medical Imaging, IEEE Transactions on, 13 (1994), 186195. 
[5] 
M. Defrise, F. Noo, R. Clackdoyle, H. Kudo, Truncated Hilbert transform and image reconstruction from limited tomographic data, Inverse Problems, 22 (2006), 10371053. 
[6] 
L. Feldkamp, L. Davis, J. Kress, Practical conebeam algorithm, JOSA A, 1 (1984), 612619. 
[7] 
P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, in Mathematical Methods in Tomography (eds. G. Herman, K. Louis and F. Natterer), Lecture Notes in Mathematics, Springer Verlag, Berlin, 1497 (1991), 6697. 
[8] 
W. Han, H. Yu, G. Wang, A general total variation minimization theorem for compressed sensing based interior tomography, Journal of Biomedical Imaging, 2009 (2009), 13. 
[9] 
S. Helgason, The Radon transform on ${R}^n$, in Integral Geometry and Radon Transforms, Springer, New York, 2011, 162. 
[10] 
W. Huda, W. Randazzo, S. Tipnis, Embryo dose estimates in body CT, AJR Am. J. Roentgenol, 194 (2010), 874880. 
[11] 
C. Kamphuis, F. Beekman, Accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm, Medical Imaging, IEEE Transactions on, 17 (1998), 11011105. 
[12] 
A. Katsevich, Theoretically exact filtered backprojectiontype inversion algorithm for spiral CT, SIAM Journal on Applied Mathematics, 62 (2002), 20122026. 
[13] 
A. Katsevich, A general scheme for constructing inversion algorithms for cone beam CT, International Journal of Mathematics and Mathematical Sciences, 2003 (2003), 13051321. 
[14] 
A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics, 32 (2004), 681697. 
[15] 
A. Katsevich, Stability estimates for helical computer tomography, Journal of Fourier Analysis and Applications, 11 (2005), 85105. 
[16] 
E. Katsevich, A. Katsevich and G. Wang, Stability of the interior problem with polynomial attenuation in the region of interest, Inverse Problems, 28(2012), 065022, 28pp. 
[17] 
J. Kim, K. Y. Kwak, S.B. Park, Z. H. Cho, Projection space iteration reconstructionreprojection, Medical Imaging, IEEE Transactions on, 4 (1985), 139143. 
[18] 
E. Klann, E. Quinto and R. Ramlau, Wavelet methods for a weighted sparsity penalty for region of interest tomography, Inverse Problems, 31(2015), 025001, 22pp. 
[19] 
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Physics in Medicine and Biology, 53(2008), 2207. 
[20] 
C. I. Lee, A. H. Haims, E. P. Monico, Diagnostic CT scans: Assessment of patient, physician, and radiologist awareness of radiation dose and possible risks, Radiology, 231 (2004), 393398. 
[21] 
S. Mallat, A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edition, Academic Press, 2008. 
[22] 
M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, Iterative reconstructionreprojection: An algorithm for limited data cardiaccomputed tomography, Biomedical Engineering, IEEE Transactions on, 29 (1982), 333341. 
[23] 
F. Natterer, The Mathematics of Computerized Tomography, SIAM: Society for Industrial and Applied Mathematics, 2001. 
[24] 
F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction, SIAM: Society for Industrial and Applied Mathematics, 2001. 
[25] 
F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, Image reconstruction from fanbeam projections on less than a short scan, Physics in Medicine and Biology, 47 (2002), 25252546. 
[26] 
A. Sen, Searchlight {CT}: A New Regularized Reconstruction Method for Highly Collimated Xray Tomography, Ph. D. thesis, University of Houston, 2012. 
[27] 
H. Tuy, An inversion formula for conebeam reconstruction, SIAM Journal on Applied Mathematics, 43 (1983), 546552. 
[28] 
G. Wang, H. Yu, The meaning of interior tomography, Physics in Medicine and Biology, 58 (2013), 161186. 
[29] 
G. Yan, J. Tian, S. Zhu, C. Qin, Y. Dai, F. Yang, D. Dong, P. Wu, Fast Katsevich algorithm based on GPU for helical conebeam computed tomography, Information Technology in Biomedicine, IEEE Transactions on, 14 (2010), 10531061. 
[30] 
J. Yang, H. Yu, M. Jiang and G. Wang, Highorder total variation minimization for interior tomography, Inverse Problems, 26(2010), 035013, 29pp. 
[31] 
Y. Ye, H. Yu and G. Wang, Exact interior reconstruction with conebeam CT, International Journal of Biomedical Imaging, 2007(2007), Article ID 10693, 5 pages. doi: 10.1155/2007/10693. 
[32] 
Y. Ye, H. Yu, Y. Wei and G. Wang, A general local reconstruction approach based on a truncated Hilbert transform, International Journal of Biomedical Imaging, 2007(2007), Article ID 63634, 8 pages. doi: 10.1155/2007/63634. 
[33] 
H. Yu, G. Wang, Studies on implementation of the Katsevich algorithm for spiral conebeam CT, Journal of XRay Science and Technology, 12 (2004), 97116. 
[34] 
H. Yu, G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 27912805. 
[35] 
G. L. Zeng, R. Clack and G. T. Gullberg, Implementation of Tuy's conebeam inversion formula, Physics in Medicine and Biology, 39 (1994), p493. doi: 10.1088/00319155/39/3/014. 
[36] 
S. Zhao, H. Yu, G. Wang, A unified framework for exact conebeam reconstruction formulas, Medical Physics, 32 (2005), 17121721. doi: 10.1118/1.1869632. 
[37] 
A. Ziegler, T. Nielsen and M. Grass, Iterative reconstruction of a region of interest for transmission tomography, Medical Imaging 2006: Physics of Medical Imaging, 6142 (2006), 614223. doi: 10.1117/12.650666. 
show all references
References:
[1] 
R. Alaifari, M. Defrise and A. Katsevich, Stability estimates for the regularized inversion of the truncated Hilbert transform, Inverse Problems, 32 (2016), 065005, 17 pp, arXiv: 1507.01141. 
[2] 
T. Aubin, Nonlinear Analysis on Manifolds: MongeAmpére Equations, Grundlehren der mathematischen Wissenschaften, Springer, New York, 1982. 
[3] 
R. Clackdoyle, F. Noo, A large class of inversion formulae for the 2d Radon transform of functions of compact support, Inverse Problems, 20 (2004), 12811291. 
[4] 
M. Defrise, R. Clack, A conebeam reconstruction algorithm using shiftvariant filtering and conebeam backprojection, Medical Imaging, IEEE Transactions on, 13 (1994), 186195. 
[5] 
M. Defrise, F. Noo, R. Clackdoyle, H. Kudo, Truncated Hilbert transform and image reconstruction from limited tomographic data, Inverse Problems, 22 (2006), 10371053. 
[6] 
L. Feldkamp, L. Davis, J. Kress, Practical conebeam algorithm, JOSA A, 1 (1984), 612619. 
[7] 
P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, in Mathematical Methods in Tomography (eds. G. Herman, K. Louis and F. Natterer), Lecture Notes in Mathematics, Springer Verlag, Berlin, 1497 (1991), 6697. 
[8] 
W. Han, H. Yu, G. Wang, A general total variation minimization theorem for compressed sensing based interior tomography, Journal of Biomedical Imaging, 2009 (2009), 13. 
[9] 
S. Helgason, The Radon transform on ${R}^n$, in Integral Geometry and Radon Transforms, Springer, New York, 2011, 162. 
[10] 
W. Huda, W. Randazzo, S. Tipnis, Embryo dose estimates in body CT, AJR Am. J. Roentgenol, 194 (2010), 874880. 
[11] 
C. Kamphuis, F. Beekman, Accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm, Medical Imaging, IEEE Transactions on, 17 (1998), 11011105. 
[12] 
A. Katsevich, Theoretically exact filtered backprojectiontype inversion algorithm for spiral CT, SIAM Journal on Applied Mathematics, 62 (2002), 20122026. 
[13] 
A. Katsevich, A general scheme for constructing inversion algorithms for cone beam CT, International Journal of Mathematics and Mathematical Sciences, 2003 (2003), 13051321. 
[14] 
A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics, 32 (2004), 681697. 
[15] 
A. Katsevich, Stability estimates for helical computer tomography, Journal of Fourier Analysis and Applications, 11 (2005), 85105. 
[16] 
E. Katsevich, A. Katsevich and G. Wang, Stability of the interior problem with polynomial attenuation in the region of interest, Inverse Problems, 28(2012), 065022, 28pp. 
[17] 
J. Kim, K. Y. Kwak, S.B. Park, Z. H. Cho, Projection space iteration reconstructionreprojection, Medical Imaging, IEEE Transactions on, 4 (1985), 139143. 
[18] 
E. Klann, E. Quinto and R. Ramlau, Wavelet methods for a weighted sparsity penalty for region of interest tomography, Inverse Problems, 31(2015), 025001, 22pp. 
[19] 
H. Kudo, M. Courdurier, F. Noo and M. Defrise, Tiny a priori knowledge solves the interior problem in computed tomography, Physics in Medicine and Biology, 53(2008), 2207. 
[20] 
C. I. Lee, A. H. Haims, E. P. Monico, Diagnostic CT scans: Assessment of patient, physician, and radiologist awareness of radiation dose and possible risks, Radiology, 231 (2004), 393398. 
[21] 
S. Mallat, A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edition, Academic Press, 2008. 
[22] 
M. Nassi, W. R. Brody, B. P. Medoff, A. Macovski, Iterative reconstructionreprojection: An algorithm for limited data cardiaccomputed tomography, Biomedical Engineering, IEEE Transactions on, 29 (1982), 333341. 
[23] 
F. Natterer, The Mathematics of Computerized Tomography, SIAM: Society for Industrial and Applied Mathematics, 2001. 
[24] 
F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction, SIAM: Society for Industrial and Applied Mathematics, 2001. 
[25] 
F. Noo, M. Defrise, R. Clackdoyle, H. Kudo, Image reconstruction from fanbeam projections on less than a short scan, Physics in Medicine and Biology, 47 (2002), 25252546. 
[26] 
A. Sen, Searchlight {CT}: A New Regularized Reconstruction Method for Highly Collimated Xray Tomography, Ph. D. thesis, University of Houston, 2012. 
[27] 
H. Tuy, An inversion formula for conebeam reconstruction, SIAM Journal on Applied Mathematics, 43 (1983), 546552. 
[28] 
G. Wang, H. Yu, The meaning of interior tomography, Physics in Medicine and Biology, 58 (2013), 161186. 
[29] 
G. Yan, J. Tian, S. Zhu, C. Qin, Y. Dai, F. Yang, D. Dong, P. Wu, Fast Katsevich algorithm based on GPU for helical conebeam computed tomography, Information Technology in Biomedicine, IEEE Transactions on, 14 (2010), 10531061. 
[30] 
J. Yang, H. Yu, M. Jiang and G. Wang, Highorder total variation minimization for interior tomography, Inverse Problems, 26(2010), 035013, 29pp. 
[31] 
Y. Ye, H. Yu and G. Wang, Exact interior reconstruction with conebeam CT, International Journal of Biomedical Imaging, 2007(2007), Article ID 10693, 5 pages. doi: 10.1155/2007/10693. 
[32] 
Y. Ye, H. Yu, Y. Wei and G. Wang, A general local reconstruction approach based on a truncated Hilbert transform, International Journal of Biomedical Imaging, 2007(2007), Article ID 63634, 8 pages. doi: 10.1155/2007/63634. 
[33] 
H. Yu, G. Wang, Studies on implementation of the Katsevich algorithm for spiral conebeam CT, Journal of XRay Science and Technology, 12 (2004), 97116. 
[34] 
H. Yu, G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 27912805. 
[35] 
G. L. Zeng, R. Clack and G. T. Gullberg, Implementation of Tuy's conebeam inversion formula, Physics in Medicine and Biology, 39 (1994), p493. doi: 10.1088/00319155/39/3/014. 
[36] 
S. Zhao, H. Yu, G. Wang, A unified framework for exact conebeam reconstruction formulas, Medical Physics, 32 (2005), 17121721. doi: 10.1118/1.1869632. 
[37] 
A. Ziegler, T. Nielsen and M. Grass, Iterative reconstruction of a region of interest for transmission tomography, Medical Imaging 2006: Physics of Medical Imaging, 6142 (2006), 614223. doi: 10.1117/12.650666. 
Density data  ROI radius  Sources locations  
Spherical  Spiral  Circle  Twin circles  
SheppLogan  45 vox  10.3%  10.9%  13.2%  14.8% 
60 vox  8.6%  9.1%  11.6%  14.7%  
75 vox  7.6%  8.3%  7.4%  8.9%  
90 vox  7.3%  8.0%  4.4%  4.8%  
Mouse tissue  45 vox  10.8%  11.4%  11.6%  12.5% 
60 vox  8.8%  9.7%  11.1%  9.4%  
75 vox  7.9%  8.8%  8.4%  8.3%  
90 vox  7.5%  8.4%  7.1%  7.8%  
Human jaw  45 vox  11.4%  11.9%  12.9%  15.0% 
60 vox  9.6%  10.8%  12.8%  13.3%  
75 vox  9.0%  9.7%  10.2%  10.2%  
90 vox  8.2%  8.5%  9.8%  9.8% 
Density data  ROI radius  Sources locations  
Spherical  Spiral  Circle  Twin circles  
SheppLogan  45 vox  10.3%  10.9%  13.2%  14.8% 
60 vox  8.6%  9.1%  11.6%  14.7%  
75 vox  7.6%  8.3%  7.4%  8.9%  
90 vox  7.3%  8.0%  4.4%  4.8%  
Mouse tissue  45 vox  10.8%  11.4%  11.6%  12.5% 
60 vox  8.8%  9.7%  11.1%  9.4%  
75 vox  7.9%  8.8%  8.4%  8.3%  
90 vox  7.5%  8.4%  7.1%  7.8%  
Human jaw  45 vox  11.4%  11.9%  12.9%  15.0% 
60 vox  9.6%  10.8%  12.8%  13.3%  
75 vox  9.0%  9.7%  10.2%  10.2%  
90 vox  8.2%  8.5%  9.8%  9.8% 
Density data  Source locations  
Spherical  Spiral  Circle  Twin circles  
SheppLogan  52 vox  56 vox  67 vox  73 vox 
Mouse tissue  52 vox  57 vox  66 vox  49 vox 
Human jaw  57 vox  70 vox  82 vox  82 vox 
Density data  Source locations  
Spherical  Spiral  Circle  Twin circles  
SheppLogan  52 vox  56 vox  67 vox  73 vox 
Mouse tissue  52 vox  57 vox  66 vox  49 vox 
Human jaw  57 vox  70 vox  82 vox  82 vox 
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