# American Institute of Mathematical Sciences

February  2018, 12(1): 29-57. doi: 10.3934/ipi.2018002

## ROI reconstruction from truncated cone-beam projections

 1 Department of Mathematics, University of Houston, Houston, TX 77204-3008, 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

* Corresponding author

Received  November 2016 Revised  September 2017 Published  December 2017

Region-of-Interest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only x-ray 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 x-ray acquisition from sources located on a smooth curve $Γ$ in $\mathbb R^3$ verifying the classical Tuy condition. In this generic situation, the non-trucated cone-beam transform of smooth density functions $f$ admits an explicit inverse $Z$ as originally shown by Grangeat. However $Z$ cannot directly reconstruct $f$ from ROI-truncated 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 ROI-truncated 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 ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region $B$.

Citation: Robert Azencott, Bernhard G. Bodmann, Tasadduk Chowdhury, Demetrio Labate, Anando Sen, Daniel Vera. ROI reconstruction from truncated cone-beam projections. Inverse Problems & Imaging, 2018, 12 (1) : 29-57. doi: 10.3934/ipi.2018002
##### References:
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Kress, Practical cone-beam algorithm, JOSA A, 1 (1984), 612-619.   Google Scholar [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), 66-97.  Google Scholar [8] W. Han, H. Yu and G. Wang, A general total variation minimization theorem for compressed sensing based interior tomography, Journal of Biomedical Imaging, 2009 (2009), 1-3.   Google Scholar [9] S. Helgason, The Radon transform on ${R}^n$, in Integral Geometry and Radon Transforms, Springer, New York, 2011, 1-62. Google Scholar [10] W. Huda, W. Randazzo and S. Tipnis, Embryo dose estimates in body CT, AJR Am. J. Roentgenol, 194 (2010), 874-880.   Google Scholar [11] C. Kamphuis and F. Beekman, Accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm, Medical Imaging, IEEE Transactions on, 17 (1998), 1101-1105.   Google Scholar [12] A. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, SIAM Journal on Applied Mathematics, 62 (2002), 2012-2026.   Google Scholar [13] A. Katsevich, A general scheme for constructing inversion algorithms for cone beam CT, International Journal of Mathematics and Mathematical Sciences, 2003 (2003), 1305-1321.   Google Scholar [14] A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics, 32 (2004), 681-697.   Google Scholar [15] A. Katsevich, Stability estimates for helical computer tomography, Journal of Fourier Analysis and Applications, 11 (2005), 85-105.   Google Scholar [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.  Google Scholar [17] J. Kim, K. Y. Kwak, S.-B. Park and Z. H. Cho, Projection space iteration reconstruction-reprojection, Medical Imaging, IEEE Transactions on, 4 (1985), 139-143.   Google Scholar [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.  Google Scholar [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. Google Scholar [20] C. I. Lee, A. H. Haims and E. P. Monico, Diagnostic CT scans: Assessment of patient, physician, and radiologist awareness of radiation dose and possible risks, Radiology, 231 (2004), 393-398.   Google Scholar [21] S. Mallat, A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edition, Academic Press, 2008.  Google Scholar [22] M. Nassi, W. R. Brody, B. P. Medoff and A. Macovski, Iterative reconstruction-reprojection: An algorithm for limited data cardiac-computed tomography, Biomedical Engineering, IEEE Transactions on, 29 (1982), 333-341.   Google Scholar [23] F. Natterer, The Mathematics of Computerized Tomography, SIAM: Society for Industrial and Applied Mathematics, 2001.  Google Scholar [24] F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction, SIAM: Society for Industrial and Applied Mathematics, 2001.  Google Scholar [25] F. Noo, M. Defrise, R. Clackdoyle and H. Kudo, Image reconstruction from fan-beam projections on less than a short scan, Physics in Medicine and Biology, 47 (2002), 2525-2546.   Google Scholar [26] A. Sen, Searchlight {CT}: A New Regularized Reconstruction Method for Highly Collimated X-ray Tomography, Ph. D. thesis, University of Houston, 2012. Google Scholar [27] H. Tuy, An inversion formula for cone-beam reconstruction, SIAM Journal on Applied Mathematics, 43 (1983), 546-552.   Google Scholar [28] G. Wang and H. Yu, The meaning of interior tomography, Physics in Medicine and Biology, 58 (2013), 161-186.   Google Scholar [29] G. Yan, J. Tian, S. Zhu, C. Qin, Y. Dai, F. Yang, D. Dong and P. Wu, Fast Katsevich algorithm based on GPU for helical cone-beam computed tomography, Information Technology in Biomedicine, IEEE Transactions on, 14 (2010), 1053-1061.   Google Scholar [30] J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26(2010), 035013, 29pp.  Google Scholar [31] Y. Ye, H. Yu and G. Wang, Exact interior reconstruction with cone-beam CT, International Journal of Biomedical Imaging, 2007(2007), Article ID 10693, 5 pages. doi: 10.1155/2007/10693.  Google Scholar [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.  Google Scholar [33] H. Yu and G. Wang, Studies on implementation of the Katsevich algorithm for spiral cone-beam CT, Journal of X-Ray Science and Technology, 12 (2004), 97-116.   Google Scholar [34] H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 2791-2805.   Google Scholar [35] G. L. Zeng, R. Clack and G. T. Gullberg, Implementation of Tuy's cone-beam inversion formula, Physics in Medicine and Biology, 39 (1994), p493. doi: 10.1088/0031-9155/39/3/014.  Google Scholar [36] S. Zhao, H. Yu and G. Wang, A unified framework for exact cone-beam reconstruction formulas, Medical Physics, 32 (2005), 1712-1721.  doi: 10.1118/1.1869632.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] T. Aubin, Nonlinear Analysis on Manifolds: Monge-Ampére Equations, Grundlehren der mathematischen Wissenschaften, Springer, New York, 1982.  Google Scholar [3] R. Clackdoyle and F. Noo, A large class of inversion formulae for the 2-d Radon transform of functions of compact support, Inverse Problems, 20 (2004), 1281-1291.   Google Scholar [4] M. Defrise and R. Clack, A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection, Medical Imaging, IEEE Transactions on, 13 (1994), 186-195.   Google Scholar [5] M. Defrise, F. Noo, R. Clackdoyle and H. Kudo, Truncated Hilbert transform and image reconstruction from limited tomographic data, Inverse Problems, 22 (2006), 1037-1053.   Google Scholar [6] L. Feldkamp, L. Davis and J. Kress, Practical cone-beam algorithm, JOSA A, 1 (1984), 612-619.   Google Scholar [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), 66-97.  Google Scholar [8] W. Han, H. Yu and G. Wang, A general total variation minimization theorem for compressed sensing based interior tomography, Journal of Biomedical Imaging, 2009 (2009), 1-3.   Google Scholar [9] S. Helgason, The Radon transform on ${R}^n$, in Integral Geometry and Radon Transforms, Springer, New York, 2011, 1-62. Google Scholar [10] W. Huda, W. Randazzo and S. Tipnis, Embryo dose estimates in body CT, AJR Am. J. Roentgenol, 194 (2010), 874-880.   Google Scholar [11] C. Kamphuis and F. Beekman, Accelerated iterative transmission ct reconstruction using an ordered subsets convex algorithm, Medical Imaging, IEEE Transactions on, 17 (1998), 1101-1105.   Google Scholar [12] A. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, SIAM Journal on Applied Mathematics, 62 (2002), 2012-2026.   Google Scholar [13] A. Katsevich, A general scheme for constructing inversion algorithms for cone beam CT, International Journal of Mathematics and Mathematical Sciences, 2003 (2003), 1305-1321.   Google Scholar [14] A. Katsevich, An improved exact filtered backprojection algorithm for spiral computed tomography, Advances in Applied Mathematics, 32 (2004), 681-697.   Google Scholar [15] A. Katsevich, Stability estimates for helical computer tomography, Journal of Fourier Analysis and Applications, 11 (2005), 85-105.   Google Scholar [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.  Google Scholar [17] J. Kim, K. Y. Kwak, S.-B. Park and Z. H. Cho, Projection space iteration reconstruction-reprojection, Medical Imaging, IEEE Transactions on, 4 (1985), 139-143.   Google Scholar [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.  Google Scholar [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. Google Scholar [20] C. I. Lee, A. H. Haims and E. P. Monico, Diagnostic CT scans: Assessment of patient, physician, and radiologist awareness of radiation dose and possible risks, Radiology, 231 (2004), 393-398.   Google Scholar [21] S. Mallat, A Wavelet Tour of Signal Processing. The Sparse Way, 3rd edition, Academic Press, 2008.  Google Scholar [22] M. Nassi, W. R. Brody, B. P. Medoff and A. Macovski, Iterative reconstruction-reprojection: An algorithm for limited data cardiac-computed tomography, Biomedical Engineering, IEEE Transactions on, 29 (1982), 333-341.   Google Scholar [23] F. Natterer, The Mathematics of Computerized Tomography, SIAM: Society for Industrial and Applied Mathematics, 2001.  Google Scholar [24] F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction, SIAM: Society for Industrial and Applied Mathematics, 2001.  Google Scholar [25] F. Noo, M. Defrise, R. Clackdoyle and H. Kudo, Image reconstruction from fan-beam projections on less than a short scan, Physics in Medicine and Biology, 47 (2002), 2525-2546.   Google Scholar [26] A. Sen, Searchlight {CT}: A New Regularized Reconstruction Method for Highly Collimated X-ray Tomography, Ph. D. thesis, University of Houston, 2012. Google Scholar [27] H. Tuy, An inversion formula for cone-beam reconstruction, SIAM Journal on Applied Mathematics, 43 (1983), 546-552.   Google Scholar [28] G. Wang and H. Yu, The meaning of interior tomography, Physics in Medicine and Biology, 58 (2013), 161-186.   Google Scholar [29] G. Yan, J. Tian, S. Zhu, C. Qin, Y. Dai, F. Yang, D. Dong and P. Wu, Fast Katsevich algorithm based on GPU for helical cone-beam computed tomography, Information Technology in Biomedicine, IEEE Transactions on, 14 (2010), 1053-1061.   Google Scholar [30] J. Yang, H. Yu, M. Jiang and G. Wang, High-order total variation minimization for interior tomography, Inverse Problems, 26(2010), 035013, 29pp.  Google Scholar [31] Y. Ye, H. Yu and G. Wang, Exact interior reconstruction with cone-beam CT, International Journal of Biomedical Imaging, 2007(2007), Article ID 10693, 5 pages. doi: 10.1155/2007/10693.  Google Scholar [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.  Google Scholar [33] H. Yu and G. Wang, Studies on implementation of the Katsevich algorithm for spiral cone-beam CT, Journal of X-Ray Science and Technology, 12 (2004), 97-116.   Google Scholar [34] H. Yu and G. Wang, Compressed sensing based interior tomography, Physics in Medicine and Biology, 54 (2009), 2791-2805.   Google Scholar [35] G. L. Zeng, R. Clack and G. T. Gullberg, Implementation of Tuy's cone-beam inversion formula, Physics in Medicine and Biology, 39 (1994), p493. doi: 10.1088/0031-9155/39/3/014.  Google Scholar [36] S. Zhao, H. Yu and G. Wang, A unified framework for exact cone-beam reconstruction formulas, Medical Physics, 32 (2005), 1712-1721.  doi: 10.1118/1.1869632.  Google Scholar [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.  Google Scholar
ROI-truncated cone-beam acquisition: projections are restricted to half-rays intersecting the ROI, which is a ball $C$ included in the target ball $B$
Visual comparison of ROI reconstruction for 3D Shepp-Logan phantom using simulated Twin Circles acquisition and truncation of projection data. ROI radius =45 voxels. Middles sections are shown from the $xy$, $yz$ and $xz$ planes. From left to right: inversion by one-step Grangeat formula; our iterative ROI reconstruction; ground truth. The last column shows intensity profiles corresponding to the middle row of the images. Green: one-step Grangeat formula; blue: our algorithm; red: ground truth
Visual comparison of ROI reconstruction for Mouse Tissue data using simulated Twin Circles acquisition and truncation of projection data. ROI radius =45 voxels. Middles sections are shown from the $xy$, $yz$ and $xz$ planes. From left to right: inversion by one-step Grangeat's formula; our iterative ROI reconstruction; ground truth. The last column shows intensity profiles corresponding to the middle row of the images. Green: one-step Grangeat formula; blue: our algorithm; red: ground truth
Visual comparison of ROI reconstruction for 3D Shepp-Logan phantom and mouse tissue using simulated spiral acquisition and truncation of projection data. A representative horizontal section from the 3D reconstructed volume is shown. From left to right: inversion by one-step Katsevich formula; our iterative ROI reconstruction; ground truth
Visual comparison of ROI reconstruction for 3D Shepp-Logan phantom and mouse tissue using simulated C-arm acquisition and truncation of projection data. A representative horizontal section from the 3D reconstructed volume is shown. From left to right: inversion by one-step FDK algorithm; our iterative ROI reconstruction; ground truth
Relative $L^1$ error of ROI reconstruction
 Density data ROI radius Sources locations Spherical Spiral Circle Twin circles Shepp-Logan 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 Shepp-Logan 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 Shepp-Logan 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 Shepp-Logan 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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