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A new nonlocal variational setting for image processing
Empirical average-case relation between undersampling and sparsity in X-ray CT
1. | Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, Building 324, 2800 Kgs. Lyngby, Denmark, Denmark |
2. | Department of Radiology, University of Chicago, 5841 South Maryland Avenue, Chicago, IL 60637, United States, United States |
References:
[1] |
H. H. Barrett and K. J. Myers, Foundations of Image Science, John Wiley & Sons, Hoboken, NJ, 2004. |
[2] |
J. Bian, J. H. Siewerdsen, X. Han, E. Y. Sidky, J. L. Prince, C. A. Pelizzari and X. Pan, Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT, Phys. Med. Biol., 55 (2010), 6575-6599.
doi: 10.1088/0031-9155/55/22/001. |
[3] |
E. Candès and J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Probl., 23 (2007), 969-985.
doi: 10.1088/0266-5611/23/3/008. |
[4] |
E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509.
doi: 10.1109/TIT.2005.862083. |
[5] |
E. J. Candès, J. K. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59 (2006), 1207-1223.
doi: 10.1002/cpa.20124. |
[6] |
A. Cohen, W. Dahmen and R. DeVore, Compressed sensing and best k-term approximation, J. Am. Math. Soc., 22 (2009), 211-231.
doi: 10.1090/S0894-0347-08-00610-3. |
[7] |
D. Donoho and J. Tanner, Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 4273-4293.
doi: 10.1098/rsta.2009.0152. |
[8] |
D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306.
doi: 10.1109/TIT.2006.871582. |
[9] |
D. L. Donoho and M. Elad, Optimally sparse representation in general (non-orthogonal) dictionaries via L1 minimization, Proc. Natl. Acad. Sci. U.S.A., 100 (2003), 2197-2202.
doi: 10.1073/pnas.0437847100. |
[10] |
D. L. Donoho and J. Tanner, Sparse nonnegative solution of underdetermined linear equations by linear programming, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 9446-9451.
doi: 10.1073/pnas.0502269102. |
[11] |
D. Donoho and J. Tanner, Neighborliness of randomly projected simplices in high dimensions, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 9452-9457.
doi: 10.1073/pnas.0502258102. |
[12] |
C. Dossal, G. Peyré and J. Fadili, A numerical exploration of compressed sampling recovery, Linear Algebra Appl., 432 (2010), 1663-1679.
doi: 10.1016/j.laa.2009.11.022. |
[13] |
L. A. Feldkamp, L. C. Davis and J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A, 1 (1984), 612-619.
doi: 10.1364/JOSAA.1.000612. |
[14] |
S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Springer, New York, NY, 2013.
doi: 10.1007/978-0-8176-4948-7. |
[15] |
E. Gouillart, F. Krzakala, M. Mézard and L. Zdeborová, Belief-propagation reconstruction for discrete tomography, Inverse Probl., 29 (2013), 035003, 22pp.
doi: 10.1088/0266-5611/29/3/035003. |
[16] |
M. Grasmair, M. Haltmeier and O. Scherzer, Necessary and sufficient conditions for linear convergence of L1-regularization, Commun. Pure Appl. Math., 64 (2011), 161-182.
doi: 10.1002/cpa.20350. |
[17] |
J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, CT, 1953.
doi: 10.1063/1.3061337. |
[18] |
X. Han, J. Bian, D. R. Eaker, T. L. Kline, E. Y. Sidky, E. L. Ritman and X. Pan, Algorithm-enabled low-dose micro-CT imaging, IEEE Trans. Med. Imaging, 30 (2011), 606-620.
doi: 10.1109/TMI.2010.2089695. |
[19] |
P. C. Hansen and M. Saxild-Hansen, AIR Tools - A MATLAB package of algebraic iterative reconstruction methods, J. Comput. Appl. Math., 236 (2012), 2167-2178.
doi: 10.1016/j.cam.2011.09.039. |
[20] |
P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical aspects of linear inversion, SIAM, Philadelphia, PA, 1998.
doi: 10.1137/1.9780898719697. |
[21] |
G. T. Herman and A. Kuba (eds.), Discrete Tomography: Foundations, Algorithms, and Applications, Springer, New York, NY, 1999.
doi: 10.1007/978-1-4612-1568-4. |
[22] |
J. S. Jørgensen and E. Y. Sidky, How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., |
[23] |
J. S. Jørgensen, E. Y. Sidky and X. Pan, Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in x-ray CT, IEEE Trans. Med. Imaging, 32 (2013), 460-473.
doi: 10.1109/TMI.2012.2230185. |
[24] |
J. Jørgensen, C. Kruschel and D. Lorenz, Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized X-ray CT, Inverse Probl. Sci. Eng., (2014), 1-23.
doi: 10.1080/17415977.2014.986724. |
[25] |
M. Li, H. Yang and H. Kudo, An accurate iterative reconstruction algorithm for sparse objects: application to 3D blood vessel reconstruction from a limited number of projections, Phys. Med. Biol., 47 (2002), 2599-2609.
doi: 10.1088/0031-9155/47/15/303. |
[26] |
H. Monajemi, S. Jafarpour, M. Gavish, Stat-330-CME-362 Collaboration and D. L. Donoho, Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices, Proc. Natl. Acad. Sci. U.S.A., 110 (2013), 1181-1186.
doi: 10.1073/pnas.1219540110. |
[27] |
MOSEK ApS, MOSEK Optimization Software, version 6.0.0.122, 2011. Available from: http://www.mosek.com. |
[28] |
F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, New York, NY, 1986. |
[29] |
D. Needell and R. Ward, Stable image reconstruction using total variation minimization, SIAM J. Imaging Sci., 6 (2013), 1035-1058.
doi: 10.1137/120868281. |
[30] |
X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Probl., 25 (2009), 123009, 36pp.
doi: 10.1088/0266-5611/25/12/123009. |
[31] |
S. Petra and C. Schnörr, Average case recovery analysis of tomographic compressive sensing, Linear Algebra Appl., 441 (2014), 168-198.
doi: 10.1016/j.laa.2013.06.034. |
[32] |
N. Pustelnik, C. Dossal, F. Turcu, Y. Berthoumieu and P. Ricoux, A greedy algorithm to extract sparsity degree for L1/L0-equivalence in a deterministic context, in Proc. EUSIPCO, Bucharest, Romania, 2012. |
[33] |
I. Reiser and R. M. Nishikawa, Task-based assessment of breast tomosynthesis: Effect of acquisition parameters and quantum noise, Med. Phys., 37 (2010), 1591-1600.
doi: 10.1118/1.3357288. |
[34] |
L. Ritschl, F. Bergner, C. Fleischmann and M. Kachelrieß, Improved total variation-based CT image reconstruction applied to clinical data, Phys. Med. Biol., 56 (2011), 1545-1561.
doi: 10.1088/0031-9155/56/6/003. |
[35] |
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[36] |
E. Y. Sidky, M. A. Anastasio and X. Pan, Image reconstruction exploiting object sparsity in boundary-enhanced X-ray phase-contrast tomography, Opt. Express, 18 (2010), 10404-10422.
doi: 10.1364/OE.18.010404. |
[37] |
E. Y. Sidky, C.-M. Kao and X. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, J. Xray Sci. Technol., 14 (2006), 119-139. |
[38] |
E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008), 4777-4807.
doi: 10.1088/0031-9155/53/17/021. |
[39] |
A. M. Tillmann and M. E. Pfetsch, The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing, IEEE Trans. Inf. Theory, 60 (2014), 1248-1259.
doi: 10.1109/TIT.2013.2290112. |
[40] |
L. Yu, X. Liu, S. Leng, J. M. Kofler, J. C. Ramirez-Giraldo, M. Qu, J. Christner, J. G. Fletcher and C. H. McCollough, Radiation dose reduction in computed tomography: Techniques and future perspective, Imaging Med., 1 (2009), 65-84.
doi: 10.2217/iim.09.5. |
show all references
References:
[1] |
H. H. Barrett and K. J. Myers, Foundations of Image Science, John Wiley & Sons, Hoboken, NJ, 2004. |
[2] |
J. Bian, J. H. Siewerdsen, X. Han, E. Y. Sidky, J. L. Prince, C. A. Pelizzari and X. Pan, Evaluation of sparse-view reconstruction from flat-panel-detector cone-beam CT, Phys. Med. Biol., 55 (2010), 6575-6599.
doi: 10.1088/0031-9155/55/22/001. |
[3] |
E. Candès and J. Romberg, Sparsity and incoherence in compressive sampling, Inverse Probl., 23 (2007), 969-985.
doi: 10.1088/0266-5611/23/3/008. |
[4] |
E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, 52 (2006), 489-509.
doi: 10.1109/TIT.2005.862083. |
[5] |
E. J. Candès, J. K. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59 (2006), 1207-1223.
doi: 10.1002/cpa.20124. |
[6] |
A. Cohen, W. Dahmen and R. DeVore, Compressed sensing and best k-term approximation, J. Am. Math. Soc., 22 (2009), 211-231.
doi: 10.1090/S0894-0347-08-00610-3. |
[7] |
D. Donoho and J. Tanner, Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 367 (2009), 4273-4293.
doi: 10.1098/rsta.2009.0152. |
[8] |
D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306.
doi: 10.1109/TIT.2006.871582. |
[9] |
D. L. Donoho and M. Elad, Optimally sparse representation in general (non-orthogonal) dictionaries via L1 minimization, Proc. Natl. Acad. Sci. U.S.A., 100 (2003), 2197-2202.
doi: 10.1073/pnas.0437847100. |
[10] |
D. L. Donoho and J. Tanner, Sparse nonnegative solution of underdetermined linear equations by linear programming, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 9446-9451.
doi: 10.1073/pnas.0502269102. |
[11] |
D. Donoho and J. Tanner, Neighborliness of randomly projected simplices in high dimensions, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 9452-9457.
doi: 10.1073/pnas.0502258102. |
[12] |
C. Dossal, G. Peyré and J. Fadili, A numerical exploration of compressed sampling recovery, Linear Algebra Appl., 432 (2010), 1663-1679.
doi: 10.1016/j.laa.2009.11.022. |
[13] |
L. A. Feldkamp, L. C. Davis and J. W. Kress, Practical cone-beam algorithm, J. Opt. Soc. Am. A, 1 (1984), 612-619.
doi: 10.1364/JOSAA.1.000612. |
[14] |
S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Springer, New York, NY, 2013.
doi: 10.1007/978-0-8176-4948-7. |
[15] |
E. Gouillart, F. Krzakala, M. Mézard and L. Zdeborová, Belief-propagation reconstruction for discrete tomography, Inverse Probl., 29 (2013), 035003, 22pp.
doi: 10.1088/0266-5611/29/3/035003. |
[16] |
M. Grasmair, M. Haltmeier and O. Scherzer, Necessary and sufficient conditions for linear convergence of L1-regularization, Commun. Pure Appl. Math., 64 (2011), 161-182.
doi: 10.1002/cpa.20350. |
[17] |
J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, CT, 1953.
doi: 10.1063/1.3061337. |
[18] |
X. Han, J. Bian, D. R. Eaker, T. L. Kline, E. Y. Sidky, E. L. Ritman and X. Pan, Algorithm-enabled low-dose micro-CT imaging, IEEE Trans. Med. Imaging, 30 (2011), 606-620.
doi: 10.1109/TMI.2010.2089695. |
[19] |
P. C. Hansen and M. Saxild-Hansen, AIR Tools - A MATLAB package of algebraic iterative reconstruction methods, J. Comput. Appl. Math., 236 (2012), 2167-2178.
doi: 10.1016/j.cam.2011.09.039. |
[20] |
P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical aspects of linear inversion, SIAM, Philadelphia, PA, 1998.
doi: 10.1137/1.9780898719697. |
[21] |
G. T. Herman and A. Kuba (eds.), Discrete Tomography: Foundations, Algorithms, and Applications, Springer, New York, NY, 1999.
doi: 10.1007/978-1-4612-1568-4. |
[22] |
J. S. Jørgensen and E. Y. Sidky, How little data is enough? Phase-diagram analysis of sparsity-regularized X-ray CT, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., |
[23] |
J. S. Jørgensen, E. Y. Sidky and X. Pan, Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in x-ray CT, IEEE Trans. Med. Imaging, 32 (2013), 460-473.
doi: 10.1109/TMI.2012.2230185. |
[24] |
J. Jørgensen, C. Kruschel and D. Lorenz, Testable uniqueness conditions for empirical assessment of undersampling levels in total variation-regularized X-ray CT, Inverse Probl. Sci. Eng., (2014), 1-23.
doi: 10.1080/17415977.2014.986724. |
[25] |
M. Li, H. Yang and H. Kudo, An accurate iterative reconstruction algorithm for sparse objects: application to 3D blood vessel reconstruction from a limited number of projections, Phys. Med. Biol., 47 (2002), 2599-2609.
doi: 10.1088/0031-9155/47/15/303. |
[26] |
H. Monajemi, S. Jafarpour, M. Gavish, Stat-330-CME-362 Collaboration and D. L. Donoho, Deterministic matrices matching the compressed sensing phase transitions of Gaussian random matrices, Proc. Natl. Acad. Sci. U.S.A., 110 (2013), 1181-1186.
doi: 10.1073/pnas.1219540110. |
[27] |
MOSEK ApS, MOSEK Optimization Software, version 6.0.0.122, 2011. Available from: http://www.mosek.com. |
[28] |
F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, New York, NY, 1986. |
[29] |
D. Needell and R. Ward, Stable image reconstruction using total variation minimization, SIAM J. Imaging Sci., 6 (2013), 1035-1058.
doi: 10.1137/120868281. |
[30] |
X. Pan, E. Y. Sidky and M. Vannier, Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?, Inverse Probl., 25 (2009), 123009, 36pp.
doi: 10.1088/0266-5611/25/12/123009. |
[31] |
S. Petra and C. Schnörr, Average case recovery analysis of tomographic compressive sensing, Linear Algebra Appl., 441 (2014), 168-198.
doi: 10.1016/j.laa.2013.06.034. |
[32] |
N. Pustelnik, C. Dossal, F. Turcu, Y. Berthoumieu and P. Ricoux, A greedy algorithm to extract sparsity degree for L1/L0-equivalence in a deterministic context, in Proc. EUSIPCO, Bucharest, Romania, 2012. |
[33] |
I. Reiser and R. M. Nishikawa, Task-based assessment of breast tomosynthesis: Effect of acquisition parameters and quantum noise, Med. Phys., 37 (2010), 1591-1600.
doi: 10.1118/1.3357288. |
[34] |
L. Ritschl, F. Bergner, C. Fleischmann and M. Kachelrieß, Improved total variation-based CT image reconstruction applied to clinical data, Phys. Med. Biol., 56 (2011), 1545-1561.
doi: 10.1088/0031-9155/56/6/003. |
[35] |
L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.
doi: 10.1016/0167-2789(92)90242-F. |
[36] |
E. Y. Sidky, M. A. Anastasio and X. Pan, Image reconstruction exploiting object sparsity in boundary-enhanced X-ray phase-contrast tomography, Opt. Express, 18 (2010), 10404-10422.
doi: 10.1364/OE.18.010404. |
[37] |
E. Y. Sidky, C.-M. Kao and X. Pan, Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT, J. Xray Sci. Technol., 14 (2006), 119-139. |
[38] |
E. Y. Sidky and X. Pan, Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization, Phys. Med. Biol., 53 (2008), 4777-4807.
doi: 10.1088/0031-9155/53/17/021. |
[39] |
A. M. Tillmann and M. E. Pfetsch, The computational complexity of the restricted isometry property, the nullspace property, and related concepts in compressed sensing, IEEE Trans. Inf. Theory, 60 (2014), 1248-1259.
doi: 10.1109/TIT.2013.2290112. |
[40] |
L. Yu, X. Liu, S. Leng, J. M. Kofler, J. C. Ramirez-Giraldo, M. Qu, J. Christner, J. G. Fletcher and C. H. McCollough, Radiation dose reduction in computed tomography: Techniques and future perspective, Imaging Med., 1 (2009), 65-84.
doi: 10.2217/iim.09.5. |
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