American Institute of Mathematical Sciences

Advanced
May  2015, 9(2): 431-446. doi: 10.3934/ipi.2015.9.431

Empirical average-case relation between undersampling and sparsity in X-ray CT

Jakob S. Jørgensen 1, , Emil Y. Sidky 2, , Per Christian Hansen 1, and  Xiaochuan Pan 2,

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

Received  July 2014 Revised  January 2015 Published  March 2015

In X-ray computed tomography (CT) it is generally acknowledged that reconstruction methods exploiting image sparsity allow reconstruction from a significantly reduced number of projections. The use of such reconstruction methods is inspired by recent progress in compressed sensing (CS). However, the CS framework provides neither guarantees of accurate CT reconstruction, nor any relation between sparsity and a sufficient number of measurements for recovery, i.e., perfect reconstruction from noise-free data. We consider reconstruction through 1-norm minimization, as proposed in CS, from data obtained using a standard CT fan-beam sampling pattern. In empirical simulation studies we establish quantitatively a relation between the image sparsity and the sufficient number of measurements for recovery within image classes motivated by tomographic applications. We show empirically that the specific relation depends on the image class and in many cases exhibits a sharp phase transition as seen in CS, i.e., same-sparsity images require the same number of projections for recovery. Finally we demonstrate that the relation holds independently of image size and is robust to small amounts of additive Gaussian white noise.
Keywords: image reconstruction, computed tomography, compressed sensing, sparse solutions., Inverse problems.
Mathematics Subject Classification: Primary: 90C90, 15A29, 44A12; Secondary: 94A0.
Citation: Jakob S. Jørgensen, Emil Y. Sidky, Per Christian Hansen, Xiaochuan Pan. Empirical average-case relation between undersampling and sparsity in X-ray CT. Inverse Problems & Imaging, 2015, 9 (2) : 431-446. doi: 10.3934/ipi.2015.9.431
References:
[1]

H. H. Barrett and K. J. Myers, Foundations of Image Science, John Wiley & Sons, Hoboken, NJ, 2004. Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

MOSEK ApS, MOSEK Optimization Software, version 6.0.0.122,, 2011. Available from: , ().   Google Scholar

[28]

F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, New York, NY, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

H. H. Barrett and K. J. Myers, Foundations of Image Science, John Wiley & Sons, Hoboken, NJ, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

MOSEK ApS, MOSEK Optimization Software, version 6.0.0.122,, 2011. Available from: , ().   Google Scholar

[28]

F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, New York, NY, 1986.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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