## Compressed sampling strategies for tomography |

JOSA A, Vol. 31, Issue 7, pp. 1369-1394 (2014)

http://dx.doi.org/10.1364/JOSAA.31.001369

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### Abstract

We investigate new sampling strategies for projection tomography, enabling one to employ fewer measurements than expected from classical sampling theory without significant loss of information. Inspired by compressed sensing, our approach is based on the understanding that many real objects are compressible in some known representation, implying that the number of degrees of freedom defining an object is often much smaller than the number of pixels/voxels. We propose a new approach based on quasi-random detector subsampling, whereas previous approaches only addressed subsampling with respect to source location (view angle). The performance of different sampling strategies is considered using object-independent figures of merit, and also based on reconstructions for specific objects, with synthetic and real data. The proposed approach can be implemented using a structured illumination of the interrogated object or the detector array by placing a coded aperture/mask at the source or detector side, respectively. Advantages of the proposed approach include (i) for structured illumination of the detector array, it leads to fewer detector pixels and allows one to integrate detectors for scattered radiation in the unused space; (ii) for structured illumination of the object, it leads to a reduced radiation dose for patients in medical scans; (iii) in the latter case, the blocking of rays reduces scattered radiation while keeping the same energy in the transmitted rays, resulting in a higher signal-to-noise ratio than that achieved by lowering exposure times or the energy of the source; (iv) compared to view-angle subsampling, it allows one to use fewer measurements for the same image quality, or leads to better image quality for the same number of measurements. The proposed approach can also be combined with view-angle subsampling.

© 2014 Optical Society of America

**OCIS Codes**

(110.6960) Imaging systems : Tomography

(110.7440) Imaging systems : X-ray imaging

(340.7430) X-ray optics : X-ray coded apertures

(340.7440) X-ray optics : X-ray imaging

(110.6955) Imaging systems : Tomographic imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: February 18, 2014

Revised Manuscript: April 19, 2014

Manuscript Accepted: April 20, 2014

Published: June 9, 2014

**Citation**

Yan Kaganovsky, Daheng Li, Andrew Holmgren, HyungJu Jeon, Kenneth P. MacCabe, David G. Politte, Joseph A. O’Sullivan, Lawrence Carin, and David J. Brady, "Compressed sampling strategies for tomography," J. Opt. Soc. Am. A **31**, 1369-1394 (2014)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-7-1369

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### References

- M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).
- F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).
- E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006). [CrossRef]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]
- R. Rangayyan, A. P. Dhawan, and R. Gordon, “Algorithms for limited-view computed-tomography—an annotated bibliography and a challenge,” Appl. Opt. 24, 4000–4012 (1985). [CrossRef]
- A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986). [CrossRef]
- E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).
- M. H. Li, H. Q. 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, 2599–2609 (2002). [CrossRef]
- J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007). [CrossRef]
- E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008). [CrossRef]
- D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982). [CrossRef]
- M. I. Sezan and H. Stark, “Image restoration by the method of convex projections: part 2 applications and numerical results,” IEEE Trans. Med. Imaging 1, 95–101 (1982). [CrossRef]
- E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006). [CrossRef]
- E. Y. Sidky and X. C. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008). [CrossRef]
- G. H. Chen, J. Tang, and S. H. Leng, “Prior image constrained compressed sensing: a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35, 660–663 (2008). [CrossRef]
- J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011). [CrossRef]
- K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010). [CrossRef]
- A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Video rate spectral imaging using a coded aperture snapshot spectral imager,” Opt. Express 17, 6368–6388 (2009). [CrossRef]
- J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express 19, 7289–7298 (2011). [CrossRef]
- K. P. MacCabe, A. D. Holmgren, M. P. Tornai, and D. J. Brady, “Snapshot 2D tomography via coded aperture x-ray scatter imaging,” Appl. Opt. 52, 4582–4589 (2013). [CrossRef]
- J. Greenberg, K. Krishnamurthy, and D. Brady, “Compressive single-pixel snapshot x-ray diffraction imaging,” Opt. Lett. 39, 111–114 (2014). [CrossRef]
- D. Brady and D. Marks, “Coding for compressive focal tomography,” Appl. Opt. 50, 4436–4449 (2011). [CrossRef]
- P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21, 10526–10545 (2013). [CrossRef]
- D. J. Brady, N. Pitsianis, and X. Sun, “Reference structure tomography,” J. Opt. Soc. Am. A 21, 1140–1147 (2004). [CrossRef]
- K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009). [CrossRef]
- R. Ning, X. Tang, and D. L. Conover, “X-ray scatter suppression algorithm for cone-beam volume CT,” Proc. SPIE 4685, 774–781 (2002). [CrossRef]
- R. Ning, X. Tang, and D. Conover, “X-ray scatter correction algorithm for cone beam CT imaging,” Med. Phys. 31, 1195–1202 (2004). [CrossRef]
- G. Harding, J. Kosanetzky, and U. Neitzel, “X-ray diffraction computed tomography,” Med. Phys. 14, 515–525 (1987). [CrossRef]
- H. Strecker, “Automatic detection of explosives in airline baggage using elastic x-ray scatter,” Medicamundi 42, 30–33 (1998).
- C. Cozzini, S. Olesinski, and G. Harding, “Modeling scattering for security applications: a multiple beam x-ray diffraction imaging system,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (IEEE, 2012), pp. 74–77.
- F. Smith, Industrial Applications of X-Ray Diffraction (CRC Press, 1999).
- S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009). [CrossRef]
- J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2009).
- E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007). [CrossRef]
- F. Krahmer and R. Ward, “Beyond incoherence: stable and robust sampling strategies for compressive imaging,” arXiv:1210.2380 (2012).
- B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,” arXiv:1302.0561 (2013).
- H. Peng and H. Stark, “Direct fourier reconstruction in fan-beam tomography,” IEEE Trans. Med. Imaging 6, 209–219 (1987). [CrossRef]
- G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. 32, 654–665 (2005). [CrossRef]
- F. Natterer, “Sampling in fan beam tomography,” SIAM J. Appl. Math. 53, 358–380 (1993). [CrossRef]
- http://www.mathworks.com/matlabcentral/fileexchange/27375-plot-wavelet-image-2d-decomposition/content/plotwavelet2.m .
- D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003). [CrossRef]
- M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008). [CrossRef]
- M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007). [CrossRef]
- M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications (Cambridge University, 2012), Chap. 1.
- J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006). [CrossRef]
- M. Sonka and J. M. Fitzpatrick, Handbook of Medical Imaging, Vol. 2 of Medical Image Processing and Analysis (SPIE, 2000).
- G. T. Herman, Image Reconstruction from Projections (Academic, 1980).
- M. E. Tipping, “Sparse Bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).
- S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008). [CrossRef]
- J. M. Bernardo and A. F. Smith, Bayesian Theory, Vol. 405 of Wiley Series in Probability and Statistics (Wiley, 2009).
- J. A. O’Sullivan and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans. Med. Imaging 26, 283–297 (2007). [CrossRef]
- M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983). [CrossRef]
- D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces,” Inf. Control 5, 279–323 (1962). [CrossRef]
- A. G. Lindgren and P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981). [CrossRef]
- S. H. Izen, “Sampling in flat detector fan beam tomography,” SIAM J. Appl. Math. 72, 61–84 (2012). [CrossRef]
- D. J. Crotty, R. L. McKinley, and M. P. Tornai, “Experimental spectral measurements of heavy k-edge filtered beams for x-ray computed mammotomography,” Phys. Med. Biol. 52, 603–616 (2007). [CrossRef]
- C. W. Dodge, A Rapid Method for the Simulation of Filtered X-Ray Spectra in Diagnostic Imaging Systems (ProQuest, 2008).

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