# American Institute of Mathematical Sciences

May  2014, 8(2): 421-457. doi: 10.3934/ipi.2014.8.421

## Compressive optical deflectometric tomography: A constrained total-variation minimization approach

 1 ICTEAM, Université catholique de Louvain, Louvain-la-Neuve, Belgium, Belgium, Belgium 2 Lambda-X SA, Rue de l'Industrie 37, Nivelles, Belgium

Received  September 2012 Revised  August 2013 Published  May 2014

Optical Deflectometric Tomography (ODT) provides an accurate characterization of transparent materials whose complex surfaces present a real challenge for manufacture and control. In ODT, the refractive index map (RIM) of a transparent object is reconstructed by measuring light deflection under multiple orientations. We show that this imaging modality can be made compressive, i.e., a correct RIM reconstruction is achievable with far less observations than required by traditional minimum energy (ME) or Filtered Back Projection (FBP) methods. Assuming a cartoon-shape RIM model, this reconstruction is driven by minimizing the map Total-Variation under a fidelity constraint with the available observations. Moreover, two other realistic assumptions are added to improve the stability of our approach: the map positivity and a frontier condition. Numerically, our method relies on an accurate ODT sensing model and on a primal-dual minimization scheme, including easily the sensing operator and the proposed RIM constraints. We conclude this paper by demonstrating the power of our method on synthetic and experimental data under various compressive scenarios. In particular, the potential compressiveness of the stabilized ODT problem is demonstrated by observing a typical gain of 24 dB compared to ME and of 30 dB compared to FBP at only 5% of 360 incident light angles for moderately noisy sensing.
Citation: Adriana González, Laurent Jacques, Christophe De Vleeschouwer, Philippe Antoine. Compressive optical deflectometric tomography: A constrained total-variation minimization approach. Inverse Problems & Imaging, 2014, 8 (2) : 421-457. doi: 10.3934/ipi.2014.8.421
##### References:
 [1] P. Antoine, E. Foumouo, J.-L. Dewandel, D. Beghuin, A. González and L. Jacques, Optical Deflection Tomography with Reconstruction Based on Sparsity,, Proceedings of OPTIMESS2012, (2012). Google Scholar [2] S. Becker and J. Fadili, A quasi-newton proximal splitting method,, Advances in Neural Information Processing Systems (NIPS), 25 (2012), 2627. Google Scholar [3] D. Beghuin, J.L. Dewandel, L. Joannes, E. Foumouo and P. Antoine, Optical deflection tomography with the phase-shifting schlieren,, Optics Letters, 35 (2010), 3745. doi: 10.1364/OL.35.003745. Google Scholar [4] M. Born, E. Wolf and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light,, Cambridge Univ Press, (1965). Google Scholar [5] E. J. Candes and T. Tao, Near-optimal signal recovery from random projections: Universal encoding strategies?,, IEEE Transactions on Information Theory, 52 (2006), 5406. doi: 10.1109/TIT.2006.885507. Google Scholar [6] E. J. Candes, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,, IEEE Transactions on Information Theory, 52 (2006), 489. doi: 10.1109/TIT.2005.862083. Google Scholar [7] A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011325.36760.1e. Google Scholar [8] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. Google Scholar [9] P. L. Combettes and J.-C. Pesquet, Proximal splitting methods in signal processing,, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (2011), 185. doi: 10.1007/978-1-4419-9569-8_10. Google Scholar [10] W. Cong, A. Momose and G. Wang, Fourier transform-based iterative method for differential phase-contrast computed tomography,, Optics Letters, 37 (2012), 1784. doi: 10.1364/OL.37.001784. Google Scholar [11] W. Cong, J. Yang and G. Wang, Differential phase-contrast interior tomography,, Phys. Med. Biol., 57 (2012), 2905. doi: 10.1088/0031-9155/57/10/2905. Google Scholar [12] D. L. Donoho, Compressed sensing,, IEEE Transactions on Information Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. Google Scholar [13] D. L. Donoho and J. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage,, Biometrika, 81 (1994), 425. doi: 10.1093/biomet/81.3.425. Google Scholar [14] M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly and R. G. Baraniuk, Single-pixel imaging via compressive sampling,, IEEE Signal Processing Magazine, 25 (2008), 83. doi: 10.1109/MSP.2007.914730. Google Scholar [15] F. X. Dupé, M. J. Fadili and J.-L. Starck, Inverse problems with poisson noise: Primal and primal-dual splitting,, 18th IEEE International Conference on Image Processing (ICIP), (2011), 1901. doi: 10.1109/ICIP.2011.6115841. Google Scholar [16] G. W. Faris and R. L. Byer, Beam-deflection optical tomography on a flame,, Optics Letters, 12 (1987), 155. doi: 10.1364/OL.12.000155. Google Scholar [17] E. Foumouo, J. L. Dewandel, L. Joannes, D. Beghuin, L. Jacques and P. Antoine, Optical tomography based on phase-shifting schlieren deflectometry,, Proceedings of SPIE, 7726 (2010). Google Scholar [18] T. Goldstein, E. Esser and R. Baraniuk, Adaptive primal-dual hybrid gradient methods for saddle-point problems, preprint,, , (). Google Scholar [19] A. González, L. Jacques and P. Antoine, Tv-$l_2$ Refractive Index Map Reconstruction from Polar Domain Deflectometry,, 1st International Traveling Workshop for Interacting Sparse models and Technology (iTWIST'12), (2012). Google Scholar [20] W. Hoeffding, Probability inequalities for sums of bounded random variables,, Journal of the American Statistical Association, 58 (1963), 13. doi: 10.1080/01621459.1963.10500830. Google Scholar [21] L. Jacques, A. González, E. Foumouo and P. Antoine, Refractive index map reconstruction in optical deflectometry using total-variation regularization,, Proceedings of SPIE, 8138 (2011). doi: 10.1117/12.892729. Google Scholar [22] L. Jacques, D. Hammond and J. Fadili, Dequantizing compressed sensing: When oversampling and non-gaussian constraints combine,, IEEE Transactions on Information Theory, 57 (2011), 559. doi: 10.1109/TIT.2010.2093310. Google Scholar [23] L. Joannes, F. Dubois and J.-C. Legros, Phase-shifting schlieren: High-resolution quantitative schlieren that uses the phase-shifting technique principle,, Applied Optics, 42 (2003), 5046. doi: 10.1364/AO.42.005046. Google Scholar [24] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging,, IEEE Press, (1988). Google Scholar [25] J. Keiner, S. Kunis and D. Potts, Using nfft 3-a software library for various nonequispaced fast fourier transforms,, ACM Transactions on Mathematical Software (TOMS), 36 (2009). doi: 10.1145/1555386.1555388. Google Scholar [26] M. Lustig, D. Donoho and J. M. Pauly, Sparse mri: The application of compressed sensing for rapid mr imaging,, Magnetic Resonance in Medicine, 58 (2007), 1182. doi: 10.1002/mrm.21391. Google Scholar [27] J. J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien. (French) [Dual convex functions and proximal points in a hilbert space],, CR Acad. Sci. (Paris), 255 (1962), 2897. Google Scholar [28] K. Morita, Applied Fourier Transform,, IOS Press, (1995). Google Scholar [29] F. Pfeiffer, C. Kottler, O. Bunk and C. David, Hard x-ray phase tomography with low-brilliance sources,, Physical Review Letters, 98 (2007). doi: 10.1103/PhysRevLett.98.108105. Google Scholar [30] T. Pock and A. Chambolle, Diagonal preconditioning for first order primal-dual algorithms in convex optimization,, IEEE International Conference on Computer Vision (ICCV), (2011), 1762. doi: 10.1109/ICCV.2011.6126441. Google Scholar [31] L. Ritschl, F. Bergner, C. Fleischmann and M. Kachelrieß, Improved total variation-based ct image reconstruction applied to clinical data,, Physics in Medicine and Biology, 56 (2011), 1545. doi: 10.1088/0031-9155/56/6/003. Google Scholar [32] L.I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [33] E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the chambolle-pock algorithm,, Phys. Med. Biol., 57 (2012), 3065. doi: 10.1088/0031-9155/57/10/3065. Google Scholar [34] G. Steidl, A note on fast fourier transforms for nonequispaced grids,, Advances in Computational Mathematics, 9 (1998), 337. doi: 10.1023/A:1018901926283. Google Scholar [35] C. Taswell, The what, how, and why of wavelet shrinkage denoising,, Computing in Science & Engineering, 2 (2000), 12. doi: 10.1109/5992.841791. Google Scholar [36] Y. Wiaux, L. Jacques, G. Puy, A. M. M. Scaife and P. Vandergheynst, Compressed sensing imaging techniques for radio interferometry,, Monthly Notices of the Royal Astronomical Society, 395 (2009), 1733. doi: 10.1111/j.1365-2966.2009.14665.x. Google Scholar [37] C. Zhang, J. Wu and W. Sun, Applications of Pseudo-polar FFT in Synthetic Aperture Radiometer Imaging,, PIERS Online, 3 (2007), 25. Google Scholar

show all references

##### References:
 [1] P. Antoine, E. Foumouo, J.-L. Dewandel, D. Beghuin, A. González and L. Jacques, Optical Deflection Tomography with Reconstruction Based on Sparsity,, Proceedings of OPTIMESS2012, (2012). Google Scholar [2] S. Becker and J. Fadili, A quasi-newton proximal splitting method,, Advances in Neural Information Processing Systems (NIPS), 25 (2012), 2627. Google Scholar [3] D. Beghuin, J.L. Dewandel, L. Joannes, E. Foumouo and P. Antoine, Optical deflection tomography with the phase-shifting schlieren,, Optics Letters, 35 (2010), 3745. doi: 10.1364/OL.35.003745. Google Scholar [4] M. Born, E. Wolf and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light,, Cambridge Univ Press, (1965). Google Scholar [5] E. J. Candes and T. Tao, Near-optimal signal recovery from random projections: Universal encoding strategies?,, IEEE Transactions on Information Theory, 52 (2006), 5406. doi: 10.1109/TIT.2006.885507. Google Scholar [6] E. J. Candes, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,, IEEE Transactions on Information Theory, 52 (2006), 489. doi: 10.1109/TIT.2005.862083. Google Scholar [7] A. Chambolle, An algorithm for total variation minimization and applications,, Journal of Mathematical Imaging and Vision, 20 (2004), 89. doi: 10.1023/B:JMIV.0000011325.36760.1e. Google Scholar [8] A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging,, Journal of Mathematical Imaging and Vision, 40 (2011), 120. doi: 10.1007/s10851-010-0251-1. Google Scholar [9] P. L. Combettes and J.-C. Pesquet, Proximal splitting methods in signal processing,, Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (2011), 185. doi: 10.1007/978-1-4419-9569-8_10. Google Scholar [10] W. Cong, A. Momose and G. Wang, Fourier transform-based iterative method for differential phase-contrast computed tomography,, Optics Letters, 37 (2012), 1784. doi: 10.1364/OL.37.001784. Google Scholar [11] W. Cong, J. Yang and G. Wang, Differential phase-contrast interior tomography,, Phys. Med. Biol., 57 (2012), 2905. doi: 10.1088/0031-9155/57/10/2905. Google Scholar [12] D. L. Donoho, Compressed sensing,, IEEE Transactions on Information Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. Google Scholar [13] D. L. Donoho and J. M. Johnstone, Ideal spatial adaptation by wavelet shrinkage,, Biometrika, 81 (1994), 425. doi: 10.1093/biomet/81.3.425. Google Scholar [14] M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly and R. G. Baraniuk, Single-pixel imaging via compressive sampling,, IEEE Signal Processing Magazine, 25 (2008), 83. doi: 10.1109/MSP.2007.914730. Google Scholar [15] F. X. Dupé, M. J. Fadili and J.-L. Starck, Inverse problems with poisson noise: Primal and primal-dual splitting,, 18th IEEE International Conference on Image Processing (ICIP), (2011), 1901. doi: 10.1109/ICIP.2011.6115841. Google Scholar [16] G. W. Faris and R. L. Byer, Beam-deflection optical tomography on a flame,, Optics Letters, 12 (1987), 155. doi: 10.1364/OL.12.000155. Google Scholar [17] E. Foumouo, J. L. Dewandel, L. Joannes, D. Beghuin, L. Jacques and P. Antoine, Optical tomography based on phase-shifting schlieren deflectometry,, Proceedings of SPIE, 7726 (2010). Google Scholar [18] T. Goldstein, E. Esser and R. Baraniuk, Adaptive primal-dual hybrid gradient methods for saddle-point problems, preprint,, , (). Google Scholar [19] A. González, L. Jacques and P. Antoine, Tv-$l_2$ Refractive Index Map Reconstruction from Polar Domain Deflectometry,, 1st International Traveling Workshop for Interacting Sparse models and Technology (iTWIST'12), (2012). Google Scholar [20] W. Hoeffding, Probability inequalities for sums of bounded random variables,, Journal of the American Statistical Association, 58 (1963), 13. doi: 10.1080/01621459.1963.10500830. Google Scholar [21] L. Jacques, A. González, E. Foumouo and P. Antoine, Refractive index map reconstruction in optical deflectometry using total-variation regularization,, Proceedings of SPIE, 8138 (2011). doi: 10.1117/12.892729. Google Scholar [22] L. Jacques, D. Hammond and J. Fadili, Dequantizing compressed sensing: When oversampling and non-gaussian constraints combine,, IEEE Transactions on Information Theory, 57 (2011), 559. doi: 10.1109/TIT.2010.2093310. Google Scholar [23] L. Joannes, F. Dubois and J.-C. Legros, Phase-shifting schlieren: High-resolution quantitative schlieren that uses the phase-shifting technique principle,, Applied Optics, 42 (2003), 5046. doi: 10.1364/AO.42.005046. Google Scholar [24] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging,, IEEE Press, (1988). Google Scholar [25] J. Keiner, S. Kunis and D. Potts, Using nfft 3-a software library for various nonequispaced fast fourier transforms,, ACM Transactions on Mathematical Software (TOMS), 36 (2009). doi: 10.1145/1555386.1555388. Google Scholar [26] M. Lustig, D. Donoho and J. M. Pauly, Sparse mri: The application of compressed sensing for rapid mr imaging,, Magnetic Resonance in Medicine, 58 (2007), 1182. doi: 10.1002/mrm.21391. Google Scholar [27] J. J. Moreau, Fonctions convexes duales et points proximaux dans un espace hilbertien. (French) [Dual convex functions and proximal points in a hilbert space],, CR Acad. Sci. (Paris), 255 (1962), 2897. Google Scholar [28] K. Morita, Applied Fourier Transform,, IOS Press, (1995). Google Scholar [29] F. Pfeiffer, C. Kottler, O. Bunk and C. David, Hard x-ray phase tomography with low-brilliance sources,, Physical Review Letters, 98 (2007). doi: 10.1103/PhysRevLett.98.108105. Google Scholar [30] T. Pock and A. Chambolle, Diagonal preconditioning for first order primal-dual algorithms in convex optimization,, IEEE International Conference on Computer Vision (ICCV), (2011), 1762. doi: 10.1109/ICCV.2011.6126441. Google Scholar [31] L. Ritschl, F. Bergner, C. Fleischmann and M. Kachelrieß, Improved total variation-based ct image reconstruction applied to clinical data,, Physics in Medicine and Biology, 56 (2011), 1545. doi: 10.1088/0031-9155/56/6/003. Google Scholar [32] L.I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259. doi: 10.1016/0167-2789(92)90242-F. Google Scholar [33] E. Y. Sidky, J. H. Jørgensen and X. Pan, Convex optimization problem prototyping for image reconstruction in computed tomography with the chambolle-pock algorithm,, Phys. Med. Biol., 57 (2012), 3065. doi: 10.1088/0031-9155/57/10/3065. Google Scholar [34] G. Steidl, A note on fast fourier transforms for nonequispaced grids,, Advances in Computational Mathematics, 9 (1998), 337. doi: 10.1023/A:1018901926283. Google Scholar [35] C. Taswell, The what, how, and why of wavelet shrinkage denoising,, Computing in Science & Engineering, 2 (2000), 12. doi: 10.1109/5992.841791. Google Scholar [36] Y. Wiaux, L. Jacques, G. Puy, A. M. M. Scaife and P. Vandergheynst, Compressed sensing imaging techniques for radio interferometry,, Monthly Notices of the Royal Astronomical Society, 395 (2009), 1733. doi: 10.1111/j.1365-2966.2009.14665.x. Google Scholar [37] C. Zhang, J. Wu and W. Sun, Applications of Pseudo-polar FFT in Synthetic Aperture Radiometer Imaging,, PIERS Online, 3 (2007), 25. Google Scholar
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