×

Compressive imaging and characterization of sparse light deflection maps. (English) Zbl 1352.94011

Summary: Light rays incident on a transparent object of uniform refractive index undergo deflections, which uniquely characterize the surface geometry of the object. Associated with each point on the surface is a deflection map (or spectrum) which describes the pattern of deflections in various directions. This article presents a novel method to efficiently acquire and reconstruct sparse deflection spectra induced by smooth object surfaces. To this end, we leverage the framework of compressed sensing (CS) in a particular implementation of a schlieren deflectometer, i.e., an optical system providing linear measurements of deflection spectra with programmable spatial light modulation patterns. In particular, we design those modulation patterns on the principle of spread spectrum CS for reducing the number of observations. Interestingly, the ability of our device to simultaneously observe the deflection spectra on a dense discretization of the object surface is related to a particular multiple measurement vector model. This scheme allows us to estimate both the noise power and the instrumental point spread function in a specific calibration procedure. We formulate the spectrum reconstruction task as the solving of a linear inverse problem regularized by an analysis sparsity prior which uses a translation invariant wavelet frame. Our results demonstrate the capability and advantages of using a CS-based approach for deflectometric imaging both on simulated data and experimental deflectometric data. Finally, the paper presents an extension of our method showing how we can extract the main deflection direction in each point of the object surface from a few compressive measurements, without needing any costly reconstruction procedures. This compressive characterization is then confirmed with experimental results on simple plano-convex and multifocal intraocular lenses studying the evolution of the main deflection as a function of the object point location.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)

Software:

UNLocBoX; SPGL1
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[2] S. Almazán-Cuéllar and D. Malacara-Hernández, {\it Two-step phase-shifting algorithm}, Optical Eng., 42 (2003), pp. 3524-3531.
[3] R. G. Baraniuk, {\it Compressive sensing}, IEEE Signal Process. Mag., 24 (2007), p. 118. · Zbl 1372.94379
[4] H. H. Bauschke and P. L. Combettes, {\it Convex Analysis and Monotone Operator Theory in Hilbert Spaces}, Springer, New York, 2011. · Zbl 1218.47001
[5] D. Beghuin, J. L. Dewandel, L. Joannes, E. Foumouo, and P. Antoine, {\it Optical deflection tomography with the phase-shifting schlieren}, Optics Lett., 35 (2010), pp. 3745-3747.
[6] J. M. Bioucas-Dias and G. Valada͂o, {\it Phase unwrapping via graph cuts}, IEEE Trans. Image Process., 16 (2007), pp. 698-709.
[7] J. Bobin, J.-L. Starck, and R. Ottensamer, {\it Compressed sensing in astronomy}, IEEE J. Selected Topics Signal Process., 2 (2008), pp. 718-726.
[8] S. P. Boyd and L. Vandenberghe, {\it Convex Optimization}, Cambridge University Press, 2004. · Zbl 1058.90049
[9] E. J. Candès, Y. C. Eldar, D. Needell and P. Randall, {\it Compressed sensing with coherent and redundant dictionaries}, Appl. Comput. Harmon. Anal., 31 (2011), pp. 59-73. · Zbl 1215.94026
[10] E. J. Candès and J. Romberg, {\it Sparsity and incoherence in compressive sampling}, Inverse Problems, 23 (2007), p. 969. · Zbl 1120.94005
[11] E. J. Candès and T. Tao, {\it Near-optimal signal recovery from random projections: Universal encoding strategies?}, IEEE Trans. Inform. Theory, 52 (2006), pp. 5406-5425. · Zbl 1309.94033
[12] A. Chambolle and T. Pock, {\it A first-order primal-dual algorithm for convex problems with applications to imaging}, J. Math. Imaging Vision, 40 (2011), pp. 120-145. · Zbl 1255.68217
[13] P. L. Combettes and J.-C. Pesquet, {\it Proximal splitting methods in signal processing}, in Springer Optim. Appl., Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer, New York, 2011, pp. 185-212. · Zbl 1242.90160
[14] M. A. Davenport, P. T. Boufounos, M. B. Wakin, and R. G. Baraniuk, {\it Signal processing with compressive measurements}, IEEE J. Selected Topics Signal Proc., 4 (2010), pp. 445-460.
[15] M. A. Davenport, M. F. Duarte, M. B. Wakin, J. N. Laska, D. Takhar, K. F. Kelly, and R. G. Baraniuk, {\it The smashed filter for compressive classification and target recognition}, in Proceedings of SPIE Computational Imaging V, San Jose, CA, 2007.
[16] T. P. Davies, {\it Schlieren photography: A short bibliography and review}, Optics Laser Technol., 13 (1981), pp. 37-42.
[17] D. L. Donoho, {\it Compressed sensing}, IEEE Trans. Inform. Theory, 52 (2006), pp. 1289-1306. · Zbl 1288.94016
[18] M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, {\it Single-pixel imaging via compressive sampling}, IEEE Signal Process. Mag., 25 (2008), pp. 83-91.
[19] M. Elad, {\it Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing}, 1st ed., Springer, New York, 2010. · Zbl 1211.94001
[20] M. Elad, P. Milanfar, and R. Rubinstein, {\it Analysis versus synthesis in signal priors}, Inverse Problems, 23 (2007), p. 947. · Zbl 1138.93055
[21] S. Foucart and H. Rauhut, {\it A Mathematical Introduction to Compressive Sensing}, Birkhäuser Ser. Appl. Numer. Harmon. Anal., Birkhäuser, Basel, 2013. · Zbl 1315.94002
[22] K. Fyhn, M. F. Duarte, and S. H. Jensen, {\it Compressive parameter estimation for sparse translation-invariant signals using polar interpolation}, arXiv:1305.3483, 2013. · Zbl 1394.94195
[23] M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, {\it Single-shot compressive spectral imaging with a dual-disperser architecture}, Optics Express, 15 (2007), pp. 14013-14027.
[24] M. Golbabaee, S. Arberet, and P. Vandergheynst, {\it Compressive source separation: Theory and methods for hyperspectral imaging}, IEEE Trans. Image Process., 22 (2013), pp. 5096-5110. · Zbl 1373.94140
[25] A. Gonzalez and L. Jacques, {\it Robust phase unwrapping by convex optimization}, in 2014 IEEE International Conference on Image Processing (ICIP), IEEE, Paris, 2014, pp. 1713-1717.
[26] A. Gonzalez, L. Jacques, C. De Vleeschouwer, and P. Antoine, {\it Compressive optical deflectometric tomography: A constrained total-variation minimization approach}, Inverse Probl. Imaging, 8 (2014), pp. 421-457. · Zbl 1302.94007
[27] P. Hariharan, B. F. Oreb, and T. Eiju, {\it Digital phase-shifting interferometry: A simple error-compensating phase calculation algorithm}, Appl. Opt., 26 (1987), pp. 2504-2506.
[28] J. Haupt, W. U. Bajwa, G. Raz, and R. Nowak, {\it Toeplitz compressed sensing matrices with applications to sparse channel estimation}, IEEE Trans. Inform. Theory, 56 (2010), pp. 5862-5875. · Zbl 1366.62110
[29] I. Ihrke and M. Magnor, {\it Image-based tomographic reconstruction of flames}, in Proceedings of the 2004 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, SCA ’04, Aire-la-Ville, Switzerland, Eurographics Association, 2004, pp. 365-373.
[30] L. Joannes, D. Beghuin, R. Ligot, S. Farinotti, and O. Dupont, {\it High-resolution shape measurements with phase-shifting schlieren (PSS)}, in Proceedings of SPIE SYS 7, 2004.
[31] L. Joannes, F. Dubois, and J. C. Legros, {\it Phase-shifting schlieren: High-resolution quantitative schlieren that uses the phase-shifting technique principle}, Appl. Opt., 42 (2003), pp. 5046-5053.
[32] S. A. Klein, {\it Understanding the diffractive bifocal contact lens}, Optometry and Vision Science, 70 (1993), pp. 439-60.
[33] G. Kutyniok, {\it Theory and applications of compressed sensing}, preprint, arXiv:1203.3815, 2012. · Zbl 1283.94018
[34] M. Lustig, D. L. Donoho, and J. M. Pauly, {\it Sparse MRI: The application of compressed sensing for rapid MR imaging}, Magnetic Resonance in Medicine, 58 (2007), pp. 1182-1195.
[35] M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, {\it Compressed Sensing MRI}, IEEE Signal Process. Mag., 25, pp. 72-82.
[36] D. Malacara, {\it Optical Shop Testing}, Wiley Ser. Pure Appl. Opt., Wiley-Interscience, New York, 2007.
[37] S. Mallat, {\it A Wavelet Tour of Signal Processing: The Sparse Way}, 3rd ed., Academic Press, New York, 2008. · Zbl 1170.94003
[38] M. Mishali and Y. C. Eldar, {\it Reduce and boost: Recovering arbitrary sets of jointly sparse vectors}, IEEE Trans. Signal Process., 56 (2008), pp. 4692-4702. · Zbl 1390.94306
[39] F. M. Naini, R. Gribonval, L. Jacques, and P. Vandergheynst, {\it Compressive sampling of pulse trains: Spread the spectrum!}, in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, 2009, ICASSP ’09, 2009, pp. 2877-2880.
[40] N. Parikh and S. Boyd, {\it Proximal algorithms}, Found. Trends Optim., 1 (2013), pp. 123-231.
[41] G. Puy, J. P. Marques, R. Gruetter, J. Thiran, D. Van De Ville, P. Vandergheynst, and Y. Wiaux, {\it Spread spectrum magnetic resonance imaging}, IEEE Trans. Med. Imaging, 31 (2012), pp. 586 –598.
[42] G. Puy, P. Vandergheynst, R. Gribonval, and Y. Wiaux, {\it Universal and efficient compressed sensing by spread spectrum and application to realistic fourier imaging techniques}, EURASIP J. Adv. Signal Process., 2012 (2012), pp. 1-13.
[43] M. Raginsky, R. M. Willett, Z. T. Harmany, and R. F. Marcia, {\it Compressed sensing performance bounds under poisson noise}, IEEE Trans. Signal Process., 58 (2010), pp. 3990-4002. · Zbl 1392.94417
[44] H. Rauhut, {\it Compressive sensing and structured random matrices}, in Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Series Comp. Appl. Math. 9, M. Fornasier ed., deGruyter 2010, pp. 1-92. · Zbl 1208.15027
[45] H. Rauhut and M. Kabanava, {\it Analysis \(ℓ_{{1}}\)-recovery with frames and Gaussian measurements}, CoRR, abs/1306.1356, 2013. · Zbl 1378.94008
[46] J. Romberg, {\it Compressive sensing by random convolution}, SIAM J. Imaging Sci., 2 (2009), pp. 1098-1128. · Zbl 1176.94017
[47] L. I. Rudin, S. Osher, and E. Fatemi, {\it Nonlinear total variation based noise removal algorithms}, Phys. D, 60 (1992), pp. 259-268. · Zbl 0780.49028
[48] G. Sauter, {\it Goniophotometry: New calibration method and instrument design}, Metrologia, 32 (1995), p. 685.
[49] G. S. Settles, {\it Colour-coding schlieren techniques for the optical study of heat and fluid flow}, Int. J. Heat Fluid Flow, 6 (1985), pp. 3-15.
[50] G. S. Settles, {\it Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transparent Media}, Springer, New York, 2001. · Zbl 0987.76002
[51] J.-L. Starck, F. Murtagh, and M.-J. Fadili, {\it Sparse Image and Signal Processing–Wavelets, Curvelets, Morphological Diversity}, Cambridge University Press, Cambridge, 2010. · Zbl 1196.94008
[52] V. Studer, J. Bobin, M. Chahid, H. S. Mousavi, E. J. Candès, and M. Dahan, {\it Compressive fluorescence microscopy for biological and hyperspectral imaging}, Proc. Nat. Acad. Sci. USA, 109 (2012), pp. E1679-E1687.
[53] P. Sudhakar, L. Jacques, X. Dubois, P. Antoine, and L. Joannes, {\it Compressive acquisition of sparse deflectometric maps}, in Proceedings of the 10th International Conference on Sampling Theory and Applications (SampTA), 2013. · Zbl 1352.94011
[54] P. Sudhakar, L. Jacques, X. Dubois, P. Antoine, and L. Joannes, {\it Compressive schlieren deflectometry}, in Proceedings of the 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2013, pp. 5999-6003.
[55] J. A. Tropp, M. B. Wakin, M. F. Duarte, D. Baron, and R. G. Baraniuk, {\it Random filters for compressive sampling and reconstruction}, in Proceedings of the 2006 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2006.
[56] Y. Tsaig and D. L. Donoho, {\it Extensions of compressed sensing}, Signal Processing, 86 (2006), pp. 549-571. · Zbl 1163.94399
[57] E. van den Berg and M. P. Friedlander, {\it SPGL1: A Solver for Large-Scale Sparse Reconstruction}, \burlhttp://www.cs.ubc.ca/labs/scl/spgl1 (2007).
[58] Y. Wiaux, L. Jacques, G. Puy, A. M. M. Scaife, and P. Vandergheynst, {\it Compressed sensing imaging techniques for radio interferometry}, Monthly Not. R. Astron. Soc., 395 (2009), pp. 1733-1742.
[59] R. M. Willett, R. F. Marcia, and J. M. Nichols, {\it Compressed sensing for practical optical imaging systems: A tutorial}, Optical Eng., 50 (2011), pp. 1-14.
[60] W. Yin, S. Morgan, J. Yang, and Y. Zhang, {\it Practical compressive sensing with Toeplitz and circulant matrices}, in Proceedings of SPIE 7744, 2010.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.