Advertisement

Compressed Sensing in Imaging and Reconstruction - An Insight Review

  • K. SreekalaEmail author
  • E. Krishna Kumar
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 941)

Abstract

Compressed sensing is a modern approach for signal sensing and sensor design which helps to reduce sampling and computation cost while acquiring signals with sparse or compressible representation. Nyquist-Shannon theorem speaks about the number of samples required to represent a bandlimited signal, but the number of samples required to represent a signal can be extensively reduced if it is sparse in a known basis. Sparsity is an essential property of signals which helps in storing the signals using few samples and recover accurately with sparse recovery techniques. Many signals are sparse in reality or when they are represented in an appropriate transform domain. Many algorithms exists in the literature for sparse signal recovery. Through this paper we make an attempt to look into the idea of compressed sensing, applications in image processing and computer vision, different basic sparse recovery algorithms and recent developments in the application of Compressed sensing in image enhancement.

Keywords

Compressive sampling Sparse signals Sparse recovery algorithms Transform domain 

References

  1. 1.
    Donoho, D.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Tsai, Y., et al.: Extensions of compressed sensing. Sig. Process. 86(3), 549–571 (2006)zbMATHGoogle Scholar
  3. 3.
    Bruckstein, A., et al.: From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1), 34–81 (2009)MathSciNetzbMATHGoogle Scholar
  4. 4.
    DeVor, R.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)Google Scholar
  5. 5.
    Elad, M.: Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer, New York (2010)zbMATHGoogle Scholar
  6. 6.
    Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37, 10–21 (1949)MathSciNetGoogle Scholar
  7. 7.
    Candes, E.J., et al.: An introduction to CS. IEEE Sig. Proc. Mag. 25, 21–30 (2008)Google Scholar
  8. 8.
    Baraniuk, R.: Compressive sensing. IEEE Sig. Proc. Mag. 24(4), 118–121 (2007)Google Scholar
  9. 9.
    Candès, E.: Compressive sampling. In: Proceedings of the International Congress of Mathematicians, Madrid, Spain (2006)Google Scholar
  10. 10.
    Candès, E., Romberg, J.: Quantitative robust uncertainty principles and optimally sparse decompositions. Found Comput. Math. 6(2), 227–254 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Candès, E., et al.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–492 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Candès, E., et al.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Candès, E., Tao, T.: Near optimal signal recovery from random projections: universal encoding strategies. IEEE Trans. Inf. Theory 52, 5406–5425 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mallat, S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  15. 15.
    Pennebaker, W., et al.: JPEG Still Image Data Compression Standard. Van Nostrand Reinhold, New York (1993)Google Scholar
  16. 16.
    Donoho, D.: Denoising by soft-thresholding. IEEE Trans. Inf. Theory 41(3), 613–627 (1995)zbMATHGoogle Scholar
  17. 17.
    Duarte, M.F., et al.: Single-pixel imaging via CS. IEEE SP Mag. 25 (2008)Google Scholar
  18. 18.
    Shin, J., et al.: Single-pixel imaging using compressed sensing and wavelength-dependent scattering. Opt. Lett. 41, 886–889 (2016)Google Scholar
  19. 19.
    Wright, G.A.: MR imaging. IEEE Sig. Proc. Mag. 14(1), 56–66 (1997)MathSciNetGoogle Scholar
  20. 20.
    Roohi, S., et al.: Super-resolution MRI images using compressive sensing. In: 20th Iranian Conference on Electrical Engineering, Tehran (2012)Google Scholar
  21. 21.
    Lustig, M., et al.: Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58(6), 1182–1195 (2007)Google Scholar
  22. 22.
    Lustig, M., et al.: Compressed sensing MRI. IEEE Sig. Proc. Mag. 25(2), 72–82 (2008)Google Scholar
  23. 23.
    Patel, V.M., et al.: Gradient-based image recovery methods from incomplete fourier measurements. IEEE Tran. Image Process. 21(1), 94–105 (2012)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Chan, T.F., et al.: Recent Developments in Total Variation Image Restoration. Springer, Berlin (2005)Google Scholar
  25. 25.
    Wang, Y., et al.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Carrara, W.G., et al.: Spotlight SAR: Signal Processing Algorithms. Artech House, Norwood (1995)zbMATHGoogle Scholar
  27. 27.
    Chen, V.C., Ling, H.: Time-Frequency Transforms for Radar Imaging and Signal Analysis. Artech House, Norwood (2002)zbMATHGoogle Scholar
  28. 28.
    Soumekh, M.: SAR Signal Processing with Matlab Algorithms. Wiley, New York (1999)zbMATHGoogle Scholar
  29. 29.
    Cumming, I.G., et al.: Digital Processing of SAR Data. Artech House, Norwood (2005)Google Scholar
  30. 30.
    Patel, V.M., et al.: Compressive passive millimeter-wave imaging with extended depth of field. Opt. Eng. 51(9), 091610 (2012)Google Scholar
  31. 31.
    Babacan, S.D., et al.: Compressive passive mm-wave imaging. In: IEEE ICIP (2011)Google Scholar
  32. 32.
    Christy, F., et al.: Millimeter-wave compressive holography. Appl. Opt. 49(19), E67–E82 (2010)Google Scholar
  33. 33.
    Fernandez, C.A., et al.: Sparse fourier sampling in millimeter-wave compressive holography. In: Digital Holography and 3 Dimensional Imaging (2010)Google Scholar
  34. 34.
    Gopalsami, N., et al.: CS in passive mm-wave imaging. In: Proceedings of SPIE, vol. 8022 (2011)Google Scholar
  35. 35.
    Noor, I., et al.: CS for a sub-millimeter wave single pixel images. In: Proceedings of SPIE, vol. 8022 (2011)Google Scholar
  36. 36.
    Peers, P., et al.: Compressive light transport sensing. ACM Trans. Graph 28, 3 (2009)Google Scholar
  37. 37.
    Cevher, V., et al.: CS for background subtraction. In: ECCV (2008)Google Scholar
  38. 38.
    Vaswani, N.: Kalman filtered CS. In: IEEE International Conference on Image Processing (2008)Google Scholar
  39. 39.
    Cossalter, M., et al.: Joint compressive video coding and analysis. IEEE Trans. Multimedia 12, 168–183 (2010)Google Scholar
  40. 40.
    Reddy, D., et al.: Compressed sensing for multi-view tracking and 3-D voxel reconstruction. In: IEEE International Conference on Image Processing (2008)Google Scholar
  41. 41.
    Wang, E., et al.: Compressive particle filtering for target tracking. In: IEEE Workshop on Statistical Signal Processing (2009)Google Scholar
  42. 42.
    Veeraraghavan, A., et al.: Coded strobing photography: CS of high speed periodic videos. IEEE Trans. Pattern Anal. Mach. Intell. 33(4), 671–686 (2011)MathSciNetGoogle Scholar
  43. 43.
    Candès, E., et al.: Decoding by linear programmingGoogle Scholar
  44. 44.
    Needell, D., Tropp, J.: CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Foucart, S.: Sparse recovery algorithms: sufficient conditions in terms of RIC. In: Approximation Theory XIII: San An tonio. Springer, New York (2012)Google Scholar
  46. 46.
    Tropp, J.A.: Algorithms for simultaneous sparse approximation. Part II: convex relaxation. Sig. Process. 86, 589–602 (2006)zbMATHGoogle Scholar
  47. 47.
    Boyd, S., et al.: Convex Optimization. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  48. 48.
    Chen, S.S., et al.: Atomic decomposition by basis pursuit. SIAM Rev. 43(1), 129–159 (2001)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Garg, R., et al.: Gradient descent with sparsification: an iterative algorithm for sparse recovery with RIP. In: Proceedings of the 26th Annual International Conference on Machine Learning. ACM (2009)Google Scholar
  50. 50.
    Lu, W., et al.: Modified basis pursuit denoising for noisy compressive sensing with partially known support. In: Proceedings of the ICASSP (2010)Google Scholar
  51. 51.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. Ser. B (Methodol.) 58, 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Efron, B., et al.: Least angle regression. Ann. Stat. 32, 407–499 (2004)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Mallat, S., et al.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Sig. Process. 41, 3397–3415 (1993)zbMATHGoogle Scholar
  54. 54.
    Tropp, J., Gilbert, A.: Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53, 4655–4666 (2007)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Dai, W., et al.: Subspace pursuit for CS signal reconstruction. IEEE Trans. Inf. Theory 55, 2230–2249 (2009)zbMATHGoogle Scholar
  56. 56.
    Blumensath, T., et al.: Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27, 265–274 (2009)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Indyket, P., et al.: Near-optimal sparse recovery in L1 norm. In: Proceedings of IEEE FOCS (2008)Google Scholar
  58. 58.
    Berinde, R., et al.: Practical near-optimal sparse recovery in the L1 norm. In: Proceedings of Allerton Conference on Communication, Control, Computing (2009)Google Scholar
  59. 59.
    Berinde, R., et al.: Sequential sparse matching pursuit. In: Proceedings of the Allerton Conference on Communication, Control, and Computing (2010)Google Scholar
  60. 60.
    Yin, W., et al.: Bregman iterative algorithms for L1-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1, 143–168 (2008)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Gilbert, A., et al.: Algorithmic linear dimension reduction in the L1 norm for sparse vectors. arXiv preprint arXiv:cs/0608079 (2006)
  62. 62.
    Gilbert, A., et al.: One sketch for all: fast algorithms for compressed sensing. In: Proceedings of the ACM Symposium on Theory of Computing (2007)Google Scholar
  63. 63.
    Gilbert, A., et al.: Improved time bounds for near-optimal sparse Fourier representations. In: Proceedings of SPIE, p. 59141 A (2005)Google Scholar
  64. 64.
    Chartrand, R.: Exact reconstruction of sparse signals via nonconvex minimization. IEEE Sig. Process. Lett. 14, 707–710 (2007)Google Scholar
  65. 65.
    Chartrand, R., et al.: Iteratively reweighted algos for CS. In: Proceedings of IEEE ICASSP (2008)Google Scholar
  66. 66.
    Murray, J., et al.: An improved FOCUSS-based learning algorithm for solving sparse linear inverse problems. In: Proceedings of Asilomar Conference on Signals, Systems and Computers, vol. 1 (2001)Google Scholar
  67. 67.
    Wipf, D., et al.: Sparse Bayesian learning for basis selection. IEEE Trans. Sig. Process. 52, 2153–2164 (2004)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Godsill, S., et al.: Bayesian computational methods for sparse audio and music processing. In: Proceedings of EURASIP Conference on Signal Processing (2007)Google Scholar
  69. 69.
    Ji, S., et al.: Bayesian compressive sensing. IEEE Trans. Sig. Process. 56, 2346–2356 (2008)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Babacan, S.D., et al.: Bayesian CS using laplace priors. IEEE Trans. Image Process. 19(1), 53–63 (2010)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Baron, D., et al.: Bayesian compressive sensing via belief propagation. IEEE Trans. Sig. Process 58(1), 269–280 (2010)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Gilbert, A., et al.: Sublinear approximation of signals. In: Proceedings of SPIE - The International Society for Optical Engineering, vol. 6232Google Scholar
  73. 73.
    Gorodnitsky, I.F., et al.: Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. Trans. Sig. Proc. 45, 600–616 (1997)Google Scholar
  74. 74.
    Zhang, K., et al.: Real-time compressive tracking. In: Conference Proceedings Computer Vision ECCV 2012. Springer, Heidelberg (2012)Google Scholar
  75. 75.
    Wang, B., et al.: 3D thermoacoustic imaging based on compressive sensing. In: International Workshop on Antenna Technology (iWAT) (2018)Google Scholar
  76. 76.
    Zarnaghi Naghsh, N., et al.: CS for microwave breast cancer imaging. IET Sig. Process. 12, 242–246 (2018)Google Scholar
  77. 77.
    Ross, D., et al.: Compressive k-space tomography. J. Lightwave Technol. 36, 4478–4485 (2018)Google Scholar
  78. 78.
    Liu, P., et al.: Compressive sensing of noisy multispectral images. IEEE Geosci. Remote Sens. Lett. 11, 1931–1935 (2014)Google Scholar
  79. 79.
    Wang, L., et al.: Compressive sensing of medical images with confidentially homomorphic aggregations. IEEE Internet Things JGoogle Scholar
  80. 80.
    Wei, Z., et al.: Wide angle SAR subaperture imaging based on modified compressive sensing. IEEE Sens. J. 18, 5439–5444 (2018)Google Scholar
  81. 81.
    Hu, X., et al.: Convolutional sparse coding for RGB+NIR imaging. IEEE Trans. Image Process. 27, 1611–1625 (2018)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Besson, A., et al.: Ultrafast ultrasound imaging as an inverse problem: matrix-free sparse image reconstruction. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65, 339–355 (2018)Google Scholar
  83. 83.
    Cheng, Q., et al.: Near-field millimeter-wave phased array imaging with CS. IEEE Access 5, 18975–18986 (2017)Google Scholar
  84. 84.
    Camlica, S., et al.: Autofocused spotlight SAR image reconstruction of off-grid sparse scenes. IEEE Trans. Aerosp. Electron. Syst. 53, 1880–1892 (2017)Google Scholar
  85. 85.
    Cheng, Q., et al.: Compressive millimeter-wave phased array imaging. IEEE Access 4, 9580–9588 (2016)Google Scholar
  86. 86.
    Djelouat, H., et al.: Joint sparsity recovery for CS based EEG system. In: IEEE 17th International Conference on Ubiquitous Wireless Broadband, Salamanca (2017)Google Scholar
  87. 87.
    Yang, Y., et al.: Pseudo-polar fourier transform-based compressed sensing MRI. IEEE Trans. Biomed. Eng. 64, 816–825 (2017)Google Scholar
  88. 88.
    Sun, Y., et al.: Super-resolution imaging using CS and binary pure-phase annular filter. IEEE Photonics J. 9, 1–10 (2017)Google Scholar
  89. 89.
    Chen, Z., et al.: Reconstruction of enhanced ultrasound images from compressed measurements using simultaneous direction method of multipliers. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63, 1525–1534 (2016)Google Scholar
  90. 90.
    Kustner, T., et al.: MR image reconstruction using a combination of CS and partial fourier acquisition: ESPReSSo. IEEE Trans. Med. Imaging 35, 2447–2458 (2016)Google Scholar
  91. 91.
    Chen, Z., et al.: Compressive deconvolution in medical ultrasound imaging. IEEE Trans. Med. Imaging 35, 728–737 (2016)Google Scholar
  92. 92.
    Sun, Y., et al.: Compressive superresolution imaging based on local and nonlocal regularizations. IEEE Photonics J. 8, 1–12 (2016)Google Scholar
  93. 93.
    Fickus, M., et al.: Compressive hyperspectral imaging for stellar spectroscopy. IEEE Sig. Process. Lett. 22, 1829–1833 (2015)Google Scholar
  94. 94.
    Gifani, P., et al.: Temporal super resolution enhancement of echocardiographic images based on sparse representation. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63, 6–19 (2016)Google Scholar
  95. 95.
    Wu, Q., et al.: High-resolution passive SAR imaging exploiting structured bayesian CS. IEEE J. Sel. Top. Sig. Process. 9, 1484–1497 (2015)Google Scholar
  96. 96.
    Yang, Y., et al.: Compressed sensing MRI via two-stage reconstruction. IEEE Trans. Biomed. Eng. 62, 110–118 (2015)Google Scholar
  97. 97.
    Zhang, S., et al.: Truncated SVD-based CS for downward-looking 3-D SAR imaging with uniform/nonuniform linear array. IEEE Geosci. Remote Sens. Lett. 12, 1–5 (2015)Google Scholar
  98. 98.
    Zhang, J., et al.: Group-based sparse representation for image restoration. IEEE Trans. Image Process. 23, 3336–3351 (2014)MathSciNetzbMATHGoogle Scholar
  99. 99.
    Li, X., et al.: Single image superresolution via directional group sparsity and directional features. IEEE Trans. Image Process. 24, 2874–2888 (2015)MathSciNetzbMATHGoogle Scholar
  100. 100.
    Bourquard, A., et al.: Binary compressed imaging. IEEE Trans. Image Process. 22, 1042–1055 (2013)MathSciNetzbMATHGoogle Scholar
  101. 101.
    Aguilera, E., et al.: A data-adaptive CS approach to polarimetric SAR tomography of forested areas. IEEE Geosci. Remote Sens. Lett. 10, 543–547 (2013)MathSciNetGoogle Scholar
  102. 102.
    Pham, D., et al.: Improved image recovery from compressed data contaminated with impulsive noise. IEEE Trans. Image Process. 21, 397–405 (2012)MathSciNetzbMATHGoogle Scholar
  103. 103.
    Guerquin-Ker, M., et al.: A fast wavelet-based reconstruction method for MR imaging. IEEE Trans. Med. Imaging 30, 1649–1660 (2011)Google Scholar
  104. 104.
    Wang, H., et al.: ISAR imaging via sparse probing frequencies. IEEE Geosci. Remote Sens. Lett. 8, 451–455 (2011)Google Scholar
  105. 105.
    Zhang, L., et al.: Resolution enhancement for inversed SAR imaging under low SNR via improved CS. IEEE Trans. Geosci. Remote Sens. 48, 3824–3838 (2010)Google Scholar
  106. 106.
    Yang, J., et al.: Image super-resolution via sparse representation. IEEE Trans. Image Process. 19, 2861–2873 (2010)MathSciNetzbMATHGoogle Scholar
  107. 107.
    Ma, J., et al.: Deblurring from highly incomplete measurements for remote sensing. IEEE Trans. Geosci. Remote Sens. 47, 792–802 (2009)Google Scholar
  108. 108.
    Ma, J.: A single-pixel imaging system for remote sensing by two-step iterative curvelet thresholding. IEEE Geosci. Remote Sens. Lett. 6, 676–680 (2009)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Visvesvaraya Technological UniversityBelgaumIndia
  2. 2.Electronics and Communications EngineeringEPCET (Visvesvaraya Technological University)BangaloreIndia

Personalised recommendations