This paper addresses the problem of reconstructing an image from 1-bit-quantized measurements, considering a simple but non-conventional optical acquisition model. Following a compressed-sensing design, a known pseudo-random phase-shifting mask is introduced at the aperture of the optical system. The associated reconstruction algorithm is tailored to this mask. Our results demonstrate the feasibility of the whole approach for reconstructing grayscale images.

© 2010 Optical Society of America

1. A. Stern, Y. Rivenson, and B. Javidi, "Optically compressed image sensing using random aperture coding," in "Proceedings of the SPIE - The International Society for Optical Engineering," (2008), pp. 69750D-1-10.
2. J. Romberg, "Sensing by random convolution," in "2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing," (2007), pp. 137-140.
3. M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, "Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras," IEEE Signal Process. Mag. 25, 83-91 (2008). [CrossRef]
4. R. F. Marcia and R. M. Willett, "Compressive coded aperture superresolution image reconstruction," in "IEEE International Conference on Acoustic, Speech and Signal Processes," (2008), pp. 833-836.
5. F. Sebert, Y. M. Zou, and L. Ying, "Toeplitz block matrices in compressed sensing and their applications in imaging," in "Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine," (2008), pp. 47-50.
6. P. T. Boufounos and R. G. Baraniuk, "1-bit compressive sensing," in "42nd Annual Conference on Information Sciences and Systems," (2008), pp. 16-21.
7. A. M. Bruckstein, D. L. Donoho, and M. Elad, "From sparse solutions of systems of equations to sparse modeling of signals and images," SIAM Rev. 51, 34-81 (2009). [CrossRef]
8. L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60, 259-268 (1992). [CrossRef]
9. W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, "Toeplitz-structured compressed sensing matrices," in "IEEE Workshop on Statistical Signal Processing Proceedings," (2007), pp. 294-298.
10. H. Rauhut, "Circulant and toeplitz matrices in compressed sensing," in "Proceedings of SPARS’09," (2009).
11. W. Yin, S. Morgan, J. Yang, and Y. Zhang, "Practical compressive sensing with toeplitz and circulant matrices," Tech. rep., CAAM, Rice University (2010).
12. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill Higher Education, 1996), 2nd ed.
13. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1959), 7th ed.
14. M. Unser, "Splines: A perfect fit for signal and image processing," IEEE Signal Process. Mag. 16, 22-38 (1999). [CrossRef]
15. R. T. Rockafellar, Convex Analysis (Princeton Mathematical Series) (Princeton University Press, 1970).
16. S. P. Lloyd, "Least squares quantization in PCM," IEEE Trans. Inf. Theory 28, 129-137 (1982). [CrossRef]
17. J. Max, "Quantizing for minimum distortion," IRE Trans. Inf. Theory IT-6, 7-12 (1960). [CrossRef]
18. L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, "The gigavision camera," in "2009 IEEE International Conference on Acoustics, Speech and Signal Processing," (2009), pp. 1093-1096.

IEEE Signal Process. Mag.

• M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, "Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras," IEEE Signal Process. Mag. 25, 83-91 (2008). [CrossRef]
• M. Unser, "Splines: A perfect fit for signal and image processing," IEEE Signal Process. Mag. 16, 22-38 (1999). [CrossRef]

IEEE Trans. Inf. Theory

• S. P. Lloyd, "Least squares quantization in PCM," IEEE Trans. Inf. Theory 28, 129-137 (1982). [CrossRef]

IRE Trans. Inf. Theory

• J. Max, "Quantizing for minimum distortion," IRE Trans. Inf. Theory IT-6, 7-12 (1960). [CrossRef]

Physica D

• L. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60, 259-268 (1992). [CrossRef]

SIAM Rev.

• A. M. Bruckstein, D. L. Donoho, and M. Elad, "From sparse solutions of systems of equations to sparse modeling of signals and images," SIAM Rev. 51, 34-81 (2009). [CrossRef]

Other

• L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, "The gigavision camera," in "2009 IEEE International Conference on Acoustics, Speech and Signal Processing," (2009), pp. 1093-1096.
• R. T. Rockafellar, Convex Analysis (Princeton Mathematical Series) (Princeton University Press, 1970).
• A. Stern, Y. Rivenson, and B. Javidi, "Optically compressed image sensing using random aperture coding," in "Proceedings of the SPIE - The International Society for Optical Engineering," (2008), pp. 69750D-1-10.
• J. Romberg, "Sensing by random convolution," in "2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing," (2007), pp. 137-140.
• W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, "Toeplitz-structured compressed sensing matrices," in "IEEE Workshop on Statistical Signal Processing Proceedings," (2007), pp. 294-298.
• H. Rauhut, "Circulant and toeplitz matrices in compressed sensing," in "Proceedings of SPARS’09," (2009).
• W. Yin, S. Morgan, J. Yang, and Y. Zhang, "Practical compressive sensing with toeplitz and circulant matrices," Tech. rep., CAAM, Rice University (2010).
• J. W. Goodman, Introduction to Fourier Optics (McGraw Hill Higher Education, 1996), 2nd ed.
• M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1959), 7th ed.
• R. F. Marcia and R. M. Willett, "Compressive coded aperture superresolution image reconstruction," in "IEEE International Conference on Acoustic, Speech and Signal Processes," (2008), pp. 833-836.
• F. Sebert, Y. M. Zou, and L. Ying, "Toeplitz block matrices in compressed sensing and their applications in imaging," in "Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine," (2008), pp. 47-50.
• P. T. Boufounos and R. G. Baraniuk, "1-bit compressive sensing," in "42nd Annual Conference on Information Sciences and Systems," (2008), pp. 16-21.

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