Abstract

We present a task-specific information (TSI) based framework for designing compressive imaging (CI) systems. The task of target detection is chosen to demonstrate the performance of the optimized CI system designs relative to a conventional imager. In our optimization framework, we first select a projection basis and then find the associated optimal photon-allocation vector in the presence of a total photon-count constraint. Several projection bases, including principal components (PC), independent components, generalized matched-filter, and generalized Fisher discriminant (GFD) are considered for candidate CI systems, and their respective performance is analyzed for the target-detection task. We find that the TSI-optimized CI system design based on a GFD projection basis outperforms all other candidate CI system designs as well as the conventional imager. The GFD-based compressive imager yields a TSI of 0.9841bits (out of a maximum possible 1bit for the detection task), which is nearly ten times the 0.0979bits achieved by the conventional imager at a signal-to-noise ratio of 5.0. We also discuss the relation between the information-theoretic TSI metric and a conventional statistical metric like probability of error in the context of the target-detection problem. It is shown that the TSI can be used to derive an upper bound on the probability of error that can be attained by any detection algorithm.

© 2008 Optical Society of America

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    [CrossRef]
  29. I. T. Jolliffe, Principal Component Analysis (Springer, 2002).
  30. D. Barber and F. V. Agakov, “The IM algorithm: a variational approach to information maximization,” in NIPS (MIT Press, 2003).
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2008 (1)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI [a look at how CS can improve on current imaging techniques],” IEEE Signal Process. Mag. 25 (2), 72-82 (2008
[CrossRef]

2007 (6)

A. Mahalanobis, “Optical systems for task specific compressed sensing and image reconstruction,” in Annual Meeting of the IEEE Lasers and Electro-Optics SocietyAnnual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), pp. 157-158.
[CrossRef]

D. P. Palomar and S. Verdu, “Representation of mutual information via input estimates,” IEEE Trans. Inf. Theory 53, 453-470 (2007).
[CrossRef]

R. Patnaik and D. Casasent, “SAR classification and confuser and clutter rejection tests on MSTAR ten-class data using Minace filters,” Proc. SPIE 6574, 657402 (2007).
[CrossRef]

M. A. Neifeld, A. Ashok, and P. K. Baheti, “Task specific information for imaging system analysis,” J. Opt. Soc. Am. A 24, 25-41 (2007).
[CrossRef]

A. Ashok and M. A. Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46, 2256-2268 (2007).
[CrossRef] [PubMed]

M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46, 5293-5303 (2007).
[CrossRef] [PubMed]

2006 (5)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

D. P. Palomar and S. Verdu, “Gradient of mutual information in linear vector Gaussian channels,” IEEE Trans. Inf. Theory 52, 141-154 (2006).
[CrossRef]

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in Vol. 3 of Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), pp. 14-19.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

M. D. Stenner, A. Ashok, and M. A. Neifeld, “Multi-domain optimization for ultra-thin cameras,” in Frontiers in Optics (2006), paper FWH5.

2005 (2)

D. Guo, S. Shamai, and S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory 51, 1261-1282 (2005).
[CrossRef]

R. Patnaik and D. Casasent, “MINACE filter classification algorithms for ATR using MSTAR data,” Proc. SPIE 5807, 100-111 (2005).
[CrossRef]

2003 (3)

2002 (3)

I. T. Jolliffe, Principal Component Analysis (Springer, 2002).

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. 41, 6080-6092 (2002).
[CrossRef] [PubMed]

M. S. Bartlett, J. R. Movellan, and T. J. Sejnowski, “Face recognition by independent component analysis,” IEEE Trans. Neural Netw. 13, 1450-1464 (2002).
[CrossRef]

2001 (2)

2000 (2)

W. Gander and W. Gautschi, “Adaptive quadrature--revisited,” BIT 40, 84-101 (2000).
[CrossRef]

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley Interscience, 2000).

1999 (1)

W. Wenzel and K. Hamacher, “A Stochastic tunneling approach for global minimization,” Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

1997 (1)

P. Belhumeur, J. Hespanha, and D. Kriegman, “Eigenfaces vs. Fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pat. Anal. Mach. Intell. 19, 711-720 (1997).
[CrossRef]

1996 (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 7.

1995 (1)

A. J. Bell and T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129-1159 (1995).
[CrossRef] [PubMed]

1993 (2)

S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice-Hall, 1993).

M. Tanner, Tools for Statistical Inference, 2nd ed. (Springer, 1993).

1991 (2)

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71-86 (1991).
[CrossRef]

T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Agakov, F. V.

D. Barber and F. V. Agakov, “The IM algorithm: a variational approach to information maximization,” in NIPS (MIT Press, 2003).

Ashok, A.

Baheti, P. K.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in Vol. 3 of Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), pp. 14-19.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Barber, D.

D. Barber and F. V. Agakov, “The IM algorithm: a variational approach to information maximization,” in NIPS (MIT Press, 2003).

Baron, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Bartlett, M. S.

M. S. Bartlett, J. R. Movellan, and T. J. Sejnowski, “Face recognition by independent component analysis,” IEEE Trans. Neural Netw. 13, 1450-1464 (2002).
[CrossRef]

Belhumeur, P.

P. Belhumeur, J. Hespanha, and D. Kriegman, “Eigenfaces vs. Fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pat. Anal. Mach. Intell. 19, 711-720 (1997).
[CrossRef]

Bell, A. J.

A. J. Bell and T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129-1159 (1995).
[CrossRef] [PubMed]

Casasent, D.

R. Patnaik and D. Casasent, “SAR classification and confuser and clutter rejection tests on MSTAR ten-class data using Minace filters,” Proc. SPIE 6574, 657402 (2007).
[CrossRef]

R. Patnaik and D. Casasent, “MINACE filter classification algorithms for ATR using MSTAR data,” Proc. SPIE 5807, 100-111 (2005).
[CrossRef]

Cathey, W. T.

Cover, T.

T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in Vol. 3 of Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), pp. 14-19.

Donoho, D. L.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI [a look at how CS can improve on current imaging techniques],” IEEE Signal Process. Mag. 25 (2), 72-82 (2008
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

Dowski, E. R.

Duarte, M. F.

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in Vol. 3 of Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), pp. 14-19.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Duda, R. O.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley Interscience, 2000).

Gander, W.

W. Gander and W. Gautschi, “Adaptive quadrature--revisited,” BIT 40, 84-101 (2000).
[CrossRef]

Gautschi, W.

W. Gander and W. Gautschi, “Adaptive quadrature--revisited,” BIT 40, 84-101 (2000).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 7.

Guo, D.

D. Guo, S. Shamai, and S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory 51, 1261-1282 (2005).
[CrossRef]

Hamacher, K.

W. Wenzel and K. Hamacher, “A Stochastic tunneling approach for global minimization,” Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

Hart, P. E.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley Interscience, 2000).

Hespanha, J.

P. Belhumeur, J. Hespanha, and D. Kriegman, “Eigenfaces vs. Fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pat. Anal. Mach. Intell. 19, 711-720 (1997).
[CrossRef]

Hyvarinen, A.

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Javidi, B.

Jolliffe, I. T.

I. T. Jolliffe, Principal Component Analysis (Springer, 2002).

Karhunen, J.

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Kay, S.

S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice-Hall, 1993).

Ke, J.

Kelly, K.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Kriegman, D.

P. Belhumeur, J. Hespanha, and D. Kriegman, “Eigenfaces vs. Fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pat. Anal. Mach. Intell. 19, 711-720 (1997).
[CrossRef]

Laska, J. N.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Lustig, M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI [a look at how CS can improve on current imaging techniques],” IEEE Signal Process. Mag. 25 (2), 72-82 (2008
[CrossRef]

Mahalanobis, A.

A. Mahalanobis, “Optical systems for task specific compressed sensing and image reconstruction,” in Annual Meeting of the IEEE Lasers and Electro-Optics SocietyAnnual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), pp. 157-158.
[CrossRef]

Movellan, J. R.

M. S. Bartlett, J. R. Movellan, and T. J. Sejnowski, “Face recognition by independent component analysis,” IEEE Trans. Neural Netw. 13, 1450-1464 (2002).
[CrossRef]

Neifeld, M. A.

Oja, E.

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

Pal, H.

Palomar, D. P.

D. P. Palomar and S. Verdu, “Representation of mutual information via input estimates,” IEEE Trans. Inf. Theory 53, 453-470 (2007).
[CrossRef]

D. P. Palomar and S. Verdu, “Gradient of mutual information in linear vector Gaussian channels,” IEEE Trans. Inf. Theory 52, 141-154 (2006).
[CrossRef]

Patnaik, R.

R. Patnaik and D. Casasent, “SAR classification and confuser and clutter rejection tests on MSTAR ten-class data using Minace filters,” Proc. SPIE 6574, 657402 (2007).
[CrossRef]

R. Patnaik and D. Casasent, “MINACE filter classification algorithms for ATR using MSTAR data,” Proc. SPIE 5807, 100-111 (2005).
[CrossRef]

Pauly, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI [a look at how CS can improve on current imaging techniques],” IEEE Signal Process. Mag. 25 (2), 72-82 (2008
[CrossRef]

Pentland, A.

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71-86 (1991).
[CrossRef]

Santos, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI [a look at how CS can improve on current imaging techniques],” IEEE Signal Process. Mag. 25 (2), 72-82 (2008
[CrossRef]

Sarvotham, S.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Sejnowski, T. J.

M. S. Bartlett, J. R. Movellan, and T. J. Sejnowski, “Face recognition by independent component analysis,” IEEE Trans. Neural Netw. 13, 1450-1464 (2002).
[CrossRef]

A. J. Bell and T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129-1159 (1995).
[CrossRef] [PubMed]

Shamai, S.

D. Guo, S. Shamai, and S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory 51, 1261-1282 (2005).
[CrossRef]

Shankar, P.

Stenner, M. D.

M. D. Stenner, A. Ashok, and M. A. Neifeld, “Multi-domain optimization for ultra-thin cameras,” in Frontiers in Optics (2006), paper FWH5.

Stork, D. G.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley Interscience, 2000).

Takhar, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Tanner, M.

M. Tanner, Tools for Statistical Inference, 2nd ed. (Springer, 1993).

Thomas, J.

T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Towghi, N.

Turk, M.

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71-86 (1991).
[CrossRef]

Verdu, S.

D. P. Palomar and S. Verdu, “Representation of mutual information via input estimates,” IEEE Trans. Inf. Theory 53, 453-470 (2007).
[CrossRef]

D. P. Palomar and S. Verdu, “Gradient of mutual information in linear vector Gaussian channels,” IEEE Trans. Inf. Theory 52, 141-154 (2006).
[CrossRef]

D. Guo, S. Shamai, and S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory 51, 1261-1282 (2005).
[CrossRef]

Wakin, M. B.

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in Vol. 3 of Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), pp. 14-19.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Wenzel, W.

W. Wenzel and K. Hamacher, “A Stochastic tunneling approach for global minimization,” Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

Appl. Opt. (4)

BIT (1)

W. Gander and W. Gautschi, “Adaptive quadrature--revisited,” BIT 40, 84-101 (2000).
[CrossRef]

IEEE Signal Process. Mag. (1)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI [a look at how CS can improve on current imaging techniques],” IEEE Signal Process. Mag. 25 (2), 72-82 (2008
[CrossRef]

IEEE Trans. Inf. Theory (4)

D. Guo, S. Shamai, and S. Verdu, “Mutual information and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory 51, 1261-1282 (2005).
[CrossRef]

D. P. Palomar and S. Verdu, “Gradient of mutual information in linear vector Gaussian channels,” IEEE Trans. Inf. Theory 52, 141-154 (2006).
[CrossRef]

D. P. Palomar and S. Verdu, “Representation of mutual information via input estimates,” IEEE Trans. Inf. Theory 53, 453-470 (2007).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

IEEE Trans. Neural Netw. (1)

M. S. Bartlett, J. R. Movellan, and T. J. Sejnowski, “Face recognition by independent component analysis,” IEEE Trans. Neural Netw. 13, 1450-1464 (2002).
[CrossRef]

IEEE Trans. Pat. Anal. Mach. Intell. (1)

P. Belhumeur, J. Hespanha, and D. Kriegman, “Eigenfaces vs. Fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pat. Anal. Mach. Intell. 19, 711-720 (1997).
[CrossRef]

J. Cogn. Neurosci. (1)

M. Turk and A. Pentland, “Eigenfaces for recognition,” J. Cogn. Neurosci. 3, 71-86 (1991).
[CrossRef]

J. Opt. Soc. Am. A (2)

Neural Comput. (1)

A. J. Bell and T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129-1159 (1995).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. Lett. (1)

W. Wenzel and K. Hamacher, “A Stochastic tunneling approach for global minimization,” Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

Proc. SPIE (3)

R. Patnaik and D. Casasent, “MINACE filter classification algorithms for ATR using MSTAR data,” Proc. SPIE 5807, 100-111 (2005).
[CrossRef]

R. Patnaik and D. Casasent, “SAR classification and confuser and clutter rejection tests on MSTAR ten-class data using Minace filters,” Proc. SPIE 6574, 657402 (2007).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 43-52 (2006).

Other (12)

T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley, 2001).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 7.

M. Tanner, Tools for Statistical Inference, 2nd ed. (Springer, 1993).

I. T. Jolliffe, Principal Component Analysis (Springer, 2002).

D. Barber and F. V. Agakov, “The IM algorithm: a variational approach to information maximization,” in NIPS (MIT Press, 2003).

S. Kay, Fundamentals of Statistical Signal Processing: Detection Theory (Prentice-Hall, 1993).

M. D. Stenner, A. Ashok, and M. A. Neifeld, “Multi-domain optimization for ultra-thin cameras,” in Frontiers in Optics (2006), paper FWH5.

M. A. Neifeld, “Multi-domain optimization,” http://ocpl.ece.arizona.edu/mdo/.

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (Wiley Interscience, 2000).

A. Mahalanobis, “Optical systems for task specific compressed sensing and image reconstruction,” in Annual Meeting of the IEEE Lasers and Electro-Optics SocietyAnnual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), pp. 157-158.
[CrossRef]

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in Vol. 3 of Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), pp. 14-19.

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Figures (16)

Fig. 1
Fig. 1

Candidate optical architectures for compressive imaging: (a) sequential and (b) parallel.

Fig. 2
Fig. 2

Block diagram of a compressive imaging system.

Fig. 3
Fig. 3

Illustration of stochastic encoding C: (a) target profile matrix T and position vector ρ and (b) clutter profile matrix V c and mixing vector β .

Fig. 4
Fig. 4

Difference mmse and mmse components versus SNR for a conventional imager.

Fig. 5
Fig. 5

Example scenes with optical blur and noise: (a) tank in the top of the scene, (b) tank in the middle of the scene.

Fig. 6
Fig. 6

Example projection vectors in the PC projection basis, clockwise from upper left, #2, #6, #16, #31.

Fig. 7
Fig. 7

TSI versus SNR for PC compressive imager.

Fig. 8
Fig. 8

Example projection vectors in the GMF projection basis, clockwise from upper left, #1, #16, #32, #64.

Fig. 9
Fig. 9

Example projection vectors in the GFD1 projection basis, clockwise from upper left, #1, #10, #11, #14.

Fig. 10
Fig. 10

Projection vector in the GFD2 projection basis.

Fig. 11
Fig. 11

Example projection vectors in the IC projection basis, clockwise from upper left, #8, #16, #22, #28.

Fig. 12
Fig. 12

Optimized compressive imagers: TSI versus SNR for the candidate CI system and a conventional imager.

Fig. 13
Fig. 13

Optimal photon allocation vectors for PC compressive imager at (a)  s = 0.5 , (b)  s = 5.0 , and (c)  s = 20.0 .

Fig. 14
Fig. 14

Optimal photon allocation vectors for GFD1 compressive imager at (a)  s = 0.5 , (b)  s = 5.0 , and (c)  s = 20.0 .

Fig. 15
Fig. 15

Lower bound on probability of error as a function of TSI.

Fig. 16
Fig. 16

Comparison of probability of error obtained via Bayes detector versus lower bound obtained by Fano’s inequality as a function of SNR.

Tables (1)

Tables Icon

Table 1 TSI (in Bits) for Candidate Compressive Imagers at Three Representative Values of SNR: Low ( s = 0.5 ), Medium ( s = 5.0 ), and High ( s = 20.0 )

Equations (35)

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TSI I ( X ; R ) = J ( X ) J ( X | R ) ,
C ( X ) = s T ρ X + c V c β ,
R = PHC ( X ) + N ,
N c = c PH V c β + N ,
R = s PHT ρ X + N c .
TSI = 1 2 0 s mmse ( s ) d s ,
where   mmse ( s ) = Tr ( H P Σ N c 1 PH ( E Y ( s ) E Y | X ( s ) ) ) .
E Y ( s ) = E [ ( Y E ( Y | R , s ) ) ( Y E ( Y | R , s ) ) T ] , E Y | X ( s ) = E [ ( Y E ( Y | R , X , s ) ) ( Y E ( Y | R , X , s ) ) T ] .
h ( i , j ) = Δ / 2 Δ / 2 Δ / 2 Δ / 2 sinc 2 ( ( x i Δ ) W ) sinc 2 ( ( y j Δ ) W ) d x d y ,
P = 1 ω P * ,
max P [ TSI ] , s.t.  max j i = 1 K | P i j | = 1.
max π [ TSI ] , s.t. max j i = 1 K | diag ( π ) P i j * | = 1.
R O O = E { o o T } ,
P GMF * = T ¯ Σ N c 1 ,
Σ i = Σ clutter = c · H V c Σ β V c T H T , i = 1 P + 1.
μ i = s H Y i + c H V c μ β i = 1 P ,
μ P + 1 = c H V c μ β .
μ GFD 1 = 1 2 P i = 1 P μ i + 1 2 μ P + 1 .
S W 1 = 1 2 P i = 1 P Σ clutter + 1 2 Σ P + 1 = 1 2 { 1 P + 1 } · Σ clutter ,
S B 1 = 1 2 P i = 1 P ( μ i μ GFD 1 ) ( μ i μ GFD 1 ) T + 1 2 ( μ P + 1 μ GFD 1 ) ( μ P + 1 μ GFD 1 ) T .
D GFD ( P GFD 1 * ) = P GFD 1 * T S B 1 P GFD 1 * P GFD 1 * T S W 1 P GFD 1 * , s.t.   P GFD 1 * T S B 1 P GFD 1 * = I .
S B 1 P GFD 1 * = S W 1 Λ 1 P GFD 1 * .
μ 1 = 1 P i = 1 P s H Y i + c H V c μ β ,
μ 2 = c H V c μ β .
μ GFD 2 = 1 2 μ 1 + 1 2 μ 2 = 1 2 P i = 1 P s H Y i + c H V c μ β .
Σ 1 = 1 P i P ( s H Y i 1 P j P s H Y j ) ( s H Y i 1 P j P s H Y j ) T + c · H V c Σ β V c T H T ,
Σ 2 = c · H V c Σ β V c T H T .
S W 2 = 1 2 Σ 1 + 1 2 Σ 2 ,
S B 2 = 1 2 · ( μ 1 μ GFD 2 ) ( μ 1 μ GFD 2 ) T + 1 2 · ( μ 2 μ GFD 2 ) ( μ 2 μ GFD 2 ) T .
I ( R 1 , R 2 ; X ) = J ( R 1 , R 2 ) J ( R 1 , R 2 | X )
= J ( R 1 ) + J ( R 2 | R 1 ) J ( R 1 , R 2 | X ) J ( R 1 ) + J ( R 2 ) J ( R 1 , R 2 | X ) .
J ( X | R ) P e log ( | X | 1 ) + J ( P e ) ,
TSI I ( X ; R ) J ( X ) P e log ( | X | 1 ) J ( P e ) .
TSI J ( X ) J ( P e ) = 1 + P e log ( P e ) + ( 1 P e ) log ( 1 P e ) .
p r ( R | X = 0 ) p r ( R | X = 1 ) X ^ = 0 X ^ = 1 p r ( X = 1 ) p r ( X = 0 ) ,

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