Abstract

We present an information-theoretic adaptive feature-specific imaging (AFSI) system for a M-class recognition task. The proposed system utilizes the recently developed task-specific information (TSI) framework to incorporate the knowledge from previous measurements and adapt the projection matrix at each step. The decision-making framework is based on sequential hypothesis testing. We quantify the number of measurements required to achieve a specified probability of misclassification (Pe), and we compare the performances of three approaches: the new TSI-based AFSI system, a previously reported statistical AFSI system, and static FSI (SFSI). The TSI-based AFSI system exhibits significant improvement compared with SFSI and statistical AFSI at low signal-to-noise ratio (SNR). It is shown that for M=4 hypotheses, SNR=20dB and desired Pe=102, TSI-based AFSI requires 3 times fewer measurements than statistical AFSI, and 16 times fewer measurements than SFSI. We also describe an extension of the proposed method that is suitable for recognition in the presence of nuisance parameters such as illumination conditions and target orientations.

© 2009 Optical Society of America

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  1. W. T. Cathey and E. R. Dowsky, “New paradigm for imaging systems,” Appl. Opt. 41, 6080-6092 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. D. J. Brady, “Multiplex sensors and the constant radiance theorem,” Opt. Lett. 27, 16-18 (2002).
    [CrossRef]
  4. M. A. Neifeld and P. Shankar, “Feature-specific imaging,” Appl. Opt. 42, 3379-3389 (2003).
    [CrossRef] [PubMed]
  5. P. K. Baheti and M. A. Neifeld, “Feature-specific structured imaging,” Appl. Opt. 45, 7382-7391 (2006).
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  6. M. A. Neifeld, A. Ashok, and P. K. Baheti, “Task specific information for imaging system analysis,” J. Opt. Soc. Am. A 24, B25-B41 (2007).
    [CrossRef]
  7. M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1991), pp. 586-591.
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  8. P. Belhumeur, J. Hespanha, and D. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711-720 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
  16. M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), Vol. 3, pp. 14-19.
  17. 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).
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    [CrossRef] [PubMed]
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  22. A. Wald, “Sequential analysis of statistical hypotheses,” Ann. Math. Stat. 16, 117-176 (1945).
    [CrossRef]
  23. P. Armitage, “Sequential analysis with more than two alternative hypotheses and its relation to discriminant function analysis,” J. R. Stat. Soc. Ser. A (Gen.) 12, 137-144 (1950).
  24. H. H. Barrett and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (2004).
  25. S. Kay, Statistical Signal Processing--Detection Theory (Prentice-Hall, 1998).
  26. A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
    [CrossRef]
  27. R. Battiti, “Using mutual information for selecting features in supervised neural net learning,” IEEE Trans. Neural Netw. 5, 537-550 (1994).
    [CrossRef] [PubMed]
  28. J. Novovicova, P. Somol, M. Haindl, and P. Pudil, “Conditional mutual information based feature selection for classification task,” in Proceedings of the 12th Iberoamerican Congress on Pattern Recognition (Springer-Verlag, 2007), pp. 417-426.
  29. N. A. Goodman, P. R. Venkata, and M. A. Neifeld, “Adaptive waveform design and sequential hypothesis testing for target recognition using cognitive radar,” IEEE J. Sel. Top. Signal Process. 1, 105-113 (2007).
    [CrossRef]
  30. J. R. Guerci and S. U. Pillai, “Adaptive transmission radar: the next wave,” in Proceedings of the IEEE National Aerospace and Electronics Conference (NAECON, 2000), pp. 779-786.
  31. M. R. Bell, “Information theory and radar waveform design,” IEEE Trans. Inf. Theory 39, 1578-1597 (1993).
    [CrossRef]
  32. E. Marszalec, B. Martinkauppi, M. Soriano, and M. Pietikinen, “A physics-based face database for color research,” J. Electron. Imaging 9, 32-38 (2000).
    [CrossRef]
  33. A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 9, 32-38 (2008).
  34. T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991).
    [CrossRef]
  35. D. P. Palomar and S. Verdu, “Gradient of mutual information in linear vector Gaussian channels,” IEEE Trans. Inf. Theory 52, 141-154 (2006).
    [CrossRef]

2008

A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 9, 32-38 (2008).

P. K. Baheti and M. A. Neifeld, “Adaptive feature-specific imaging: a face recognition example,” Appl. Opt. 47, B21-B31 (2008), feature issue on Computational Optical Sensing and Imaging.
[CrossRef] [PubMed]

2007

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

M. A. Neifeld, A. Ashok, and P. K. Baheti, “Task specific information for imaging system analysis,” J. Opt. Soc. Am. A 24, B25-B41 (2007).
[CrossRef]

N. A. Goodman, P. R. Venkata, and M. A. Neifeld, “Adaptive waveform design and sequential hypothesis testing for target recognition using cognitive radar,” IEEE J. Sel. Top. Signal Process. 1, 105-113 (2007).
[CrossRef]

2006

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. 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).

P. K. Baheti and M. A. Neifeld, “Feature-specific structured imaging,” Appl. Opt. 45, 7382-7391 (2006).
[CrossRef] [PubMed]

2005

2003

M. A. Neifeld and P. Shankar, “Feature-specific imaging,” Appl. Opt. 42, 3379-3389 (2003).
[CrossRef] [PubMed]

W. Zhao, R. Chellapa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399-458 (2003).
[CrossRef]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

2002

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

D. J. Brady, “Multiplex sensors and the constant radiance theorem,” Opt. Lett. 27, 16-18 (2002).
[CrossRef]

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

2000

E. Marszalec, B. Martinkauppi, M. Soriano, and M. Pietikinen, “A physics-based face database for color research,” J. Electron. Imaging 9, 32-38 (2000).
[CrossRef]

1997

S. Lawrence, C. L. Giles, A. C. Tsoi, and A. D. Back, “Face recognition: A convolutional neural network approach,” IEEE Trans. Neural Netw. 8, 98-113 (1997).
[CrossRef] [PubMed]

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

1994

R. Battiti, “Using mutual information for selecting features in supervised neural net learning,” IEEE Trans. Neural Netw. 5, 537-550 (1994).
[CrossRef] [PubMed]

A. Mahanalobis, B. V. K. Kumar, S. R. F. Sims, and J. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751-3759 (1994).
[CrossRef]

1993

1992

H. H. Barrett, T. Gooley, K. Girodias, J. Rolland, T. White, and J. Lao, “Linear discriminants and image quality,” Image Vis. Comput. 10, 451-460 (1992).
[CrossRef]

1964

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139-145 (1964).
[CrossRef]

1950

P. Armitage, “Sequential analysis with more than two alternative hypotheses and its relation to discriminant function analysis,” J. R. Stat. Soc. Ser. A (Gen.) 12, 137-144 (1950).

1945

A. Wald, “Sequential analysis of statistical hypotheses,” Ann. Math. Stat. 16, 117-176 (1945).
[CrossRef]

Armitage, P.

P. Armitage, “Sequential analysis with more than two alternative hypotheses and its relation to discriminant function analysis,” J. R. Stat. Soc. Ser. A (Gen.) 12, 137-144 (1950).

Ashok, A.

A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 9, 32-38 (2008).

M. A. Neifeld, A. Ashok, and P. K. Baheti, “Task specific information for imaging system analysis,” J. Opt. Soc. Am. A 24, B25-B41 (2007).
[CrossRef]

Back, A. D.

S. Lawrence, C. L. Giles, A. C. Tsoi, and A. D. Back, “Face recognition: A convolutional neural network approach,” IEEE Trans. Neural Netw. 8, 98-113 (1997).
[CrossRef] [PubMed]

Baheti, P. K.

Baraniuk, R. G.

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

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).

Barrett, H. H.

H. H. Barrett, T. Gooley, K. Girodias, J. Rolland, T. White, and J. Lao, “Linear discriminants and image quality,” Image Vis. Comput. 10, 451-460 (1992).
[CrossRef]

H. H. Barrett and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (2004).

Battiti, R.

R. Battiti, “Using mutual information for selecting features in supervised neural net learning,” IEEE Trans. Neural Netw. 5, 537-550 (1994).
[CrossRef] [PubMed]

Belhumeur, P.

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

Bell, M. R.

M. R. Bell, “Information theory and radar waveform design,” IEEE Trans. Inf. Theory 39, 1578-1597 (1993).
[CrossRef]

Brady, D. J.

D. J. Brady, “Multiplex sensors and the constant radiance theorem,” Opt. Lett. 27, 16-18 (2002).
[CrossRef]

N. P. Pitsianis, D. J. Brady, and X. Sun, “The quantized cosine transform for sensor-layer image compression,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JMA4.

Cathey, W. T.

Chellapa, R.

W. Zhao, R. Chellapa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399-458 (2003).
[CrossRef]

Cohen, D.

A. Yuille, D. Cohen, and P. Hallinan, “Feature extraction from faces using deformable templates,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1989), pp. 104-109.

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 Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP, 2006), Vol. 3, pp. 14-19.

Dowsky, E. R.

Duarte, M. F.

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

Epperson, J.

Ganotra, D.

Giles, C. L.

S. Lawrence, C. L. Giles, A. C. Tsoi, and A. D. Back, “Face recognition: A convolutional neural network approach,” IEEE Trans. Neural Netw. 8, 98-113 (1997).
[CrossRef] [PubMed]

Girodias, K.

H. H. Barrett, T. Gooley, K. Girodias, J. Rolland, T. White, and J. Lao, “Linear discriminants and image quality,” Image Vis. Comput. 10, 451-460 (1992).
[CrossRef]

Goodman, N. A.

N. A. Goodman, P. R. Venkata, and M. A. Neifeld, “Adaptive waveform design and sequential hypothesis testing for target recognition using cognitive radar,” IEEE J. Sel. Top. Signal Process. 1, 105-113 (2007).
[CrossRef]

Gooley, T.

H. H. Barrett, T. Gooley, K. Girodias, J. Rolland, T. White, and J. Lao, “Linear discriminants and image quality,” Image Vis. Comput. 10, 451-460 (1992).
[CrossRef]

Guerci, J. R.

J. R. Guerci and S. U. Pillai, “Adaptive transmission radar: the next wave,” in Proceedings of the IEEE National Aerospace and Electronics Conference (NAECON, 2000), pp. 779-786.

Haindl, M.

J. Novovicova, P. Somol, M. Haindl, and P. Pudil, “Conditional mutual information based feature selection for classification task,” in Proceedings of the 12th Iberoamerican Congress on Pattern Recognition (Springer-Verlag, 2007), pp. 417-426.

Hallinan, P.

A. Yuille, D. Cohen, and P. Hallinan, “Feature extraction from faces using deformable templates,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1989), pp. 104-109.

Hespanha, J.

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

Jain, A.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Javidi, B.

Kay, S.

S. Kay, Statistical Signal Processing--Detection Theory (Prentice-Hall, 1998).

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. Pattern Anal. Mach. Intell. 19, 711-720 (1997).
[CrossRef]

Kumar, B. V. K.

Lao, J.

H. H. Barrett, T. Gooley, K. Girodias, J. Rolland, T. White, and J. Lao, “Linear discriminants and image quality,” Image Vis. Comput. 10, 451-460 (1992).
[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).

Lawrence, S.

S. Lawrence, C. L. Giles, A. C. Tsoi, and A. D. Back, “Face recognition: A convolutional neural network approach,” IEEE Trans. Neural Netw. 8, 98-113 (1997).
[CrossRef] [PubMed]

Mahanalobis, A.

Marszalec, E.

E. Marszalec, B. Martinkauppi, M. Soriano, and M. Pietikinen, “A physics-based face database for color research,” J. Electron. Imaging 9, 32-38 (2000).
[CrossRef]

Martinkauppi, B.

E. Marszalec, B. Martinkauppi, M. Soriano, and M. Pietikinen, “A physics-based face database for color research,” J. Electron. Imaging 9, 32-38 (2000).
[CrossRef]

Miller, M. I.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Moulin, P.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science, Wiley Series in Pure and Applied Optics (2004).

Neifeld, M. A.

Novovicova, J.

J. Novovicova, P. Somol, M. Haindl, and P. Pudil, “Conditional mutual information based feature selection for classification task,” in Proceedings of the 12th Iberoamerican Congress on Pattern Recognition (Springer-Verlag, 2007), pp. 417-426.

Pal, H. S.

Palomar, D. P.

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

Pauca, V. P.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Pentland, A. P.

M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1991), pp. 586-591.
[CrossRef]

Phillips, P. J.

W. Zhao, R. Chellapa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399-458 (2003).
[CrossRef]

Pietikinen, M.

E. Marszalec, B. Martinkauppi, M. Soriano, and M. Pietikinen, “A physics-based face database for color research,” J. Electron. Imaging 9, 32-38 (2000).
[CrossRef]

Pillai, S. U.

J. R. Guerci and S. U. Pillai, “Adaptive transmission radar: the next wave,” in Proceedings of the IEEE National Aerospace and Electronics Conference (NAECON, 2000), pp. 779-786.

Pitsianis, N. P.

N. P. Pitsianis, D. J. Brady, and X. Sun, “The quantized cosine transform for sensor-layer image compression,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JMA4.

Plemmons, R. J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Prasad, S.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Pudil, P.

J. Novovicova, P. Somol, M. Haindl, and P. Pudil, “Conditional mutual information based feature selection for classification task,” in Proceedings of the 12th Iberoamerican Congress on Pattern Recognition (Springer-Verlag, 2007), pp. 417-426.

Ramchandran, K.

A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran, “Information-theoretic bounds on target recognition performance based on degraded data,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 1153-1166 (2002).
[CrossRef]

Réfrégier, P.

Rolland, J.

H. H. Barrett, T. Gooley, K. Girodias, J. Rolland, T. White, and J. Lao, “Linear discriminants and image quality,” Image Vis. Comput. 10, 451-460 (1992).
[CrossRef]

Rosenfeld, A.

W. Zhao, R. Chellapa, P. J. Phillips, and A. Rosenfeld, “Face recognition: a literature survey,” ACM Comput. Surv. 35, 399-458 (2003).
[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).

Shankar, P.

Sims, S. R. F.

Somol, P.

J. Novovicova, P. Somol, M. Haindl, and P. Pudil, “Conditional mutual information based feature selection for classification task,” in Proceedings of the 12th Iberoamerican Congress on Pattern Recognition (Springer-Verlag, 2007), pp. 417-426.

Soriano, M.

E. Marszalec, B. Martinkauppi, M. Soriano, and M. Pietikinen, “A physics-based face database for color research,” J. Electron. Imaging 9, 32-38 (2000).
[CrossRef]

Sun, X.

N. P. Pitsianis, D. J. Brady, and X. Sun, “The quantized cosine transform for sensor-layer image compression,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, Technical Digest (Optical Society of America, 2005), paper JMA4.

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).

Thomas, J.

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

Torgersen, T. C.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Tsoi, A. C.

S. Lawrence, C. L. Giles, A. C. Tsoi, and A. D. Back, “Face recognition: A convolutional neural network approach,” IEEE Trans. Neural Netw. 8, 98-113 (1997).
[CrossRef] [PubMed]

Turk, M. A.

M. A. Turk and A. P. Pentland, “Face recognition using eigenfaces,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1991), pp. 586-591.
[CrossRef]

van der Gracht, J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139-145 (1964).
[CrossRef]

Venkata, P. R.

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Figures (13)

Fig. 1
Fig. 1

(a) Schematic representation of optical feature extraction using a feature-specific imager. (b) Example optical implementation for FSI system.

Fig. 2
Fig. 2

(a) Example of M = 4 objects considered in the recognition problem. (b) Study of number of measurements required K as a function of T int for SFSI with M = 4 , P e = 10 2 , and σ 2 = 10 .

Fig. 3
Fig. 3

Comparison of AFSI and SFSI for M = 4 , P e = 10 2 , and σ 2 = 10 .

Fig. 4
Fig. 4

Flowchart for CI-AFSI system.

Fig. 5
Fig. 5

CI-AFSI system results: (a) Example to demonstrate the update for a sample experiment where object 4 was actually chosen. (b) Updated projection vector at each iteration corresponding to the experiment in (a).

Fig. 6
Fig. 6

CI-AFSI system results with M = 2 : (a) Example to demonstrate the update for a sample experiment where object 1 was actually chosen. (b) Updated projection vector at each iteration corresponding to the experiment in (a).

Fig. 7
Fig. 7

Comparison of AFSI and CI-AFSI performance for M = 4 , P e = 10 2 , and σ 2 = 10 .

Fig. 8
Fig. 8

CI-AFSI example with M = 4 , L = 3 , T int = 0.1 , P e = 10 2 , and σ 2 = 10 ; the rows represent the iteration index and the columns represent the 3 TSI-optimal projection vectors at each iteration.

Fig. 9
Fig. 9

Average detection time for AFSI and CI-AFSI for M = 4 , P e = 10 2 , and σ 2 = 10 .

Fig. 10
Fig. 10

CI-AFSI extension to include nuisance parameters: (a) Example perspectives for M = 4 hypotheses, (b) example to demonstrate the update of L = 1 projection vector for CI-AFSI with S = 5 , and (c) example to demonstrate the update of L = 1 projection vector for CI-AFSI with S = 11 .

Fig. 11
Fig. 11

Plots of E [ K ] as a function of T int for CI-AFSI with S = 1 , 5, 11, and 21 ( M = 4 , L = 1 , P e = 10 2 , and σ 2 = 10 ).

Fig. 12
Fig. 12

CI-AFSI example with M = 4 , S = 5 , L = 3 , T int = 0.5 , P e = 10 2 , and σ 2 = 10 ; the rows represent the iteration index and the columns represent the 3 TSI-optimal projection vectors at each iteration.

Fig. 13
Fig. 13

Comparisons of CI-AFSI with S = 1 , 5, and 21, L = 1 , 2, and 3 ( M = 4 , P e = 10 2 , and σ 2 = 10 ).

Equations (54)

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r = P T G i + n ,
max l { j = 1 L [ P ] l j } = T int ,
P e = i = 1 M Pr ( H 1 , , H i 1 , H i + 1 , , H M H i ) P i ,
R = j = 1 M i = 1 M , i j P i P j ( G i G j ) ( G i G j ) T .
r k = P k T G i + n ,
λ i , j k = m = 1 k p i , m l = 1 k p j , l P i P j ,
p i , l = 1 ( 4 π σ 2 ) L 2 exp [ 1 4 σ 2 ( r l P l T G i ) T ( r l P l T G i ) ]
R k + 1 = j = 1 M i = 1 ; i j M P j k P i k ( G j G i ) ( G j G i ) T .
P j k = Pr ( r k H j ) P j k 1 m = 1 M Pr ( r k H m ) P m k 1 ,
R ̃ k = ν k V k V k T ,
r k = P k T G x + n ,
J k = I ( r ( k ) ; x ) ,
J k = J k 1 + J k k 1 ,
max P k [ J k k 1 ] , such that max l j = 1 L P k l j = T int .
P k ( i + 1 ) = P k ( i ) γ Z ( i ) .
Z = [ J k k 1 P k ] = P k { I ( P k T G x + n ; x r ( k 1 ) ) } = [ E x { p [ r ( k 1 ) x ] [ G x ] [ G x ] T } E x , r k { p [ r ( k 1 ) x ] { E x r ( k ) [ G x ] } { E x r ( k ) [ G x ] } T } ] P k 2 σ 2 ,
p [ r ( k 1 ) x ] = [ 1 ( 4 π σ 2 ) ] ( k 1 ) L 2 exp ( { r ( k 1 ) [ P ( k 1 ) ] T G x } T { r ( k 1 ) [ P ( k 1 ) ] T G x } 4 σ 2 ) .
E x r ( k ) [ G x ] = j = 1 M [ G e j ] p [ r ( k ) x = e j ] P j m = 1 M p [ r ( k ) x = e m ] P m ,
Z = [ J 1 P 1 ] = { E x [ ( G x ) ( G x ) T ] E x , r 1 [ ( E x r 1 [ G x ] ) ( E x r 1 [ G x ] ) T ] } P 1 2 σ 2 .
Z r 0 , x 0 ( p ( r ( k 1 ) x 0 ) { ( G x 0 ) ( G x 0 ) T [ E x r ( k 1 ) , r 0 [ G x ] ] [ E x r ( k 1 ) , r 0 [ G x ] ] T } ) P k 2 σ 2 K MC ,
E x r ( k 1 ) , r 0 [ G x ] = j = 1 M [ G e j ] p [ r ( k 1 ) x = e j ] p ( r 0 x = e j ) P j m = 1 M p [ r ( k 1 ) x = e m ] p ( r 0 x = e m ) P m .
Z ( i ) μ Z ( i 1 ) + ( 1 μ ) Z mod ( i ) ,
r k = P k T G ¯ ρ ¯ x + n ,
ρ ¯ = [ ρ 0 0 0 ρ 0 0 0 ρ ] ,
J k k 1 P k = ( E r k , x , ρ { p [ r ( k 1 ) x , ρ ] [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] T } E r k , x , ρ { p [ r ( k 1 ) x , ρ ] [ E x , ρ r ( k ) [ G ¯ ρ ¯ x ] ] [ E x , ρ r ( k ) [ G ¯ ρ ¯ x ] ] T } ) P k 2 σ 2 .
p [ r ( k 1 ) x , ρ ] = exp ( 1 4 σ 2 { r ( k 1 ) [ P ( k 1 ) ] T G ¯ ρ ¯ x } T { r ( k 1 ) [ P ( k 1 ) ] T G ¯ ρ ¯ x } ) ( 4 π σ 2 ) [ ( k 1 ) L ] 2 .
E x , ρ r ( k ) [ G ¯ ρ ¯ x ] = j = 1 M s = 1 S [ G ¯ c s e j ] p [ r ( k ) ρ = c s , x = e j ] ( P j S ) m = 1 M l = 1 S p [ r ( k ) ρ = c l , x = e m ] ( P m S ) ,
E ρ x , r ( k ) [ G ¯ ρ ¯ x ] = s = 1 M [ G ¯ c s x ] p [ r ( k ) ρ = c s , x ] ( 1 S ) l = 1 M p [ r ( k ) ρ = c l , x ] ( 1 S ) .
J 1 P 1 = { E r 1 , x , ρ [ ( E ρ x , r 1 [ G ¯ ρ ¯ x ] ) ( E ρ x , r 1 [ G ¯ ρ ¯ x ] ) T ] E r 1 , x , ρ [ ( E x , ρ r 1 [ G ¯ ρ ¯ x ] ) ( E x , ρ r 1 [ G ¯ ρ ¯ x ] ) T ] } P 1 2 σ 2 .
J k k 1 = I ( r k ; x r ( k 1 ) ) = E { log p [ x r k , r ( k 1 ) ] p [ x r ( k 1 ) ] } = E [ log p ( x r ( k ) ) ] E [ log p ( x r ( k 1 ) ) ] .
J k k 1 = E { log p [ r ( k ) x ] } + E [ log p ( x ) ] E { log p [ r ( k ) ] } E { log p [ x r ( k 1 ) ] } .
p ( r ( k ) x ) = [ 1 ( 4 π σ 2 ) ] k L 2 exp ( { r ( k ) [ P ( k ) ] T G x } T { r ( k ) [ P ( k ) ] T G x } 4 σ 2 )
E { log p [ r ( k ) x ] } = k L 2 log ( 4 π e σ 2 ) .
J k k 1 P k = P k { E [ log p ( r ( k ) ) ] } = P k [ p [ r ( k ) ] log p [ r ( k ) ] d r k ] = { 1 + log p [ r ( k ) ] } P k { p [ r ( k ) ] } d r k .
P k { p [ r ( k ) ] } = j = 1 M p [ r ( k 1 ) e j ] p ( r k e j ) P j [ 2 4 σ 2 ( G e j ) ( r k P k T G e j ) T ] = j = 1 M P j [ G e j ] r k T { p [ r ( k ) e j ] } = E x [ [ G x ] r k T { p [ r ( k ) x ] } ] .
J k k 1 P k = { 1 + log p [ r ( k ) ] } E x ( [ G x ] r k T { p [ r ( k ) x ] } ) d r k = E x [ [ G x ] ( { 1 + log p [ r ( k ) ] } r k T { p [ r ( k ) x ] } d r k ) ] .
J k k 1 P k = E x ( [ G x ] { p [ r ( k ) x ] p [ r ( k ) ] } p [ r ( k ) ] r k T d r k ) .
J k k 1 P k = E x { p [ r ( k ) x ] p [ r ( k ) ] [ G x ] } { p [ r ( k ) ] r k T } d r k = E x { p [ x r ( k ) ] p ( x ) [ G x ] } { p [ r ( k ) ] r k T } d r k = [ E x r ( k ) [ G x ] ] [ r k T ( E x { p [ r ( k ) x ] } ) ] d r k = [ E x r ( k ) [ G x ] ] [ E x ( r k T { p [ r ( k ) x ] } ) ] d r k = [ E x r ( k ) [ G x ] ] ( E x { p [ r ( k ) x ] 2 σ 2 [ r k P k T G x ] T } ) d r k = 1 2 σ 2 [ E x r ( k ) [ G x ] ] ( p [ r ( k ) ] { r k T [ E x r ( k ) [ G x ] ] T P k } ) d r k = 1 2 σ 2 ( { j = 1 M [ G e j r k T ] p [ x = e j , r ( k ) ] } d r k p [ r ( k ) ] [ E x r ( k ) [ G x ] ] [ E x r ( k ) [ G x ] ] T P k d r k ) = 1 2 σ 2 { j = 1 M p [ r ( k 1 ) e j ] P j G e j r k T p ( r k e j ) d r k p [ r ( k ) ] [ E x r ( k ) [ G x ] ] [ E x r ( k ) [ G x ] ] T P k d r k } = ( j = 1 M p [ r ( k 1 ) e j ] P j [ G e j ] [ G e j ] T E x , r k { p [ r ( k 1 ) x ] [ E x r ( k ) [ G x ] ] [ E x r ( k ) [ G x ] ] T } ) P k 2 σ 2 = ( E x { p [ r ( k 1 ) x ] [ G x ] [ G x ] T } E x , r k { p [ r ( k 1 ) x ] [ E x r ( k ) [ G x ] ] [ E x r ( k ) [ G x ] ] T } ) P k 2 σ 2 .
J 1 P 1 = P 1 { E [ log p ( r 1 ) ] } = P 1 [ p ( r 1 ) log p ( r 1 ) d r 1 ] = [ 1 + log p ( r 1 ) ] P 1 [ p ( r 1 ) ] d r 1 .
Z = { E x [ ( G x ) ( G x ) T ] E x , r 1 [ ( E x r 1 [ G x ] ) ( E x r 1 [ G x ] ) T ] } P 1 2 σ 2 .
J k k 1 = E { log p [ r ( k ) x ] } + E [ log p ( x ) ] E { log p [ r ( k ) ] } E { log p [ x r ( k 1 ) ] } .
Z = [ J k k 1 P k ] = P k ( E { log p [ r ( k ) x ] } ) P k ( E { log p [ r ( k ) ] } ) .
P k ( E { log p [ r ( k ) x ] } ) = P k ( E x { p [ r ( k ) x ] log p [ r ( k ) x ] d r k } ) = E x ( { 1 + log p [ r ( k ) x ] } p [ r ( k ) x ] P k d r k ) .
P k ( E { log p [ r ( k ) ] } ) = { 1 + log p [ r ( k ) ] } p [ r ( k ) ] P k d r k .
p [ r ( k ) ] = E x { p [ r ( k ) x ] } = j = 1 M p [ r ( k ) x = e j ] P j = j = 1 M s = 1 S p [ r ( k ) e j , ρ = c s ] Pr ( ρ = c s ) P j = E x , ρ { p [ r ( k ) x , ρ ] } ,
where p [ r ( k ) x , ρ ] = ( 1 4 π σ 2 ) k L 2 exp ( { r ( k ) [ P ( k ) ] T G ¯ ρ ¯ x } T { r ( k ) [ P ( k ) ] T G ¯ ρ ¯ x } 4 σ 2 ) .
p ( r ( k ) x ) P k = s = 1 S Pr ( ρ = c s ) ( P k { p [ r ( k ) x , ρ = c s ] } ) = E ρ x { [ G ¯ ρ ¯ x ] p [ r ( k ) x , ρ ] r k T } ,
p [ r ( k ) ] P k = j = 1 M s = 1 S Pr ( ρ = c s ) P j ( P k { p [ r ( k ) e j , ρ = c s ] } ) = E x , ρ { [ G ¯ ρ ¯ x ] p [ r ( k ) x , ρ ] r k T } .
P k ( E { log p [ r ( k ) x ] } ) = E x ( { 1 + log p [ r ( k ) x ] } E ρ x { [ G ¯ ρ ¯ x ] p [ r ( k ) x , ρ ] r k T } d r k ) = E x , ρ ( [ G ¯ ρ ¯ x ] { 1 + log p [ r ( k ) x ] } p [ r ( k ) x , ρ ] r k T d r k ) = E x , ρ { [ G ¯ ρ ¯ x ] p [ r ( k ) x , ρ ] p ( r ( k ) x ) p [ r ( k ) x ] r k T d r k } = E x ( E ρ x { p [ ρ x , r ( k ) ] p ( ρ x ) [ G ¯ ρ ¯ x ] } p [ r ( k ) x ] r k T d r k ) = E x ( [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] E ρ x { p [ r ( k ) x , ρ ] r k T } d r k ) = 1 2 σ 2 E x ( [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] E ρ x { p [ r ( k ) x , ρ ] ( r k P k T G ¯ ρ ¯ x ) T } d r k ) = 1 2 σ 2 E x ( [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] p [ r ( k ) x ] { r k T [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] T P k } d r k ) .
P k ( E { log p [ r ( k ) x ] } ) = 1 2 σ 2 E x , ρ { p [ r ( k 1 ) x , ρ ] [ G ¯ ρ ¯ x ] [ G ¯ ρ ¯ x ] T } + 1 2 σ 2 ( E r k , x , ρ { p [ r ( k 1 ) x , ρ ] [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] T } ) P k .
P k ( E { log p [ r ( k ) ] } ) = 1 2 σ 2 E x , ρ { p [ r ( k 1 ) x , ρ ] [ G ¯ ρ ¯ x ] [ G ¯ ρ ¯ x ] T } + 1 2 σ 2 ( E r k , x , ρ { p [ r ( k 1 ) x , ρ ] [ E x , ρ r ( k ) [ G ¯ ρ ¯ x ] ] [ E x , ρ r ( k ) [ G ¯ ρ ¯ x ] ] T } ) P k .
J k k 1 P k = ( E r k , x , ρ { p [ r ( k 1 ) x , ρ ] [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] [ E ρ x , r ( k ) [ G ¯ ρ ¯ x ] ] T } E r k , x , ρ { p [ r ( k 1 ) x , ρ ] [ E x , ρ r ( k ) [ G ¯ ρ ¯ x ] ] [ E x , ρ r ( k ) [ G ¯ ρ ¯ x ] ] T } ) P k 2 σ 2 .
J 1 P 1 = P 1 { E [ log p ( r 1 x ) ] } P 1 { E [ log p ( r 1 ) ] } = E x { [ 1 + log p ( r 1 x ) ] p ( r 1 x ) P 1 d r 1 } [ 1 + log p ( r 1 ) ] P 1 [ p ( r 1 ) ] d r 1 .
J 1 P 1 = { E r 1 , x , ρ [ ( E ρ x , r 1 [ G ¯ ρ ¯ x ] ) ( E ρ x , r 1 [ G ¯ ρ ¯ x ] ) T ] E r 1 , x , ρ [ ( E x , ρ r 1 [ G ¯ ρ ¯ x ] ) ( E x , ρ r 1 [ G ¯ ρ ¯ x ] ) T ] } P 1 2 σ 2 .

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