Abstract

Compressive imaging systems typically exploit the spatial correlation of the scene to facilitate a lower dimensional measurement relative to a conventional imaging system. In natural time-varying scenes there is a high degree of temporal correlation that may also be exploited to further reduce the number of measurements. In this work we analyze space–time compressive imaging using Karhunen–Loève (KL) projections for the read-noise-limited measurement case. Based on a comprehensive simulation study, we show that a KL-based space–time compressive imager offers higher compression relative to space-only compressive imaging. For a relative noise strength of 10% and reconstruction error of 10%, we find that space–time compressive imaging with 8×8×16 spatiotemporal blocks yields about 292× compression compared to a conventional imager, while space-only compressive imaging provides only 32× compression. Additionally, under high read-noise conditions, a space–time compressive imaging system yields lower reconstruction error than a conventional imaging system due to the multiplexing advantage. We also discuss three electro-optic space-time compressive imaging architecture classes, including charge-domain processing by a smart focal plane array (FPA). Space–time compressive imaging using a smart FPA provides an alternative method to capture the nonredundant portions of time-varying scenes.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
    [CrossRef]
  5. E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
    [CrossRef]
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    [CrossRef]
  8. J. A. Decker, “Experimental realization of the multiplex advantage with a Hadamard-transform spectrometer,” Appl. Opt. 10, 510–514 (1971).
    [CrossRef]
  9. D. J. Brady, “Multiplex sensors and the constant radiance theorem,” Opt. Lett. 27, 16–18 (2002).
    [CrossRef]
  10. M. A. Neifeld and P. Shankar, “Feature-specific imaging,” Appl. Opt. 42, 3379–3389 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. T. Sikora, “MPEG digital video-coding standards,” IEEE Signal Process. Mag. 14, 82–100 (1997).
    [CrossRef]
  16. G. K. Wallace, “The JPEG still picture compression standard,” IEEE Trans. Consum. Electron. 38, xviii–xxxiv (1992).
    [CrossRef]
  17. R. J. Clarke, “Relation between the Karhunen-Loève and cosine transforms,” IEE Proc. Commun. Radar Signal Process.128, 359–360 (1981).
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    [CrossRef]
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    [CrossRef]

2008 (1)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

2007 (1)

2006 (5)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

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

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
[CrossRef]

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

2003 (1)

2002 (2)

1997 (1)

T. Sikora, “MPEG digital video-coding standards,” IEEE Signal Process. Mag. 14, 82–100 (1997).
[CrossRef]

1995 (1)

D. Dong and J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

1992 (1)

G. K. Wallace, “The JPEG still picture compression standard,” IEEE Trans. Consum. Electron. 38, xviii–xxxiv (1992).
[CrossRef]

1991 (1)

E. R. Fossum, S. E. Kemeny, R. A. Bredthauer, and M. LaShell, “Digitally programmable gain control circuit for charge-domain signal processing,” IEEE J. Solid-State Circuits 26, 683–686 (1991).
[CrossRef]

1989 (1)

E. R. Fossum, “Charge-domain analog signal processing for detector arrays,” Nucl. Instrum. Methods Phys. Res. A 275, 530–535 (1989).
[CrossRef]

1987 (1)

E. R. Fossum, “Charge-coupled computing for focal plane image preprocessing,” Opt. Eng. 26, 916–922 (1987).

1976 (1)

1975 (1)

A. G. Marshall and M. B. Comisarow, “Fourier and Hadamard transform methods in spectroscopy,” Anal. Chem. 47, 491A–504A (1975).
[CrossRef]

1971 (1)

Atick, J.

D. Dong and J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Baraniuk, R.

M. Wakin, M. Duarte, S. Sarvotham, D. Baron, and R. Baraniuk, “Recovery of jointly sparse signals from few random projections,” in Advances in Neural Information Processing Systems 18, Y. Weiss, B. Schölkopf, and J. Platt, eds. (MIT Press, 2006), pp. 1433–1440.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

Baron, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

M. Wakin, M. Duarte, S. Sarvotham, D. Baron, and R. Baraniuk, “Recovery of jointly sparse signals from few random projections,” in Advances in Neural Information Processing Systems 18, Y. Weiss, B. Schölkopf, and J. Platt, eds. (MIT Press, 2006), pp. 1433–1440.

Brady, D. J.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

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

Bredthauer, R. A.

E. R. Fossum, S. E. Kemeny, R. A. Bredthauer, and M. LaShell, “Digitally programmable gain control circuit for charge-domain signal processing,” IEEE J. Solid-State Circuits 26, 683–686 (1991).
[CrossRef]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Cathey, W. T.

Clarke, R. J.

R. J. Clarke, “Relation between the Karhunen-Loève and cosine transforms,” IEE Proc. Commun. Radar Signal Process.128, 359–360 (1981).

Comisarow, M. B.

A. G. Marshall and M. B. Comisarow, “Fourier and Hadamard transform methods in spectroscopy,” Anal. Chem. 47, 491A–504A (1975).
[CrossRef]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Decker, J. A.

Dong, D.

D. Dong and J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Donoho, D. L.

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

Dowski, E. R.

Duarte, M.

M. Wakin, M. Duarte, S. Sarvotham, D. Baron, and R. Baraniuk, “Recovery of jointly sparse signals from few random projections,” in Advances in Neural Information Processing Systems 18, Y. Weiss, B. Schölkopf, and J. Platt, eds. (MIT Press, 2006), pp. 1433–1440.

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

Feldman, M. R.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Fiddy, M. A.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Fossum, E. R.

E. R. Fossum, S. E. Kemeny, R. A. Bredthauer, and M. LaShell, “Digitally programmable gain control circuit for charge-domain signal processing,” IEEE J. Solid-State Circuits 26, 683–686 (1991).
[CrossRef]

E. R. Fossum, “Charge-domain analog signal processing for detector arrays,” Nucl. Instrum. Methods Phys. Res. A 275, 530–535 (1989).
[CrossRef]

E. R. Fossum, “Charge-coupled computing for focal plane image preprocessing,” Opt. Eng. 26, 916–922 (1987).

Haupt, J.

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
[CrossRef]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Vol. I (Prentice-Hall, 1993).

Ke, J.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

Kemeny, S. E.

E. R. Fossum, S. E. Kemeny, R. A. Bredthauer, and M. LaShell, “Digitally programmable gain control circuit for charge-domain signal processing,” IEEE J. Solid-State Circuits 26, 683–686 (1991).
[CrossRef]

LaShell, M.

E. R. Fossum, S. E. Kemeny, R. A. Bredthauer, and M. LaShell, “Digitally programmable gain control circuit for charge-domain signal processing,” IEEE J. Solid-State Circuits 26, 683–686 (1991).
[CrossRef]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

Marcellin, M. W.

D. S. Taubman and M. W. Marcellin, JPEG 2000: Image Compression Fundamentals, Standards and Practice (Kluwer Academic, 2001).

Marshall, A. G.

A. G. Marshall and M. B. Comisarow, “Fourier and Hadamard transform methods in spectroscopy,” Anal. Chem. 47, 491A–504A (1975).
[CrossRef]

Neifeld, M. A.

Nowak, R.

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
[CrossRef]

Oliver, C. J.

Pitsianis, N. P.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Portnoy, A.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Sarvotham, S.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

M. Wakin, M. Duarte, S. Sarvotham, D. Baron, and R. Baraniuk, “Recovery of jointly sparse signals from few random projections,” in Advances in Neural Information Processing Systems 18, Y. Weiss, B. Schölkopf, and J. Platt, eds. (MIT Press, 2006), pp. 1433–1440.

Shankar, P.

Sikora, T.

T. Sikora, “MPEG digital video-coding standards,” IEEE Signal Process. Mag. 14, 82–100 (1997).
[CrossRef]

Suleski, T.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

Sun, X.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

Taubman, D. S.

D. S. Taubman and M. W. Marcellin, JPEG 2000: Image Compression Fundamentals, Standards and Practice (Kluwer Academic, 2001).

TeKolste, R. D.

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Wakin, M.

M. Wakin, M. Duarte, S. Sarvotham, D. Baron, and R. Baraniuk, “Recovery of jointly sparse signals from few random projections,” in Advances in Neural Information Processing Systems 18, Y. Weiss, B. Schölkopf, and J. Platt, eds. (MIT Press, 2006), pp. 1433–1440.

Wakin, M. B.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

Wallace, G. K.

G. K. Wallace, “The JPEG still picture compression standard,” IEEE Trans. Consum. Electron. 38, xviii–xxxiv (1992).
[CrossRef]

Anal. Chem. (1)

A. G. Marshall and M. B. Comisarow, “Fourier and Hadamard transform methods in spectroscopy,” Anal. Chem. 47, 491A–504A (1975).
[CrossRef]

Appl. Opt. (5)

IEEE J. Solid-State Circuits (1)

E. R. Fossum, S. E. Kemeny, R. A. Bredthauer, and M. LaShell, “Digitally programmable gain control circuit for charge-domain signal processing,” IEEE J. Solid-State Circuits 26, 683–686 (1991).
[CrossRef]

IEEE Signal Process. Mag. (2)

T. Sikora, “MPEG digital video-coding standards,” IEEE Signal Process. Mag. 14, 82–100 (1997).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[CrossRef]

IEEE Trans. Consum. Electron. (1)

G. K. Wallace, “The JPEG still picture compression standard,” IEEE Trans. Consum. Electron. 38, xviii–xxxiv (1992).
[CrossRef]

IEEE Trans. Inf. Theory (3)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[CrossRef]

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

J. Haupt and R. Nowak, “Signal reconstruction from noisy random projections,” IEEE Trans. Inf. Theory 52, 4036–4048(2006).
[CrossRef]

Network Comput. Neural Syst. (1)

D. Dong and J. Atick, “Statistics of natural time-varying images,” Network Comput. Neural Syst. 6, 345–358 (1995).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (1)

E. R. Fossum, “Charge-domain analog signal processing for detector arrays,” Nucl. Instrum. Methods Phys. Res. A 275, 530–535 (1989).
[CrossRef]

Opt. Eng. (1)

E. R. Fossum, “Charge-coupled computing for focal plane image preprocessing,” Opt. Eng. 26, 916–922 (1987).

Opt. Lett. (1)

Proc. SPIE (2)

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE, 6065, 606509 (2006).
[CrossRef]

N. P. Pitsianis, D. J. Brady, A. Portnoy, X. Sun, T. Suleski, M. A. Fiddy, M. R. Feldman, and R. D. TeKolste, “Compressive imaging sensors,” Proc. SPIE 6232, 62320A (2006)
[CrossRef]

Other (4)

M. Wakin, M. Duarte, S. Sarvotham, D. Baron, and R. Baraniuk, “Recovery of jointly sparse signals from few random projections,” in Advances in Neural Information Processing Systems 18, Y. Weiss, B. Schölkopf, and J. Platt, eds. (MIT Press, 2006), pp. 1433–1440.

R. J. Clarke, “Relation between the Karhunen-Loève and cosine transforms,” IEE Proc. Commun. Radar Signal Process.128, 359–360 (1981).

D. S. Taubman and M. W. Marcellin, JPEG 2000: Image Compression Fundamentals, Standards and Practice (Kluwer Academic, 2001).

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory, Vol. I (Prentice-Hall, 1993).

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Figures (18)

Fig. 1.
Fig. 1.

Object metaframe with K×K×L pixel object blocks over time T.

Fig. 2.
Fig. 2.

Sample frames from the video database.

Fig. 3.
Fig. 3.

Example KL projection masks for K=8 and L=8 spatiotemporal blocks.

Fig. 4.
Fig. 4.

Reconstruction RMSE for K=8 and L={1,8,16} blocks with σn=10% relative noise strength for KL and random projections.

Fig. 5.
Fig. 5.

Normalized cumulative energy for K=8 and selected L.

Fig. 6.
Fig. 6.

Compression factor for a desired reconstruction error with K=8 and σn=10% relative noise strength.

Fig. 7.
Fig. 7.

Reconstruction RMSE for selected spatiotemporal block sizes with σn=15% noise strength and KL basis.

Fig. 8.
Fig. 8.

Measurement error term of Eq. (8) for K=8 and selected L with σn=10%.

Fig. 9.
Fig. 9.

Reconstruction RMSE for K=8 and L=8 blocks with 5%, 10%, and 15% relative noise strengths.

Fig. 10.
Fig. 10.

Selected frames with 15% relative noise strength from (a) reconstructed STFSI metaframe with 12× compression using K=8, L=8 blocks, RMSE=7.8%, and (b) conventional imager, RMSE=15%. Magnified central part of frame 8 from (c) STFSI and (d) conventional imager.

Fig. 11.
Fig. 11.

Fully optical polarization-based pipeline architecture.

Fig. 12.
Fig. 12.

Fully optical lenslet-array-based parallel architecture.

Fig. 13.
Fig. 13.

Charge-level diagram of sFPA1 functionality for complete charge-domain processing.

Fig. 14.
Fig. 14.

Number of pixels supported by a single multiplier unit versus spatial extent with clkmax=20MHz.

Fig. 15.
Fig. 15.

Number of pixels supported versus spatial extent and selected temporal extents with FPA Amax=400mm2 and clkmax=20MHz.

Fig. 16.
Fig. 16.

Number of pixels supported versus spatial extent for various maximum clocks rates with L=16 and Amax=400mm2.

Fig. 17.
Fig. 17.

Interleaved feature collection scheme with sFPA2.

Fig. 18.
Fig. 18.

Hybrid optical and charge-domain processing: (a) optical portion and (b) charge-domain portion (single detector of sFPA2).

Tables (3)

Tables Icon

Table 1. Normalized Cumulative Energy for Selected Fixed Percentage of Measurements and K=8

Tables Icon

Table 2. Percentage of Total Features to Capture Fixed Normalized Cumulative Energy for K=8

Tables Icon

Table 3. Important Architecture Parameters (K×K×L Object)

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

m=PCx+n,
P=[P1|P2||PL],
m+=kC+P+x+n+,
m=1kCPx+n,
f^=C+km+C1km=Px+n.
x^=Wf^=RxPT(PRxPT+Rn)1f^,
RMSE=1DRmax[1NE[xx^22]]1/2=1DRmax[1NE[i=1N|xix^i|2]]1/2,
RMSEKL=1DRmax[1Ni=M+1Nλi+1Ni=1Mλiσn2λi+σn2]1/2,

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