Abstract

Hadamard multiplexing provides a considerable SNR boost over additive random noise but Poisson noise such as photon noise reduces the boost. We develop the theory for full H-matrix Hadamard transform imaging under additive and Poisson noise effects. We show that H-matrix encoding results in no effect on average on the noise level due to Poisson noise sources while preferentially reducing additive noise. We use this result to explain the wavelength-dependent varying SNR boost in a Hadamard hyperspectral imager and argue that such a preferential boost is useful when the main noise source is indeterminant or varying.

© 2009 Optical Society of America

Full Article  |  PDF Article

Corrections

L. Streeter, M. J. Cree, and G. R. Burling-Claridge, "Optical full Hadamard matrix multiplexing and noise effects: errata," Appl. Opt. 50, 6092-6093 (2011)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-50-32-6092

References

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  1. M. Harwit and N. Sloan, Hadamard Transform Optics (Academic, 1979).
  2. R. Damaschini, “Limitation of the multiplex gain in HADAMARD transform spectroscopy,” Pure Appl. Opt. 2, 173-177(1993).
    [CrossRef]
  3. G. Nitzsche and R. Riesenberg, “Noise, fluctuation, and HADAMARD-transform spectrometry,” Proc. SPIE 5111, 273-282 (2003).
    [CrossRef]
  4. A. Wuttig, “Optimal transformations for optical multiplex measurements in the presence of photon noise,” Appl. Opt. 44, 2710-2719 (2005).
    [CrossRef] [PubMed]
  5. N. Ratner, Y. Schechner, and F. Goldberg, “Optimal multiplexed sensing: bounds, conditions and a graph theory link,” Opt. Express 15, 17072-17092 (2007).
    [CrossRef] [PubMed]
  6. D. S. Davis, “Multiplexed imaging by means of optically generated Kronecker products: 1. the basic concept,” Appl. Opt. 34, 1170-1176 (1995).
    [CrossRef] [PubMed]
  7. W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
    [CrossRef]
  8. L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Visible/near infrared hyperspectral imaging via spatial illumination source modulation,” J. Near Infrared Spectrosc. 15, 395-399 (2007).
    [CrossRef]
  9. Q. Hanley, P. Verveer, D. Arndt-Jovin, and T. Jovin, “Three-dimensional spectral imaging by HADAMARD transform spectroscopy in a programmable array microscope,” J. Microsc. 197, 5-14 (2000).
    [CrossRef] [PubMed]
  10. R. DeVerse, R. Hammaker, and W. Fateley, “Realization of the HADAMARD multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751-1758 (2000).
    [CrossRef]
  11. W. Fateley, R. Hammaker, and R. DeVerse, “Modulations used to transmit information in spectrometry and imaging,” J. Mol. Struct. 550-551, 117-122 (2000).
    [CrossRef]
  12. R. DeVerse, R. Hammaker, and W. Fateley, “Hadamard transform raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177-186 (1999).
    [CrossRef]
  13. D. Takhar, J. Laska, M. Wakin, M. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc SPIE 6065, 606509 (2006).
    [CrossRef]
  14. M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.
  15. L. Nguyen, B. Aazhang, and J. F. Young, “All-optical CDMA with bipolar codes,” Electron. Lett. 31, 469-470 (1995).
    [CrossRef]
  16. L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Reference beam method for source modulated HADAMARD multiplexing,” Proc SPIE 6812, 68160J1 (2008).
  17. K. Hassler, T. Anhut, and T. Lasser, “Time-resolved HADAMARD fluorescence imaging,” Appl. Opt. 44, 7564-7572 (2005).
    [CrossRef] [PubMed]
  18. K. Pearson, “Mathematical contributions to the theory of evolution--on a form of spurious correlation which may arise when indices are used in the measurement of organs,” Proc. R. Soc. London 60, 489-498 (1897).
  19. L. Eldeń, Fundamentals of Algorithms: Matrix Methods in Data Mining and Pattern Recognition (SIAM, 2007), Chap. 6.
    [CrossRef]

2008 (1)

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Reference beam method for source modulated HADAMARD multiplexing,” Proc SPIE 6812, 68160J1 (2008).

2007 (2)

N. Ratner, Y. Schechner, and F. Goldberg, “Optimal multiplexed sensing: bounds, conditions and a graph theory link,” Opt. Express 15, 17072-17092 (2007).
[CrossRef] [PubMed]

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Visible/near infrared hyperspectral imaging via spatial illumination source modulation,” J. Near Infrared Spectrosc. 15, 395-399 (2007).
[CrossRef]

2006 (1)

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

2005 (2)

2003 (1)

G. Nitzsche and R. Riesenberg, “Noise, fluctuation, and HADAMARD-transform spectrometry,” Proc. SPIE 5111, 273-282 (2003).
[CrossRef]

2002 (1)

W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
[CrossRef]

2000 (3)

Q. Hanley, P. Verveer, D. Arndt-Jovin, and T. Jovin, “Three-dimensional spectral imaging by HADAMARD transform spectroscopy in a programmable array microscope,” J. Microsc. 197, 5-14 (2000).
[CrossRef] [PubMed]

W. Fateley, R. Hammaker, and R. DeVerse, “Modulations used to transmit information in spectrometry and imaging,” J. Mol. Struct. 550-551, 117-122 (2000).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Realization of the HADAMARD multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751-1758 (2000).
[CrossRef]

1999 (1)

R. DeVerse, R. Hammaker, and W. Fateley, “Hadamard transform raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177-186 (1999).
[CrossRef]

1995 (2)

L. Nguyen, B. Aazhang, and J. F. Young, “All-optical CDMA with bipolar codes,” Electron. Lett. 31, 469-470 (1995).
[CrossRef]

D. S. Davis, “Multiplexed imaging by means of optically generated Kronecker products: 1. the basic concept,” Appl. Opt. 34, 1170-1176 (1995).
[CrossRef] [PubMed]

1993 (1)

R. Damaschini, “Limitation of the multiplex gain in HADAMARD transform spectroscopy,” Pure Appl. Opt. 2, 173-177(1993).
[CrossRef]

1897 (1)

K. Pearson, “Mathematical contributions to the theory of evolution--on a form of spurious correlation which may arise when indices are used in the measurement of organs,” Proc. R. Soc. London 60, 489-498 (1897).

Aazhang, B.

L. Nguyen, B. Aazhang, and J. F. Young, “All-optical CDMA with bipolar codes,” Electron. Lett. 31, 469-470 (1995).
[CrossRef]

Anhut, T.

Arndt-Jovin, D.

Q. Hanley, P. Verveer, D. Arndt-Jovin, and T. Jovin, “Three-dimensional spectral imaging by HADAMARD transform spectroscopy in a programmable array microscope,” J. Microsc. 197, 5-14 (2000).
[CrossRef] [PubMed]

Baraniuk, R.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Baron, D.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Burling-Claridge, G.

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Reference beam method for source modulated HADAMARD multiplexing,” Proc SPIE 6812, 68160J1 (2008).

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Visible/near infrared hyperspectral imaging via spatial illumination source modulation,” J. Near Infrared Spectrosc. 15, 395-399 (2007).
[CrossRef]

Coifman, R.

W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
[CrossRef]

Cree, M.

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Reference beam method for source modulated HADAMARD multiplexing,” Proc SPIE 6812, 68160J1 (2008).

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Visible/near infrared hyperspectral imaging via spatial illumination source modulation,” J. Near Infrared Spectrosc. 15, 395-399 (2007).
[CrossRef]

Damaschini, R.

R. Damaschini, “Limitation of the multiplex gain in HADAMARD transform spectroscopy,” Pure Appl. Opt. 2, 173-177(1993).
[CrossRef]

Davis, D. S.

DeVerse, R.

W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
[CrossRef]

W. Fateley, R. Hammaker, and R. DeVerse, “Modulations used to transmit information in spectrometry and imaging,” J. Mol. Struct. 550-551, 117-122 (2000).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Realization of the HADAMARD multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751-1758 (2000).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Hadamard transform raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177-186 (1999).
[CrossRef]

Duarte, M.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Elden, L.

L. Eldeń, Fundamentals of Algorithms: Matrix Methods in Data Mining and Pattern Recognition (SIAM, 2007), Chap. 6.
[CrossRef]

Fateley, W.

W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
[CrossRef]

W. Fateley, R. Hammaker, and R. DeVerse, “Modulations used to transmit information in spectrometry and imaging,” J. Mol. Struct. 550-551, 117-122 (2000).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Realization of the HADAMARD multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751-1758 (2000).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Hadamard transform raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177-186 (1999).
[CrossRef]

Geshwind, F.

W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
[CrossRef]

Goldberg, F.

Hammaker, R.

W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
[CrossRef]

W. Fateley, R. Hammaker, and R. DeVerse, “Modulations used to transmit information in spectrometry and imaging,” J. Mol. Struct. 550-551, 117-122 (2000).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Realization of the HADAMARD multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54, 1751-1758 (2000).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Hadamard transform raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177-186 (1999).
[CrossRef]

Hanley, Q.

Q. Hanley, P. Verveer, D. Arndt-Jovin, and T. Jovin, “Three-dimensional spectral imaging by HADAMARD transform spectroscopy in a programmable array microscope,” J. Microsc. 197, 5-14 (2000).
[CrossRef] [PubMed]

Harwit, M.

M. Harwit and N. Sloan, Hadamard Transform Optics (Academic, 1979).

Hassler, K.

Jovin, T.

Q. Hanley, P. Verveer, D. Arndt-Jovin, and T. Jovin, “Three-dimensional spectral imaging by HADAMARD transform spectroscopy in a programmable array microscope,” J. Microsc. 197, 5-14 (2000).
[CrossRef] [PubMed]

Kelly, K.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Künnemeyer, R.

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Reference beam method for source modulated HADAMARD multiplexing,” Proc SPIE 6812, 68160J1 (2008).

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Visible/near infrared hyperspectral imaging via spatial illumination source modulation,” J. Near Infrared Spectrosc. 15, 395-399 (2007).
[CrossRef]

Laska, J.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Lasser, T.

Nguyen, L.

L. Nguyen, B. Aazhang, and J. F. Young, “All-optical CDMA with bipolar codes,” Electron. Lett. 31, 469-470 (1995).
[CrossRef]

Nitzsche, G.

G. Nitzsche and R. Riesenberg, “Noise, fluctuation, and HADAMARD-transform spectrometry,” Proc. SPIE 5111, 273-282 (2003).
[CrossRef]

Pearson, K.

K. Pearson, “Mathematical contributions to the theory of evolution--on a form of spurious correlation which may arise when indices are used in the measurement of organs,” Proc. R. Soc. London 60, 489-498 (1897).

Ratner, N.

Riesenberg, R.

G. Nitzsche and R. Riesenberg, “Noise, fluctuation, and HADAMARD-transform spectrometry,” Proc. SPIE 5111, 273-282 (2003).
[CrossRef]

Sarvotham, S.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Schechner, Y.

Sloan, N.

M. Harwit and N. Sloan, Hadamard Transform Optics (Academic, 1979).

Streeter, L.

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Reference beam method for source modulated HADAMARD multiplexing,” Proc SPIE 6812, 68160J1 (2008).

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Visible/near infrared hyperspectral imaging via spatial illumination source modulation,” J. Near Infrared Spectrosc. 15, 395-399 (2007).
[CrossRef]

Takhar, D.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Verveer, P.

Q. Hanley, P. Verveer, D. Arndt-Jovin, and T. Jovin, “Three-dimensional spectral imaging by HADAMARD transform spectroscopy in a programmable array microscope,” J. Microsc. 197, 5-14 (2000).
[CrossRef] [PubMed]

Wakin, M.

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

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

Wuttig, A.

Young, J. F.

L. Nguyen, B. Aazhang, and J. F. Young, “All-optical CDMA with bipolar codes,” Electron. Lett. 31, 469-470 (1995).
[CrossRef]

Appl. Opt. (3)

Appl. Spectrosc. (1)

Electron. Lett. (1)

L. Nguyen, B. Aazhang, and J. F. Young, “All-optical CDMA with bipolar codes,” Electron. Lett. 31, 469-470 (1995).
[CrossRef]

J. Microsc. (1)

Q. Hanley, P. Verveer, D. Arndt-Jovin, and T. Jovin, “Three-dimensional spectral imaging by HADAMARD transform spectroscopy in a programmable array microscope,” J. Microsc. 197, 5-14 (2000).
[CrossRef] [PubMed]

J. Mol. Struct. (1)

W. Fateley, R. Hammaker, and R. DeVerse, “Modulations used to transmit information in spectrometry and imaging,” J. Mol. Struct. 550-551, 117-122 (2000).
[CrossRef]

J. Near Infrared Spectrosc. (1)

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Visible/near infrared hyperspectral imaging via spatial illumination source modulation,” J. Near Infrared Spectrosc. 15, 395-399 (2007).
[CrossRef]

Opt. Express (1)

Proc SPIE (2)

L. Streeter, G. Burling-Claridge, M. Cree, and R. Künnemeyer, “Reference beam method for source modulated HADAMARD multiplexing,” Proc SPIE 6812, 68160J1 (2008).

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

Proc. R. Soc. London (1)

K. Pearson, “Mathematical contributions to the theory of evolution--on a form of spurious correlation which may arise when indices are used in the measurement of organs,” Proc. R. Soc. London 60, 489-498 (1897).

Proc. SPIE (1)

G. Nitzsche and R. Riesenberg, “Noise, fluctuation, and HADAMARD-transform spectrometry,” Proc. SPIE 5111, 273-282 (2003).
[CrossRef]

Pure Appl. Opt. (1)

R. Damaschini, “Limitation of the multiplex gain in HADAMARD transform spectroscopy,” Pure Appl. Opt. 2, 173-177(1993).
[CrossRef]

Vib. Spectrosc. (2)

W. Fateley, R. Hammaker, R. DeVerse, R. Coifman, and F. Geshwind, “The other spectroscopy: demonstration of a new de-dispersion imaging spectrograph,” Vib. Spectrosc. 29, 163-170 (2002).
[CrossRef]

R. DeVerse, R. Hammaker, and W. Fateley, “Hadamard transform raman imagery with a digital micro-mirror array,” Vib. Spectrosc. 19, 177-186 (1999).
[CrossRef]

Other (3)

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, “An architecture for compressive imaging,” in 2006 IEEE International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.

M. Harwit and N. Sloan, Hadamard Transform Optics (Academic, 1979).

L. Eldeń, Fundamentals of Algorithms: Matrix Methods in Data Mining and Pattern Recognition (SIAM, 2007), Chap. 6.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Illustration of splitting the Hadamard matrix into H + and H components. In H + the 1 ’s of the original H-matrix are converted to 0’s. In H the + 1 ’s are converted to 0’s and the 1 ’s are converted to + 1 ’s.

Fig. 2
Fig. 2

Theoretical SNR boosts as a function of light source intensity for two different values of mean squared error of the additive noise.

Fig. 3
Fig. 3

Plan view diagram of the optics. Arrows indicate the path of light collected by the spectrometer.

Fig. 4
Fig. 4

Spectra from the eighth row of the acrylic and PCB fiberglass image, distinguishable by the gross intensity difference, i.e., acrylic has more reflectance.

Fig. 5
Fig. 5

Spectra from the eighth row of the wood and polystyrene image, distinguishable by their respective spectral shape. Wood has pronounced absorbance bands near 600 nm and around 1500 nm .

Fig. 6
Fig. 6

Images from the simple application. (a) Image of the acrylic and PCB fiberglass sample at 1533 nm . (b) Image of the wood and polystyrene sample at 1533 nm . (c) Segmented image classifying the bare acrylic in (a) as black and the PCB fiberglass over acrylic as white. (d) Segmented image classifying the wood in (b) as white and polystyrene as black.

Fig. 7
Fig. 7

SNR boost via Hadamard multiplexing. The horizontal line is the theoretical boost of 16.

Equations (27)

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

H T H = H 2 = N I ,
H 1 = 1 N H .
a = H p + e .
p ^ = p + 1 N H e .
σ H 2 = 1 N σ 0 2 .
H + = 1 + H 2 , H = 1 H 2 .
a + = α s ( R + + E P + ) H + p + T + e + , a = α s ( R + E P ) H p + T + e .
var ( { E P } j , j ) = { R } j , j = r j ,
a = a + a = α s R H p + e a , P + e ,
e a , P = α s ( E P + H + E P H ) p
p ^ = 1 N H a = α s N H R H p + α s N H ( E P + H + E P H ) p + 1 N H e .
α s N H r I H = α s r .
σ j , a , P 2 = ( σ j , P + ) 2 + ( σ j , P ) 2 = α s r j N p .
σ p ^ , P 2 = α s r p .
σ point , i 2 = α s r i p i .
σ point , t , i 2 = α s r i p i + σ 2 ,
σ t 2 = α s r p + 1 N σ 2 .
SNR point , i 2 = ( α s r i p i ) 2 α s r t p i + σ 2 ,
SNR i 2 = N ( α s r p i ) 2 N α s r p + σ 2 .
boost i = SNR i SNR point , i = N α s r i p i + σ 2 N α s r p + σ 2 ,
{ R r } i , i = α r { R + E P , r } i , i + { e r } i ,
p ^ = 1 N H 1 α r R 1 [ α s R H p ] + e t = α s α r p + e t ,
σ t 2 = 1 N var ( a R ) = 1 N R r 2 ( σ a 2 + a 2 R r 2 σ r 2 2 a R r σ a , r 2 ) ,
σ t 2 = 1 N ( α r r ) 2 ( α s r N p + σ 2 + α s 2 r 2 N 2 p 2 α r 2 r 2 α r r 2 α s r N p α r r σ a , r 2 ) .
σ t 2 = N α s r p + σ 2 N α r 2 r 2 ( 1 + 2 N α s p α r ) + N p 2 α r r α s 2 α r 2 .
σ t 2 α s r N p + σ 2 N α r 2 r 2 .
SNR i 2 N ( α s r p i ) 2 N α s r p + σ 2 .

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