Abstract

Measuring an array of variables is central to many systems, including imagers (array of pixels), spectrometers (array of spectral bands) and lighting systems. Each of the measurements, however, is prone to noise and potential sensor saturation. It is recognized by a growing number of methods that such problems can be reduced by multiplexing the measured variables. In each measurement, multiple variables (radiation channels) are mixed (multiplexed) by a code. Then, after data acquisition, the variables are decoupled computationally in post processing. Potential benefits of the use of multiplexing include increased signal-to-noise ratio and accommodation of scene dynamic range. However, existing multiplexing schemes, including Hadamard-based codes, are inhibited by fundamental limits set by sensor saturation and Poisson distributed photon noise, which is scene dependent. There is thus a need to find optimal codes that best increase the signal to noise ratio, while accounting for these effects. Hence, this paper deals with the pursuit of such optimal measurements that avoid saturation and account for the signal dependency of noise. The paper derives lower bounds on the mean square error of demultiplexed variables. This is useful for assessing the optimality of numerically-searched multiplexing codes, thus expediting the numerical search. Furthermore, the paper states the necessary conditions for attaining the lower bounds by a general code. We show that graph theory can be harnessed for finding such ideal codes, by the use of strongly regular graphs.

© 2007 Optical Society of America

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2007 (3)

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang. “Polarization multiplexing and demultiplexing for appearance-based modeling.” IEEE Trans. PAMI 29:362–367 (2007).
[Crossref]

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “Multiplexing for optimal lighting.” IEEE Trans. PAMI 29:1339–1354 (2007).
[Crossref]

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. “Longwave infrared (LWIR) coded aperture dispersive spectrometer.” Opt. Express 15:5742–5753 (2007).
[Crossref] [PubMed]

2006 (4)

C. Liu, W. T. Freeman, R. Szeliski, and S. B. Kang. “Noise estimation from a single image.” In Proc. CVPR Vol. 1 pages 901–908 (2006).

F. Alter, Y. Matsushita, and X. Tang. “An intensity similarity measure in low-light conditions.” In Proc. ECCV Vol. 4, pages 267–280 (2006).

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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy.” Appl. Opt. 43:2965–2974 (2006).
[Crossref]

2005 (3)

K. C. Lee, J. Ho, and D. J. Kriegman. “Acquiring linear subspaces for face recognition under variable lighting.” IEEE Trans. PAMI 27:684–698 (2005).
[Crossref]

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

A. Wuttig. “Optimal transformations for optical multiplex measurements in the presence of photon noise.” Appl. Opt. 44:2710–2719 (2005).
[Crossref] [PubMed]

2004 (2)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

A. M. Bronstein, M. M. Bronstein, E. Gordon, and R. Kimmel. “Fusion of 2d and 3d data in three-dimensional face recognition.” In Proc. IEEE ICIP Vol. 1, pages 87–90 (2004).

2003 (2)

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “A theory of multiplexed illumination.” In Proc. IEEE ICCV Vol. 2, pages 808–815 (2003).

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

2002 (2)

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. “Spectrally resolved fluorescence lifetime imaging microscopy.” Appl. Spectrosc. 56:63–84 (2002).
[Crossref]

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

2000 (2)

Y. A. Shutova. “Optimization of binary masks for Hadamard-transform optical spectrometers”. J. Opt. Technol. 67:50–53 (2000).
[Crossref]

M. T. Chu. “A fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values.” SIAM J. on Numerical Analysis 37(3):1004–1020 (2000).
[Crossref]

1997 (2)

M. Alicacute, B. Mond, J. Pecbreve aricacute, and V. Volenec. “The arithmetic-geometric-harmonic-mean and related matrix inequalities.” Linear Algebra and its Applications 264(1):55–62 (1997).
[Crossref]

A. Busboom, H. D. Schotten, and H. Elders-Boll. “Coded aperture imaging with multiple measurements.” J. Opt. Soc. Am. A 14(5):1058–1065 (1997).
[Crossref]

1996 (1)

J. F. Turner and P. J. Treado. “Adaptive filtering and hadamard transform imaging spectroscopy with an acousto-optic tunable filter (AOTF).” In Proc. SPIE volume 2599, pages 285–293 (1996).
[Crossref]

1992 (1)

V. P. Kozlov and E. V. Sedunov. “Optimization of multiplex measuring systems in the presence of statistical signal fluctuations.” Cybernetics and Systems Analysis 28:830–839 (1992).
[Crossref]

1988 (1)

G. K. Skinner. “X-ray imaging with coded masks.” Scientific American 259:84–89 (1988).
[Crossref] [PubMed]

1979 (1)

R. J. Proctor, G. K. Skinner, and A. P. Willmore. “The design of optimum coded mask X-ray telescopes.” Royal Astronomical Society, Monthly Notices 187:633–643 (1979).

1978 (2)

1974 (1)

T. M. Palmieri. “Multiplex methods and advantages in X-ray astronomy.” Astrophysics and Space Science 28:277–287 (1974).
[Crossref]

1968 (1)

J. J. Seidel. “Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3.” Linear Algebra Appl. 1:281–289 (1968).
[Crossref]

Alicacute, M.

M. Alicacute, B. Mond, J. Pecbreve aricacute, and V. Volenec. “The arithmetic-geometric-harmonic-mean and related matrix inequalities.” Linear Algebra and its Applications 264(1):55–62 (1997).
[Crossref]

Alter, F.

F. Alter, Y. Matsushita, and X. Tang. “An intensity similarity measure in low-light conditions.” In Proc. ECCV Vol. 4, pages 267–280 (2006).

Arndt-Jovin, D. J.

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. “Spectrally resolved fluorescence lifetime imaging microscopy.” Appl. Spectrosc. 56:63–84 (2002).
[Crossref]

Baraniuk, R. G.

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.” In Proc. SPIE volume 6065 (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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

Barrett, H. H.

H. H. Barrett and W. Swindell. Radiological Imaging, volume 1. Academic press, New York (1981).

Belhumeur, P. N.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “Multiplexing for optimal lighting.” IEEE Trans. PAMI 29:1339–1354 (2007).
[Crossref]

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “A theory of multiplexed illumination.” In Proc. IEEE ICCV Vol. 2, pages 808–815 (2003).

F. Moreno-Noguer, S. K. Nayar, and P. N. Belhumeur. “Optimal illumination for image and video relighting.” In Proc. CVMP pages 201–210 (2005).

Bolas, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

Brady, D. J.

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. “Longwave infrared (LWIR) coded aperture dispersive spectrometer.” Opt. Express 15:5742–5753 (2007).
[Crossref] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy.” Appl. Opt. 43:2965–2974 (2006).
[Crossref]

Bronstein, A. M.

A. M. Bronstein, M. M. Bronstein, E. Gordon, and R. Kimmel. “Fusion of 2d and 3d data in three-dimensional face recognition.” In Proc. IEEE ICIP Vol. 1, pages 87–90 (2004).

Bronstein, M. M.

A. M. Bronstein, M. M. Bronstein, E. Gordon, and R. Kimmel. “Fusion of 2d and 3d data in three-dimensional face recognition.” In Proc. IEEE ICIP Vol. 1, pages 87–90 (2004).

Busboom, A.

Cameron, P. J.

P. J. Cameron and J. H. V. Lint. Designs, Graphs, Codes, and Their Links. Cambridge University Press, New York, NY, USA (1991).
[Crossref]

Cannon, T. M.

Chen, B.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

Chu, M. T.

M. T. Chu. “A fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values.” SIAM J. on Numerical Analysis 37(3):1004–1020 (2000).
[Crossref]

Coifman, R. R.

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

Coolsaet, K.

K. Coolsaet and J. Degraer. “The strongly regular (45,12,3,3) graphs.” Elec. Journ. Combin13(1) (2006).

Cula, O. G.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang. “Polarization multiplexing and demultiplexing for appearance-based modeling.” IEEE Trans. PAMI 29:362–367 (2007).
[Crossref]

Dana, K. J.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang. “Polarization multiplexing and demultiplexing for appearance-based modeling.” IEEE Trans. PAMI 29:362–367 (2007).
[Crossref]

Debevec, P.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

Degraer, J.

K. Coolsaet and J. Degraer. “The strongly regular (45,12,3,3) graphs.” Elec. Journ. Combin13(1) (2006).

DeVerse, R. A.

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

Diestel, R.

R. Diestel. Graph Theory. Springer, 3rd edition (2000).

Duarte, M. F.

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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

Elders-Boll, H.

Fateley, W. G.

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

Fenimore, E. E.

Fernandez, C.

Freeman, W. T.

C. Liu, W. T. Freeman, R. Szeliski, and S. B. Kang. “Noise estimation from a single image.” In Proc. CVPR Vol. 1 pages 901–908 (2006).

Gardner, A.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

Geballe, T.

P. Puxley and T. Geballe. “Transmission Spectra” (1999) http://www.gemini.edu/sciops/ObsProcess/obsConstraints/ocTransSpectra.html

Gehm, M. E.

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. “Longwave infrared (LWIR) coded aperture dispersive spectrometer.” Opt. Express 15:5742–5753 (2007).
[Crossref] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy.” Appl. Opt. 43:2965–2974 (2006).
[Crossref]

Geshwind, F. B.

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

Gordon, E.

A. M. Bronstein, M. M. Bronstein, E. Gordon, and R. Kimmel. “Fusion of 2d and 3d data in three-dimensional face recognition.” In Proc. IEEE ICIP Vol. 1, pages 87–90 (2004).

Guenther, B. D.

Haemers, W.

W. Haemers. “Matrix techniques for strongly regular graphs and related geometries.” Intensive Course on Finite Geometry and its Applications, University of Ghent (2000).

Hammaker, R. M.

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

Hanley, Q. S.

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. “Spectrally resolved fluorescence lifetime imaging microscopy.” Appl. Spectrosc. 56:63–84 (2002).
[Crossref]

Harwit, M.

M. Harwit and N. J. A. Sloane. Hadamard Transform Optics. Academic Press, New York (1979).

Hawkins, T.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

Ho, J.

K. C. Lee, J. Ho, and D. J. Kriegman. “Acquiring linear subspaces for face recognition under variable lighting.” IEEE Trans. PAMI 27:684–698 (2005).
[Crossref]

Horn, R. A.

R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge, New York (1985).

Horowitz, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

Ioué, S.

S. Ioué and K. R. Spring. Video Microscopy, 2nd ed.ch. 6,7,8, Plenum Press, New York. (1997).

Johnson, C. R.

R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge, New York (1985).

Jovin, T. M.

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. “Spectrally resolved fluorescence lifetime imaging microscopy.” Appl. Spectrosc. 56:63–84 (2002).
[Crossref]

Kang, S. B.

C. Liu, W. T. Freeman, R. Szeliski, and S. B. Kang. “Noise estimation from a single image.” In Proc. CVPR Vol. 1 pages 901–908 (2006).

Kelly, K. F.

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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

Kimmel, R.

A. M. Bronstein, M. M. Bronstein, E. Gordon, and R. Kimmel. “Fusion of 2d and 3d data in three-dimensional face recognition.” In Proc. IEEE ICIP Vol. 1, pages 87–90 (2004).

Kozlov, V. P.

V. P. Kozlov and E. V. Sedunov. “Optimization of multiplex measuring systems in the presence of statistical signal fluctuations.” Cybernetics and Systems Analysis 28:830–839 (1992).
[Crossref]

Kriegman, D. J.

K. C. Lee, J. Ho, and D. J. Kriegman. “Acquiring linear subspaces for face recognition under variable lighting.” IEEE Trans. PAMI 27:684–698 (2005).
[Crossref]

Laska, J. N.

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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

Lee, K. C.

K. C. Lee, J. Ho, and D. J. Kriegman. “Acquiring linear subspaces for face recognition under variable lighting.” IEEE Trans. PAMI 27:684–698 (2005).
[Crossref]

Levoy, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

Lint, J. H. V.

P. J. Cameron and J. H. V. Lint. Designs, Graphs, Codes, and Their Links. Cambridge University Press, New York, NY, USA (1991).
[Crossref]

Liu, C.

C. Liu, W. T. Freeman, R. Szeliski, and S. B. Kang. “Noise estimation from a single image.” In Proc. CVPR Vol. 1 pages 901–908 (2006).

Matsushita, Y.

F. Alter, Y. Matsushita, and X. Tang. “An intensity similarity measure in low-light conditions.” In Proc. ECCV Vol. 4, pages 267–280 (2006).

McCain, S. T.

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy.” Appl. Opt. 43:2965–2974 (2006).
[Crossref]

McDowall, I.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

Meyer, C. D.

C. D. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM (2000).
[Crossref]

Mond, B.

M. Alicacute, B. Mond, J. Pecbreve aricacute, and V. Volenec. “The arithmetic-geometric-harmonic-mean and related matrix inequalities.” Linear Algebra and its Applications 264(1):55–62 (1997).
[Crossref]

Moreno-Noguer, F.

F. Moreno-Noguer, S. K. Nayar, and P. N. Belhumeur. “Optimal illumination for image and video relighting.” In Proc. CVMP pages 201–210 (2005).

Nayar, S. K.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “Multiplexing for optimal lighting.” IEEE Trans. PAMI 29:1339–1354 (2007).
[Crossref]

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “A theory of multiplexed illumination.” In Proc. IEEE ICCV Vol. 2, pages 808–815 (2003).

F. Moreno-Noguer, S. K. Nayar, and P. N. Belhumeur. “Optimal illumination for image and video relighting.” In Proc. CVMP pages 201–210 (2005).

Nitzsche, G.

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

Pai, D. K.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang. “Polarization multiplexing and demultiplexing for appearance-based modeling.” IEEE Trans. PAMI 29:362–367 (2007).
[Crossref]

Palmieri, T. M.

T. M. Palmieri. “Multiplex methods and advantages in X-ray astronomy.” Astrophysics and Space Science 28:277–287 (1974).
[Crossref]

Pecbreve aricacute, J.

M. Alicacute, B. Mond, J. Pecbreve aricacute, and V. Volenec. “The arithmetic-geometric-harmonic-mean and related matrix inequalities.” Linear Algebra and its Applications 264(1):55–62 (1997).
[Crossref]

Pitsianis, N. P.

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy.” Appl. Opt. 43:2965–2974 (2006).
[Crossref]

Potuluri, P.

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy.” Appl. Opt. 43:2965–2974 (2006).
[Crossref]

Proctor, R. J.

R. J. Proctor, G. K. Skinner, and A. P. Willmore. “The design of optimum coded mask X-ray telescopes.” Royal Astronomical Society, Monthly Notices 187:633–643 (1979).

Puxley, P.

P. Puxley and T. Geballe. “Transmission Spectra” (1999) http://www.gemini.edu/sciops/ObsProcess/obsConstraints/ocTransSpectra.html

Ratner, N.

N. Ratner and Y. Y. Schechner. “Illumination multiplexing within fundamental limits.” In Proc. IEEE CVPR (2007).

Riesenberg, R.

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

Royle, G.

G. Royle. “Strongly regular graphs” (1996) http://people.csse.uwa.edu.au/gordon/remote/srgs/index.html

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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

Schechner, Y. Y.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “Multiplexing for optimal lighting.” IEEE Trans. PAMI 29:1339–1354 (2007).
[Crossref]

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “A theory of multiplexed illumination.” In Proc. IEEE ICCV Vol. 2, pages 808–815 (2003).

N. Ratner and Y. Y. Schechner. “Illumination multiplexing within fundamental limits.” In Proc. IEEE CVPR (2007).

Schotten, H. D.

Sedunov, E. V.

V. P. Kozlov and E. V. Sedunov. “Optimization of multiplex measuring systems in the presence of statistical signal fluctuations.” Cybernetics and Systems Analysis 28:830–839 (1992).
[Crossref]

Seidel, J. J.

J. J. Seidel. “Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3.” Linear Algebra Appl. 1:281–289 (1968).
[Crossref]

Shutova, Y. A.

Skinner, G. K.

G. K. Skinner. “X-ray imaging with coded masks.” Scientific American 259:84–89 (1988).
[Crossref] [PubMed]

R. J. Proctor, G. K. Skinner, and A. P. Willmore. “The design of optimum coded mask X-ray telescopes.” Royal Astronomical Society, Monthly Notices 187:633–643 (1979).

Sloane, N. J. A.

M. Harwit and N. J. A. Sloane. Hadamard Transform Optics. Academic Press, New York (1979).

Spence, T.

T. Spence. “Strongly Regular Graphs on at most 64 vertices” http://www.maths.gla.ac.uk/es/srgraphs.html

Spring, K. R.

S. Ioué and K. R. Spring. Video Microscopy, 2nd ed.ch. 6,7,8, Plenum Press, New York. (1997).

Sullivan, M. E.

C. Fernandez, B. D. Guenther, M. E. Gehm, D. J. Brady, and M. E. Sullivan. “Longwave infrared (LWIR) coded aperture dispersive spectrometer.” Opt. Express 15:5742–5753 (2007).
[Crossref] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan. “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy.” Appl. Opt. 43:2965–2974 (2006).
[Crossref]

Swindell, W.

H. H. Barrett and W. Swindell. Radiological Imaging, volume 1. Academic press, New York (1981).

Szeliski, R.

C. Liu, W. T. Freeman, R. Szeliski, and S. B. Kang. “Noise estimation from a single image.” In Proc. CVPR Vol. 1 pages 901–908 (2006).

Takhar, 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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

Tang, X.

F. Alter, Y. Matsushita, and X. Tang. “An intensity similarity measure in low-light conditions.” In Proc. ECCV Vol. 4, pages 267–280 (2006).

Tchou, C.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

Treado, P. J.

J. F. Turner and P. J. Treado. “Adaptive filtering and hadamard transform imaging spectroscopy with an acousto-optic tunable filter (AOTF).” In Proc. SPIE volume 2599, pages 285–293 (1996).
[Crossref]

Turner, J. F.

J. F. Turner and P. J. Treado. “Adaptive filtering and hadamard transform imaging spectroscopy with an acousto-optic tunable filter (AOTF).” In Proc. SPIE volume 2599, pages 285–293 (1996).
[Crossref]

Unger, J.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

Vaish, V.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

Volenec, V.

M. Alicacute, B. Mond, J. Pecbreve aricacute, and V. Volenec. “The arithmetic-geometric-harmonic-mean and related matrix inequalities.” Linear Algebra and its Applications 264(1):55–62 (1997).
[Crossref]

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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

Wang, D.

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang. “Polarization multiplexing and demultiplexing for appearance-based modeling.” IEEE Trans. PAMI 29:362–367 (2007).
[Crossref]

Wenger, A.

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

Willmore, A. P.

R. J. Proctor, G. K. Skinner, and A. P. Willmore. “The design of optimum coded mask X-ray telescopes.” Royal Astronomical Society, Monthly Notices 187:633–643 (1979).

Wuttig, A.

ACM TOG (2)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas. “Synthetic aperture confocal imaging.” ACM TOG 23:825–834 (2004).

A. Wenger, A. Gardner, C. Tchou, J. Unger, T. Hawkins, and P. Debevec. “Performance relighting and reflectance transformation with time-multiplexed illumination.” ACM TOG 24:756–764 (2005).

Appl. Opt. (4)

Appl. Spectrosc. (1)

Q. S. Hanley, D. J. Arndt-Jovin, and T. M. Jovin. “Spectrally resolved fluorescence lifetime imaging microscopy.” Appl. Spectrosc. 56:63–84 (2002).
[Crossref]

Astrophysics and Space Science (1)

T. M. Palmieri. “Multiplex methods and advantages in X-ray astronomy.” Astrophysics and Space Science 28:277–287 (1974).
[Crossref]

Cybernetics and Systems Analysis (1)

V. P. Kozlov and E. V. Sedunov. “Optimization of multiplex measuring systems in the presence of statistical signal fluctuations.” Cybernetics and Systems Analysis 28:830–839 (1992).
[Crossref]

IEEE Trans. PAMI (3)

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “Multiplexing for optimal lighting.” IEEE Trans. PAMI 29:1339–1354 (2007).
[Crossref]

O. G. Cula, K. J. Dana, D. K. Pai, and D. Wang. “Polarization multiplexing and demultiplexing for appearance-based modeling.” IEEE Trans. PAMI 29:362–367 (2007).
[Crossref]

K. C. Lee, J. Ho, and D. J. Kriegman. “Acquiring linear subspaces for face recognition under variable lighting.” IEEE Trans. PAMI 27:684–698 (2005).
[Crossref]

In Proc. CVPR (1)

C. Liu, W. T. Freeman, R. Szeliski, and S. B. Kang. “Noise estimation from a single image.” In Proc. CVPR Vol. 1 pages 901–908 (2006).

In Proc. ECCV (1)

F. Alter, Y. Matsushita, and X. Tang. “An intensity similarity measure in low-light conditions.” In Proc. ECCV Vol. 4, pages 267–280 (2006).

In Proc. IEEE ICCV (1)

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur. “A theory of multiplexed illumination.” In Proc. IEEE ICCV Vol. 2, pages 808–815 (2003).

In Proc. IEEE ICIP (1)

A. M. Bronstein, M. M. Bronstein, E. Gordon, and R. Kimmel. “Fusion of 2d and 3d data in three-dimensional face recognition.” In Proc. IEEE ICIP Vol. 1, pages 87–90 (2004).

In Proc. SPIE (3)

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

J. F. Turner and P. J. Treado. “Adaptive filtering and hadamard transform imaging spectroscopy with an acousto-optic tunable filter (AOTF).” In Proc. SPIE volume 2599, pages 285–293 (1996).
[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.” In Proc. SPIE volume 6065 (2006).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Technol. (1)

Linear Algebra and its Applications (1)

M. Alicacute, B. Mond, J. Pecbreve aricacute, and V. Volenec. “The arithmetic-geometric-harmonic-mean and related matrix inequalities.” Linear Algebra and its Applications 264(1):55–62 (1997).
[Crossref]

Linear Algebra Appl. (1)

J. J. Seidel. “Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3.” Linear Algebra Appl. 1:281–289 (1968).
[Crossref]

Opt. Express (1)

Royal Astronomical Society, Monthly Notices (1)

R. J. Proctor, G. K. Skinner, and A. P. Willmore. “The design of optimum coded mask X-ray telescopes.” Royal Astronomical Society, Monthly Notices 187:633–643 (1979).

Scientific American (1)

G. K. Skinner. “X-ray imaging with coded masks.” Scientific American 259:84–89 (1988).
[Crossref] [PubMed]

SIAM J. on Numerical Analysis (1)

M. T. Chu. “A fast recursive algorithm for constructing matrices with prescribed eigenvalues and singular values.” SIAM J. on Numerical Analysis 37(3):1004–1020 (2000).
[Crossref]

Vib. Spectrosc. (1)

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

Other (14)

M. Harwit and N. J. A. Sloane. Hadamard Transform Optics. Academic Press, New York (1979).

F. Moreno-Noguer, S. K. Nayar, and P. N. Belhumeur. “Optimal illumination for image and video relighting.” In Proc. CVMP pages 201–210 (2005).

R. Diestel. Graph Theory. Springer, 3rd edition (2000).

P. J. Cameron and J. H. V. Lint. Designs, Graphs, Codes, and Their Links. Cambridge University Press, New York, NY, USA (1991).
[Crossref]

K. Coolsaet and J. Degraer. “The strongly regular (45,12,3,3) graphs.” Elec. Journ. Combin13(1) (2006).

T. Spence. “Strongly Regular Graphs on at most 64 vertices” http://www.maths.gla.ac.uk/es/srgraphs.html

G. Royle. “Strongly regular graphs” (1996) http://people.csse.uwa.edu.au/gordon/remote/srgs/index.html

P. Puxley and T. Geballe. “Transmission Spectra” (1999) http://www.gemini.edu/sciops/ObsProcess/obsConstraints/ocTransSpectra.html

S. Ioué and K. R. Spring. Video Microscopy, 2nd ed.ch. 6,7,8, Plenum Press, New York. (1997).

C. D. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM (2000).
[Crossref]

R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge, New York (1985).

N. Ratner and Y. Y. Schechner. “Illumination multiplexing within fundamental limits.” In Proc. IEEE CVPR (2007).

W. Haemers. “Matrix techniques for strongly regular graphs and related geometries.” Intensive Course on Finite Geometry and its Applications, University of Ghent (2000).

H. H. Barrett and W. Swindell. Radiological Imaging, volume 1. Academic press, New York (1981).

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Figures (7)

Fig. 1.
Fig. 1.

An example of a strongly regular graph (Peterson) [31]. This graph has the parameters (N=10;k=3;α=0;β=1).

Fig. 2.
Fig. 2.

MSE ˜ as a function of {µf } N f =1. Each green dot marks the vector μ that minimizes MSE ˜ , when S is fixed. The highlighted line marks the ideal value of S. The green dot along this line marks the global minimum of MSE ˜ . This global minimum is derived in closed form and is thus unaffected by local minima.

Fig. 3.
Fig. 3.

The curve 1/µ represents elements summed in Eq. (34). The black dashed lines mark a state of the SVs of W. If the largest squared-SV, µ N , is reduces by Δµ N and in return the smallest squared-SV, µ 1, is increased by Δµ N , the sum in Eq. (34) is reduced.

Fig. 4.
Fig. 4.

The bound B min(C), for N=63. Here C varies from 1 to 63. The minimum of B min(C) is at C opt. There may exist C sat, above which saturation occurs, inhibiting multiplexing.

Fig. 5.
Fig. 5.

A semi-logarithmic plot of Eq. (62).

Fig. 6.
Fig. 6.

An example for an adjacency matrix of an SRG with parameters (45; 12; 3;3) developed by Ref. [35]. Here 1s and 0s are represented by white and black squares, respectively. Here α=β=3, hence this graph satisfies the conditions of Theorem 7.

Fig. 7.
Fig. 7.

(a) Graylevel spectral radiance of light transmitted through the atmosphere [38]. It is used to simulate the ground truth i. (b) Absolute error values |î-i| of simulated estimates based either on trivial sensing (red plot) or on SRG-based multiplexing (green plot).

Equations (89)

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

a = W i + υ .
i ̂ = W 1 a .
MSE i ̂ = σ a 2 N trace [ ( W t W ) 1 ] .
trace ( RQ ) = trace ( QR ) .
trace ( W ) = f = 1 N λ f .
ξ f = μ f .
ξ m = λ m m .
m = f N ξ m m = f N λ m f { 2 , , N }
m=1 N ξ m = m=1 N | λ m |.
ω p , q = { 1 if p and q are neighbors 0 otherwise
ω ̅ p , q = { 1 if ω p , q = 0 and p q 0 otherwise .
q = 1 N ω p , q = k p .
k ( k α 1 ) = ( N k 1 ) β .
λ 1 Ω = ( α β ) + Δ 2
λ 2 Ω = ( α β ) Δ 2
λ 3 Ω = k ,
Δ ( α β ) 2 + 4 ( k β ) .
ξ f Ω = λ f Ω f .
C = C Had N + 1 2 .
s = 1 N w m , s = C m { 1 , 2 , , N } .
0 w m , s 1 m , s { 1 , 2 , , N } .
MSE ˜ MSE i ̂ σ a 2 = 1 N trace [ ( W t W ) 1 ] .
min W MSE ˜ min W 1 N trace [ ( W t W ) 1 ]
s . t . s = 1 N w m , s C = 0 m { 1 , , N }
w m , s 0 m , s { 1 , , N }
w m , s 1 0 m , s { 1 , , N } .
MSE ˜ 1 N trace [ ( W t W ) 1 ] = 1 N f = 1 N 1 μ f ,
μ f ξ f 2 f { 1 , , N } .
ξ N λ N .
λ N = C .
μ N ξ N 2 C 2 .
MSE ˜ = 1 N μ N + 1 N f = 1 N 1 1 μ f .
S f = 1 N μ f
min μ N { 1 N μ N + 1 N f = 1 N 1 1 μ f } = 1 N C 2 + 1 N f = 1 N 1 1 μ f .
Δ MSE ˜ = MSE ˜ μ f Δ μ N = 1 N μ f 2 Δ μ N < 0 .
Δ MSE ˜ = Δ μ N N ( 1 μ 1 2 1 μ N 2 ) < 0 .
ξ N = C
μ N = C 2 .
1 N f = 1 N 1 1 μ f N 1 N ( 1 N 1 f = 1 N 1 1 μ f ) .
1 N 1 f = 1 N 1 1 μ f N 1 f = 1 N 1 μ f .
1 N 1 f = 1 N 1 1 μ f N 1 S C 2 .
MSE ˜ B ,
B = 1 N C 2 + ( N 1 ) 2 N ( S C 2 ) .
trace ( W t W ) = trace ( W W t ) .
( W W t ) m , m = s = 1 N ( w m , s ) 2 .
( w m , s ) 2 w m , s .
( W W t ) m , m s = 1 N w m , s = C .
S = k = 1 N μ k = trace ( W t W ) .
S = trace ( W W t ) = m = 1 N ( W W t ) m , m N C .
0 S N C .
S ideal = N C .
B B min
B min ( C ) = [ 1 N C 2 + ( N 1 ) 2 N ( N C C 2 ) ] .
MSE i ̂ σ a 2 B min ( C ) .
Ψ { [ 1 , N ] + } .
C opt free arg min C Ψ B min ( C ) .
B min C 2 N C 3 ( N 1 ) 2 ( N 2 C ) N ( N C C 2 ) 2 = 0 .
C opt free = N 2 2 N 3 ± ( N 1 ) ( N 2 2 N + 9 ) 4 N 8 .
C opt int ROUND ( C opt free )
Ψ int { [ 1 , N ] + } .
C opt int = arg min C Ψ int B min ( C ) .
C Had C opt free = N 1 N 2 . N + 1 ( N 1 ) 2 + 8 4 .
lim N ( C Had C opt free ) = 0.5 .
C Had > C opt free > C Had 0.5 .
C opt int = C Had ,
Ψ sat = { Ψ [ 1 , C sat ] } ,
C opt sat = arg min C Ψ sat B min ( C ) .
C opt sat = { C opt free if C opt free C sat C sat otherwise .
ξ 1 Ω = C β
ξ 2 Ω = C .
α = k ( k 1 ) ( N 1 ) = C ( C 1 ) ( N 1 ) .
VAR ( n electr photo ) = ( n electr photo ) ,
a = n electr photo Q electr .
( n electr photo ) Q electr 2 = a Q electr .
σ a 2 = κ gray 2 + a Q electr .
a i s C .
σ a 2 = κ gray 2 + C η 2 .
MSE i ̂ = σ a 2 MSE ˜ .
MSE i ̂ = ( κ gray 2 + C η 2 ) MSE ˜ .
C opt final = arg min C MSE i ̂ ( C ) .
W offset W C I .
s = 1 N w m , s offset = 0 m { 1 , 2 , , N } .
max m { 1 , , N } u m f = 1 .
u m f = 1
w m , s u m f 1 m , s { 1 , , N } .
( W u f ) m = s = 1 N w m , s u s f s = 1 N w m , s u s f C .
( W u f ) m = λ f u f m = λ f u m f .
( W u f ) m = λ f .
λ f C .

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