Abstract

We experimentally evaluate diverse static independent column codes in a coded aperture spectrometer. The performance of each code is evaluated based on the signal-to-noise ratio (SNR), defined as the peak value in the spectrum to the standard deviation of the background noise, as a function of subpixel vertical misalignments. Among the code families tested, an S-matrix-based code produces spectral reconstructions with the highest SNR. The SNR is least sensitive to vertical subpixel misalignments on the detector with a Hadamard-matrix-based code. Finally, the increased sensitivity of a spectrometer using a coded aperture instead of a slit is demonstrated.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, "Hadamard spectroscopy with a 2-dimensional detecting array," Appl. Opt. 32, 7095-7105 (1993).
    [CrossRef] [PubMed]
  2. M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, "Static 2D aperture coding for multimodal multiplex spectroscopy," Appl. Opt. 45, 2857-3183 (2006).
    [CrossRef]
  3. S. McCain, M. Gehm, Y. Wang, N. Pitsianis, and D. Brady, "Coded aperture Raman spectroscopy for quantitative measurements of ethanol in a tissue phatom," Appl. Spectrosc. 60, 663-671 (2006).
    [CrossRef] [PubMed]
  4. J. A. Decker and M. O. Harwitt, "Sequential encoding with multislit spectrometers," Appl. Opt. 7, 2205-2209 (1968).
    [CrossRef] [PubMed]
  5. M. Harwit and N. Sloane, Hadamard Transform Optics (Academic, 1979), pp. 50-59.
  6. D. Schroeder, Astronomical Optics (Academic, 1987).
  7. C. Lawson and R. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).
  8. Y. Zhou and C. K. Rushforth, "Least-squares reconstruction of spatially limited objects using smoothness and non-negativity constraints," Appl. Opt. 21, 1249-1252 (1982).
    [CrossRef] [PubMed]
  9. A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, 2005).
    [CrossRef]
  10. L. Pan, D. Edwards, J. Gille, M. Smith, and J. Drummond, "Satellite remote sensing of tropospheric CO2 and CH4: forward model studies of the MOPITT instrument," Appl. Opt. 34, 6976-6988 (1995).
    [CrossRef] [PubMed]
  11. J. A. Decker, "Experimental realization of multiplex advantage with a Hadamard transform spectrometer," Appl. Opt. 10, 510-514 (1971).
    [CrossRef] [PubMed]

2006 (2)

2005 (1)

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, 2005).
[CrossRef]

1995 (1)

1993 (1)

1987 (1)

D. Schroeder, Astronomical Optics (Academic, 1987).

1982 (1)

1979 (1)

M. Harwit and N. Sloane, Hadamard Transform Optics (Academic, 1979), pp. 50-59.

1974 (1)

C. Lawson and R. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

1971 (1)

1968 (1)

Brady, D.

Brady, D. J.

Claflin, E. S.

Decker, J. A.

Drummond, J.

Edwards, D.

Gehm, M.

Gehm, M. E.

Gille, J.

Hanson, R.

C. Lawson and R. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

Harwit, M.

M. Harwit and N. Sloane, Hadamard Transform Optics (Academic, 1979), pp. 50-59.

Harwitt, M. O.

Lawson, C.

C. Lawson and R. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

McCain, S.

McCain, S. T.

Mende, S. B.

Pan, L.

Pitsianis, N.

Pitsianis, N. P.

Potuluri, P.

Rairden, R. L.

Rushforth, C. K.

Schroeder, D.

D. Schroeder, Astronomical Optics (Academic, 1987).

Sloane, N.

M. Harwit and N. Sloane, Hadamard Transform Optics (Academic, 1979), pp. 50-59.

Smith, M.

Sullivan, M. E.

Swenson, G. R.

Tarantola, A.

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, 2005).
[CrossRef]

Wang, Y.

Zhou, Y.

Appl. Opt. (6)

Appl. Spectrosc. (1)

Other (4)

M. Harwit and N. Sloane, Hadamard Transform Optics (Academic, 1979), pp. 50-59.

D. Schroeder, Astronomical Optics (Academic, 1987).

C. Lawson and R. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

A. Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation (Society for Industrial and Applied Mathematics, 2005).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (23)

Fig. 1
Fig. 1

Shuffled order-48 S-matrix with dead rows ( 3456   μm tall, 2592   μm wide).

Fig. 2
Fig. 2

Shuffled order-48 S-matrix without dead rows ( 3456   μm tall, 2592   μm wide).

Fig. 3
Fig. 3

Shuffled, row-doubled order-24 Hadamard with dead rows ( 3456   μm tall, 1296   μm wide).

Fig. 4
Fig. 4

Shuffled, row-doubled order-24 Hadamard without dead rows ( 3456   μm tall, 1296   μm wide).

Fig. 5
Fig. 5

Shuffled, row-doubled, stacked order-24 Hadamard with dead rows ( 3456   μm tall, 1296   μm wide).

Fig. 6
Fig. 6

Shuffled, row-doubled, stacked order-24 Hadamard without dead rows ( 3456   μm tall, 1296   μm wide).

Fig. 7
Fig. 7

Half-toned harmonic ( 3456   μm tall, 2592   μm wide).

Fig. 8
Fig. 8

Half-toned, reordered harmonic ( 3456   μm tall, 2592   μm wide).

Fig. 9
Fig. 9

Half-toned Legendre ( 3456   μm tall, 2592   μm wide).

Fig. 10
Fig. 10

Half-toned, reordered Legendre ( 3456   μm tall, 2592   μm wide).

Fig. 11
Fig. 11

(Color online) Experimental setup used to test the performance of a static multimodal multiplex coded aperture spectrometer with different aperture codes. Each aperture code mask is placed and secured inside the mask holder. The rotation stage allows rotational alignment of the mask images on the CCD. The two-axis translation stage provides control over the vertical subpixel misalignments of the mask images on the CCD and allows proper positioning of the mask along the optical axis to ensure that the mask images are in focus.

Fig. 12
Fig. 12

Example of raw CCD data from the MMS spectrometer.

Fig. 13
Fig. 13

Foward model development flow chart.

Fig. 14
Fig. 14

Comparing the raw and simulated CCD images.

Fig. 15
Fig. 15

Comparing the raw and simulated CCD images at the pixel level.

Fig. 16
Fig. 16

Comparing the ideal coding scheme to the coding scheme determined to be physically implemented in the spectrometer through forward model development.

Fig. 17
Fig. 17

(Color online) Spectral estimate before and after forward model development. Note that the spectral peaks are dramatically enhanced when the system is inverted using the coding matrix generated by the forward model, rather than the basic pattern printed on the aperture mask.

Fig. 18
Fig. 18

(Color online) Sample spectral estimate and the noise mask (dashed line) used to evaluate the SNR of the spectral estimate.

Fig. 19
Fig. 19

(Color online) Applied subpixel shift versus tracked subpixel shift. Using the forward model process to incorporate different subpixel shifts into the decoding matrix and choosing the one that maximizes the SNR allows excellent tracking of the applied subpixel shift.

Fig. 20
Fig. 20

(Color online) SNR versus applied subpixel shift. The subpixel correction plot demonstrates that incorporating subpixel shifts into the decoding matrix results in the ability to reconstruct spectra with the same SNR at any vertical subpixel misalignment of the aperture images on the detector. When subpixel shifts are not incorporated into the decoding matrix, the SNR degrades with increasing subpixel misalignment.

Fig. 21
Fig. 21

(Color online) Mean SNR versus mask type. The error bars indicate standard deviation. The S-matrix aperture codes produce spectral estimates with the highest SNR. Discrete codes without dead rows have higher mean SNRs than their counterparts containing dead rows. The addition of dead rows decrease the sensitivity of the spectrometer to vertical subpixel misalignments. RDHD, row-doubled Hadamard.

Fig. 22
Fig. 22

(Color online) Fractional SNR deviation with 1 pixel misalignment of each aperture code. The discrete codes with dead rows perform considerably better than any of the continuous column codes. The addition of dead rows to a discrete code greatly decreases the sensitivity of the spectrometer to vertical subpixel misalignments of the aperture images on the detector.

Fig. 23
Fig. 23

(Color online) (Top) Spectra reconstructed with slit at increasing exposure times. (Bottom) Spectra reconstructed with coded aperture at increasing exposure times. The coded aperture makes the spectrometer considerably more sensitive to the spectral peaks of xenon in the spectral range of the instrument.

Equations (7)

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

H ( m , : ) [ 1 2 ( 1 + H ( m , : ) ) 1 2 ( 1 H ( m , : ) ) ] .
T x { 1 2 ( 1 + cos ( m y π Y ) ) } , m [ 0 ,   …   ,   48 ] .
T x { 1 2 ( 1 + P m ( y Y ) ) } , m [ 0 , ,   48 ] ,
P m ( y ) = 1 2 m n = 0 m / 2 ( 1 ) n ( m n ) ( 2 m 2 n m ) y m 2 n .
( A ) X × Y ( S ) Y × Z = ( C ) X × Z ,
arg min x | M x C i | 2 , x 0 ,
SNR = peak   value   of   the   spectrum σ   of   background   noise   under   the   noise   mask .

Metrics