Abstract

Laser-based optical diagnostics, such as planar laser-induced fluorescence and, especially, Raman imaging, often require selective spectral filtering. We advocate the use of an imaging spectrograph with a broad entrance slit as a spectral filter for two-dimensional imaging. A spectrograph in this mode of operation produces output that is a convolution of the spatial and spectral information that is present in the incident light. We describe an analytical deconvolution procedure, based on Bayesian statistics, that retrieves the spatial information while it avoids excessive noise blowup. The method permits direct imaging through a spectrograph, even under broadband illumination. We introduce the formalism and discuss the underlying assumptions. The performance of the procedure is demonstrated on an artificial but pathological example. In a companion paper [Appl. Opt. 43, 5682–5690 (2004)] the method is applied to the practical case of fuel equivalence ratio Raman imaging in a combustible methane–air mixture.

© 2004 Optical Society of America

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References

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  1. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon Breach, Amsterdam, 1996).
  2. K. Kohse-Höinghaus, J. B. Jeffries, Applied Combustion Diagnostics (Taylor Francis, New York, 2002).
  3. W. Merzkirch, Flow Visualization (Academic, Orlando, Fla., 1987).
  4. W. Demtröder, Laser Spectroscopy—Basic Concepts and Instrumentation, 2nd ed., Vol. 5 of Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1996).
  5. E. Hecht, Optics, 4th ed. (Addison-Wesley, San Francisco, Calif., 2002).
  6. S. P Nandula, T. M. Brown, P. A. Skaggs, R. W. Pitz, P. A. DeBarber, “Multi-species line Raman measurements in H2-air turbulent flames,” paper AIAA-94-0227, presented at the 32nd AIAA Aerospace Sciences Meeting, Reno, Nev., 10–13 January, 1994 (American Institute for Aeronautics and Astronautics, Reston, Va., 1994).
  7. M. Mansour, Y. Chen, “Line Raman, Rayleigh, and laser-induced predissociation fluorescence technique for combustion with a tunable KrF excimer laser,” Appl. Opt. 35, 4252–4260 (1996).
    [CrossRef] [PubMed]
  8. F. Rabenstein, A. Leipertz, “One-dimensional, time-resolved Raman measurements in a sooting flame made with 355-nm excitation,” Appl. Opt. 37, 4937–4943 (1998).
    [CrossRef]
  9. G. Grünefeld, H. Schlüter, P. Andresen, “Simultaneous multiple-line Rayleigh–Raman/LIF measurements in combustion,” Appl. Phys. B 70, 309–313 (2000).
    [CrossRef]
  10. J. O. Gilmore, S. Sharma, D. Fletcher, D. Bershader, “Single-pulse spontaneous Raman scattering measurements in an expanding nitrogen/oxygen admixture,” paper AIAA-95-2125, presented at the 30th AIAA Thermophysics Conference, San Diego, Calif., 19–22 June, 1995 (American Institute for Aeronautics and Astronautics, Reston, Va., 1995).
  11. G. Tejeda, J. M. Fernández-Sánchez, S. Montero, “High-performance dual Raman spectrometer,” Appl. Spectrosc. 51, 265–276 (1997).
    [CrossRef]
  12. N. M. Sijtsema, R. A. L. Tolboom, N. J. Dam, J. J. ter Meulen, “Two-dimensional multispecies imaging of a supersonic nozzle flow,” Opt. Lett. 24, 664–666 (1999).
    [CrossRef]
  13. R. A. L. Tolboom, N. J. Dam, N. M. Sijtsema, J. J. ter Meulen, “Quantitative spectrally resolved imaging through a spectrograph,” Opt. Lett. 28, 2046–2048 (2003).
    [CrossRef] [PubMed]
  14. R. Tolboom, “Expanding laser diagnostics in non-seeded compressible flow research,” Ph.D. dissertation (University of Nijmegen, Nijmegen, The Netherlands, 2002), available from http://webdoc.ubn.kun.nl/mono/t/tolboom_r/expaladii.pdf .
  15. R. A. L. Tolboom, N. J. Dam, J. J. ter Meulen, “Quantitative imaging through a spectrograph. 2. Stoichiometry mapping by Raman scattering,” Appl. Opt. 43, 5682–5690 (2004).
    [CrossRef] [PubMed]
  16. A. Thorne, U. Litzén, S. Johansson, Spectrophysics: Principles and Applications (Springer-Verlag, Berlin, 1999).
  17. R. A. L. Tolboom, N. M. Sijtsema, N. J. Dam, J. J. ter Meulen, “Raman imaging for combustion diagnostics,” paper AIAA-00-0956, presented at the 38th AIAA Aerospace Sciences Meeting, Reno, Nev., 14–19 January, 2000 (American Institute for Aeronautics and Astronautics, Reston, Va., 2000).
  18. Unless explicitly stated otherwise, the integration limits will be from -∞ to +∞ for (reciprocal) space coordinates and from 0 to +∞ for wavelengths and frequencies.
  19. Compare this to rolling two dice, λ and xin, the sum of their results being xout. There are various combinations of (λ, xin) that lead, for example, to xout = 7. These combinations are described by f(λ, xin). Once outcome xout is chosen to be 7, however, there is only one corresponding λ for every xin, namely, f̂(xin; xout).
  20. The line at λ = 579.40 nm is an unresolved doublet. In fact, the exact wavelengths used do not matter for the processing of OMA graphs.
  21. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Prentice-Hall, Upper Saddle River, N.J., 2002).
  22. K. C. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  23. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  24. As input signal S(x) of the FT is purely real, the Fourier-transformed data S̃(k) are even under complex conjugation; i.e., S̃(-k) = S̃*(k). Therefore, only the positive k components need to be plotted to represent all power information.
  25. See, e.g., P. M. Lee, Bayesian Statistics: An Introduction, 2nd ed. (Arnold, London, 1997).
  26. R. Durrett, Probability: Theory and Examples (Duxbury, Belmont, Calif., 1991).
  27. D. J. C. MacKay, “Information theory, inference, and learning algorithms,” draft 2.4.1, 2002, available from http://www.inference.phy.cam.ac.uk/itprnn/book.pdf .
  28. M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 2nd ed. (World Scientific, Singapore, 1994).
  29. H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
    [CrossRef]
  30. This is a general property of Fourier transformation. For visual clarity, the reconstructions presented in this paper (Figs. 4 and 6) have been recentered in the image frames.
  31. B. Buck, V. A. Macaulay, Maximum Entropy in Action (Clarendon, Oxford, 1991).
  32. See, e.g., Ref. 1, Sec. 3.6.
  33. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computation, 2nd ed. (Cambridge U. Press, Cambridge, Mass., 1992).
  34. In this paper the convention is that the power or the absolute square of a signal S is defined as the signal times its complex conjugate (i.e., |S|2 = S × S*), and the norm of the signal ‖S‖ is the integral over S.

2004

2003

2000

G. Grünefeld, H. Schlüter, P. Andresen, “Simultaneous multiple-line Rayleigh–Raman/LIF measurements in combustion,” Appl. Phys. B 70, 309–313 (2000).
[CrossRef]

1999

1998

1997

1996

Andresen, P.

G. Grünefeld, H. Schlüter, P. Andresen, “Simultaneous multiple-line Rayleigh–Raman/LIF measurements in combustion,” Appl. Phys. B 70, 309–313 (2000).
[CrossRef]

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Bergersen, B.

M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 2nd ed. (World Scientific, Singapore, 1994).

Bershader, D.

J. O. Gilmore, S. Sharma, D. Fletcher, D. Bershader, “Single-pulse spontaneous Raman scattering measurements in an expanding nitrogen/oxygen admixture,” paper AIAA-95-2125, presented at the 30th AIAA Thermophysics Conference, San Diego, Calif., 19–22 June, 1995 (American Institute for Aeronautics and Astronautics, Reston, Va., 1995).

Brown, T. M.

S. P Nandula, T. M. Brown, P. A. Skaggs, R. W. Pitz, P. A. DeBarber, “Multi-species line Raman measurements in H2-air turbulent flames,” paper AIAA-94-0227, presented at the 32nd AIAA Aerospace Sciences Meeting, Reno, Nev., 10–13 January, 1994 (American Institute for Aeronautics and Astronautics, Reston, Va., 1994).

Buck, B.

B. Buck, V. A. Macaulay, Maximum Entropy in Action (Clarendon, Oxford, 1991).

Castleman, K. C.

K. C. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Chen, Y.

Dam, N. J.

DeBarber, P. A.

S. P Nandula, T. M. Brown, P. A. Skaggs, R. W. Pitz, P. A. DeBarber, “Multi-species line Raman measurements in H2-air turbulent flames,” paper AIAA-94-0227, presented at the 32nd AIAA Aerospace Sciences Meeting, Reno, Nev., 10–13 January, 1994 (American Institute for Aeronautics and Astronautics, Reston, Va., 1994).

Demtröder, W.

W. Demtröder, Laser Spectroscopy—Basic Concepts and Instrumentation, 2nd ed., Vol. 5 of Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1996).

Durrett, R.

R. Durrett, Probability: Theory and Examples (Duxbury, Belmont, Calif., 1991).

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon Breach, Amsterdam, 1996).

Engl, H. W.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

Fernández-Sánchez, J. M.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computation, 2nd ed. (Cambridge U. Press, Cambridge, Mass., 1992).

Fletcher, D.

J. O. Gilmore, S. Sharma, D. Fletcher, D. Bershader, “Single-pulse spontaneous Raman scattering measurements in an expanding nitrogen/oxygen admixture,” paper AIAA-95-2125, presented at the 30th AIAA Thermophysics Conference, San Diego, Calif., 19–22 June, 1995 (American Institute for Aeronautics and Astronautics, Reston, Va., 1995).

Gilmore, J. O.

J. O. Gilmore, S. Sharma, D. Fletcher, D. Bershader, “Single-pulse spontaneous Raman scattering measurements in an expanding nitrogen/oxygen admixture,” paper AIAA-95-2125, presented at the 30th AIAA Thermophysics Conference, San Diego, Calif., 19–22 June, 1995 (American Institute for Aeronautics and Astronautics, Reston, Va., 1995).

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Prentice-Hall, Upper Saddle River, N.J., 2002).

Grünefeld, G.

G. Grünefeld, H. Schlüter, P. Andresen, “Simultaneous multiple-line Rayleigh–Raman/LIF measurements in combustion,” Appl. Phys. B 70, 309–313 (2000).
[CrossRef]

Hanke, M.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, San Francisco, Calif., 2002).

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Jeffries, J. B.

K. Kohse-Höinghaus, J. B. Jeffries, Applied Combustion Diagnostics (Taylor Francis, New York, 2002).

Johansson, S.

A. Thorne, U. Litzén, S. Johansson, Spectrophysics: Principles and Applications (Springer-Verlag, Berlin, 1999).

Kohse-Höinghaus, K.

K. Kohse-Höinghaus, J. B. Jeffries, Applied Combustion Diagnostics (Taylor Francis, New York, 2002).

Lee, P. M.

See, e.g., P. M. Lee, Bayesian Statistics: An Introduction, 2nd ed. (Arnold, London, 1997).

Leipertz, A.

Litzén, U.

A. Thorne, U. Litzén, S. Johansson, Spectrophysics: Principles and Applications (Springer-Verlag, Berlin, 1999).

Macaulay, V. A.

B. Buck, V. A. Macaulay, Maximum Entropy in Action (Clarendon, Oxford, 1991).

Mansour, M.

Merzkirch, W.

W. Merzkirch, Flow Visualization (Academic, Orlando, Fla., 1987).

Montero, S.

Nandula, S. P

S. P Nandula, T. M. Brown, P. A. Skaggs, R. W. Pitz, P. A. DeBarber, “Multi-species line Raman measurements in H2-air turbulent flames,” paper AIAA-94-0227, presented at the 32nd AIAA Aerospace Sciences Meeting, Reno, Nev., 10–13 January, 1994 (American Institute for Aeronautics and Astronautics, Reston, Va., 1994).

Neubauer, A.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

Pitz, R. W.

S. P Nandula, T. M. Brown, P. A. Skaggs, R. W. Pitz, P. A. DeBarber, “Multi-species line Raman measurements in H2-air turbulent flames,” paper AIAA-94-0227, presented at the 32nd AIAA Aerospace Sciences Meeting, Reno, Nev., 10–13 January, 1994 (American Institute for Aeronautics and Astronautics, Reston, Va., 1994).

Plischke, M.

M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 2nd ed. (World Scientific, Singapore, 1994).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computation, 2nd ed. (Cambridge U. Press, Cambridge, Mass., 1992).

Rabenstein, F.

Schlüter, H.

G. Grünefeld, H. Schlüter, P. Andresen, “Simultaneous multiple-line Rayleigh–Raman/LIF measurements in combustion,” Appl. Phys. B 70, 309–313 (2000).
[CrossRef]

Sharma, S.

J. O. Gilmore, S. Sharma, D. Fletcher, D. Bershader, “Single-pulse spontaneous Raman scattering measurements in an expanding nitrogen/oxygen admixture,” paper AIAA-95-2125, presented at the 30th AIAA Thermophysics Conference, San Diego, Calif., 19–22 June, 1995 (American Institute for Aeronautics and Astronautics, Reston, Va., 1995).

Sijtsema, N. M.

R. A. L. Tolboom, N. J. Dam, N. M. Sijtsema, J. J. ter Meulen, “Quantitative spectrally resolved imaging through a spectrograph,” Opt. Lett. 28, 2046–2048 (2003).
[CrossRef] [PubMed]

N. M. Sijtsema, R. A. L. Tolboom, N. J. Dam, J. J. ter Meulen, “Two-dimensional multispecies imaging of a supersonic nozzle flow,” Opt. Lett. 24, 664–666 (1999).
[CrossRef]

R. A. L. Tolboom, N. M. Sijtsema, N. J. Dam, J. J. ter Meulen, “Raman imaging for combustion diagnostics,” paper AIAA-00-0956, presented at the 38th AIAA Aerospace Sciences Meeting, Reno, Nev., 14–19 January, 2000 (American Institute for Aeronautics and Astronautics, Reston, Va., 2000).

Skaggs, P. A.

S. P Nandula, T. M. Brown, P. A. Skaggs, R. W. Pitz, P. A. DeBarber, “Multi-species line Raman measurements in H2-air turbulent flames,” paper AIAA-94-0227, presented at the 32nd AIAA Aerospace Sciences Meeting, Reno, Nev., 10–13 January, 1994 (American Institute for Aeronautics and Astronautics, Reston, Va., 1994).

Tejeda, G.

ter Meulen, J. J.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computation, 2nd ed. (Cambridge U. Press, Cambridge, Mass., 1992).

Thorne, A.

A. Thorne, U. Litzén, S. Johansson, Spectrophysics: Principles and Applications (Springer-Verlag, Berlin, 1999).

Tolboom, R. A. L.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computation, 2nd ed. (Cambridge U. Press, Cambridge, Mass., 1992).

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Prentice-Hall, Upper Saddle River, N.J., 2002).

Appl. Opt.

Appl. Phys. B

G. Grünefeld, H. Schlüter, P. Andresen, “Simultaneous multiple-line Rayleigh–Raman/LIF measurements in combustion,” Appl. Phys. B 70, 309–313 (2000).
[CrossRef]

Appl. Spectrosc.

Opt. Lett.

Other

J. O. Gilmore, S. Sharma, D. Fletcher, D. Bershader, “Single-pulse spontaneous Raman scattering measurements in an expanding nitrogen/oxygen admixture,” paper AIAA-95-2125, presented at the 30th AIAA Thermophysics Conference, San Diego, Calif., 19–22 June, 1995 (American Institute for Aeronautics and Astronautics, Reston, Va., 1995).

R. Tolboom, “Expanding laser diagnostics in non-seeded compressible flow research,” Ph.D. dissertation (University of Nijmegen, Nijmegen, The Netherlands, 2002), available from http://webdoc.ubn.kun.nl/mono/t/tolboom_r/expaladii.pdf .

A. Thorne, U. Litzén, S. Johansson, Spectrophysics: Principles and Applications (Springer-Verlag, Berlin, 1999).

R. A. L. Tolboom, N. M. Sijtsema, N. J. Dam, J. J. ter Meulen, “Raman imaging for combustion diagnostics,” paper AIAA-00-0956, presented at the 38th AIAA Aerospace Sciences Meeting, Reno, Nev., 14–19 January, 2000 (American Institute for Aeronautics and Astronautics, Reston, Va., 2000).

Unless explicitly stated otherwise, the integration limits will be from -∞ to +∞ for (reciprocal) space coordinates and from 0 to +∞ for wavelengths and frequencies.

Compare this to rolling two dice, λ and xin, the sum of their results being xout. There are various combinations of (λ, xin) that lead, for example, to xout = 7. These combinations are described by f(λ, xin). Once outcome xout is chosen to be 7, however, there is only one corresponding λ for every xin, namely, f̂(xin; xout).

The line at λ = 579.40 nm is an unresolved doublet. In fact, the exact wavelengths used do not matter for the processing of OMA graphs.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Prentice-Hall, Upper Saddle River, N.J., 2002).

K. C. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

As input signal S(x) of the FT is purely real, the Fourier-transformed data S̃(k) are even under complex conjugation; i.e., S̃(-k) = S̃*(k). Therefore, only the positive k components need to be plotted to represent all power information.

See, e.g., P. M. Lee, Bayesian Statistics: An Introduction, 2nd ed. (Arnold, London, 1997).

R. Durrett, Probability: Theory and Examples (Duxbury, Belmont, Calif., 1991).

D. J. C. MacKay, “Information theory, inference, and learning algorithms,” draft 2.4.1, 2002, available from http://www.inference.phy.cam.ac.uk/itprnn/book.pdf .

M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 2nd ed. (World Scientific, Singapore, 1994).

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
[CrossRef]

This is a general property of Fourier transformation. For visual clarity, the reconstructions presented in this paper (Figs. 4 and 6) have been recentered in the image frames.

B. Buck, V. A. Macaulay, Maximum Entropy in Action (Clarendon, Oxford, 1991).

See, e.g., Ref. 1, Sec. 3.6.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C—The Art of Scientific Computation, 2nd ed. (Cambridge U. Press, Cambridge, Mass., 1992).

In this paper the convention is that the power or the absolute square of a signal S is defined as the signal times its complex conjugate (i.e., |S|2 = S × S*), and the norm of the signal ‖S‖ is the integral over S.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon Breach, Amsterdam, 1996).

K. Kohse-Höinghaus, J. B. Jeffries, Applied Combustion Diagnostics (Taylor Francis, New York, 2002).

W. Merzkirch, Flow Visualization (Academic, Orlando, Fla., 1987).

W. Demtröder, Laser Spectroscopy—Basic Concepts and Instrumentation, 2nd ed., Vol. 5 of Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1996).

E. Hecht, Optics, 4th ed. (Addison-Wesley, San Francisco, Calif., 2002).

S. P Nandula, T. M. Brown, P. A. Skaggs, R. W. Pitz, P. A. DeBarber, “Multi-species line Raman measurements in H2-air turbulent flames,” paper AIAA-94-0227, presented at the 32nd AIAA Aerospace Sciences Meeting, Reno, Nev., 10–13 January, 1994 (American Institute for Aeronautics and Astronautics, Reston, Va., 1994).

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the diffraction of light by a grating.

Fig. 2
Fig. 2

Cross section of a spectrograph including a first-order ray trace for bichromatic light. The heights of the entrance slit and the exit port as well as the grooves of the grating are perpendicular to the plane of the picture. Also shown are coordinates x in and x out that appear in the formalism of Section 2.

Fig. 3
Fig. 3

Three OMA graphs recorded with different widths of the entrance slit of the spectrograph, as indicated in the photographs. The horizontal axes contain both spectral and spatial information, and the vertical axes are purely spatial. (a) Imaged object, a 5.0 mm × 5.0 mm grid printed on white paper with a linewidth of 0.5 mm; (b) spectrum of the light source [a Hg(Ar) calibration lamp; λ (in nanometers) indicated], recorded by reflection off white paper; (c), (d) OMA graphs of the object shown in (a) under illumination by the same source as in (b). The traces on top of the images are single-strip cross sections of the images at the positions of the arrows, cutting the lower circles of the “68” on the grid. All images are scaled individually. The curvature of the images and the horizontal extrusion of the grid are artifacts of the spectrograph.

Fig. 4
Fig. 4

Strip-by-strip deconvolution of Fig. 3(d) with the spectrum of Fig. 3(b) by means of the unfiltered Fourier-transformation algorithm [Eq. (A6) below]. Purely spatial images (a) and (c) show data in direct space; the corresponding power spectra are shown in (b) and (d) (first half of the k components only). (a), (b) Direct deconvolution; note the large high-k components in (b). (c), (d) Direct convolution but with two-pixel binning; the high-k components partly cancel (d). Top, single-strip cross sections at the positions of the arrows. [Zero baseline indicated in (a) and (c); left ordinates omitted in (b) and (d) to emphasize the low-k components.]

Fig. 5
Fig. 5

Power spectra (single strips at the location of the arrows in Fig. 3). (a) Fourier-transformed spectral reference function k of Fig. 3(b), (b) nonfiltered deconvolution function, (c) filtering of (b) (note the logarithmic scale), (d) linear Bayesian filtered deconvolution function. The power of the filtering function (c) is the prefactor filter of Eq. (34). (c), (d) Calculated for σ/τ = 6 counts-1.

Fig. 6
Fig. 6

Strip-by-strip linear Bayesian deconvolution of Fig. 3(d) with the spectrum of Fig. 3(b). Right to left, results for three ratios σ/τ, ranging from (a) too low (σ/τ = 1 count-1) to (b) best (σ/τ = 6 counts-1) to (c) too high (σ/τ = 800 counts-1). Above the images are the single-strip cross sections (similar to those in Figs. 3 and 4), and on top of them are their power spectra for the first halves of k components. The images are scaled individually, but the traces are all on the same linear gray scale. The left ordinates of the power spectra are omitted to show the similarity of the barely filtered, low-k components.

Fig. 7
Fig. 7

Contrast (defined in the text) as a function of σ/τ. The maximum in the curve is taken as the best σ/τ for the deconvolution as it minimizes the relative power in the physically dark region. The corresponding value for the nonfiltered results [Fig. 4(a)] is 〈P(in)〉/〈P(out)〉 = 1.35.

Fig. 8
Fig. 8

Reconvolved image of the data that were obtained with a deconvolution for σ/τ = 6 counts-1 (solid curve) and its difference from the original, measured data (residual; gray curve). The difference is 0.04 count on average and has a standard deviation of more than 7 kcounts.

Equations (43)

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

dsin α±sin β=mλ,
Txout=λxinslit Tλ, xin; xoutSinλ, xindλdxin,
Tλ, xin; xout=ηλδxout-fλ, xin,
Sinλ, xin=Sλλ×Sxin.
Txout=λxin ηλδxout-fλ, xin×SλλSxindλdxin=xin ηfˆxin; xoutSλfˆxin; xoutSxindxin,
Msxin,0-xin-xout,0-xout=ζλ.
fˆxin; xoutλ=xout-xout,0-Msxin-xin,0ζ.
Txout=xin ηxout-xout,0-Msxin-xin,0ζ×Sλxout-xout,0-Msxin-xin,0ζ×Sxindxin
=η×Sλ * Sxout.
Sδxin=S0δxin-xin,0,
Tδxout=ηxout-xout,0ζSλxout-xout,0ζS0,
ηxout-xout,0-Msxin-xin,0ζ×Sλxout-xout,0-Msxin-xin,0ζ =Rxout-Msxin-xin,0S0.
Txout=xin Rxout-Msxin-xin,0SxinS0dxin.
Txout=xin R[xout-Msxin-xin,0]Gxindxin+Nxout,
FT[Fx]F˜k=defx Fxexp-ikxdx,
FT-1[F˜k]Fx=def12πk F˜kexpikxdk,
T˜k=R˜kexpiMskxin,0G˜Msk+N˜k
G˜Msk=T˜k-N˜kR˜kexpiMskxin,0,
EG|T=tt
E|Gn-nt|2n,
Gn=c+σXnn, Nm=b+τYmm,
Xm  Xnmn, i.e., the pdf of the optics beforethe spectrograph is neglected;Ym  Ynmn, i.e., the noise is accounted for per individual pixel; andXm  Ynm,n, i.e., the noise is not correlated to the signal at all.
Gngn=12πσexp-12gn-cσ2n
Nmνm=12πτexp-12νm-bτ2m.
Tm=n Rm-n+n0Gn+Nmm,
Tmtm|G=g=12πτ×exp-12tm-n Rm-n+n0gn+bτ2m.
EGq|T=t=gq gqGq=gq|T=t=g gqG=g|T=t q.
Gqa, b|T=t=ab Gqgq|T=tdgq.
G=g|T=t=T=t|G=g×G=gT=t
EGq|T=t=g gqT=t|G=g×G=gT=tq.
T=t=g T=t|G=g×G=g,
EGq|T=t=g gqT=t|G=g×G=gg T=t|G=g×G=gq,
EGq|T=t=N gqTt|G=g×GgdgN Tt|G=g×Ggdg=N gq exp-12t-R * g-bτ2exp-12g-cσ2dgN exp-12t-R * g-bτ2exp-12g-cσ2dg=N gq exp-12mtm-n Rm-n+n0gn-bτ2+gm-cσ2dgNexp-12mtm-n Rm-n+n0gn-bτ2+gm-cσ2dg,
EGq|T=t=cτ/σ2-bR˜k=0R˜k=02+τ/σ2+FT-1R˜k,n0*t˜k|R˜k,n0|2+τ/σ2
R˜k,n0*t˜k|R˜k,n0|2+τ/σ2=|R˜k,n0|2|R˜k,n0|2+τ/σ2t˜kR˜k,n0=11+τ/σ2|R˜k,n0|-2t˜kR˜k,n0.
R˜k,n0*|R˜k,n0|2+τ/σ22filtered deconvolution function=|R˜k,n0|2|R˜k,n0|2+τ/σ22filter×1R˜k,n02deconvolution function.
EGq|T=a1t1+a2t2=a1+a2cτ/σ2-bR˜k=0R˜k=02+τ/σ2+FT-1R˜k,n0*a1t1k+a2t2k|R˜k,n0|2+τ/σ2=a1EGq|T=t1+a2EGq|T=t2.
Tnout=nin Rnout-Msnin+nin,0×Gnin+Nnout,
Tm=n=0N-1 Rm-n+n0×Gn+Nm.
F˜k=n=0N-1 Fn exp-2πiknN,Fn=1Nk=0N-1 F˜k exp2πiknN,
T˜k=R˜k,n0×G˜k+N˜k, R˜k,n0 =def R˜k exp2πikn0N,
G˜k=T˜k-N˜kR˜k,n0.
Gn=FT-1G˜k=FT-1T˜k-N˜kR˜k,n0=1Nk=0N-1T˜k-N˜kR˜k,n0exp2πiknN,

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