Abstract

A procedure for deconvolution of multiple images of the same object with space-variant point-spread functions (PSFs) is presented. It is based on expressing deconvolution with inverse filtering as convolution with kernels corresponding to inverse PSFs. Sets of basis functions are made from these inverse PSFs, given at discrete sample points, through Karhunen–Loève (K–L) decomposition. The entire field of view can then be convolved with the K–L kernels. Coadding the results using continuous maps of expansion weights, interpolated for every pixel between the sample points, results in an image that is deconvolved with smoothly varying PSFs that match the discrete measurements. A demonstration data set is used to show how the transition between the grid points improves deconvolutions compared to piecewise deconvolution and mosaicking by avoiding the blending of discontinuities at the interfaces between adjacent subfields.

© 2007 Optical Society of America

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References

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  1. M. G. Löfdahl, "Multi-frame blind deconvolution with linear equality constraints," in Image Reconstruction from Incomplete Data II, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 4792, 146-155 (2002).
    [CrossRef]
  2. M. G. Löfdahl and G. B. Scharmer, "Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE 4853, 567-575 (2003).
    [CrossRef]
  3. R. A. Gonsalves, "Phase retreival and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).
  4. M. van Noort, L. Rouppe van der Voort, and M. G. Löfdahl, "Solar image restoration by use of multi-frame blind deconvolution with multiple objects and phase diversity," Solar Phy. 228, 191-215 (2005).
    [CrossRef]
  5. R. G. Paxman, B. J. Thelen, and J. H. Seldin, "Correction of anisoplanatic blur by using phase diversity," in Adaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE 2201, 1066-1067 (1994).
    [CrossRef]
  6. R. A. Gonsalves, "Nonisoplanatic imaging by phase diversity," Opt. Lett. 19, 493-495 (1994).
    [CrossRef] [PubMed]
  7. O. von der Lühe, "Speckle imaging of solar small structure: I. Methods," Astron. Astrophy. 268, 374-390 (1993).
  8. G. B. Scharmer, K. Bjelksjö, T. K. Korhonen, B. Lindberg, and B. Pettersson, "The 1-meter Swedish solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE 4853, 341-350 (2003).
    [CrossRef]
  9. M. G. Löfdahl and G. B. Scharmer, "Wavefront sensing and image restoration from focused and defocused solar images," Astron. Astrophy. Suppl. Ser. 107, 243-264 (1994).
  10. G. B. Scharmer, D. S. Brown, L. Pettersson, and J. Rehn, "Concepts for the Swedish 50-cm Vacuum Solar Telescope," Appl. Opt. 24, 2558-2564 (1985).
    [CrossRef] [PubMed]
  11. R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, and C. U. Keller, "Evaluation of phase-diversity techniques for solar-image restoration," Astrophys. J. 466, 1087-1099 (1996).
    [CrossRef]
  12. J. G. Nagy and D. P. O'Leary, "Fast iterative image restoration with a spatially varying PSF," in Advanced Signal Processing: Algorithms, Architectures, and Implementations VII, F. T. Luk, ed., Proc. SPIE 3162, 388-399 (1997).
    [CrossRef]
  13. T. R. Lauer, "Deconvolution with a spatially-variant PSF," in Astronomical Data Analysis II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE 4847, 167-173 (2002).
    [CrossRef]
  14. C. Alard, "Image subtraction using a space-varying kernel," Astron. Astrophy. Suppl. Ser. 144, 363-370 (2000).
    [CrossRef]
  15. R. Lupton, J. E. Gunn, Z. Ivezic, and G. R. Knapp, "The SDSS imaging pipelines," in Astronomical Data Analysis Software and Systems X, F. R. Harnden, Jr., F. A. Primini, and H. E. Payne, eds., ASP Conf. Ser. 238, 269-278 (2001).
  16. R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations by using phase diversity," Opt. Soc. Am. A 9, 1072-1085 (1992).
    [CrossRef]
  17. N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
    [CrossRef]
  18. M. S. Vogeley and A. S. Szalay, "Eigenmode analysis of galaxy redshift surveys. I. Theory and methods," Astrophys. J. 465, 34-53 (1996).
    [CrossRef]
  19. S. K. Park and R. A. Schowengerdt, "Image reconstruction by parametric cubic convolution," Comput. Vis. Graph. Image Process. 23, 258-272 (1983).
    [CrossRef]
  20. R. A. Shine, L. Z. Porter, Z. Frank, J. B. Gurman, D. Pothier, and S. Ferguson, A User's Guide to ANA (Lockheed Palo Alto Research Laboratory, 1988); see also ana.lmsal.com.
  21. Institute for Astronomy of the University of Hawaii, "AR map 2 May, 2003," www.solar.ifa.hawaii.edu/ARMaps/archive.html.
  22. G. B. Scharmer, P. Dettori, M. G. Löfdahl, and M. Shand, "Adaptive optics system for the new Swedish solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE4853,370-380 (2003).
  23. M. van Noort and M. Löfdahl, "Multi-object multi-frame blind deconvolution," public domain C++ code available at www.momfbd.org (2005).
  24. M. G. Löfdahl, T. E. Berger, R. A. Shine, and A. M. Title, "Preparation of a dual wavelength sequence of high-resolution solar photospheric images using phase diversity," Astrophys. J. 495, 965-972 (1998).
    [CrossRef]

2005 (1)

M. van Noort, L. Rouppe van der Voort, and M. G. Löfdahl, "Solar image restoration by use of multi-frame blind deconvolution with multiple objects and phase diversity," Solar Phy. 228, 191-215 (2005).
[CrossRef]

2003 (2)

M. G. Löfdahl and G. B. Scharmer, "Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE 4853, 567-575 (2003).
[CrossRef]

G. B. Scharmer, K. Bjelksjö, T. K. Korhonen, B. Lindberg, and B. Pettersson, "The 1-meter Swedish solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE 4853, 341-350 (2003).
[CrossRef]

2002 (2)

M. G. Löfdahl, "Multi-frame blind deconvolution with linear equality constraints," in Image Reconstruction from Incomplete Data II, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 4792, 146-155 (2002).
[CrossRef]

T. R. Lauer, "Deconvolution with a spatially-variant PSF," in Astronomical Data Analysis II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE 4847, 167-173 (2002).
[CrossRef]

2001 (1)

R. Lupton, J. E. Gunn, Z. Ivezic, and G. R. Knapp, "The SDSS imaging pipelines," in Astronomical Data Analysis Software and Systems X, F. R. Harnden, Jr., F. A. Primini, and H. E. Payne, eds., ASP Conf. Ser. 238, 269-278 (2001).

2000 (1)

C. Alard, "Image subtraction using a space-varying kernel," Astron. Astrophy. Suppl. Ser. 144, 363-370 (2000).
[CrossRef]

1998 (1)

M. G. Löfdahl, T. E. Berger, R. A. Shine, and A. M. Title, "Preparation of a dual wavelength sequence of high-resolution solar photospheric images using phase diversity," Astrophys. J. 495, 965-972 (1998).
[CrossRef]

1997 (1)

J. G. Nagy and D. P. O'Leary, "Fast iterative image restoration with a spatially varying PSF," in Advanced Signal Processing: Algorithms, Architectures, and Implementations VII, F. T. Luk, ed., Proc. SPIE 3162, 388-399 (1997).
[CrossRef]

1996 (2)

M. S. Vogeley and A. S. Szalay, "Eigenmode analysis of galaxy redshift surveys. I. Theory and methods," Astrophys. J. 465, 34-53 (1996).
[CrossRef]

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, and C. U. Keller, "Evaluation of phase-diversity techniques for solar-image restoration," Astrophys. J. 466, 1087-1099 (1996).
[CrossRef]

1994 (3)

M. G. Löfdahl and G. B. Scharmer, "Wavefront sensing and image restoration from focused and defocused solar images," Astron. Astrophy. Suppl. Ser. 107, 243-264 (1994).

R. G. Paxman, B. J. Thelen, and J. H. Seldin, "Correction of anisoplanatic blur by using phase diversity," in Adaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE 2201, 1066-1067 (1994).
[CrossRef]

R. A. Gonsalves, "Nonisoplanatic imaging by phase diversity," Opt. Lett. 19, 493-495 (1994).
[CrossRef] [PubMed]

1993 (1)

O. von der Lühe, "Speckle imaging of solar small structure: I. Methods," Astron. Astrophy. 268, 374-390 (1993).

1992 (1)

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations by using phase diversity," Opt. Soc. Am. A 9, 1072-1085 (1992).
[CrossRef]

1990 (1)

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

1985 (1)

1983 (1)

S. K. Park and R. A. Schowengerdt, "Image reconstruction by parametric cubic convolution," Comput. Vis. Graph. Image Process. 23, 258-272 (1983).
[CrossRef]

1982 (1)

R. A. Gonsalves, "Phase retreival and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

Appl. Opt. (1)

ASP Conf. Ser. (1)

R. Lupton, J. E. Gunn, Z. Ivezic, and G. R. Knapp, "The SDSS imaging pipelines," in Astronomical Data Analysis Software and Systems X, F. R. Harnden, Jr., F. A. Primini, and H. E. Payne, eds., ASP Conf. Ser. 238, 269-278 (2001).

Astron. Astrophy. (1)

O. von der Lühe, "Speckle imaging of solar small structure: I. Methods," Astron. Astrophy. 268, 374-390 (1993).

Astron. Astrophy. Suppl. Ser. (2)

C. Alard, "Image subtraction using a space-varying kernel," Astron. Astrophy. Suppl. Ser. 144, 363-370 (2000).
[CrossRef]

M. G. Löfdahl and G. B. Scharmer, "Wavefront sensing and image restoration from focused and defocused solar images," Astron. Astrophy. Suppl. Ser. 107, 243-264 (1994).

Astrophys. J. (3)

R. G. Paxman, J. H. Seldin, M. G. Löfdahl, G. B. Scharmer, and C. U. Keller, "Evaluation of phase-diversity techniques for solar-image restoration," Astrophys. J. 466, 1087-1099 (1996).
[CrossRef]

M. S. Vogeley and A. S. Szalay, "Eigenmode analysis of galaxy redshift surveys. I. Theory and methods," Astrophys. J. 465, 34-53 (1996).
[CrossRef]

M. G. Löfdahl, T. E. Berger, R. A. Shine, and A. M. Title, "Preparation of a dual wavelength sequence of high-resolution solar photospheric images using phase diversity," Astrophys. J. 495, 965-972 (1998).
[CrossRef]

Comput. Vis. Graph. Image Process. (1)

S. K. Park and R. A. Schowengerdt, "Image reconstruction by parametric cubic convolution," Comput. Vis. Graph. Image Process. 23, 258-272 (1983).
[CrossRef]

Opt. Eng. (2)

R. A. Gonsalves, "Phase retreival and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

Opt. Lett. (1)

Opt. Soc. Am. A (1)

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations by using phase diversity," Opt. Soc. Am. A 9, 1072-1085 (1992).
[CrossRef]

Proc. SPIE (6)

M. G. Löfdahl, "Multi-frame blind deconvolution with linear equality constraints," in Image Reconstruction from Incomplete Data II, P. J. Bones, M. A. Fiddy, and R. P. Millane, eds., Proc. SPIE 4792, 146-155 (2002).
[CrossRef]

M. G. Löfdahl and G. B. Scharmer, "Phase diverse speckle inversion applied to data from the Swedish 1-meter solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE 4853, 567-575 (2003).
[CrossRef]

J. G. Nagy and D. P. O'Leary, "Fast iterative image restoration with a spatially varying PSF," in Advanced Signal Processing: Algorithms, Architectures, and Implementations VII, F. T. Luk, ed., Proc. SPIE 3162, 388-399 (1997).
[CrossRef]

T. R. Lauer, "Deconvolution with a spatially-variant PSF," in Astronomical Data Analysis II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE 4847, 167-173 (2002).
[CrossRef]

R. G. Paxman, B. J. Thelen, and J. H. Seldin, "Correction of anisoplanatic blur by using phase diversity," in Adaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE 2201, 1066-1067 (1994).
[CrossRef]

G. B. Scharmer, K. Bjelksjö, T. K. Korhonen, B. Lindberg, and B. Pettersson, "The 1-meter Swedish solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE 4853, 341-350 (2003).
[CrossRef]

Solar Phy. (1)

M. van Noort, L. Rouppe van der Voort, and M. G. Löfdahl, "Solar image restoration by use of multi-frame blind deconvolution with multiple objects and phase diversity," Solar Phy. 228, 191-215 (2005).
[CrossRef]

Other (4)

R. A. Shine, L. Z. Porter, Z. Frank, J. B. Gurman, D. Pothier, and S. Ferguson, A User's Guide to ANA (Lockheed Palo Alto Research Laboratory, 1988); see also ana.lmsal.com.

Institute for Astronomy of the University of Hawaii, "AR map 2 May, 2003," www.solar.ifa.hawaii.edu/ARMaps/archive.html.

G. B. Scharmer, P. Dettori, M. G. Löfdahl, and M. Shand, "Adaptive optics system for the new Swedish solar telescope," in Innovative Telescopes and Instrumentation for Solar Astrophysics, S. Keil and S. Avakyan, eds., Proc. SPIE4853,370-380 (2003).

M. van Noort and M. Löfdahl, "Multi-object multi-frame blind deconvolution," public domain C++ code available at www.momfbd.org (2005).

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Figures (7)

Fig. 1
Fig. 1

Sunspot image restored using the method described in this paper. PSFs were estimated with phase diversity from five pairs of simultaneous exposures in and out of focus. The image was restored using only the focused data frames along with their PSFs. The tick marks are 1 apart.

Fig. 2
Fig. 2

Blurring and deblurring kernels from one of the focused frames, magnified and mosaiced to show their position in the FOV of Fig. 1. (Left) The estimated PSFs, s j k , displayed with gamma set to emphasize the wings. (Right) The corresponding inverse PSFs, q j k .

Fig. 3
Fig. 3

First 20 K–L kernels, g j m , corresponding to the inverse PSFs of Fig. 2. Top row, left–right: m = 1 , 2, 3, 4. Second row: m = 5 , 6, 7, 8; etc. The kernels are individually scaled to use the available dynamical range.

Fig. 4
Fig. 4

One inverse PSF, q j k , from Fig. 2, bottom right tile, framed. The other tiles show expansions, left–right: M = 1 , 2, 3, 4; etc. Compare Fig. 3.

Fig. 5
Fig. 5

Normalized eigenvalues corresponding to the K–L kernels in Fig. 3. The dotted lines indicate the M = 20 cutoff used for the demonstration.

Fig. 6
Fig. 6

Sample expansion coefficient surface, a j m ( x , y ) , coded as shades of gray and covering the entire FOV of Fig. 1. These maps correspond to the inverse PSFs of Fig. 2 and the fourth K–L kernel of Fig. 3. (Left) Piecewise constant. (Right) Damped bicubic interpolation.

Fig. 7
Fig. 7

Restoration along the interface between adjacent rows of mosaic subfields. The FOV is part of Fig. 1. (1)–(5) Raw data, (a) Restoration using multiframe, space-variant inverse filtering; (b) mosaic of inverse-filtered subfields; (c) single-frame restoration using space-variant inverse filtering; (d) mosaic of single-frame, space-variant inverse filtering using window functions without taper or overlap; (e) difference between (c) and (d), emphasizing the discontinuities between adjacent rows of subfields in the mosaic. Note mismatch both in position and in contrast over the subfield boundary. (There are similar mismatches between columns, although not as noticeable here.) The tick marks are 1 apart.

Equations (14)

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

d j k ( x , y ) = [ f k s j k ] ( x , y ) + n j k ( x , y ) ,
F ^ k = H k j D j k S j k * j | S j k | 2 = j D j k H k S j k * j | S j k | 2 ,
Q j k = H k S j k * j | S j k | 2 ,
F ^ k = j D j k Q j k .
q j k = F 1 { Q j k } ,
f ^ k = j f ^ j k = j d j k q j k .
f ^ j k = d j k q j k = d j k ( m α j m k g j m ) = m ( α j m k α j k ) g j m ,
f ^ j ( x , y ) = m [ a j m ( x , y ) d j ( x , y ) ] g j m ,
f ^ = j m ( a j m d j ) g j m .
C k k = x , y q j k q j k .
C = Q T Q ,
1 = ( Λ 1 / 2 E T Q T ) ( Q E Λ 1 / 2 ) = G T G ,
g j m = λ m m 1 / 2 k E k q j k ,
1 m M [ a j m g j m ] ( x , y ) ; j , x , y .

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