Abstract

A modified method for maximum-likelihood deconvolution of astronomical adaptive optics images is presented. By parametrizing the anisoplanatic character of the point-spread function (PSF), a simultaneous optimization of the spatially variant PSF and the deconvolved image can be performed. In the ideal case of perfect information, it is shown that the algorithm is able to perfectly cancel the adverse effects of anisoplanatism down to the level of numerical precision. Exploring two different modes of deconvolution (using object bases of pixel values or stellar field parameters), we then quantify the performance of the algorithm in the presence of Poissonian noise for crowded and noncrowded stellar fields.

© 2005 Optical Society of America

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References

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  1. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]
  2. R. J. Sasiela, “Strehl ratios with various types of anisoplanatism,” J. Opt. Soc. Am. A 9, 1398–1406 (1992).
    [CrossRef]
  3. E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
    [CrossRef]
  4. T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
    [CrossRef]
  5. T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
    [CrossRef]
  6. T. Lauer, “Deconvolution with a spatially-variant PSF,” in Astronomical Data Analysis II., J.-L. Starck, F. D. Murtagh, eds., Proc. SPIE4847, 167–173 (2002).
    [CrossRef]
  7. C. Alard, R. H. Lupton, “A method for optimal image subtraction,” Astrophys. J. 503, 325 (1998).
    [CrossRef]
  8. W. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  9. L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astrophys. J. 79, 745–754 (1974).
  10. E. Thiébaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
    [CrossRef]
  11. R. G. Lane, “Methods for maximum-likelihood deconvolution,” J. Opt. Soc. Am. A 13, 1992–1998 (1996).
    [CrossRef]
  12. P. Magain, F. Courbin, S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
    [CrossRef]
  13. T. Fusco, J.-P. Véran, J.-M. Conan, L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys., Suppl. Ser. 134, 193–200 (1999).
    [CrossRef]
  14. F. J. Rigaut, J. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
    [CrossRef]
  15. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, New York, 2000).
  16. R. C. Flicker, F. J. Rigaut, “Hokupa’a anisoplanatism study and Mauna Kea turbulence characterization,” Publ. Astron. Soc. Pac. 114, 1006–1015 (2002).
    [CrossRef]
  17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

2002

R. C. Flicker, F. J. Rigaut, “Hokupa’a anisoplanatism study and Mauna Kea turbulence characterization,” Publ. Astron. Soc. Pac. 114, 1006–1015 (2002).
[CrossRef]

2000

T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
[CrossRef]

1999

T. Fusco, J.-P. Véran, J.-M. Conan, L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys., Suppl. Ser. 134, 193–200 (1999).
[CrossRef]

1998

C. Alard, R. H. Lupton, “A method for optimal image subtraction,” Astrophys. J. 503, 325 (1998).
[CrossRef]

P. Magain, F. Courbin, S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

1996

1995

1992

1982

1974

L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astrophys. J. 79, 745–754 (1974).

1972

Alard, C.

C. Alard, R. H. Lupton, “A method for optimal image subtraction,” Astrophys. J. 503, 325 (1998).
[CrossRef]

Bendinelli, O.

E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
[CrossRef]

Bonaccini, D.

E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, New York, 2000).

Chauvin, G.

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Close, L. M.

E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
[CrossRef]

Conan, J.

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Conan, J.-M.

T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
[CrossRef]

T. Fusco, J.-P. Véran, J.-M. Conan, L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys., Suppl. Ser. 134, 193–200 (1999).
[CrossRef]

E. Thiébaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
[CrossRef]

Courbin, F.

P. Magain, F. Courbin, S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

Currie, D. G.

E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
[CrossRef]

Diolati, E.

E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
[CrossRef]

Flannery, B. P.

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

Flicker, R. C.

R. C. Flicker, F. J. Rigaut, “Hokupa’a anisoplanatism study and Mauna Kea turbulence characterization,” Publ. Astron. Soc. Pac. 114, 1006–1015 (2002).
[CrossRef]

Fried, D. L.

Fusco, T.

T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
[CrossRef]

T. Fusco, J.-P. Véran, J.-M. Conan, L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys., Suppl. Ser. 134, 193–200 (1999).
[CrossRef]

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Lagrange, A.

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Lai, O.

F. J. Rigaut, J. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Lane, R. G.

Lauer, T.

T. Lauer, “Deconvolution with a spatially-variant PSF,” in Astronomical Data Analysis II., J.-L. Starck, F. D. Murtagh, eds., Proc. SPIE4847, 167–173 (2002).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astrophys. J. 79, 745–754 (1974).

Lupton, R. H.

C. Alard, R. H. Lupton, “A method for optimal image subtraction,” Astrophys. J. 503, 325 (1998).
[CrossRef]

Magain, P.

P. Magain, F. Courbin, S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

Marchis, F.

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Michau, V.

T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
[CrossRef]

Mouillet, D.

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Mugnier, L. M.

T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
[CrossRef]

T. Fusco, J.-P. Véran, J.-M. Conan, L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys., Suppl. Ser. 134, 193–200 (1999).
[CrossRef]

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Parmeggiani, G.

E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
[CrossRef]

Press, W. H.

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

Richardson, W.

Rigaut, F. J.

R. C. Flicker, F. J. Rigaut, “Hokupa’a anisoplanatism study and Mauna Kea turbulence characterization,” Publ. Astron. Soc. Pac. 114, 1006–1015 (2002).
[CrossRef]

F. J. Rigaut, J. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Roddier, F. J.

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Rousset, G.

T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
[CrossRef]

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

Sasiela, R. J.

Sohy, S.

P. Magain, F. Courbin, S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

Teukolsky, S. A.

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

Thiébaut, E.

Veran, J.

F. J. Rigaut, J. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

Véran, J.-P.

T. Fusco, J.-P. Véran, J.-M. Conan, L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys., Suppl. Ser. 134, 193–200 (1999).
[CrossRef]

Vetterling, W. T.

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

Astron. Astrophys., Suppl. Ser.

T. Fusco, J.-M. Conan, L. M. Mugnier, V. Michau, G. Rousset, “Characterization of adaptive optics point spread function for anisoplanatic imaging. Application to stellar field deconvolution,” Astron. Astrophys., Suppl. Ser. 142, 149–156 (2000).
[CrossRef]

T. Fusco, J.-P. Véran, J.-M. Conan, L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys., Suppl. Ser. 134, 193–200 (1999).
[CrossRef]

Astrophys. J.

P. Magain, F. Courbin, S. Sohy, “Deconvolution with correct sampling,” Astrophys. J. 494, 472–477 (1998).
[CrossRef]

C. Alard, R. H. Lupton, “A method for optimal image subtraction,” Astrophys. J. 503, 325 (1998).
[CrossRef]

L. B. Lucy, “An iterative technique for rectification of observed distributions,” Astrophys. J. 79, 745–754 (1974).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Publ. Astron. Soc. Pac.

R. C. Flicker, F. J. Rigaut, “Hokupa’a anisoplanatism study and Mauna Kea turbulence characterization,” Publ. Astron. Soc. Pac. 114, 1006–1015 (2002).
[CrossRef]

Other

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

F. J. Rigaut, J. Veran, O. Lai, “Analytical model for Shack–Hartmann-based adaptive optics systems,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 1038–1048 (1998).
[CrossRef]

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, New York, 2000).

E. Diolati, O. Bendinelli, D. Bonaccini, L. M. Close, D. G. Currie, G. Parmeggiani, “Starfinder: an IDL GUI-based code to analyze crowded fields with isoplanatic correcting PSF fitting,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 879–887 (2000).
[CrossRef]

T. Fusco, L. M. Mugnier, J. Conan, F. Marchis, G. Chauvin, G. Rousset, A. Lagrange, D. Mouillet, F. J. Roddier, “Deconvolution of astronomical images obtained from ground-based telescopes with adaptive optics,” in Adaptive Optical System Technologies II, P. L. Wizinowich, D. Bonaccini, eds., Proc. SPIE4839, 1065–1075 (2003).
[CrossRef]

T. Lauer, “Deconvolution with a spatially-variant PSF,” in Astronomical Data Analysis II., J.-L. Starck, F. D. Murtagh, eds., Proc. SPIE4847, 167–173 (2002).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

First nine modes p i (top) and amplitude fields a i (bottom) from a sample KL decomposition. The guide star is in the lower left corner of the field (0, 0).

Fig. 2
Fig. 2

Convergence of the parametrized stellar field deconvolution in the ideal case (lightface) and with PSF fitting errors included (bold). The quantities plotted are the cost function ℳ (solid curves), the normalized image error E (dashed curves), the α error (long dashed curves), the field-averaged photon count E ϕ (dotted curves), and position E x (dotted–dashed curves) errors reported in photons and pixels on the common ordinate.

Fig. 3
Fig. 3

Simulation of 200 images with independent realizations of photon noise and star magnitudes, deconvolved with the parametrized stellar field model. Curves indicate the median fit and the one-sigma deviations as computed for the data (solid curves with pluses) and as predicted by a theoretical model (solid curves) for a Strehl ratio of 0.2.

Fig. 4
Fig. 4

Data from Fig. 3 seperated into domains of the Strehl ratio. Curves indicate the median fit and the one-sigma deviations as computed for the data (solid curve with pluses) and as predicted by a theoretical model (solid curve) for the mean Strehl ratio of each bin. Note the varying scale of the ordinates.

Fig. 5
Fig. 5

Simulation of 200 images with independent realizations of photon noise and star magnitudes, deconvolved with the pixelized method. Curves indicate the median fit and the one-sigma deviations as computed for the data (solid curve with pluses) and as predicted by a theoretical model (solid curve) for a Strehl ratio of 0.2.

Fig. 6
Fig. 6

Data from Fig. 5 seperated into domains of the Strehl ratio. Curves indicate the median fit and the one-sigma deviations as computed for the data (solid curve with pluses) and as predicted by a theoretical model (solid curve) for the mean Strehl ratio of each bin. Note the varying scale of the ordinates.

Fig. 7
Fig. 7

Pixelized deconvolution (right) of a simulated 512 × 512   K-band image (left). The field of view is 10 arcsec for the full image (bottom) and 2.2 arcsec for the closeups of the top panel. The image stretch is logarithmic.

Fig. 8
Fig. 8

Cross sections of the stars indicated in Fig. 7, where pluses show the original profile and diamonds show the profile of the deconvolved star. In the upper panel the cut is along the x axis, and in the lower panel are shown the cut along the guide star vector (e.g., the radial direction) (solid curves) and the orthogonal (azimuthal) direction (dashed curves).

Fig. 9
Fig. 9

Magnitude (upper panel) and position (lower panel) errors (in arc seconds) for the parametrized deconvolution of the K-band image in Fig. 7.

Equations (26)

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

d ( x ) = d x h ( x ,   x ) o ( x ) + n ( x ) ,
h ( x ,   x ) = i = 0 N - 1 a i ( x ) p i ( x - x ) ,
d ( x ) = i = 0 N - 1 d x a i ( x ) o ( x ) p i ( x - x ) + n ( x )
= i = 0 N - 1 [ ( a i o ) * p i ] ( x ) + n ( x ) ,
Φ aniso ( κ ,   θ ) = Π ( κ ) × 2 Φ ( κ ) l = 0 N l - 1 f l [ 1 - cos ( 2 π h l θ     κ ) ] ,
o ( x ) = g ( x ) + k = 0 N s - 1 ϕ k δ ( x - x k ) ,
o * = arg   min o ˆ   M ( o ˆ )
P ( d | o ˆ ) = i = 0 N - 1   d ˆ ( x i ) d ( x i ) exp [ - d ˆ ( x i ) ] d ( x i ) ! ,
M ( o ˆ ) = - ln P ( d | o ˆ ) = i = 0 N - 1 [ d ˆ ( x i ) - d ( x i ) ln d ˆ ( x i ) ] ,
δ M ( o ˆ ) = i = 0 m - 1 a i ( α x ) ( ˆ * p i ) ( x ) ,
M α = d x o ˆ ( x ) i = 0 m - 1 [ x     a i ( α x ) ] ( ˆ * p i ) ( x ) ,
M α = k = 0 N s - 1 ϕ k i = 0 m - 1 [ x k     a i ( α x k ) ] ( ˆ * p i ) ( x k ) .
k M = 2 α ϕ k i = 0 m - 1 a i ( α x k ) ( ˆ * p i ) ( x k )
+ 2 ϕ k i = 0 m - 1 a i ( α x k ) ( ˆ * p i ) ( x k ) .
FT ( ˆ * p i ) = 2 π u - 1 × FT ( ˆ ) × FT ( p i ) .
M c q = d x ˆ ( x ) ( h     x i y j ) ( x ) .
M ( o ˆ ) = 1 2 σ 2   i = 0 N - 1 [ d ˆ ( x i ) - d ( x i ) ] 2 ,
E = d x | d ( x ) - d ˆ ( x ) | 2 d x | d ( x ) | 2 ,
E ϕ = 1 N s   k = 0 N s - 1 | ϕ k - ϕ ˆ k | , E x = 1 N s   k = 0 N s - 1 | x k - x ˆ k | .
δ M ( o ˆ ;   v ) = lim λ 0   M ( o ˆ + λ v ) - M ( o ˆ ) λ ,
δ M ( o ˆ ;   v ) = lim λ 0   1 λ   i = 0 N - 1 [ h     ( o ˆ + λ v ) ] ( x i ) - lim λ 0   1 λ   i = 0 N - 1 d ( x i ) ln { [ h     ( o ˆ + λ v ) ] ( x i ) } - lim λ 0   1 λ   i = 0 N - 1 { ( h     o ˆ ) ( x i ) - d ( x i ) ln [ ( h     o ˆ ) ( x i ) ] } = i = 0 N - 1 ( h     v ) ( x i ) - i = 0 N - 1 d ( x i ) lim λ 0 1 λ ln 1 + λ   ( h     v ) ( x i ) ( h     o ˆ ) ( x i ) .
lim λ 0   a ( λ ) b ( λ ) = lim λ 0   a ( λ ) b ( λ ) = ( h     v ) ( x i ) ( h     o ˆ ) ( x i ) .
δ M ( o ˆ ;   v ) = i = 0 N - 1 ( h     v ) ( x i ) - d ( x i )   ( h     v ) ( x i ) ( h     o ˆ ) ( x i )
= i = 0 N - 1 ˆ ( x i ) ( h     v ) ( x i ) ,
h     δ ( x - y ) = i = 0 m - 1 d x a i ( α x ) δ ( x - y ) p i ( x - x ) = i = 0 m - 1 a i ( α y ) p i ( x - y ) .
δ M ( o ˆ ) = i = 0 m - 1 a i ( α y ) ( ˆ     *     p i ) ( y ) .

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