Abstract

We show that x-ray computer tomography algorithms can be applied with minimal alteration to the three-dimensional reconstruction of visible sources. Diffraction and opacity affect visible systems more severely than x-ray systems. For camera-based tomography, diffraction can be neglected for objects within the depth of field. We show that, for convex objects, opacity has the effect of windowing the angular observation range and thus blurring the reconstruction. For concave objects, opacity leads to nonlinearity in the transformation from object to reconstruction and may cause multiple objects to map to the same reconstruction. In x-ray tomography, the contribution of an object point to a line integral is independent of the orientation of the line. In optical tomography, however, a Lambertian assumption may be more realistic. We derive an expression for the blur function (the patch response) for a Lambertian source. We present experimental results showing cone-beam reconstruction of an incoherently illuminated opaque object.

© 2001 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  6. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  7. J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence tomography in scattering media,” Opt. Lett. 19, 590–592 (1994).
    [CrossRef] [PubMed]
  8. B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency modulation/continuous wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
    [CrossRef]
  9. D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  21. A. J. Johnson, D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, “Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography,” in Proceedings of the IEEE Conference on image processing (Institute for Electrical and Electronics Engineers, New York, 1999), pp. 663–667.

1999 (2)

D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[CrossRef] [PubMed]

D. L. Marks, R. A. Stack, D. J. Brady, J. van der Gracht, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. 24, 253–255 (1999).
[CrossRef]

1998 (1)

1997 (1)

1996 (3)

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency modulation/continuous wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

J. Rosen, A. Yariv, “Reconstruction of longitudinal distributed incoherent sources,” Opt. Lett. 21, 1803–1806 (1996).
[CrossRef] [PubMed]

S. K. Nayar, M. Watanabe, M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1186–1198 (1996).
[CrossRef]

1995 (1)

1994 (3)

1993 (1)

N. Ahuja, A. L. Abbott, “Active stereo: integrating disparity, vergence, focus, aperture, and calibration for surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1007–1029 (1993).
[CrossRef]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1984 (1)

1983 (1)

H. K. Tuy, “An inversion formula for cone-beam tomography,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

1981 (1)

P. A. Rattey, A. G. Lindgren, “Sampling the 2-D Radon transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 994–1002 (1981).
[CrossRef]

Abbott, A. L.

N. Ahuja, A. L. Abbott, “Active stereo: integrating disparity, vergence, focus, aperture, and calibration for surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1007–1029 (1993).
[CrossRef]

Ahuja, N.

N. Ahuja, A. L. Abbott, “Active stereo: integrating disparity, vergence, focus, aperture, and calibration for surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1007–1029 (1993).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1980).

Bradburn, S.

Brady, D. J.

D. L. Marks, R. A. Stack, D. J. Brady, J. van der Gracht, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. 24, 253–255 (1999).
[CrossRef]

D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[CrossRef] [PubMed]

D. I. Marks, D. J. Brady, “Three-dimensional source reconstruction with a scanned pinhole camera,” Opt. Lett. 23, 820–822 (1998).
[CrossRef]

A. J. Johnson, D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, “Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography,” in Proceedings of the IEEE Conference on image processing (Institute for Electrical and Electronics Engineers, New York, 1999), pp. 663–667.

Brady, R. B.

D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[CrossRef] [PubMed]

Cathey, W. T.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Darrell, T.

Davis, L. C.

Dowski, E. R.

Feldkamp, L. A.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence tomography in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Girod, B.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence tomography in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Izatt, J. A.

Johnson, A. J.

A. J. Johnson, “Patch response of cone-beam tomography,” M. S. thesis (University of Illinois at Urbana-Champaign, Urbana, Illinois, 1999).

A. J. Johnson, D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, “Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography,” in Proceedings of the IEEE Conference on image processing (Institute for Electrical and Electronics Engineers, New York, 1999), pp. 663–667.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Kress, J. W.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Lindgren, A. G.

P. A. Rattey, A. G. Lindgren, “Sampling the 2-D Radon transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 994–1002 (1981).
[CrossRef]

Marks, D. I.

Marks, D. L.

D. L. Marks, R. A. Stack, D. J. Brady, J. van der Gracht, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. 24, 253–255 (1999).
[CrossRef]

D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[CrossRef] [PubMed]

A. J. Johnson, D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, “Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography,” in Proceedings of the IEEE Conference on image processing (Institute for Electrical and Electronics Engineers, New York, 1999), pp. 663–667.

Munson, D.

D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[CrossRef] [PubMed]

Munson, D. C.

A. J. Johnson, D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, “Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography,” in Proceedings of the IEEE Conference on image processing (Institute for Electrical and Electronics Engineers, New York, 1999), pp. 663–667.

Nakagama, Y.

S. K. Nayar, Y. Nakagama, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

Nayar, S. K.

S. K. Nayar, M. Watanabe, M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1186–1198 (1996).
[CrossRef]

S. K. Nayar, Y. Nakagama, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

Noguchi, M.

S. K. Nayar, M. Watanabe, M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1186–1198 (1996).
[CrossRef]

Owen, G. M.

Pentland, A.

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Rattey, P. A.

P. A. Rattey, A. G. Lindgren, “Sampling the 2-D Radon transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 994–1002 (1981).
[CrossRef]

Rosen, J.

Ruff, W. C.

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency modulation/continuous wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Scherock, S.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Stack, R. A.

D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[CrossRef] [PubMed]

D. L. Marks, R. A. Stack, D. J. Brady, J. van der Gracht, “Three-dimensional tomography using a cubic-phase plate extended depth-of-field system,” Opt. Lett. 24, 253–255 (1999).
[CrossRef]

A. J. Johnson, D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, “Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography,” in Proceedings of the IEEE Conference on image processing (Institute for Electrical and Electronics Engineers, New York, 1999), pp. 663–667.

Stann, B. L.

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency modulation/continuous wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, J. G. Fujimoto, “Optical coherence tomography in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sztankay, Z. G.

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency modulation/continuous wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Tuy, H. K.

H. K. Tuy, “An inversion formula for cone-beam tomography,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

van der Gracht, J.

Watanabe, M.

S. K. Nayar, M. Watanabe, M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1186–1198 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1980).

Yariv, A.

Appl. Opt. (2)

IEEE Trans. Acoust. Speech Signal Process. (1)

P. A. Rattey, A. G. Lindgren, “Sampling the 2-D Radon transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 994–1002 (1981).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

N. Ahuja, A. L. Abbott, “Active stereo: integrating disparity, vergence, focus, aperture, and calibration for surface estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 1007–1029 (1993).
[CrossRef]

S. K. Nayar, Y. Nakagama, “Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 824–831 (1994).
[CrossRef]

S. K. Nayar, M. Watanabe, M. Noguchi, “Real-time focus range sensor,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1186–1198 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

B. L. Stann, W. C. Ruff, Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency modulation/continuous wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Opt. Lett. (4)

Science (2)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

D. L. Marks, R. A. Stack, D. J. Brady, D. Munson, R. B. Brady, “Visible cone-beam tomography with a lensless interferometric camera,” Science 284, 2164–2166 (1999).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

H. K. Tuy, “An inversion formula for cone-beam tomography,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

Other (5)

A. J. Johnson, “Patch response of cone-beam tomography,” M. S. thesis (University of Illinois at Urbana-Champaign, Urbana, Illinois, 1999).

A. J. Johnson, D. L. Marks, R. A. Stack, D. J. Brady, D. C. Munson, “Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography,” in Proceedings of the IEEE Conference on image processing (Institute for Electrical and Electronics Engineers, New York, 1999), pp. 663–667.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1980).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

T. Wilson, ed., Confocal Microscopy (Academic, San Diego, Calif., 1990).

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Figures (12)

Fig. 1
Fig. 1

Optical imaging in cone-beam tomography.

Fig. 2
Fig. 2

Coordinate system that defines the projections. t is the vertex point for a family of projections and β is a direction vector along a particular projection.

Fig. 3
Fig. 3

General stigmatic optical system.

Fig. 4
Fig. 4

Illustration of coordinate systems used on midplane in Feldkamp’s algorithm. and are the dimensions in the midplane, with pointing at the vertex point and pointing along the midplane; ŷ points out of the midplane; Y and Z are the coordinates on the plane of rotation; d is the radius of the vertex circle; ϕ is the angle from the x axis to the vector from the origin to the vertex point; and θ is the angle between the x axis and the axis.

Fig. 5
Fig. 5

Coordinate system used for the projection-slice theorem 2-D integral. 2l is the length of the Lambertian line, r and θ are the polar coordinates of the 2-D density reconstruction, and ϕ is the angle of the projections of the Lambertian line.

Fig. 6
Fig. 6

Density plot of a 2-D patch response function.

Fig. 7
Fig. 7

Three constant-brightness Lambertian objects that have the same silhouettes and therefore the same reconstructed power density.

Fig. 8
Fig. 8

Density plot of the patch response with various sampling densities.

Fig. 9
Fig. 9

2-D patch response with various projection angles.

Fig. 10
Fig. 10

Setup of cone-beam data acquisition.

Fig. 11
Fig. 11

Toy bear reconstructed by Feldkamp’s algorithm: (a) and (b) ray-cast renderings of the side and front views of the power density of the bear, respectively; (c) a lateral cross section through the bear’s head.

Fig. 12
Fig. 12

Laplacian-filtered reconstructed bear: (a) and (b) ray-cast renderings of the side and front views of the power density of the bear, respectively; (c) a lateral cross section through the bear’s head.

Equations (22)

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

pt, β=1|β|  Dt+αβdα,
p0, r=0Dκzρx, κzρy, zdz,
ψr=expjπ|r|2λz  expjπ|r|2λz×ψDrhr+zzr, zdr,
hr, z= trexpj2πλzr·r×expjπλ1z+1z-1f|r|2dr.
Pr= Drhr+zzr, z2dr,
A2λ1z+1z-1f<1,
gyξy=-ωy0ωy0dω|ω|expiωξy-2|ω|/ωy0, gzξz=sinξzωz0πξz.
P¯ϕξy, ξz=-dξy -dξzgyξy-ξygzξz-ξz×Pϕξy, ξz1+ξy2+ξz2-1/2.
DEr=14π2  d2d+r·xˆ2 P¯ϕr·yˆd+r·xˆ, r·zˆd+r·xˆdϕ,
PRFx, y=14π22xx2+y23/2logx2+y21/2+xx2+y21/2-x-4x2+y2,
fr, θ=002π s expi2πrs cosθ-ϕ-ks×Ps, ϕdϕds,
pl, θ=rectl-lϕ/2/lϕ]+rectl+lϕ+π/2/lϕ+π,
Ps, ϕ=sin πslϕπsexp-iπslϕ+sin πslϕ+ππs×expiπslϕ+π.
fr, θ=002π s expi2πrs cosθ-ϕ-ks×sin πslϕπsexp-iπslϕ+sin πslϕ+ππsexpiπslϕ+πdϕds.
fr, θ=0-ππexpi2πrs cosθ-ϕ-ksiπ×1-expi2πslϕdϕds.
fr, θ=-ππ2π2r cosθ-ϕ+iπk-1-2π2r cosθ-ϕ-lϕ+iπk-1dϕ.
fr, θ=-ππ2π2lθ-r cosθ-ϕ-iπk-1dϕ,
fr, θ; l=-π/2π/22π2l cos ϕ-r cosθ-ϕ-iπk-1dϕ+π/23π/22π2-l cos ϕ-r cosθ-ϕ-iπk-1dϕ.
fx, y; l=-π/2π/2ikπ-2π2l+xcos ϕ+y sin ϕ-1-ikπ+2π2l+xcos ϕ+y sin ϕ-1dϕ.
fx, y=-π/2π/2k-2πix cos ϕ-2πiy sin ϕ-2+k+2πix cos ϕ+2πiy sin ϕ-2cos ϕdϕ.
fx, y=4k2+4π2x2+y21/2+2πx logk2+4π2x2+y21/2-2πx-y2+k2k2+4π2x2+y21/2+2πx-y2+k2+2πx logk2+4π2x2+y21/2+2πx+y2+k2k2+4π2x2+y21/2-2πx+y2+k2k2+4π2x2+y2-3/2.
PRFx, y=14π22xx2+y23/2logx2+y21/2+xx2+y21/2-x-4x2+y2.

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