Abstract

Light reflection from a surface is described by the bidirectional reflectance distribution function (BRDF). In this paper, BRDF effects in reflection tomography are studied using modeled range-resolved reflection from well-characterized geometrical surfaces. It is demonstrated that BRDF effects can cause a darkening at the interior boundary of the reconstructed surface analogous to the well-known beam hardening artifact in x-ray transmission computed tomography (CT). This artifact arises from reduced reflection at glancing incidence angles to the surface. It is shown that a purely Lambertian surface without shadowed components is perfectly reconstructed from range-resolved measurements. This result is relevant to newly fabricated carbon nanotube materials. Shadowing is shown to cause crossed streak artifacts similar to limited-angle effects in CT reconstruction. In tomographic reconstruction, these effects can overwhelm highly diffuse components in proximity to specularly reflecting elements. Diffuse components can be recovered by specialized processing, such as reducing glints via thresholded measurements.

© 2009 Optical Society of America

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2008 (2)

Z.-P. Yang, L. Ci, J. A. Bur, S.-Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8, 446-451 (2008).
[CrossRef]

C. Srinivasan, “The blackest black material from carbon nanotubes,” Curr. Sci. 94, 974-975 (2008).

2006 (1)

Z. Li, G. Hewei, L. Shuanglei, C. Zhiqiang, and X. Yuxiang, “Cupping artifacts analysis and correction for a FPD-based cone-beam CT, Proc. SPIE 6065, 60650K (2006).
[CrossRef]

2003 (1)

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

2002 (1)

2001 (2)

R. M. Watson and P. N. Raven, “Comparison of Measured BRDF Data with Parameterized Reflectance Models,” Proc. SPIE 4370, 159-168 (2001).
[CrossRef]

S. A. Hanes, V. N. Benham, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micronreflection tomography imaging system,” Proc. SPIE 4167, 230-241 (2001).
[CrossRef]

1998 (1)

1997 (1)

1995 (1)

C. L. Matson, “Tomographic satellite image reconstruction using ladar E-field or intensity projections: computer simulation results,” Proc. SPIE 2566, 166-176 (1995).
[CrossRef]

1991 (1)

F. K. Knight, D. L. Klick, D. P. Ryan-Howard, and J. R. Theriault, “Visible laser radar: range tomography and angle-angle-range,” Opt. Eng. 30, 55-65 (1991).
[CrossRef]

1989 (1)

F. K. Knight, S. R. Kulkarni, R. M. Marino, and J. K. Parker, “Tomographic techniques applied to laser radar reflective measurements,” Lincoln Lab. J. 2, 143-159 (1989).

1988 (3)

1985 (1)

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252-255(1985).
[CrossRef]

1981 (1)

M. E. Davison and F. A. Grunbaum, “Tomographic reconstruction with arbitrary directions,” Commun. Pure Appl. Math. 34, 77-120 (1981).
[CrossRef]

1980 (2)

A. K. Louis, “Picture reconstruction from projections in restricted range,” Math. Meth. Appl. Sci. 2, 209-220 (1980).
[CrossRef]

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl. Sci. 27, 445-458 (1980).
[CrossRef]

1979 (1)

G. T. Herman, “Correction for beam hardening in computed tomography,” Phys. Med. Biol. 24, 81-106 (1979).
[CrossRef]

1978 (1)

P. K. Kijewski and B. E. Bjarngard, “Correction for beam hardening in computed tomography,” Med. Phys. 5, 209-214(1978).
[CrossRef]

1976 (2)

R. E. Alverez and A. Macovski, “Energy-selective reconstructions in x-ray computerized tomography,” Phys. Med. Biol. 21, 733-744 (1976).
[CrossRef]

R. A. Brooks and G. D. Chiro, “Beam hardening in X-ray reconstruction tomography,” Phys. Med. Biol. 21, 390-398(1976).
[CrossRef]

1975 (1)

1974 (1)

L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. 21, 228-236 (1974).
[CrossRef]

1971 (1)

G. N. Ramachandran and A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236-2240 (1971).
[CrossRef]

1965 (1)

Adams, J. S.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Ajayan, P. M.

Z.-P. Yang, L. Ci, J. A. Bur, S.-Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8, 446-451 (2008).
[CrossRef]

Alverez, R. E.

R. E. Alverez and A. Macovski, “Energy-selective reconstructions in x-ray computerized tomography,” Phys. Med. Biol. 21, 733-744 (1976).
[CrossRef]

Benham, V. N.

S. A. Hanes, V. N. Benham, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micronreflection tomography imaging system,” Proc. SPIE 4167, 230-241 (2001).
[CrossRef]

Bjarngard, B. E.

P. K. Kijewski and B. E. Bjarngard, “Correction for beam hardening in computed tomography,” Med. Phys. 5, 209-214(1978).
[CrossRef]

Boger, J.

Brooks, R. A.

R. A. Brooks and G. D. Chiro, “Beam hardening in X-ray reconstruction tomography,” Phys. Med. Biol. 21, 390-398(1976).
[CrossRef]

Bur, J. A.

Z.-P. Yang, L. Ci, J. A. Bur, S.-Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8, 446-451 (2008).
[CrossRef]

Chiro, G. D.

R. A. Brooks and G. D. Chiro, “Beam hardening in X-ray reconstruction tomography,” Phys. Med. Biol. 21, 390-398(1976).
[CrossRef]

Ci, L.

Z.-P. Yang, L. Ci, J. A. Bur, S.-Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8, 446-451 (2008).
[CrossRef]

Craig, E. B.

Davis, W. R.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Davison, M. E.

M. E. Davison and F. A. Grunbaum, “Tomographic reconstruction with arbitrary directions,” Commun. Pure Appl. Math. 34, 77-120 (1981).
[CrossRef]

Forman, S. E.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Gouldin, F. C.

Grunbaum, F. A.

M. E. Davison and F. A. Grunbaum, “Tomographic reconstruction with arbitrary directions,” Commun. Pure Appl. Math. 34, 77-120 (1981).
[CrossRef]

Hanes, S. A.

S. A. Hanes, V. N. Benham, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micronreflection tomography imaging system,” Proc. SPIE 4167, 230-241 (2001).
[CrossRef]

Hatch, R. E.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Herman, G. T.

G. T. Herman, “Correction for beam hardening in computed tomography,” Phys. Med. Biol. 24, 81-106 (1979).
[CrossRef]

Hewei, G.

Z. Li, G. Hewei, L. Shuanglei, C. Zhiqiang, and X. Yuxiang, “Cupping artifacts analysis and correction for a FPD-based cone-beam CT, Proc. SPIE 6065, 60650K (2006).
[CrossRef]

Kachelmyer, A. L.

A. L. Kachelmyer, “Range-Doppler imaging: waveforms and receiver design,” Proc. SPIE 999, 138-161 (1988).

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Kijewski, P. K.

P. K. Kijewski and B. E. Bjarngard, “Correction for beam hardening in computed tomography,” Med. Phys. 5, 209-214(1978).
[CrossRef]

Klick, D. I.

Klick, D. L.

F. K. Knight, D. L. Klick, D. P. Ryan-Howard, and J. R. Theriault, “Visible laser radar: range tomography and angle-angle-range,” Opt. Eng. 30, 55-65 (1991).
[CrossRef]

Knight, F. K.

F. K. Knight, D. L. Klick, D. P. Ryan-Howard, and J. R. Theriault, “Visible laser radar: range tomography and angle-angle-range,” Opt. Eng. 30, 55-65 (1991).
[CrossRef]

F. K. Knight, S. R. Kulkarni, R. M. Marino, and J. K. Parker, “Tomographic techniques applied to laser radar reflective measurements,” Lincoln Lab. J. 2, 143-159 (1989).

J. K. Parker, E. B. Craig, D. I. Klick, F. K. Knight, S. R. Kulkarni, R. M. Marino, J. R. Senning, and B. K. Tussey, “Reflective tomography: images from range-resolved laser radar measurements,” Appl. Opt. 27, 2642-2643 (1988).
[CrossRef]

Knowlton, R. C.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Koenderink, J. J.

Kulkarni, S. R.

F. K. Knight, S. R. Kulkarni, R. M. Marino, and J. K. Parker, “Tomographic techniques applied to laser radar reflective measurements,” Lincoln Lab. J. 2, 143-159 (1989).

J. K. Parker, E. B. Craig, D. I. Klick, F. K. Knight, S. R. Kulkarni, R. M. Marino, J. R. Senning, and B. K. Tussey, “Reflective tomography: images from range-resolved laser radar measurements,” Appl. Opt. 27, 2642-2643 (1988).
[CrossRef]

Lakshminarayanan, A. V.

G. N. Ramachandran and A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236-2240 (1971).
[CrossRef]

Lasche, J. B.

S. A. Hanes, V. N. Benham, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micronreflection tomography imaging system,” Proc. SPIE 4167, 230-241 (2001).
[CrossRef]

Li, Z.

Z. Li, G. Hewei, L. Shuanglei, C. Zhiqiang, and X. Yuxiang, “Cupping artifacts analysis and correction for a FPD-based cone-beam CT, Proc. SPIE 6065, 60650K (2006).
[CrossRef]

Lin, S.-Y.

Z.-P. Yang, L. Ci, J. A. Bur, S.-Y. Lin, and P. M. Ajayan, “Experimental observation of an extremely dark material made by a low-density nanotube array,” Nano Lett. 8, 446-451 (2008).
[CrossRef]

Logan, B. F.

L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. 21, 228-236 (1974).
[CrossRef]

Louis, A. K.

A. K. Louis, “Picture reconstruction from projections in restricted range,” Math. Meth. Appl. Sci. 2, 209-220 (1980).
[CrossRef]

Macdonald, B.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl. Sci. 27, 445-458 (1980).
[CrossRef]

Macovski, A.

R. E. Alverez and A. Macovski, “Energy-selective reconstructions in x-ray computerized tomography,” Phys. Med. Biol. 21, 733-744 (1976).
[CrossRef]

Marino, R. M.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

F. K. Knight, S. R. Kulkarni, R. M. Marino, and J. K. Parker, “Tomographic techniques applied to laser radar reflective measurements,” Lincoln Lab. J. 2, 143-159 (1989).

J. K. Parker, E. B. Craig, D. I. Klick, F. K. Knight, S. R. Kulkarni, R. M. Marino, J. R. Senning, and B. K. Tussey, “Reflective tomography: images from range-resolved laser radar measurements,” Appl. Opt. 27, 2642-2643 (1988).
[CrossRef]

Matson, C. L.

C. L. Matson and J. Boger, “Laboratory validation of range-resolved reflective tomography signal-to-noise expressions,” Appl. Opt. 36, 3165-3173 (1997).
[CrossRef]

C. L. Matson, “Tomographic satellite image reconstruction using ladar E-field or intensity projections: computer simulation results,” Proc. SPIE 2566, 166-176 (1995).
[CrossRef]

McLaughlin, J. L.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Mooney, J. G.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986), Chap. 6, pp. 160-166.

Nicodemus, F. E.

O'Brian, M. E.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Parker, J. K.

F. K. Knight, S. R. Kulkarni, R. M. Marino, and J. K. Parker, “Tomographic techniques applied to laser radar reflective measurements,” Lincoln Lab. J. 2, 143-159 (1989).

J. K. Parker, E. B. Craig, D. I. Klick, F. K. Knight, S. R. Kulkarni, R. M. Marino, J. R. Senning, and B. K. Tussey, “Reflective tomography: images from range-resolved laser radar measurements,” Appl. Opt. 27, 2642-2643 (1988).
[CrossRef]

Perez-Mendez, V.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl. Sci. 27, 445-458 (1980).
[CrossRef]

Ramachandran, G. N.

G. N. Ramachandran and A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236-2240 (1971).
[CrossRef]

Raven, P. N.

R. M. Watson and P. N. Raven, “Comparison of Measured BRDF Data with Parameterized Reflectance Models,” Proc. SPIE 4370, 159-168 (2001).
[CrossRef]

Ravichandran, M.

Reitz, K. P.

Robertson, D. C.

B. P. Sandford and D. C. Robertson, Infrared reflectance properties of aircraft paints, U. S. Air Force Research Laboratory PreprintAFRL/VSBT ESC-94-1004, approved for public release August 1994.

Rowe, G. S.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Rowland, K. B.

S. A. Hanes, V. N. Benham, J. B. Lasche, and K. B. Rowland, “Field demonstration and characterization of a 10.6 micronreflection tomography imaging system,” Proc. SPIE 4167, 230-241 (2001).
[CrossRef]

Ryan-Howard, D. P.

F. K. Knight, D. L. Klick, D. P. Ryan-Howard, and J. R. Theriault, “Visible laser radar: range tomography and angle-angle-range,” Opt. Eng. 30, 55-65 (1991).
[CrossRef]

Sandford, B. P.

B. P. Sandford and D. C. Robertson, Infrared reflectance properties of aircraft paints, U. S. Air Force Research Laboratory PreprintAFRL/VSBT ESC-94-1004, approved for public release August 1994.

Senning, J. R.

Shepp, L. A.

L. A. Shepp and B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. 21, 228-236 (1974).
[CrossRef]

Shuanglei, L.

Z. Li, G. Hewei, L. Shuanglei, C. Zhiqiang, and X. Yuxiang, “Cupping artifacts analysis and correction for a FPD-based cone-beam CT, Proc. SPIE 6065, 60650K (2006).
[CrossRef]

Siddon, R. L.

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252-255(1985).
[CrossRef]

Skelly, L.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Snyder, W. C.

Srinivasan, C.

C. Srinivasan, “The blackest black material from carbon nanotubes,” Curr. Sci. 94, 974-975 (2008).

Stephens, T.

R. M. Marino, T. Stephens, R. E. Hatch, J. L. McLaughlin, J. G. Mooney, M. E. O'Brian, G. S. Rowe, J. S. Adams, L. Skelly, R. C. Knowlton, S. E. Forman, and W. R. Davis, “A compact 3D imaging laser radar system using geiger-mode APD arrays: systems and measurements,” Proc. SPIE 5086, 1-15 (2003).
[CrossRef]

Tam, K. C.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Limited angle 3-D reconstructions from continuous and pinhole projections,” IEEE Trans. Nucl. Sci. 27, 445-458 (1980).
[CrossRef]

Theriault, J. R.

F. K. Knight, D. L. Klick, D. P. Ryan-Howard, and J. R. Theriault, “Visible laser radar: range tomography and angle-angle-range,” Opt. Eng. 30, 55-65 (1991).
[CrossRef]

Trowbridge, T. S.

Tussey, B. K.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry for ladar-based ranging and surface reconstruction.

Fig. 2
Fig. 2

Reconstructed region: area between surface #1 and surface #2.

Fig. 3
Fig. 3

Ladar-based radiance bins for an elliptical phantom. Location of radiance asymmetry indicated.

Fig. 4
Fig. 4

Ladar-based radiance bins for adjacent disks phantom. Location of radiance gap indicated.

Fig. 5
Fig. 5

Ladar range-resolved radiance for a two-dimensional disk of radius A with A / 256 bin resolution, and grazing surface reflectivities b = 0 (solid curve) and b = 0.99 (dotted curve) in the diffuse SR BRDF model in Eq. (A1). Inset: disk illumination geometry. Note that the lower half of the disk is shadowed by the upper half.

Fig. 6
Fig. 6

Tomographic reconstruction of a disk with A / 256 resolved range bins for disk radius A = 0.8 using the diffuse SR BRDF model in Eq. (A1). (a)  b = 0 Lambertian, (b)  b = 0.99 narrowly reflective, and (c) slices through the center of the reconstructions for b = 0 (solid curve) and b = 0.99 (dotted curve). The dip in the center of (a) is a sampling artifact.

Fig. 7
Fig. 7

Tomographic reconstructions of an elliptical disk with semimajor axis A = 0.6 and semiminor axis B = 0.3 . Input data are range-resolved ladar returned radiance computed using the SR diffuse BRDF model in Eq. (A1) with (a)  b = 0 , (b)  b = 0.99 .

Fig. 8
Fig. 8

Tomographic reconstruction of a disk using ladar range-resolved radiance with the specular SR BRDF model in Eq. (A5) with (a)  e = 1 , (b)  e = 0.007 .

Fig. 9
Fig. 9

Tomographic reconstruction of two adjacent disk phantoms with radii 0.4 and 0.3 from range-resolved ladar reflection with the full SR BRDF model in Eqs. (A1, A2, A3, A4, A5, A6, A7, A8): (a) dark gray coatings on both disks, (b) commercial aluminum coatings on both disks.

Fig. 10
Fig. 10

Tomographic reconstruction of two adjacent disks with radii 0.4 and 0.3 from range-resolved ladar reflection with the full SR BRDF model in Eqs. (A1, A2, A3, A4, A5, A6, A7, A8): (a) dark gray coating on the large disk and commercial aluminum surface on the small disk, (b) commercial aluminum surface on the large disk and dark gray coating on the small disk.

Fig. 11
Fig. 11

Tomographic reconstruction of two adjacent disks in Fig. 10a with the full SR BRDF model in Eqs. (A1, A2, A3, A4, A5, A6, A7, A8): (a) input logarithm of the binned radiance, (b) clipped radiance to remove glints at a threshold of 0.008 times the maximum value.

Fig. 12
Fig. 12

Sandford–Robertson model: specular reflection lobe showing glint angle α defined as the direction of the glint vector g. Weighting function h ( α ) shown in shade.

Equations (20)

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F ( λ ; θ i , ϕ i ; θ r , ϕ r ) = d L r ( λ ; θ r , ϕ r ) L i ( λ ; θ i , ϕ i ) cos θ i d ω i ,
ρ ( λ ; θ i , ϕ i ) = 0 π / 2 d θ r sin θ r 0 2 π d φ r F ( λ ; θ i , ϕ i ; θ r , ϕ r ) cos θ r .
ε ( λ ; θ i , ϕ i ) + ρ ( λ ; θ i , ϕ i ) = 1 ,
F ( λ ; θ i , θ r ) = F d ( λ ; θ i , θ r ) + F s ( λ ; θ i , θ r ) ,
n = 1 1 + f ( x ) 2 [ f ( x ) x + y ] ,
d l = d x 2 + d y 2 = d y 1 + f ( x ) 2 | f ( x ) | .
n · y = cos ( θ ) = 1 / 1 + f ( x ) 2 ,
d l cos ( θ ) × F ( λ ; θ ; θ ) = d y | f ( x ) | × F ( λ ; θ ; θ ) .
( f ( x ) y ) d x = Δ ( x ) d x = Δ ( x ) d y | f ( x ) | .
Δ ( x ) = F ( λ , θ ; θ ) ,
F d ( λ ; θ i , ϕ i ; θ r , ϕ r ) = g ( θ i ) ρ d ( λ ) g ( θ r ) π G ( b ) 2 ,
g ( θ ) = 1 1 + b 2 tan 2 θ
G ( b ) = 1 π 0 π / 2 d θ i sin θ i 0 2 π d ϕ i g ( θ i ) cos θ i = 1 1 b 2 [ 1 + b 2 1 b 2 ln ( b 2 ) ] .
ρ d ( λ ; θ i , ϕ i ) = g ( θ i ) G ( b ) ρ d ( λ ) .
F s ( λ ; θ i , ϕ i ; θ r , ϕ r ) = 1 4 π ρ s ( λ , θ i ) h ( α ) H ( θ i ) cos θ r ,
h ( α ) = 1 ( e 2 cos 2 α + sin 2 α ) 2 ,
ε ( λ , θ ) = ε ( λ ) g ( θ ) G ( b ) .
ρ s ( λ , θ i ) = 1 [ ε ( λ ) + ρ d ( λ ) ] g ( θ i ) G ( b ) .
ρ s ( λ , θ i ) = 0 π / 2 d θ r sin θ r 0 2 π d ϕ r F s cos θ r ,
H ( θ i ) = 1 4 π 0 π / 2 d θ r sin θ r 0 2 π d φ r h ( α ) 1 2 e 2 { ( 1 e 2 ) cos θ i + ( 2 e 2 + ( 1 e 2 ) 2 cos 2 θ i ) 4 e 2 + ( 1 e 2 ) 2 cos 2 θ i } .

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