Abstract

A geometric optical model for three-dimensional radiative transfer capable of handling arbitrary arrangements of surfaces within anisotropic scattering media is described. The model operates by discretizing surfaces and volumes into patches and voxels and establishing the radiative transfer relationship between every pair of elements. In a plane-parallel configuration results for directional radiance agree closely with the numerical integration invariant imbedded method. Model accuracy for two examples incorporating surface water waves and complex benthic structures were assessed by conservation of energy, errors were less than 1%. Potential applications in remote sensing or photobiological studies of structurally complex benthos in shallow water environments are illustrated.

© 2008 Optical Society of America

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2005

K. R. N. Anthony, M. O. Hoogenboom, and S. R. Connolly, "Adaptive variation in coral geometry and the optimization of internal colony light climates," Func. Ecol. 19, 17-26 (2005).
[CrossRef]

S. Enriquez, E. R. Mendez, and R. Iglesias-Prieto, "Multiple scattering on coral skeletons enhances light absorption by symbiotic algae," Limnol. Oceanogr. 50, 1025-1032 (2005).
[CrossRef]

S. Enriquez and N. I. Pantoja-Reyes, "Form-function analysis of the effect of canopy morphology on leaf selfshading in the seagrass Thalassia testudinum," Oecologia 145, 235-243 (2005).
[CrossRef] [PubMed]

J. D. Hedley, A. R. Harborne, and P. J. Mumby, "Simple and robust removal of sun glint for mapping shallowwater benthos," Int. J. Remote Sens. 26, 2107-2112 (2005).
[CrossRef]

2004

J. D. Hedley, P. J. Mumby, K. E. Joyce, and S. R. Phinn, "Spectral unmixing of coral reef benthos under ideal conditions," Coral Reefs 23, 60-73 (2004).
[CrossRef]

2003

C. Mobley, H. Zhang, and K. Voss, "Effects of optically shallow bottoms on upwelling radiances: Bidirectional reflectance distribution function effects," Limnol. Oceanogr. 48, 337-345 (2003).
[CrossRef]

C. Mobley and L. Sundman, "Effects of optically shallow bottoms on upwelling radiances: Inhomogeneous and sloping bottoms," Limnol. Oceanogr. 48, 329-336 (2003).
[CrossRef]

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

J. Zaneveld and E. Boss, "The influence of bottom morphology on reflectance: Theory and two-dimensional geometry model," Limnol. Oceanogr. 48, 374-379 (2003).
[CrossRef]

C. Soler, F. X. Sillion, F. Blaise, and P. Dereffye, "An efficient instantiation algorithm for simulating radiant energy transfer in plant models," ACM Trans. Graphics. 22, 204-233 (2003).
[CrossRef]

K. R. N. Anthony and O. Hoegh-Guldberg, "Variation in coral photosynthesis, respiration and growth characteristics in contrasting light microhabitats: an analogue to plants in forest gaps and understoreys?," Func. Ecol. 17, 246-259 (2003).
[CrossRef]

2002

J. D. Hedley and P. J. Mumby, "Biological and remote sensing perspectives of pigmentation in coral reef organisms," Adv. Mar. Biol. 43, 277-317 (2002).
[CrossRef] [PubMed]

C. Mobley, L. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002).
[CrossRef] [PubMed]

2001

D. Lubin, W. Li, P. Dustan, C. Mazel, and K. Stamnes, "Spectral signatures of coral reefs: Features from space," Remote Sens. Environ. 75, 127-137 (2001).
[CrossRef]

1998

K. F. Evans, "The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

1996

R. H. Grant, G. M. Heisler, and W. Gao, "Photosynthetically-active radiation: Sky radiance distributions under clear and overcast conditions," Agr. For. Met. 82, 267-292 (1996).
[CrossRef]

1993

C. D. Mobley, B. Gentili, H. R. Gordon, Z. H. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, "Comparison of numerical-models for computing underwater light fields," Appl. Opt. 32, 7484-7504 (1993).
[CrossRef] [PubMed]

D. Stramski, G. Rosenberg, and L. Legendre, "Photosynthetic and optical-properties of the marine chlorophyte Dunaliella-tertiolecta grown under fluctuating light caused by surface-wave focusing," Mar. Biol. 115, 363-372 (1993).
[CrossRef]

1991

C. C. Borel, S. A.W. Gerstl, and B. J. Powers, "The radiosity method in optical remote sensing of structured 3-D surfaces," Remote Sens. Environ. 36, 13-44 (1991).
[CrossRef]

1988

1981

J. Kirk, "Monte-Carlo study of the nature of the underwater light-field in, and the relationships between opticalproperties of, turbid yellow waters," Aus. J. Mar. Fresh. Res. 32, 517-532 (1981).
[CrossRef]

Anthony, K. R. N.

K. R. N. Anthony, M. O. Hoogenboom, and S. R. Connolly, "Adaptive variation in coral geometry and the optimization of internal colony light climates," Func. Ecol. 19, 17-26 (2005).
[CrossRef]

K. R. N. Anthony and O. Hoegh-Guldberg, "Variation in coral photosynthesis, respiration and growth characteristics in contrasting light microhabitats: an analogue to plants in forest gaps and understoreys?," Func. Ecol. 17, 246-259 (2003).
[CrossRef]

Blaise, F.

C. Soler, F. X. Sillion, F. Blaise, and P. Dereffye, "An efficient instantiation algorithm for simulating radiant energy transfer in plant models," ACM Trans. Graphics. 22, 204-233 (2003).
[CrossRef]

Borel, C. C.

C. C. Borel, S. A.W. Gerstl, and B. J. Powers, "The radiosity method in optical remote sensing of structured 3-D surfaces," Remote Sens. Environ. 36, 13-44 (1991).
[CrossRef]

Boss, E.

J. Zaneveld and E. Boss, "The influence of bottom morphology on reflectance: Theory and two-dimensional geometry model," Limnol. Oceanogr. 48, 374-379 (2003).
[CrossRef]

C. Mobley, L. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002).
[CrossRef] [PubMed]

Carder, K. L.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Chen, F. R.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Connolly, S. R.

K. R. N. Anthony, M. O. Hoogenboom, and S. R. Connolly, "Adaptive variation in coral geometry and the optimization of internal colony light climates," Func. Ecol. 19, 17-26 (2005).
[CrossRef]

Davis, C. O.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Dereffye, P.

C. Soler, F. X. Sillion, F. Blaise, and P. Dereffye, "An efficient instantiation algorithm for simulating radiant energy transfer in plant models," ACM Trans. Graphics. 22, 204-233 (2003).
[CrossRef]

Dustan, P.

D. Lubin, W. Li, P. Dustan, C. Mazel, and K. Stamnes, "Spectral signatures of coral reefs: Features from space," Remote Sens. Environ. 75, 127-137 (2001).
[CrossRef]

English, D. C.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Enriquez, S.

S. Enriquez and N. I. Pantoja-Reyes, "Form-function analysis of the effect of canopy morphology on leaf selfshading in the seagrass Thalassia testudinum," Oecologia 145, 235-243 (2005).
[CrossRef] [PubMed]

S. Enriquez, E. R. Mendez, and R. Iglesias-Prieto, "Multiple scattering on coral skeletons enhances light absorption by symbiotic algae," Limnol. Oceanogr. 50, 1025-1032 (2005).
[CrossRef]

Evans, K. F.

K. F. Evans, "The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer," J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

Gao, W.

R. H. Grant, G. M. Heisler, and W. Gao, "Photosynthetically-active radiation: Sky radiance distributions under clear and overcast conditions," Agr. For. Met. 82, 267-292 (1996).
[CrossRef]

Gentili, B.

Gerstl, S. A.W.

C. C. Borel, S. A.W. Gerstl, and B. J. Powers, "The radiosity method in optical remote sensing of structured 3-D surfaces," Remote Sens. Environ. 36, 13-44 (1991).
[CrossRef]

Gordon, H. R.

Grant, R. H.

R. H. Grant, G. M. Heisler, and W. Gao, "Photosynthetically-active radiation: Sky radiance distributions under clear and overcast conditions," Agr. For. Met. 82, 267-292 (1996).
[CrossRef]

Harborne, A. R.

J. D. Hedley, A. R. Harborne, and P. J. Mumby, "Simple and robust removal of sun glint for mapping shallowwater benthos," Int. J. Remote Sens. 26, 2107-2112 (2005).
[CrossRef]

Hedley, J. D.

J. D. Hedley, A. R. Harborne, and P. J. Mumby, "Simple and robust removal of sun glint for mapping shallowwater benthos," Int. J. Remote Sens. 26, 2107-2112 (2005).
[CrossRef]

J. D. Hedley, P. J. Mumby, K. E. Joyce, and S. R. Phinn, "Spectral unmixing of coral reef benthos under ideal conditions," Coral Reefs 23, 60-73 (2004).
[CrossRef]

J. D. Hedley and P. J. Mumby, "Biological and remote sensing perspectives of pigmentation in coral reef organisms," Adv. Mar. Biol. 43, 277-317 (2002).
[CrossRef] [PubMed]

Heisler, G. M.

R. H. Grant, G. M. Heisler, and W. Gao, "Photosynthetically-active radiation: Sky radiance distributions under clear and overcast conditions," Agr. For. Met. 82, 267-292 (1996).
[CrossRef]

Hoegh-Guldberg, O.

K. R. N. Anthony and O. Hoegh-Guldberg, "Variation in coral photosynthesis, respiration and growth characteristics in contrasting light microhabitats: an analogue to plants in forest gaps and understoreys?," Func. Ecol. 17, 246-259 (2003).
[CrossRef]

Hoogenboom, M. O.

K. R. N. Anthony, M. O. Hoogenboom, and S. R. Connolly, "Adaptive variation in coral geometry and the optimization of internal colony light climates," Func. Ecol. 19, 17-26 (2005).
[CrossRef]

Iglesias-Prieto, R.

S. Enriquez, E. R. Mendez, and R. Iglesias-Prieto, "Multiple scattering on coral skeletons enhances light absorption by symbiotic algae," Limnol. Oceanogr. 50, 1025-1032 (2005).
[CrossRef]

Ivey, J. E.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Jayaweera, K.

Jin, Z. H.

Joyce, K. E.

J. D. Hedley, P. J. Mumby, K. E. Joyce, and S. R. Phinn, "Spectral unmixing of coral reef benthos under ideal conditions," Coral Reefs 23, 60-73 (2004).
[CrossRef]

Kattawar, G. W.

Kirk, J.

J. Kirk, "Monte-Carlo study of the nature of the underwater light-field in, and the relationships between opticalproperties of, turbid yellow waters," Aus. J. Mar. Fresh. Res. 32, 517-532 (1981).
[CrossRef]

Lee, Z. P.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Legendre, L.

D. Stramski, G. Rosenberg, and L. Legendre, "Photosynthetic and optical-properties of the marine chlorophyte Dunaliella-tertiolecta grown under fluctuating light caused by surface-wave focusing," Mar. Biol. 115, 363-372 (1993).
[CrossRef]

Li, W.

D. Lubin, W. Li, P. Dustan, C. Mazel, and K. Stamnes, "Spectral signatures of coral reefs: Features from space," Remote Sens. Environ. 75, 127-137 (2001).
[CrossRef]

Liu, C. C.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Lubin, D.

D. Lubin, W. Li, P. Dustan, C. Mazel, and K. Stamnes, "Spectral signatures of coral reefs: Features from space," Remote Sens. Environ. 75, 127-137 (2001).
[CrossRef]

Mazel, C.

D. Lubin, W. Li, P. Dustan, C. Mazel, and K. Stamnes, "Spectral signatures of coral reefs: Features from space," Remote Sens. Environ. 75, 127-137 (2001).
[CrossRef]

Mendez, E. R.

S. Enriquez, E. R. Mendez, and R. Iglesias-Prieto, "Multiple scattering on coral skeletons enhances light absorption by symbiotic algae," Limnol. Oceanogr. 50, 1025-1032 (2005).
[CrossRef]

Mobley, C.

C. Mobley and L. Sundman, "Effects of optically shallow bottoms on upwelling radiances: Inhomogeneous and sloping bottoms," Limnol. Oceanogr. 48, 329-336 (2003).
[CrossRef]

C. Mobley, H. Zhang, and K. Voss, "Effects of optically shallow bottoms on upwelling radiances: Bidirectional reflectance distribution function effects," Limnol. Oceanogr. 48, 337-345 (2003).
[CrossRef]

C. Mobley, L. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002).
[CrossRef] [PubMed]

Mobley, C. D.

Morel, A.

Mumby, P. J.

J. D. Hedley, A. R. Harborne, and P. J. Mumby, "Simple and robust removal of sun glint for mapping shallowwater benthos," Int. J. Remote Sens. 26, 2107-2112 (2005).
[CrossRef]

J. D. Hedley, P. J. Mumby, K. E. Joyce, and S. R. Phinn, "Spectral unmixing of coral reef benthos under ideal conditions," Coral Reefs 23, 60-73 (2004).
[CrossRef]

J. D. Hedley and P. J. Mumby, "Biological and remote sensing perspectives of pigmentation in coral reef organisms," Adv. Mar. Biol. 43, 277-317 (2002).
[CrossRef] [PubMed]

Pantoja-Reyes, N. I.

S. Enriquez and N. I. Pantoja-Reyes, "Form-function analysis of the effect of canopy morphology on leaf selfshading in the seagrass Thalassia testudinum," Oecologia 145, 235-243 (2005).
[CrossRef] [PubMed]

Patten, J.

K. L. Carder, C. C. Liu, Z. P. Lee, D. C. English, J. Patten, F. R. Chen, J. E. Ivey, and C. O. Davis, "Illumination and turbidity effects on observing faceted bottom elements with uniform Lambertian albedos," Limnol. Oceanogr. 48, 355-363 (2003).
[CrossRef]

Phinn, S. R.

J. D. Hedley, P. J. Mumby, K. E. Joyce, and S. R. Phinn, "Spectral unmixing of coral reef benthos under ideal conditions," Coral Reefs 23, 60-73 (2004).
[CrossRef]

Powers, B. J.

C. C. Borel, S. A.W. Gerstl, and B. J. Powers, "The radiosity method in optical remote sensing of structured 3-D surfaces," Remote Sens. Environ. 36, 13-44 (1991).
[CrossRef]

Reinersman, P.

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Supplementary Material (9)

» Media 1: AVI (3681 KB)     
» Media 2: AVI (3681 KB)     
» Media 3: AVI (4056 KB)     
» Media 4: AVI (4056 KB)     
» Media 5: AVI (3549 KB)     
» Media 6: AVI (3549 KB)     
» Media 7: AVI (3681 KB)     
» Media 8: AVI (3681 KB)     
» Media 9: AVI (3681 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Overview of the 3D model. Surfaces are spatially discretized into patches while regions of space containing scattering media are discretized into cubic voxels (only a subset of the voxels are illustrated).

Fig. 2.
Fig. 2.

Directional discretization schemes. (a) HydroLight standard spherical partition, (b) the cubic 8-partition and (c) as a spherical projection, (d) the corresponding hemicube 8-partition for surface directional functions and (e) its unfolded visualization as used in Fig. 4.

Fig. 3.
Fig. 3.

Voxelization of volumetric radiative transfer. (a) Scattered energy resulting from a beam of radiance passing through the voxel through solid angular cell q in and the view-point p is found by integrating radiance along the in-voxel path length d. The resulting directional path radiances L *(q out) are assumed uniform throughout the voxel volume, and the radiance leaving the voxel is found by evaluating the integral of Eq. (4) along the exitant direction path length. (b) When surfaces and voxels intersect the path lengths are truncated and the view-point is shifted to the center of the remaining voxel segment.

Fig. 4.
Fig. 4.

Processing chain for one surface patch from the model example run of Sec. 5.2. Sequential rendering processes are shown on an unfolded hemicubic 128-partition, an identical process occurs for voxels but the rendering occurs on a cubic partition data structure.

Fig. 5.
Fig. 5.

Modeled substrate-incident radiance distribution in the plane of the sun, for optical depth 5.0, substrate diffuse reflectance R D=0.7 and single scattering albedos, 0≤ω 0≤1, with Petzold’s phase function. Sun zenith angle is 30° in a HydroLight idealized sky model (C =0, R dif=0.3, irradiance=1.0 for all bands) [29]. The water surface is perfectly flat. (a) Comparison between invariant imbedded method (II) and 3D model algorithm (3Dpp) with 10 layers and 20 iterations. (b) 3D model convergence rate to invariant imbedded solution for substrate incident radiance at 60° from nadir. (c) Effect of vertical spatial discretization (no. of layers) on modeled substrate-incident radiance at 60° and 87° from nadir, runs of 100 iterations.

Fig. 6.
Fig. 6.

Input vector meshes (a) and (c), and example camera sensor outputs (b) and (d) for the domes high sun and branches low sun model setups. On the left, red pyramids show a sample of the virtual camera FOV locations and blue rectangles are remote sensing pixel sensors above and below the water surface. Animations over the 24 time step solutions show effects of water movement (Media 1), (Media 2), and the 72 circularly placed cameras give a 360° rotational view of the 3D light field for a single time point (Media 3), (Media 4). Solution algorithm hemicubic and cubic resolutions (Fig. 4) were 96 and 48 respectively. Solution stability threshold was 0.001 with runs limited to 5 iterations. For clarity volumetric scattering is directionally interpolated in these visualizations.

Fig. 7.
Fig. 7.

Upwelling radiance, L u integrated over 1 m2 above the water surface by the virtual remote sensing pixel sensor for (a) domes high sun (Media 5) and (b) branches low sun (Media 6). The third panel (c) shows minimum and maximum percentage energy loss over the 24 runs for the domes and branches setups, ‘isec’ is one run of the branches setup using a patch intersection algorithm.

Fig. 8.
Fig. 8.

Hemispherical ‘fish-eye’ projections of downwelling radiance, L d, from (a) the sky, (b) directly under the water surface (Media 7), (c) on the substrate under the branching structures without water column scattering included in the visualization (Media 8) and (d) with scattering (Media 9).

Equations (17)

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b ( λ , p ) = 2 π 0 π β ( λ , p , ψ ) sin ψ d ψ
c ( λ , p ) = a ( λ , p ) + b ( λ , p )
L * ( λ , p , v out ) = L E * ( λ , p , v out ) + Ξ L ( λ , p , v in ) β ( λ , p , cos 1 [ v in · v out ] ) d Ω ( v in )
L ( λ , p , v ) = K ( λ , p s , p ) L EX G ( λ , p s , v ) + p p s K ( λ , p , p ) L * ( λ , p , v ) d x ( p )
K ( λ , p 1 , p 2 ) = exp ( p 1 p 2 c ( λ , p ' ) d x ( p ' ) )
= exp ( 0 d c ( λ , p 1 + x d ( p 1 p 2 ) ) d x )
M G L ( p ) = ( X x ( p ) X y ( p ) X z ( p ) Y x ( p ) Y y ( p ) Y z ( p ) n x ( p ) n y ( p ) n z ( p ) )
s ( λ , p , u in , u out ) = { r 1 ( λ , p , u in , u out ) , if u in Ξ z and u out Ξ z + r 2 ( λ , p , u in , u out ) , if u in Ξ z + and u out Ξ z t 12 ( λ , p , u in , u out ) , if u in Ξ z and u out Ξ z t 21 ( λ , p , u in , u out ) , if u in Ξ z + and u out Ξ z +
L EX L ( λ , p , u out ) = L E ( λ , p , u out ) + Ξ L IN L ( λ , p , u in ) s ( λ , p , u in , u out ) d Ω ( u in )
L IN L ( λ , p , u in ) = L [ λ , p , M L G ( p ) u in ]
r 1 ( λ , p , u in , u out ) = R ( λ , p , u in , u out ) cos θ
q ( u ) = 1 + { 0 + n f x u y + h + f x u z + h , for u x < u y and u x < u z n 2 + n f x u y + h + f x u z + h , for u x > u y and u x > u z 2 n 2 + n f y u x + h + f y u z + h , for u y < u x and u y < u z 3 n 2 + n f y u x + h + f y u z + h , for u y > u x and u y > u z 4 n 2 + n f z u x + h + f z u y + h , for u z < u x and u z < u y 5 n 2 + n f z u x + h + f z u y + h , for u z > u x and u z > u y
L * ( λ , i , q out ) = q in = 1 6 n 2 β ( λ , i , q in , q out ) [ L EXT ( λ , i , q in ) + L SELF ( λ , i , q in ) ] Ω ( q in )
L EXT ( λ , i , q in ) = d 2 d 2 L ( λ , i , q in ) exp [ c ( λ , i ) x ] d x
L SELF ( λ , i , q in ) = 0 d ( 0 x L * ( λ , i , q in ) exp [ c ( λ , i ) t ] d t ) exp [ c ( λ , t ) x ] d x
L EXT ( λ , i , q in ) = L ( λ , i , q in ) c ( λ , i ) x ( q in ) ( exp [ c ( λ , i ) x ( q in ) 2 ] exp [ c ( λ , i ) x ( q in ) 2 ] )
L SELF ( λ , i , q in ) = L * ( λ , i , q in ) c ( λ , i ) 2 x ( q in ) ( c ( λ , i ) x ( q in ) + exp [ c ( λ , i ) x ( q in ) ] 1 )

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