Abstract

Many shape recovery algorithms—in particular, shape from shading (SFS)—are based on a point source at infinity or a uniform hemispheric source. It will be convenient and useful if we can perform SFS under indoor lights that may be spherical, cylindrical, or flat (ceiling lights) in shape in an uncontrolled environment. As a first step toward this goal we propose a light source model for each of the above shapes for performing SFS. In this study we give the derivation of the rectangular, spherical, and cylindrical light source models. In indoor environments the positions (usually on the ceiling or walls) of the light sources are known. Assuming that the target object is small relative to the distances from the sources, we have derived a reflectance map for the Lambertian surface of an object under a mixture of light sources of the above shapes. Hence the shape recovery can be performed by using the SFS technique. This is a significant step toward the application of SFS in uncontrolled practical environments under household or office lighting. This technique of shape recovery is verified by many examples of simulations and real experiments, and the results are good.

© 1999 Optical Society of America

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References

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  1. B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. H. Winston, ed. (McGraw-Hill, New York, 1975), pp. 115–155.
  2. B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989), pp. 53–87.
  3. M. S. Langer, S. W. Zucker, “Shape-from-shading on a cloudy day,” J. Opt. Soc. Am. A 11, 467–478 (1994).
    [CrossRef]
  4. K. Ikeuchi, “Determining surface orientations of specular surfaces by using the photometric stereo,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 661–669 (1981).
    [CrossRef]
  5. M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 172–178.
  6. S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
    [CrossRef]
  7. Y. Sato, K. Ikeuchi, “Temporal-color space analysis of reflection,” J. Opt. Soc. Am. A 11, 2990–3002 (1994).
    [CrossRef]
  8. Y. L. Tian, H. T. Tsui, “3D shape recovery from two color image sequences using a genetic algorithm,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 674–678.
  9. Y. L. Tian, H. T. Tsui, “Shape recovery from a color image for non-Lambertian surface,” J. Opt. Soc. Am. A 14, 397–404 (1997).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).
  11. Y. M. Zhang, X. Z. Sun, Applied Optics (Mechanical Industry Publishing House (Beijing, 1982).
  12. M. Bichsel, A. P. Pentland, “A simple algorithm for shape from shading,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif.1992), pp. 459–465.
  13. J. Oliensis, “Shape from shading as a partially well-constrained problem,” Comput. Vis. Graph. Image Process. 54, 163–183 (1991).
  14. P. Dupuis, J. Oliensis, “Direct method for reconstructing shape from shading,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 563–571.
  15. J. Oliensis, P. Dupuis, “A global algorithm for shape from shading,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 692–701.
  16. R. Zhang, P. Tsai, J. E. Cryer, M. Shah, “Analysis of shape from shading techniques,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 377–384.
  17. S. A. Shafer, T. Kanade, “Using shadows in finding surface orientations,” Comput. Vis. Graph. Image Process. 22, 145–146 (1983).
    [CrossRef]
  18. J. R. Kender, E. M. Smith, “Shape from darkness: deriving surface information from dynamic shadows,” in Proceedings of the 1st International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif.1987), pp. 539–546.
  19. L. Wang, J. Clark, “Active shape and depth extraction from shadow images,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 550–553.

1997

1994

1991

J. Oliensis, “Shape from shading as a partially well-constrained problem,” Comput. Vis. Graph. Image Process. 54, 163–183 (1991).

1990

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

1983

S. A. Shafer, T. Kanade, “Using shadows in finding surface orientations,” Comput. Vis. Graph. Image Process. 22, 145–146 (1983).
[CrossRef]

1981

K. Ikeuchi, “Determining surface orientations of specular surfaces by using the photometric stereo,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 661–669 (1981).
[CrossRef]

Bichsel, M.

M. Bichsel, A. P. Pentland, “A simple algorithm for shape from shading,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif.1992), pp. 459–465.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).

Brooks, M. J.

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989), pp. 53–87.

Clark, J.

L. Wang, J. Clark, “Active shape and depth extraction from shadow images,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 550–553.

Cryer, J. E.

R. Zhang, P. Tsai, J. E. Cryer, M. Shah, “Analysis of shape from shading techniques,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 377–384.

Dupuis, P.

J. Oliensis, P. Dupuis, “A global algorithm for shape from shading,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 692–701.

P. Dupuis, J. Oliensis, “Direct method for reconstructing shape from shading,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 563–571.

Horn, B. K. P.

B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. H. Winston, ed. (McGraw-Hill, New York, 1975), pp. 115–155.

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989), pp. 53–87.

Ikeuchi, K.

Y. Sato, K. Ikeuchi, “Temporal-color space analysis of reflection,” J. Opt. Soc. Am. A 11, 2990–3002 (1994).
[CrossRef]

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

K. Ikeuchi, “Determining surface orientations of specular surfaces by using the photometric stereo,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 661–669 (1981).
[CrossRef]

Kanade, T.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

S. A. Shafer, T. Kanade, “Using shadows in finding surface orientations,” Comput. Vis. Graph. Image Process. 22, 145–146 (1983).
[CrossRef]

Kender, J. R.

J. R. Kender, E. M. Smith, “Shape from darkness: deriving surface information from dynamic shadows,” in Proceedings of the 1st International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif.1987), pp. 539–546.

Langer, M. S.

M. S. Langer, S. W. Zucker, “Shape-from-shading on a cloudy day,” J. Opt. Soc. Am. A 11, 467–478 (1994).
[CrossRef]

M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 172–178.

Nayar, S. K.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

Oliensis, J.

J. Oliensis, “Shape from shading as a partially well-constrained problem,” Comput. Vis. Graph. Image Process. 54, 163–183 (1991).

J. Oliensis, P. Dupuis, “A global algorithm for shape from shading,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 692–701.

P. Dupuis, J. Oliensis, “Direct method for reconstructing shape from shading,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 563–571.

Pentland, A. P.

M. Bichsel, A. P. Pentland, “A simple algorithm for shape from shading,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif.1992), pp. 459–465.

Sato, Y.

Shafer, S. A.

S. A. Shafer, T. Kanade, “Using shadows in finding surface orientations,” Comput. Vis. Graph. Image Process. 22, 145–146 (1983).
[CrossRef]

Shah, M.

R. Zhang, P. Tsai, J. E. Cryer, M. Shah, “Analysis of shape from shading techniques,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 377–384.

Smith, E. M.

J. R. Kender, E. M. Smith, “Shape from darkness: deriving surface information from dynamic shadows,” in Proceedings of the 1st International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif.1987), pp. 539–546.

Sun, X. Z.

Y. M. Zhang, X. Z. Sun, Applied Optics (Mechanical Industry Publishing House (Beijing, 1982).

Tian, Y. L.

Y. L. Tian, H. T. Tsui, “Shape recovery from a color image for non-Lambertian surface,” J. Opt. Soc. Am. A 14, 397–404 (1997).
[CrossRef]

Y. L. Tian, H. T. Tsui, “3D shape recovery from two color image sequences using a genetic algorithm,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 674–678.

Tsai, P.

R. Zhang, P. Tsai, J. E. Cryer, M. Shah, “Analysis of shape from shading techniques,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 377–384.

Tsui, H. T.

Y. L. Tian, H. T. Tsui, “Shape recovery from a color image for non-Lambertian surface,” J. Opt. Soc. Am. A 14, 397–404 (1997).
[CrossRef]

Y. L. Tian, H. T. Tsui, “3D shape recovery from two color image sequences using a genetic algorithm,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 674–678.

Wang, L.

L. Wang, J. Clark, “Active shape and depth extraction from shadow images,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 550–553.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).

Zhang, R.

R. Zhang, P. Tsai, J. E. Cryer, M. Shah, “Analysis of shape from shading techniques,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 377–384.

Zhang, Y. M.

Y. M. Zhang, X. Z. Sun, Applied Optics (Mechanical Industry Publishing House (Beijing, 1982).

Zucker, S. W.

M. S. Langer, S. W. Zucker, “Shape-from-shading on a cloudy day,” J. Opt. Soc. Am. A 11, 467–478 (1994).
[CrossRef]

M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 172–178.

Comput. Vis. Graph. Image Process.

J. Oliensis, “Shape from shading as a partially well-constrained problem,” Comput. Vis. Graph. Image Process. 54, 163–183 (1991).

S. A. Shafer, T. Kanade, “Using shadows in finding surface orientations,” Comput. Vis. Graph. Image Process. 22, 145–146 (1983).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell.

K. Ikeuchi, “Determining surface orientations of specular surfaces by using the photometric stereo,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 661–669 (1981).
[CrossRef]

IEEE Trans. Rob. Autom.

S. K. Nayar, K. Ikeuchi, T. Kanade, “Determining shape and reflectance of hybrid surfaces by photometric sampling,” IEEE Trans. Rob. Autom. 6, 418–431 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Other

M. S. Langer, S. W. Zucker, “What is a light source?” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 172–178.

J. R. Kender, E. M. Smith, “Shape from darkness: deriving surface information from dynamic shadows,” in Proceedings of the 1st International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif.1987), pp. 539–546.

L. Wang, J. Clark, “Active shape and depth extraction from shadow images,” in Proceedings of the International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 550–553.

B. K. P. Horn, “Obtaining shape from shading information,” in The Psychology of Computer Vision, P. H. Winston, ed. (McGraw-Hill, New York, 1975), pp. 115–155.

B. K. P. Horn, M. J. Brooks, Shape from Shading (MIT Press, Cambridge, Mass., 1989), pp. 53–87.

Y. L. Tian, H. T. Tsui, “3D shape recovery from two color image sequences using a genetic algorithm,” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1996), pp. 674–678.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959).

Y. M. Zhang, X. Z. Sun, Applied Optics (Mechanical Industry Publishing House (Beijing, 1982).

M. Bichsel, A. P. Pentland, “A simple algorithm for shape from shading,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif.1992), pp. 459–465.

P. Dupuis, J. Oliensis, “Direct method for reconstructing shape from shading,” in Proceedings of the Image Understanding Workshop (Morgan Kaufmann, San Mateo, Calif., 1992), pp. 563–571.

J. Oliensis, P. Dupuis, “A global algorithm for shape from shading,” in Proceedings of the 4th International Conference on Computer Vision (IEEE Computer Society Press, Los Alamitos, Calif., 1993), pp. 692–701.

R. Zhang, P. Tsai, J. E. Cryer, M. Shah, “Analysis of shape from shading techniques,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society Press, Los Alamitos, Calif., 1994), pp. 377–384.

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Figures (22)

Fig. 1
Fig. 1

Geometry of the rectangular light source in a 3D coordinate system.

Fig. 2
Fig. 2

Geometry of the spherical light source in a 3D coordinate system.

Fig. 3
Fig. 3

Geometry of the cylindrical light source in a 3D coordinate system.

Fig. 4
Fig. 4

Cross section of the cylindrical light source in a 3D coordinate system.

Fig. 5
Fig. 5

Diagram of light sources 1–4.

Fig. 6
Fig. 6

Shape recovery under four rectangular light sources (simulations): (a), (c) synthetic images; (b), (d) depth maps.

Fig. 7
Fig. 7

Diagram of (a) two spherical and (b) two cylindrical light sources.

Fig. 8
Fig. 8

Shape recovery under two spherical light sources (simulations): (a), (c) synthetic images; (b), (d) depth maps.

Fig. 9
Fig. 9

Shape recovery under two cylindrical light sources (simulations): (a), (c) synthetic images; (b), (d) depth maps.

Fig. 10
Fig. 10

Diagram of light sources (a) 1 and 2 and (b) 3 and 4 in real experiments.

Fig. 11
Fig. 11

Shape recovery under the two rectangular light sources 1 and 2 (real experiments): (a), (c) real images; (b), (d) depth maps.

Fig. 12
Fig. 12

Shape recovery under the two rectangular light sources 3 and 4 (real experiments): (a), (c) real images; (b), (d) depth maps.

Fig. 13
Fig. 13

Shape recovery under two spherical light sources (real experiment): (a) real image, (b) depth map.

Fig. 14
Fig. 14

Shape recovery under a cylindrical light source (real experiments): (a), (c) real images; (b), (d) depth maps.

Fig. 15
Fig. 15

Shape recovery under a rectangular light source for a complex object (real experiment): (a) real image, (b) depth map.

Fig. 16
Fig. 16

Experimental setup of mixed light sources.

Fig. 17
Fig. 17

Shape recovery under mixed light sources (real experiments): (a), (c), (e), (g) real images; (b), (d), (f), (h) depth maps.

Fig. 18
Fig. 18

Relative depth error for different light source models, where the light source direction is (0, 0, 20): (a) point light source, (b) spherical light source, (c) rectangular light source, (b) spherical light source, (c) rectangular light source, (d) cylindrical light source.

Fig. 19
Fig. 19

Relative depth error for different light source models, where the light source direction is (5, 5, 20): (a) point light source, (b) spherical light source, (c) rectangular light source, (d) cylindrical light source.

Fig. 20
Fig. 20

Diagram of an object under different light sources: (a) light source 1, (b) light source 2, (c) light sources 1 and 2.

Fig. 21
Fig. 21

(a)–(c) Images of Mozart and (d)–(f) depth maps under rectangular light sources: (a), (d) light source 1; (b), (e) light source 2; (c), (f) light sources 1 and 2.

Fig. 22
Fig. 22

Geometry of the polar coordinate system.

Tables (4)

Tables Icon

Table 1 Error Analyses for Different Light Source Models (Synthetic Images)

Tables Icon

Table 2 Error Analyses for Noise with Light Source Direction in (0, 0, 20) cm (Synthetic Images)

Tables Icon

Table 3 Error Analyses for Light Source Position with Light Source Direction in (5, 5, 20) cm (Synthetic Images)

Tables Icon

Table 4 Error Analyses for Interreflections (Real Images)

Equations (49)

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

L(θ)=E cos β,
cos β=zD,
L(θ)=E cos β=EzD.
B=L(θ)cos ΔθD2,Δθ<90°0,otherwise,
cos Δθ=sin θ cos φ sin θn cos φn+sin θ×sin φ sin θn sin φn+cos θ cos θn,
B=x1x2y1y2B dxdy=x1x2y1y2 L(θ)cos ΔθD2dxdy=Ezx1x2y1y2 1D3(sin θ cos φ sin θn cos φn+sin θ sin φ sin θn sin φn+cos θ cos θn)dxdy=(Ez sin θn cos φn)x1x2y1y2 x(x2+y2+z2)2dxdy+(Ez sinθn sinφn)x1x2y1y2 y(x2+y2+z2)2dxdy+(Ez2 cos θn)x1x2y1y2 1(x2+y2+z2)2dxdy=(Ez sin θn cos φn)×(part1)+(Ez sin θn sin φn)×(part2)+(Ez2 cos θn)×(part3),
B=Ar sin θn cos φn+Br sin θn sin φn+Cr cos θn,
Ar=Ezγ4-γ12x12+z2+γ2-γ32x22+z2,
Br=Ezδ2-δ12y12+z2+δ4-δ32y22+z2,
Cr=Ey2(δ3-δ4)2y22+z2+y1(δ1-δ2)2y12+z2+x2(γ3-γ2)2x22+z2+x1(γ1-γ4)2x12+z2.
α=sin-1 rH.
E=ρI0r2,
L(θ)=E cos β.
sin β=H sin θr,
D=H cos θ-(r2-H2 sin2 θ)1/2.
cos β=(r2-H2 sin2 θ)1/2r,
B=L(θ)cos ΔθD2,Δθ<90°0,otherwise,
cos Δθ=sin θ cos φ sin θn cos φn+sin θ sin φ sin θn sin φn+cos θ cos θn.
B=0α02πB dθdφ=0α02π L(θ)cos ΔθD2dθdφ=k cos θns,
B=k cos θns=ksin θcs sin φcs sin θn sin φn-sin θcs cos θcs cos φcs sin θn cos φn(1-sin2 θcs cos2 φcs)1/2+cos θcs cos θn=As sin θn cos φn+Bs sin θn sin φn+Cs cos θn,
As=k sin θcs sin φcs(1-sin2 θcs cos2 φcs)1/2,
Bs=-k sin θcs cos θcs cos φcs(1-sin2 θcs cos2 φcs)1/2,
Cs=k cos θcs(1-sin2 θcs cos2 φcs)1/2.
cos β=1D(x, y, z)·(rx, ry, rz).
α1=π+arctan z0x0-arccos rD,
α2=π+arctan z0x0+arccos rD.
B=L(θ)cos ΔθD2,Δθ<90°0,otherwise,
B=y1y2α1α2B dαdy=y1y2α1α2 L(θ)cos ΔθD2α dy=y1y2α1α2 L(θ)D21D(x, y, z)·(nx, ny, nz)dαdy=Ac sin θn cos φn+Bc sin θn sin φn+Cc cos θn,
B=Br+Bs+Bc=A* sin θn cos φn+B* sin θn sin φn+C* cos θn,
R(n(x, y))=1πn·i=A*nx(x, y)+B*ny(x, y)+C*nz(x, y)ifA*nx(x, y)+B*ny(x, y)+C*nz(x, y)>00otherwise,
R(p, q)=1π(A*, B*, C*)·(-p,-q, 1)=ρ -A*p-B*q+C*(1+p2+q2)1/2,
ρ=1π(A*2+B*2+C*2)1/2.
n(nx, ny, nz)=-p(p2+q2+1)1/2, -q(p2+q2+1)1/2,×1(p2+q2+1)1/2.
p=-A*2C*2±[(1-R2)(R2-B*2)]1/2R2-A*2-B*2,
q=A*B*p-B*C*R2-B*2,
|truedepth-estimateddepth|truedepth×100%.
ME=1NiNEi=1NiN|truedepth-estimateddepth|i,
σE=1NiN(truedepth-estimateddepth)i21/2,
part1=x1x2y1y2 x(x2+y2+z2)2dxdy=t=x2y1y2dyx1x2 12(t+y2+z2)2dt=-12y1y21x22+y2+z2-1x12+y2+z2dy=12y1y2 1x12+y2+z2dy-12y1y2 1x22+y2+z2dy=12x12+z2arctan y2x12+z2-arctan y1x12+z2+12x22+z2arctan y1x22+z2-arctan y2x22+z2=γ4-γ12x12+z2+γ2-γ32x22+z2,
part2=x1x2y1y2 y(x2+y2+z2)2dxdy=t=y2x1x2dxy1y2 12(t+x2+z2)2dt=-12x1x21x2+y22+z2-1x2+y12+z2dx=12x1x2 1x2+y12+z2dx-12x1x2 1x2+y22+z2dx=12y12+z2arctan x2y12+z2-arctan x1y12+z2+12y22+z2arctan x1y22+z2-arctan x2y22+z2=δ2-δ12y12+z2+δ4-δ32y22+z2.
part3=x1x2y1y2 1(x2+y2+z2)2dxdy=x1x2dxy1y2 1(x2+y2+z2)2dy=12x1x2y2(x2+z2)(x2+y22+z2)-y1(x2+z2)(x2+y12+z2)+arctan y2/(x2+z2)(x2+z2)x2+z2-arctan y1/(x2+z2)(x2+z2)x2+z2dx.
part3=S 1(x2+y2+z2)2dxdy=S=S1+S2+S3 ρ(ρ2+z2)2dρdθ=S1 ρ(ρ2+z2)2dρdθ+S2 ρ(ρ2+z2)2dρdθ+S3 ρ(ρ2+z2)2dρdθ.
S1 ρ(ρ2+z2)2dρdθ=θ1θ2dθρ1ρ2 ρ(ρ2+z2)2dρ=-12θ1θ2 1ρ22+z2-1ρ12+z2dθ=-12θ1θ2cos2 θx22+z2 cos2 θ-sin2 θy12+z2 sin2 θdθ.
S1 ρ(ρ2+z2)2dρdθ=12tan θ1tan θ2t2(1+t2)y12+z2t2-1(1+t2)x22+z2×11+t2dt=12tan θ1tan θ21(1+t2)z2-y12[y12+(y12+z2)t2]z2dt-12tan θ1tan θ21(1+t2)z2-x22(x22+z2+x22t2)z2dt=x22z2x22+z2arctan y2x22+z2-arctan y1x22+z2-y12z2y12+z2arctan y2y12+z2x2y1-arctan y12+z2x2.
S2 ρ(ρ2+z2)2dρdθ=θ1θ2dθρ1ρ2 ρ(ρ2+z2)2dρ=-12θ1θ21ρ22+z2-1ρ12+z2dθ=-12θ1θ2sin2 θy22+z2 sin2 θ-sin2 θy12+z2 sin2 θdθ=12tan θ1tan θ2t2(1+t2)y12+z2t2-t2(1+t2)y22+z2t2×11+t2dt=12tan θ1tan θ21(1+t2)z2-y12[y12+(y12+z2)t2]z2dt-12tan θ1tan θ21(1+t2)z2-y22[y22+(y22+z2)t2]z2dt=y22z2y22+z2arctan y1y22+z2x1y2-arctan y22+z2x2-y12z2y12+z2arctan y12+z2x1-arctan y2y12+z2x2y1.
S3 ρ(ρ2+z2)2dρdθ
=θ1θ2 dθρ1ρ2 ρ(ρ2+z2)2dρ=-12θ1θ21ρ22+z2-1ρ12+z2dθ=-12θ1θ2sin2 θy22+z2 sin2 θ-cos2 θx12+z2 cos2 θdθ=-12tan θ1tan θ2t2(1+t2)y22+z2t2-1(1+t2)x12+z2×11+t2dt=12tan θ1tan θ2 1[(1+t2)x12+z2](1+t2)dt-12tan θ1tan θ2 t2[(y22+z2)t2+y22](1+t2)dt
=12tan θ1tan θ21(1+t2)z2-x12(x12+z2+x12t2)z2dt-12tan θ1tan θ21(1+t2)z2-y22[y22+(y22+z2)t2]z2dt=y222z2tan θ1tan θ2 1y22+(y22+z2)t2dt-x122z2tan θ1tan θ2 1x12+z2+x12t2dt=y22z2y22+z2arctan y22+z2x1-arctan y1y22+z2x1y2-x12z2x12+z2arctan y2x12+z2-arctan y1x12+z2.
part3=S=S1+S2+S3 ρ(ρ2+z2)2dρdθ=S1 ρ(ρ2+z2)2dρdθ+S2 ρ(ρ2+z2)2dρdθ+S3 ρ(ρ2+z2)2dρdθ=y22z2y22+z2arctan y22+z2x1-arctan y22+z2x2+y12z2y12+z2arctan y12+z2x2-arctan y12+z2x1+x22z2x22+z2arctan y2x22+z2-arctan y1x22+z2+x12z2x12+z2arctan y1x12+z2-arctan y2x12+z2=y2(δ3-δ4)2z2y22+z2+y1(δ1-δ2)2z2y12+z2+x2(γ3-γ2)2z2x22+z2+x1(γ1-γ4)2z2x12+z2.

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