Abstract

Many algorithms for deriving surface shape from shading require an estimate of the direction of illumination. This paper presents a new estimator for illuminant direction, which also generates an estimate of the degree of surface relief, that is measured by the variance of surface orientation (the partial derivatives of surface depth). Surfaces are considered to be samples of a stochastic process representing depth as a function of position in the image plane. We derive an estimator for illuminant tilt that is based only on some general assumptions about the process. The assumptions are that the process is wide-sense stationary, strictly isotropic, and mean-square differentiable and that the second partial derivatives of surface depth are locally independent of the first partial derivatives. We develop an estimator of illuminant slant and degree of surface relief in two stages. In the first, we develop a general format for an estimator based on the same assumptions that are used for the tilt estimator. The second stage is the actual implementation of the estimator and requires the specification of a functional form for the local probability distribution of surface orientations. This approach contrasts with previous ones, which begin their development with an assumption of a particular distribution for surfaces. The approach has the advantage that it separates the problems of surface modeling and light-source estimation, permiting one to easily implement specific estimators for different surface models. We implement the illuminant slant estimator for surfaces that have a Gaussian distribution of surface orientations and show simulation results. Degraded performance in the presence of self-shadowing is discussed.

© 1990 Optical Society of America

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References

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  1. K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
    [CrossRef]
  2. C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
    [CrossRef]
  3. D. C. Knill, D. Kersten, “Learning a near optimal estimator for surface shape from shading,” Comput. Vision Graphics Image Process. (to be published).
  4. A. P. Pentland, “The visual inference of shape: computation from local features,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1982).
  5. A. P. Pentland, “Finding the illuminant direction,” J. Opt. Soc. Am. 72, 448–455 (1982).
    [CrossRef]
  6. This is a weaker condition than sample differentiability, which would require that all samples of the process were differentiable.
  7. The variance of a random variable X is defined as Var [X] = E[X2] − [X]2. For most cases presented in this paper, E[X] = 0, and the variance reduces to Var [X] = E[X2]. To be more compact, we use E[X2] to refer to variance when the mean of the random variable (or stochastic process) in question is zero.
  8. Staff of the Research and Education Association, Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms (Research and Education Association, New York, 1984).
  9. H. J. Larson, B. O. Shubert, Random Noise, Signals and Dynamic Systems Vol. 2 of Probabilistic Models in Engineering Science (Wiley, New York, 1979).

1985 (1)

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

1982 (1)

1981 (1)

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Horn, B. K. P.

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Ikeuchi, K.

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

Kersten, D.

D. C. Knill, D. Kersten, “Learning a near optimal estimator for surface shape from shading,” Comput. Vision Graphics Image Process. (to be published).

Knill, D. C.

D. C. Knill, D. Kersten, “Learning a near optimal estimator for surface shape from shading,” Comput. Vision Graphics Image Process. (to be published).

Larson, H. J.

H. J. Larson, B. O. Shubert, Random Noise, Signals and Dynamic Systems Vol. 2 of Probabilistic Models in Engineering Science (Wiley, New York, 1979).

Lee, C. H.

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

Pentland, A. P.

A. P. Pentland, “Finding the illuminant direction,” J. Opt. Soc. Am. 72, 448–455 (1982).
[CrossRef]

A. P. Pentland, “The visual inference of shape: computation from local features,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1982).

Rosenfeld, A.

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

Shubert, B. O.

H. J. Larson, B. O. Shubert, Random Noise, Signals and Dynamic Systems Vol. 2 of Probabilistic Models in Engineering Science (Wiley, New York, 1979).

Artif. Intell. (2)

K. Ikeuchi, B. K. P. Horn, “Numerical shape from shading and occluding boundaries,” Artif. Intell. 17, 141–184 (1981).
[CrossRef]

C. H. Lee, A. Rosenfeld, “Improved methods of estimating shape from shading using the light source coordinate system,” Artif. Intell. 26, 125–143 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (6)

This is a weaker condition than sample differentiability, which would require that all samples of the process were differentiable.

The variance of a random variable X is defined as Var [X] = E[X2] − [X]2. For most cases presented in this paper, E[X] = 0, and the variance reduces to Var [X] = E[X2]. To be more compact, we use E[X2] to refer to variance when the mean of the random variable (or stochastic process) in question is zero.

Staff of the Research and Education Association, Handbook of Mathematical, Scientific, and Engineering Formulas, Tables, Functions, Graphs, Transforms (Research and Education Association, New York, 1984).

H. J. Larson, B. O. Shubert, Random Noise, Signals and Dynamic Systems Vol. 2 of Probabilistic Models in Engineering Science (Wiley, New York, 1979).

D. C. Knill, D. Kersten, “Learning a near optimal estimator for surface shape from shading,” Comput. Vision Graphics Image Process. (to be published).

A. P. Pentland, “The visual inference of shape: computation from local features,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1982).

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Figures (9)

Fig. 1
Fig. 1

Imaging geometry assumed for the discussion. All vectors are represented in a three-dimensional coordinate system, in which the z axis points toward the viewer. The x–y plane perpendicular to this direction is referred to as the image plane. Local surface orientation is represented by a normal vector N . The unit vector L points toward the light source. For matte surfaces, the percentage of light energy reflected to the viewer from a point is given by the cosine of the angle β between N and L . Orthographic projection of the surface to the image is assumed.

Fig. 2
Fig. 2

Image of a smoothed fractal surface illuminated by a point light source at 135° tilt and 30° slant (from the upper-left-hand corner). The surface has a fractal dimension of 2.2 and has been smoothed by low-pass filtering of the depth values.

Fig. 3
Fig. 3

a, Smoothed fractal surface illuminated by a point light source at 135° tilt and 30° slant. b, This image shows the same surface after being stretched along the viewing direction (z axis) (the depths were scaled by a factor of 2) and illuminated by a source at a shallower slant (15°). The differences in the degree of relief in the surfaces and in the slant of the illuminant conspire to keep the average contrasts in the two images approximately equal.

Fig. 4
Fig. 4

Images used as examples in the simulations. a, The first test image is a smoothed fractal surface. b, The second test image, a sphere, is shown. Both images are generated by using a point light source at 135° tilt and 30° slant.

Fig. 5
Fig. 5

Plot of the average estimated illuminant tilt generated by the tilt estimator, when applied to images of smoothed fractal surfaces, versus the actual illuminant tilt. The error bars represent the standard deviation of the estimates (Table 2).

Fig. 6
Fig. 6

Plot of the standard deviation of estimated illuminant tilt generated by the tilt estimator, when applied to images of smoothed fractal surfaces, versus the illuminant slant used in generating the images.

Fig. 7
Fig. 7

a, Plot of the variance of the partial derivative of luminance as a function of the direction in which the derivative is computed for images of smoothed fractal surfaces. The images are generated by using a light source at a tilt of 45° and slants of 0°, 20°, and 40°. These data reflect an ensemble average, and, as expected, the peak of the function is found at 45° (see text for discussion). b, Plot of the sample variance as a function of direction for images of a sample fractal surface. The peaks of the functions are shifted away from 45° owing to random variations of the surface. Note that the accuracy of the peak is highest for the images generated with larger slants.

Fig. 8
Fig. 8

Plot of the average estimated illuminant slant generated by the slant and surface-relief estimator, when applied to images of smoothed fractal surfaces, versus the actual illuminant slant. The error bars represent the standard deviation of the estimates (Table 5). The ideal performance is shown as a dotted line.

Fig. 9
Fig. 9

Scatter plot of the estimated standard deviation of P and Q Q ( σ ̂ n ) generated by the slant and surface-relief estimator, when applied to images of smoothed fractal surfaces, versus the actual standard deviation (σp). The ideal performance is shown as a solid line.

Tables (6)

Tables Icon

Table 1 Tilt Estimates for Smoothed Fractal Surfaces: Simulation 1a

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Table 2 Tilt Estimates for Spheres: Simulation 1a

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Table 3 Tilt Estimates for Smoothed Fractal Surfaces: Simulation 2a

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Table 4 Tilt Estimates for Spheres: Simulation 2a

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Table 5 Tilt Estimates for Smoothed Fractal Surfaces: Simulation 3a

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Table 6 Tilt Estimates for Spheres: Simulation 4a

Equations (153)

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I = ρ λ ( N L ) ,
n x = P P 2 + q 2 + 1 ,
n y = q P 2 + q 2 + 1 ,
n z = 1 P 2 + q 2 + 1 ,
P = z x , q = z y .
I = ρ λ ( n x l x + n y l y + n z l z ) .
I = ρ λ cos β ,
s = cos 1 ( n z ) ,
τ = tan 1 ( n y / n x ) ,
s l = cos 1 ( l z ) ,
τ l = tan 1 ( l y / l x ) .
P = S x ,
Q = S y ,
P x = P x = 2 S x 2 ,
Q y = Q y = 2 S y 2 ,
P y = P y = 2 S x y = Q x = Q x .
N = ( n x , n y , n z ) T ,
n x = P P 2 + Q 2 + 1 ,
n y = Q P 2 + Q 2 + 1 ,
n z = 1 P 2 + Q 2 + 1 .
= cos 1 ( n z ) = cos 1 1 P 2 + Q 2 + 1 ,
T = tan 1 ( n y n x ) = tan 1 ( Q P ) .
E [ ( I r θ ) 2 ] = E [ ( I x ) 2 ] cos 2 θ + E [ ( I y ) 2 ] sin 2 θ + 2 E [ ( I x ) ( I y ) ] sin θ cos θ ,
τ ̂ l = θ ̂ = 1 2 tan 1 2 E [ ( I / x ) ( I / y ) ] E [ ( I / x ) 2 ] E [ ( I / x ) 2 ] .
L = ( l x , 0 , l z ) T .
I = ρ λ ( n x l x + n z l z ) ,
I r θ = ρ λ [ ( n x r θ ) l x + ( n z r θ ) l z ] .
E [ ( I r θ ) 2 ] = ρ 2 λ 2 E [ ( n x r θ l x + n z r θ l z ) 2 ] .
E [ ( I r θ ) 2 ] = ρ 2 λ 2 { l x 2 E [ ( n x r θ ) 2 ] + l z 2 E [ ( n z r θ ) 2 ] } .
f ( θ ) = ρ 2 λ 2 l x 2 E [ ( n x r θ ) 2 ] .
f ( θ ) = ρ 2 λ 2 l x 2 E [ ( cos θ n x x + sin θ n x y ) 2 ] .
f ( θ ) = ρ 2 λ 2 l x 2 { cos 2 θ E [ ( n x x ) 2 ] + sin 2 θ E [ ( n x y ) 2 ] } .
sin θ cos θ E [ ( n x x ) 2 ] = sin θ cos θ E [ ( n x y ) 2 ] .
f ( 0 ) = ρ 2 λ 2 l x 2 E [ ( n x x ) 2 ] ,
f ( π / 2 ) = ρ 2 λ 2 l x 2 E [ ( n x y ) 2 ] .
E [ ( I r θ ) 2 ] = E [ ( I x cos θ + I y sin θ ) 2 ] , E [ ( I r θ ) 2 ] = E [ ( I x ) 2 ] cos 2 θ + E [ ( I y ) 2 ] sin 2 θ + 2 E [ ( I x ) ( I y ) ] sin θ cos θ .
sin θ ̂ cos θ ̂ cos 2 θ ̂ sin 2 θ ̂ = E [ ( I / x ) ( I / y ) ] E [ ( I / x ) 2 ] E [ ( I / y ) 2 ] , 1 2 tan 2 θ ̂ = E [ ( I / x ) ( I / y ) ] E [ ( I / x ) 2 ] [ E ( I / y ) 2 ] , θ ̂ = 1 2 tan 1 2 E [ ( I / x ) ( I / y ) ] E [ ( I / x ) 2 ] [ E ( I / y ) 2 ] .
C = E [ ( I E [ I ] ) 2 ] E ( I ) 2 , C = E [ I 2 ] E [ I ] 2 E ( I ) 2 .
E [ I ] = E [ ρ λ ( l x n x + l y n y + l z n z ) ] , E [ I ] = ρ λ { l x E [ n x ] + l y E [ n y ] + l z E [ n z ] } .
E [ I ] = ρ λ l z E [ n z ] .
E [ I 2 ] = E [ ρ 2 λ 2 ( l x n x + l y n y + l z n z ) 2 ] , E [ I 2 ] = ρ 2 λ 2 { l x 2 E [ n x 2 ] + l y 2 E [ n y 2 ] + l z 2 E [ n z 2 ] + 2 l x l y E [ n x n y ] + 2 l x l z E [ n x n z ] + 2 l y l z E [ n y n z ] } .
E [ I 2 ] = ρ 2 λ 2 { l x 2 E [ n x 2 ] + l y 2 E [ n y 2 ] + l z 2 E [ n z 2 ] } .
n x = 1 n z 2 cos T ,
n y = 1 n z 2 sin T ,
E [ I 2 ] = ρ 2 λ 2 { l x 2 E [ 1 n z 2 ] E [ cos 2 T ] + l y 2 E [ 1 n z 2 ] × E [ sin 2 T ] + l z 2 E [ n z 2 ] } .
E [ I 2 ] = ½ ρ 2 λ 2 { 1 l z 2 + ( 3 l z 2 1 ) E [ n z 2 ] } .
C = 1 l z 2 + ( 3 l z 2 1 ) E [ n z 2 ] 2 l z 2 E [ n z ] 2 1 .
R = E [ ( I / x ) 2 ] E [ ( I / y ) 2 ] ,
I r θ = ρ λ ( l x n x r θ + l z n z r θ ) .
E [ ( I r θ ) 2 ] = ρ 2 λ 2 { l x 2 E [ ( n x r θ ) 2 ] + l z 2 E [ ( n z r θ ) 2 ] } .
E [ ( I r θ ) 2 ] = ρ 2 λ 2 { ( 1 l z 2 ) E [ ( n x r θ ) 2 ] + l z 2 E [ ( n z r θ ) 2 ] } .
R = ( 1 l z 2 ) E [ ( n x / x ) 2 ] + l z 2 E [ ( n z / x ) 2 ] ( 1 l z 2 ) E [ ( n x / y ) 2 ] + l z 2 E [ ( n z / y ) 2 ] .
E [ ( n x x ) 2 ] = { 5 4 E [ n z 2 ] + 1 2 E [ n z 4 ] + 5 4 E [ n z 6 ] } E [ P y 2 ] ,
E [ ( n x y ) 2 ] = { 3 4 E [ n z 2 ] 1 2 E [ n z 4 ] + 3 4 E [ n z 6 ] } E [ P y 2 ] ,
E [ ( n z x ) 2 ] = E [ ( n z y ) 2 ] = { 2 E [ n z 4 ] 2 E [ n z 6 ] } E [ P y 2 ] .
R = 5 E [ n z 2 ] + 2 E [ n z 4 ] + 5 E [ n z 6 ] l z 2 { 5 E [ n z 2 ] 6 E [ n z 4 ] + 13 E [ n z 6 ] } 3 E [ n z 2 ] 2 E [ n z 4 ] + 3 E [ n z 6 ] l z 2 { 3 E [ n z 2 ] 10 E [ n z 4 ] + 11 E [ n z 6 ] } .
E [ n z i ] = g i ( σ p ) .
R = P 2 + Q 2 .
f R ( r ) = p ( R = r ) = r σ p 2 exp [ r 2 / ( 2 σ p 2 ) ] ( r 0 ) .
n z = g ( R ) = 1 R 2 + 1 .
f n z ( n z ) = f R [ g 1 ( n z ) ] | d g 1 ( n z ) d n z | .
f n z ( n z ) = exp [ 1 / ( 2 σ p 2 ) ] σ p 2 n z 3 exp [ 1 / ( 2 n z 2 σ p 2 ) ] ( 0 < n z 1 ) .
E [ n z ] = 2 π 2 σ p exp [ 1 / ( 2 σ p 2 ) ] [ 1 erf ( 1 2 σ p ) ] ,
E [ n z 2 ] = exp [ 1 / ( 2 σ p 2 ) ] 2 σ p 2 E 1 ( 1 2 σ p 2 ) ,
E [ n z 4 ] = 1 2 σ p 2 ( 1 E [ n z 2 ] ) ,
E [ n z 6 ] = 1 4 σ P 2 ( 1 E [ n z 4 ] ) .
D = ( 0.0577 , 0.215 , 0.804 , 0 , 0.804 , 0.215 , 0.0577 ) T .
E ( l ̂ z , σ ̂ P ) = [ C m C ( l ̂ z , σ ̂ p ) ] 2 + [ R m R ( l ̂ z , σ ̂ p ) ] 2 ,
R s ( x , y ) = E [ S ( x , y ) S ( 0 , 0 ) ] = E [ S ( x 0 + x , y 0 + y ) S ( x 0 , y 0 ) ] .
E [ P ] = E [ Q ] = E [ P x ] = E [ P y ] = E [ Q x ] = E [ Q y ] = 0 .
R p ( x , y ) = 2 x 2 R s ( x , y ) ,
R q ( x , y ) = 2 y 2 R s ( x , y ) ,
R p x ( x , y ) = 4 x 4 R s ( x , y ) ,
R q y ( x , y ) = 4 y 4 R s ( x , y ) ,
R p y ( x , y ) = R q x ( x , y ) = 4 x 2 y 2 R s ( x , y ) ,
R s p ( x , y ) = x R s ( x , y ) ,
R s q ( x , y ) = y R s ( x , y ) ,
R pq ( x , y ) = 2 x y R s ( x , y ) ,
R pp x ( x , y ) = 3 x 3 R s ( x , y ) ,
R qq y ( x , y ) = 3 y 3 R s ( x , y ) ,
R pp y ( x , y ) = 3 x 2 y R s ( x , y ) ,
R qq x ( x , y ) = 3 x y 2 R s ( x , y ) ,
R p x p y ( x , y ) = 4 x 3 y R s ( x , y ) ,
R q x q y ( x , y ) = 4 x y 3 R s ( x , y ) .
f s [ s ( x 1 , y 1 ) , , s ( x k , y k ) ] = f s [ s ( x 1 , y 1 ) , , s ( x k , y k ) ] ,
x i = x i cos θ + y i cos θ ,
y i = x i sin θ + y i cos θ
E [ SP ] = R sp ( 0 , 0 ) = 0 ,
E [ SQ ] = R sq ( 0 , 0 ) = 0 ,
E [ PP x ] = R pp x ( 0 , 0 ) = 0 ,
E [ PP y ] = R pp y ( 0 , 0 ) = 0 ,
E [ QQ x ] = R qq x ( 0 , 0 ) = 0 ,
E [ QQ y ] = R qq y ( 0 , 0 ) = 0 .
R sp ( x , y ) = x R s ( x , y ) .
R s ( x , y ) = R s ( x , y ) .
R sp ( 0 , 0 ) = x R s ( x , y ) x = 0 , y = 0 = 0 .
E [ PQ ] = R pq ( 0 , 0 ) = 0 ,
E [ P x P y ] = E [ P x Q x ] = R P x P y ( 0 , 0 ) = 0 ,
E [ Q x Q y ] = E [ P y Q y ] = R q x q y ( 0 , 0 ) = 0 .
R s ( x , y ) = F s ( f x , f y ) exp ( i 2 π f x x ) exp ( i 2 π f y y ) d f x d f y .
R pq ( x , y ) = 2 x y R s ( x , y ) ,
R pq ( x , y ) = 2 x y F s ( f x , f y ) exp ( i 2 π f x x ) × exp ( i 2 π f y y ) d f x d f y , R pq ( 0 , 0 ) f x f y F s ( f x , f y ) d f x d f y .
R pq ( 0 , 0 ) = 0 .
E [ P x 2 ] = E [ Q y 2 ] = 3 E [ P y 2 ] = 3 E [ Q x 2 ] .
R p x ( x , y ) = 4 x 4 R s ( x , y ) .
R p x ( x , y ) = 4 x 4 F s ( f x , f y ) exp ( i 2 π f x x ) × exp ( i 2 π f y y ) d f x d f y , R p x ( x , y ) = 16 π 4 f x 4 F s ( f x , f y ) exp ( i 2 π f x x ) × exp ( i 2 π f y y ) d f x d f y .
E [ P x 2 ] = R p x ( 0 , 0 ) = 16 π 4 f x 4 F s ( f x , f y ) d f x d f y .
E [ Q y 2 ] = R q y ( 0 , 0 ) = 16 π 4 f y 4 F s ( f x , f y ) d f x d f y , E [ P y 2 ] = E [ Q x 2 ] = R p ν ( 0 , 0 ) = 16 π 4 f x 2 f y 2 F s ( f x , f y ) d f x d f y .
R p x ( 0 , 0 ) = 0 2 π 0 16 π 4 f r 5 cos 4 θ F s ( f r ) d f r d θ ) = 0 2 π cos 4 θ d θ 0 16 π 4 f r 5 F s ( f r ) d f r = 3 π 4 0 2 π 16 π 4 f r 5 F s ( f r ) d f r , R q y ( 0 , 0 ) = 0 2 π 0 16 π 4 f r 5 sin 4 θ F s ( f r ) d f r d θ = 0 2 π sin 4 θ d θ 0 16 π 4 f r 5 F s ( f r ) d f r = 3 π 4 0 16 π 4 f r 5 F s ( f r ) d f r , R p y ( 0 , 0 ) = 0 2 π 0 16 π 4 f r 5 sin 2 θ cos 2 θ F s ( f r ) d f r d θ = 0 2 π sin 2 θ cos 2 d θ 0 16 π 4 f r 5 F s ( f r ) d f r = π 4 0 16 π 4 f r 5 F s ( f r ) d f r .
T = tan 1 Q P ,
Σ = cos 1 ( 1 P 2 + Q 2 + 1 ) .
p [ T ( x , y ) = τ ] = 1 2 π , 0 τ < 2 π ,
p ( T = τ ) = p ( T = τ ) = p ( T = τ θ ) 0 θ < 2 π .
p ( Σ T = τ ) = p ( Σ T = τ ) = p ( Σ T = τ θ ) 0 θ < 2 π .
p ( Σ T ) = p ( Σ ) ,
E [ n x n y ] = E [ n x n z ] = E [ n y n z ] = 0 .
n z = 1 ( n x 2 + n y 2 ) .
p ( n x , n z ) = p ( n x , n z ) , p ( n y , n z ) = p ( n y , n z ) ,
E [ n x n z ] = E [ n y n z ] = 0 .
E [ n x r θ n x r θ ] = E [ n y r θ n y r θ ] = 0 .
E [ n x x n x y ] = E [ n y x n y y ] = E [ n z x n z y ] = 0 .
n x r θ = n x P P r θ + n x Q Q r θ = [ 1 ( P 2 + Q 2 + 1 ) 1 / 2 P 2 ( P 2 + Q 2 + 1 ) 3 / 2 ] P r θ + [ PQ ( P 2 + Q 2 + 1 ) 3 / 2 ] Q r θ = ( n z + n x 2 n z ) P r θ + ( n x n y n z ) Q r θ .
n x = 1 n z 2 cos T ,
n y = 1 n z 2 sin T .
n x r θ = [ n z + ( n z n z 3 ) cos 2 T ] P r θ + [ ( n z n z 3 ) sin T cos T ] Q r θ .
n z r θ = ( n z 2 1 n z 2 cos T ) P r θ + ( n z 2 1 n z 2 sin T ) Q r θ .
E [ n x r θ n z r θ ] = E [ n z 3 1 n z 2 cos ( T ) P r θ 2 ] + E [ 1 n z 2 ( n z 3 n z 5 ) cos 3 ( T ) P r θ 2 ] + E [ 1 n z 2 ( n z 3 n z 5 ) sin 2 ( T ) cos ( T ) Q r θ 2 ] + E [ n z 3 1 n z 2 sin ( T ) P r θ Q r θ ] + 2 E [ 1 n z 2 ( n z 3 n z 5 ) sin ( T ) cos 2 ( T ) P r θ Q r θ ] .
E [ n x r θ n z r θ ] = E [ n z 3 1 n z 2 ] E [ cos T ] E [ P r θ 2 ] + E [ 1 n z 2 ( n z 3 n z 5 ) ] E [ cos 3 T ] E [ P r θ 2 ] + E [ 1 n z 2 ( n z 3 n z 5 ) ] E [ sin 2 T cos T ] E [ Q r θ 2 ] + E [ n z 3 1 n z 2 ] E [ sin T ] E [ P r θ Q r θ ] + 2 E [ 1 n z 2 ( n z 3 n z 5 ) ] E [ sin T cos 2 T ] × E [ P r θ Q r θ ] .
E [ n x x n z x ] = 0 .
E [ n x x n x y ] = E [ { n z + ( n z n z 3 ) cos 2 T } 2 ] E [ P x P y ] + E [ { ( n z n z 3 ) sin T cos T } 2 ] E [ Q x Q y ] + E [ ( n z 2 n z 4 ) ] E [ sin T cos T ] E [ P x Q y ] + E [ ( n z n z 3 ) 2 ] E [ sin T cos 3 T ] E [ P x Q y ] + E [ ( n z 2 n z 4 ) ] E [ sin T cos T ] E [ Q x 2 ] + E [ ( n z n z 3 ) 3 ] E [ sin T cos 2 T ] E [ Q x 2 ] .
E [ n x x n x y ] = 0 .
E [ ( n x x ) 2 ] = E [ { n z + ( n z n z 3 ) cos 2 T } 2 ] E [ P x 2 ] + E [ { ( n z n z 3 ) sin T cos T } 2 ] E [ Q x 2 ] = { E [ n z 2 ] 2 E [ n z 2 ] E [ cos 2 T ] + 2 E [ n z 4 ] E [ cos 2 T ] + E [ n z 2 ] E [ cos 4 T ] 2 E [ n z 4 ] E [ cos 4 T ] + E [ n z 6 ] E [ cos 4 T ] } E [ P x 2 ] + { E [ n z 2 ] E [ sin 2 T cos 2 T ] 2 E [ n z 4 ] × E [ sin 2 T cos 2 T ] + E [ n z 6 ] E [ sin 2 T cos 2 T ] } E [ Q x 2 ] .
E [ cos 2 T ] = 1 2 ,
E [ cos 4 T ] = 3 8 ,
E [ sin 2 T cos 2 T ] = 1 8 .
E [ ( n x x ) 2 ] = { 3 8 E [ n z 2 ] + 1 4 E [ n z 4 ] + 3 8 E [ n z 6 ] } E [ P x 2 ] + { 1 8 E [ n z 2 ] 1 4 E [ n z 4 ] + 1 8 E [ n z 6 ] } E [ Q x 2 ] .
E [ ( n x x ) 2 ] = { 3 8 E [ n z 2 ] + 1 4 E [ n z 4 ] + 3 8 E [ n z 6 ] } E [ P y 2 ] + { 1 8 E [ n z 2 ] 1 4 E [ n z 4 ] + 1 8 E [ n z 6 ] } E [ Q y 2 ] .
E [ ( n x x ) 2 ] = { 5 4 E [ n z 2 ] + 1 2 E [ n z 4 ] + 5 4 E [ n z 6 ] } E [ P y 2 ] ,
E [ ( n x y ) 2 ] = { 3 4 E [ n z 2 ] 1 2 E [ n z 4 ] + 3 4 E [ n z 6 ] } E [ P y 2 ] .
E [ ( n z x ) 2 ] = E [ n z 4 ( 1 n z 2 ) cos 2 T ] E [ P x 2 ] + E [ n z 4 ( 1 n z 2 ) sin 2 T ] E [ Q x 2 ] = { [ E ( n z 4 ) E ( n z 6 ) ] E [ cos 2 T ] } E [ P x 2 ] + { ( E [ n z 4 ] E [ n z 6 ] ) E [ sin 2 T ] } E [ Q x 2 ] .
E [ ( n z x ) 2 ] = { 1 2 E [ n z 4 ] 1 2 E [ n z 6 ] } E [ P x 2 ] + { 1 2 E [ n z 4 ] 1 2 E [ n z 6 ] } E [ Q x 2 ] .
E [ ( n z x ) 2 ] = E [ ( n z y ) 2 ] = { 2 E [ n z 4 ] 2 E [ n z 6 ] } E [ P y 2 ] .
f P ( p ) = 1 2 π σ p exp [ p 2 / ( 2 σ p 2 ) ] ,
f Q ( q ) = 1 2 π σ p exp [ q 2 / ( 2 σ p 2 ) ] ,
f n z ( n z ) = exp [ 1 / ( 2 σ p 2 ) ] σ p 2 n z 3 exp [ 1 / ( 2 n z 2 σ p 2 ) ] , 0 < n z 1 .
E [ n z ] = 0 1 n z f n z ( n z ) d n z = 0 1 exp [ 1 / ( 2 σ P ) ] σ p 2 n z 2 exp [ 1 / ( 2 n z 2 σ p 2 ) ] d n z = 2 π σ p exp ( 1 / 2 σ P 2 ) [ 1 erf ( 1 2 σ p ) ] ,
E [ n z 2 ] = 0 1 n z 2 f n z ( n z ) d n z = 0 1 exp [ 1 / ( 2 σ p ) ] σ p 2 n z exp [ 1 / ( 2 n z 2 σ p 2 ) ] d n z = exp [ 1 / ( 2 σ p 2 ) ] 2 σ p 2 E 1 ( 1 2 σ p 2 ) ,
E n ( t ) = t x n e x d x .
E [ n z 2 n ] = 0 1 n z 2 n f n z ( n z ) d n z = 0 1 exp [ 1 / ( 2 σ p ) ] σ p 2 n z 2 n 3 exp [ 1 / ( 2 n z 2 σ p 2 ) ] d n z = exp [ 1 / ( 2 σ p 2 ) ] 2 n σ p 2 n E n ( 1 2 σ p 2 ) .
E n 1 ( t ) = 1 t n 1 e t ( n 1 ) E n ( t ) .
E [ n z 2 n ] = 1 ( 2 n 2 ) σ p 2 { 1 E [ n z 2 n 2 ] } , n > 1 .
E [ n z 4 ] = 1 2 σ p 2 { 1 E [ n z 2 ] } ,
E [ n z 6 ] = 1 4 σ p 2 { 1 E [ n z 4 ] } .

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