Abstract

Cinema viewed from a location other than a canonical viewing point (CVP) presents distortions to the viewer in both its static and its dynamic aspects. Past works have investigated mainly the static aspect of this problem and attempted to explain why viewers still seem to perceive the scene very well. The dynamic aspect of depth perception, which is known as structure from motion, and its possible distortion, have not been well investigated. We derive the dynamic depth cues perceived by the viewer and use the so-called isodistortion framework to understand its distortion. The result is that viewers seated at a reasonably central position experience a shift in the intrinsic parameters of their visual systems. Despite this shift, the key properties of the perceived depths remain largely the same, being determined in the main by the accuracy to which extrinsic motion parameters can be recovered. For a viewer seated at a noncentral position and watching the movie screen at a slant angle, the view is related to the view at the CVP by a homography, resulting in various aberrations such as noncentral projection.

© 2007 Optical Society of America

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2005 (3)

M. S. Banks, H. F. Rose, D. Vishwanath, and A. R. Girshick, "Where should you sit to watch a movie?" Proc. SPIE 5666, 316-325 (2005).
[CrossRef]

M. J. A. Doumen, A. M. L. Kappers, and J. J. Koenderink, "Visual space under free viewing conditions," Percept. Psychophys. 67, 1177-1189 (2005).
[CrossRef]

J. Trommershauser, S. Gepshtein, L. T. Maloney, M. S. Landy, and M. S. Banks, "Optimal compensation for changes in task-relevant movement variability," J. Neurosci. 25, 7169-7178 (2005).
[CrossRef] [PubMed]

2004 (4)

J. M. Hillis, S. J. Watt, M. S. Landy, and M. S. Banks, "Slant from texture and disparity cues: optimal cue combination," J. Vision 4, 967-992 (2004).
[CrossRef]

D. Kersten, P. Mamassian, and A. Yuille, "Object perception as Bayesian inference," Annu. Rev. Psychol. 55, 271-304 (2004).
[CrossRef] [PubMed]

L. F. Cheong and C. H. Peh, "Depth distortion under calibration uncertainty," Comput. Vis. Image Underst. 93, 221-244 (2004).
[CrossRef]

M. A. Goodale and D. A. Westwood, "An evolving view of duplex vision: separate but interacting cortical pathways for perception and action," Curr. Opin. Neurobiol. 14, 203-211 (2004).
[CrossRef] [PubMed]

2003 (1)

D. C. Knill and J. Saunders, "Do humans optimally integrate stereo and texture information for judgments of surface slant?" Vision Res. 43, 2539-2558 (2003).
[CrossRef] [PubMed]

2002 (1)

V. Cornilleau-Pèrés, M. Wexler, J. Droulez, E. Marin, C. Miège, and B. Bourdoncle, "Visual perception of planar orientation: dominance of static depth cues over motion cues," Vision Res. 42, 1403-1412 (2002).
[CrossRef] [PubMed]

2001 (1)

L. F. Cheong and T. Xiang, "Characterizing depth distortion under different generic motions," Int. J. Comput. Vis. 44, 199-217 (2001).
[CrossRef]

2000 (2)

E. Grossmann and J. Santos-Victor, "Uncertainty analysis of 3-D reconstruction from uncalibrated views," Image Vis. Comput. 18, 685-696 (2000).
[CrossRef]

J. Oliensis, "A new structure-from-motion ambiguity," IEEE Trans. Pattern Anal. Mach. Intell. 22, 685-700 (2000).
[CrossRef]

1999 (3)

J. T. Todd and V. J. Perotti, "The visual perception of surface orientation from optical flow," Percept. Psychophys. 61, 1577-1589 (1999).
[CrossRef] [PubMed]

J. M. Loomis and J. W. Philbeck, "Is the anisotropy of perceived 3-d shape invariant across scale?" Percept. Psychophys. 61, 397-402 (1999).
[CrossRef] [PubMed]

A. Guirao and P. Artal, "Off-axis monochromatic aberrations estimated from double pass measurements in the human eye," Vision Res. 39, 207-217 (1999).
[CrossRef] [PubMed]

1998 (4)

K. A. Stevens and A. Brookes, "Integrating stereopsis with monocular interpretations of planar surfaces," Vision Res. 28, 371-386 (1998).
[CrossRef]

J. E. Sparrow and W. M. Stine, "The perceived rigidity of rotating eight-vertex geometric forms; extracting nonrigid structure from rigid motion," Vision Res. 38, 541-556 (1998).
[CrossRef] [PubMed]

F. Domini, C. Caudek, and S. Richmann, "Distortions of depth-order relations and parallelism in structure from motion," Percept. Psychophys. 60, 1164-1174 (1998).
[CrossRef] [PubMed]

L. F. Cheong, C. Fermuller, and Y. Aloimonos, "Effects of errors in the viewing geometry on shape estimation," Comput. Vis. Image Underst. 71, 356-372 (1998).
[CrossRef]

1997 (2)

R. Szeliski and S. B. Kang, "Shape ambiguities in structure-from-motion," IEEE Trans. Pattern Anal. Mach. Intell. 19, 506-512 (1997).
[CrossRef]

T. S. Meese and M. G. Harris, "Computation of surface slant from optic flow: orthogonal components of speed gradient can be combined," Vision Res. 37, 2369-2379 (1997).
[CrossRef] [PubMed]

1996 (2)

S. J. Galvin, D. R. Williams, and N. J. Coletta, "The spatial grain of motion perception in human peripheral vision," Vision Res. 36, 2283-2296 (1996).
[CrossRef] [PubMed]

T. Viéville, O. D. Faugeras, and Q. T. Luong, "Motion of points and lines in the uncalibrated case," Int. J. Comput. Vis. 17, 7-41 (1996).
[CrossRef]

1995 (4)

P. Artal, A. Derrington, and E. Colombo, "Refraction, aliasing, and the absence of motion reversals in peripheral vision," Vision Res. 35, 939-947 (1995).
[CrossRef] [PubMed]

J. S. Tittle, J. T. Todd, V. J. Perotti, and J. F. Norman, "Systematic distortion of perceived three-dimensional structure from motion and binocular stereopsis," J. Exp. Psychol. Hum. Percept. Perform. 21, 663-678 (1995).
[CrossRef] [PubMed]

J. J. Koenderink and A. J. van Doorn, "Relief: pictorial and otherwise," Image Vis. Comput. 13, 321-334 (1995).
[CrossRef]

O. D. Faugeras, "Stratification of 3D vision: projective, affine, and metric representations," J. Opt. Soc. Am. A 12, 465-484 (1995).
[CrossRef]

1994 (2)

W. J. M. Damme, F. H. Oosterhoff, and W. A. van de Grind, "Discrimination of 3-D shape and 3-D curvature from motion in active vision," Percept. Psychophys. 55, 340-349 (1994).
[CrossRef] [PubMed]

V. Cornilleau-Pérès and J. Droulez, "The visual perception of 3D shape from self-motion and object-motion," Vision Res. 34, 2331-2336 (1994).
[CrossRef] [PubMed]

1993 (1)

W. J. M. Damme and W. A. van de Grind, "Active vision and the identification of 3D shape," Vision Res. 11, 1581-1587 (1993).

1992 (2)

J. F. Norman and J. S. Lappin, "The detection of surface curvatures defined by optical motion," Percept. Psychophys. 51, 386-396 (1992).
[CrossRef] [PubMed]

G. S. Young and R. Chellappa, "Statistical analysis of inherent ambiguities in recovering 3-D motion from a noisy flow field," IEEE Trans. Pattern Anal. Mach. Intell. 14, 995-1013 (1992).
[CrossRef]

1990 (3)

B. Caprile and V. Torre, "Using vanishing points for camera calibration," Int. J. Comput. Vis. 4, 127-140 (1990).
[CrossRef]

J. T. Todd and P. Bressan, "The perception of 3-dimensional affine structure from minimal apparent motion sequences," Percept. Psychophys. 48, 419-430 (1990).
[CrossRef] [PubMed]

J. Droulez and V. Cornilleau-Pérès, "Visual perception of surface curvature, the spin variation and its physiological implications," Biol. Cybern. 62, 211-224 (1990).
[CrossRef] [PubMed]

1989 (3)

V. Cornilleau-Pérès and J. Droulez, "Visual perception of surface curvature: Psychophysics of curvature detection induced by motion parallax," Percept. Psychophys. 46, 351-364 (1989).
[CrossRef] [PubMed]

G. Adiv, "Inherent ambiguities in recovering 3-D motion and structure from a noisy flow field," IEEE Trans. Pattern Anal. Mach. Intell. 11, 477-489 (1989).
[CrossRef]

R. C. Nelson and J. Aloimonos, "Obstacle avoidance using flow field divergence," IEEE Trans. Pattern Anal. Mach. Intell. 11, 1102-1106 (1989).
[CrossRef]

1987 (1)

J. E. Cutting, "Rigidity of cinema seen from the front row, side aisle," J. Exp. Psychol. Hum. Percept. Perform. 13, 323-334 (1987).
[CrossRef] [PubMed]

1985 (1)

M. Wagner, "The metric of visual space," Percept. Psychophys. 38, 483-495 (1985).
[CrossRef] [PubMed]

1983 (1)

B. J. Rogers and M. E. Graham, "Anisotropies in the perception of three-dimensional surfaces," Science 221, 1409-1411 (1983).
[CrossRef] [PubMed]

1981 (1)

H. C. Longuet-Higgins, "A computer algorithm for reconstructing a scene from two projections," Nature (London) 293, 133-135 (1981).
[CrossRef]

1980 (1)

D. N. Lee, "The optic flow field: The foundation of vision," Philos. Trans. R. Soc. London, Ser. B 290, 169-178 (1980).
[CrossRef] [PubMed]

1973 (1)

D. N. Perkins, "Compensating for distortion in viewing pictures obliquely," Percept. Psychophys. 14, 13-18 (1973).
[CrossRef]

1966 (1)

W. C. Hoffman, "The lie algebra of visual perception," J. Math. Psychol. 3, 65-98 (1966).
[CrossRef]

Adiv, G.

G. Adiv, "Inherent ambiguities in recovering 3-D motion and structure from a noisy flow field," IEEE Trans. Pattern Anal. Mach. Intell. 11, 477-489 (1989).
[CrossRef]

Aloimonos, J.

R. C. Nelson and J. Aloimonos, "Obstacle avoidance using flow field divergence," IEEE Trans. Pattern Anal. Mach. Intell. 11, 1102-1106 (1989).
[CrossRef]

Aloimonos, Y.

L. F. Cheong, C. Fermuller, and Y. Aloimonos, "Effects of errors in the viewing geometry on shape estimation," Comput. Vis. Image Underst. 71, 356-372 (1998).
[CrossRef]

Artal, P.

A. Guirao and P. Artal, "Off-axis monochromatic aberrations estimated from double pass measurements in the human eye," Vision Res. 39, 207-217 (1999).
[CrossRef] [PubMed]

P. Artal, A. Derrington, and E. Colombo, "Refraction, aliasing, and the absence of motion reversals in peripheral vision," Vision Res. 35, 939-947 (1995).
[CrossRef] [PubMed]

Åström, K.

A. Heyden and K. Åström, "Euclidean reconstruction from image sequences with varying and unknown focal length and principal point," In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1997), pp. 438-443.

Banks, M. S.

J. Trommershauser, S. Gepshtein, L. T. Maloney, M. S. Landy, and M. S. Banks, "Optimal compensation for changes in task-relevant movement variability," J. Neurosci. 25, 7169-7178 (2005).
[CrossRef] [PubMed]

M. S. Banks, H. F. Rose, D. Vishwanath, and A. R. Girshick, "Where should you sit to watch a movie?" Proc. SPIE 5666, 316-325 (2005).
[CrossRef]

J. M. Hillis, S. J. Watt, M. S. Landy, and M. S. Banks, "Slant from texture and disparity cues: optimal cue combination," J. Vision 4, 967-992 (2004).
[CrossRef]

Bourdoncle, B.

V. Cornilleau-Pèrés, M. Wexler, J. Droulez, E. Marin, C. Miège, and B. Bourdoncle, "Visual perception of planar orientation: dominance of static depth cues over motion cues," Vision Res. 42, 1403-1412 (2002).
[CrossRef] [PubMed]

Bressan, P.

J. T. Todd and P. Bressan, "The perception of 3-dimensional affine structure from minimal apparent motion sequences," Percept. Psychophys. 48, 419-430 (1990).
[CrossRef] [PubMed]

Brookes, A.

K. A. Stevens and A. Brookes, "Integrating stereopsis with monocular interpretations of planar surfaces," Vision Res. 28, 371-386 (1998).
[CrossRef]

Caprile, B.

B. Caprile and V. Torre, "Using vanishing points for camera calibration," Int. J. Comput. Vis. 4, 127-140 (1990).
[CrossRef]

Caudek, C.

F. Domini, C. Caudek, and S. Richmann, "Distortions of depth-order relations and parallelism in structure from motion," Percept. Psychophys. 60, 1164-1174 (1998).
[CrossRef] [PubMed]

Charman, W. N.

W. N. Charman, Visual Optics and Instrumentation (Macmillan, 1991).

Chellappa, R.

G. S. Young and R. Chellappa, "Statistical analysis of inherent ambiguities in recovering 3-D motion from a noisy flow field," IEEE Trans. Pattern Anal. Mach. Intell. 14, 995-1013 (1992).
[CrossRef]

Cheong, L. F.

L. F. Cheong and C. H. Peh, "Depth distortion under calibration uncertainty," Comput. Vis. Image Underst. 93, 221-244 (2004).
[CrossRef]

L. F. Cheong and T. Xiang, "Characterizing depth distortion under different generic motions," Int. J. Comput. Vis. 44, 199-217 (2001).
[CrossRef]

L. F. Cheong, C. Fermuller, and Y. Aloimonos, "Effects of errors in the viewing geometry on shape estimation," Comput. Vis. Image Underst. 71, 356-372 (1998).
[CrossRef]

L. F. Cheong, T. Xiang, V. Cornilleau-Pérès, and L. C. Tai, "Not all motions are equivalent in terms of depth recovery," in Computer Vision and Robotics, J.X.Liu, ed. (Nova, 2005), pp. 99-134.

L. F. Cheong and X. Xiang, "Error characteristics of sfm with unknown focal length," in Proceedings of Asian Conference on Computer Vision, P.J.Narayanan, S.K.Nayar and H.Y.Shum, eds. (Springer, 2006).

Coletta, N. J.

S. J. Galvin, D. R. Williams, and N. J. Coletta, "The spatial grain of motion perception in human peripheral vision," Vision Res. 36, 2283-2296 (1996).
[CrossRef] [PubMed]

Colombo, E.

P. Artal, A. Derrington, and E. Colombo, "Refraction, aliasing, and the absence of motion reversals in peripheral vision," Vision Res. 35, 939-947 (1995).
[CrossRef] [PubMed]

Cornilleau-Pérès, V.

V. Cornilleau-Pérès and J. Droulez, "The visual perception of 3D shape from self-motion and object-motion," Vision Res. 34, 2331-2336 (1994).
[CrossRef] [PubMed]

J. Droulez and V. Cornilleau-Pérès, "Visual perception of surface curvature, the spin variation and its physiological implications," Biol. Cybern. 62, 211-224 (1990).
[CrossRef] [PubMed]

V. Cornilleau-Pérès and J. Droulez, "Visual perception of surface curvature: Psychophysics of curvature detection induced by motion parallax," Percept. Psychophys. 46, 351-364 (1989).
[CrossRef] [PubMed]

L. F. Cheong, T. Xiang, V. Cornilleau-Pérès, and L. C. Tai, "Not all motions are equivalent in terms of depth recovery," in Computer Vision and Robotics, J.X.Liu, ed. (Nova, 2005), pp. 99-134.

Cornilleau-Pèrés, V.

V. Cornilleau-Pèrés, M. Wexler, J. Droulez, E. Marin, C. Miège, and B. Bourdoncle, "Visual perception of planar orientation: dominance of static depth cues over motion cues," Vision Res. 42, 1403-1412 (2002).
[CrossRef] [PubMed]

Cutting, J. E.

J. E. Cutting, "Rigidity of cinema seen from the front row, side aisle," J. Exp. Psychol. Hum. Percept. Perform. 13, 323-334 (1987).
[CrossRef] [PubMed]

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J. S. Tittle, J. T. Todd, V. J. Perotti, and J. F. Norman, "Systematic distortion of perceived three-dimensional structure from motion and binocular stereopsis," J. Exp. Psychol. Hum. Percept. Perform. 21, 663-678 (1995).
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J. E. Cutting, "Rigidity of cinema seen from the front row, side aisle," J. Exp. Psychol. Hum. Percept. Perform. 13, 323-334 (1987).
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W. C. Hoffman, "The lie algebra of visual perception," J. Math. Psychol. 3, 65-98 (1966).
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J. Trommershauser, S. Gepshtein, L. T. Maloney, M. S. Landy, and M. S. Banks, "Optimal compensation for changes in task-relevant movement variability," J. Neurosci. 25, 7169-7178 (2005).
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J. Opt. Soc. Am. A (1)

J. Vision (1)

J. M. Hillis, S. J. Watt, M. S. Landy, and M. S. Banks, "Slant from texture and disparity cues: optimal cue combination," J. Vision 4, 967-992 (2004).
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Percept. Psychophys. (10)

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D. N. Perkins, "Compensating for distortion in viewing pictures obliquely," Percept. Psychophys. 14, 13-18 (1973).
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Figures (8)

Fig. 1
Fig. 1

Image formation model: O is the optical center. The optical axis is aligned with the Z axis and the horizontal and vertical image axes are aligned with the X and the Y axis, respectively. A world point P = ( X , Y , Z ) T is projected to its image pixel coordinate ( x , y ) . The focal length is denoted by f and the principal point by ( o x , o y ) .

Fig. 2
Fig. 2

Three-D camera motion with a translational velocity v = ( U , V , W ) T and a rotational velocity w = ( α , β , γ ) T . The motion induces a relative motion between the static scene point P and the camera.

Fig. 3
Fig. 3

Simple cinema viewing configuration with the optical axes of the viewer and the projector parallel, a situation applicable to most viewers who are seated not near the side or right at the front. x p , x s , x v represent, respectively, the feature points on the projector film, screen, and viewer’s retina corresponding to the same world point. The distances (along the Z axis) from the screen to the projector and to the viewer are D p and D v , respectively. (a) Optical axes of viewer and projector are coincident. (b) Optical axes of viewer and projector are not coincident but are parallel to each other.

Fig. 4
Fig. 4

Configuration in which the viewer’s and projector’s optical axes are parallel but not coincident. The viewer is displaced from the CVP by ( t x , t y , t z ) .

Fig. 5
Fig. 5

General configuration with a slant angle ϕ between the viewer’s optical axis and the vertical axis. The view of a viewer seated at CVP (with feature x v ) is related to that of a viewer seated at a general position (with feature x v ) via a homography induced by the screen plane.

Fig. 6
Fig. 6

Camera operations: (a) Basic terminologies for translational and rotational operations. (b) Typical camera operation on rail that results in lateral translation and rotation.

Fig. 7
Fig. 7

Families of isodistortion contours for lateral motion obtained by intersecting the isodistortion surfaces with the x Z plane. F o V = 53 ° , f = f v = 309.0 , U = V = 0.81 , β = 0.002 , α = 0.002 . (a) Viewer at CVP with errors only in the 3-D motion estimates: U ̂ = 1.0 , β ̂ = 0.001 . (b) Viewer with optical axis parallel to and coincident with the projector’s optical axis: U ̂ = 1.0 , β ̂ = 0.001 , f ̂ v = 303.0 . (c) Viewer with optical axis parallel to but not coincident with the projector’s optical axis: U ̂ = 1.0 , V ̂ = 1.0 , α ̂ = 0.001 , β ̂ = 0.001 , f ̂ v = 303.0 , o x = o y = 10,000 .

Fig. 8
Fig. 8

Families of isodistortion contours for forward motion. (a) Viewer seated at CVP, f v = 309.0 , β e = 0.001 , α e = 0.001 . (b) Viewer seated on the optical axis of the projector with D v < D c , f v = 309.0 , f ̂ v = 303.0 , β = 0.002 , β ̂ = 0.001 , α = 0.002 , α ̂ = 0.001 . INF stands for infinity.

Equations (47)

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p = ( x y 1 ) = K Π 0 P = [ f s θ o x 0 f o y 0 0 1 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 ] ( X Y Z 1 ) ,
P ̇ = v w × p ,
u = W Z x f U Z + x y f α f ( 1 + x 2 f 2 ) β + γ y ,
v = W Z y f V Z x y f β + f ( 1 + y 2 f 2 ) α γ x ,
Z = ( x x 0 , y y 0 ) n ( u u rot , v v rot ) n .
x p = f p x s D p ,
x v = f v x s D v = x p k ,
k = D v f p D p f v .
u p = f p u s D p ,
u v = f v u s D v = f v D p u p D v f p = u p k .
u p = W Z ( x p f c U W ) + α x p y p f c β ( x p 2 f c + f c ) + γ y ,
v p = W Z ( y p f c V W ) β x p y p f c + α ( y p 2 f c + f c ) γ y ,
u v = W Z ( x v f c U k W ) + α x v y v f c k β ( x v 2 f c k + f c k ) + γ y v .
u v = W Z ( x v + t x f v D v ) f v U Z + ( x v + t x f v D v ) ( y v + t y f v D v ) f v α f v ( 1 + ( x v + t x f v D v ) 2 f v 2 ) β + γ ( y v + t y f v D v ) ,
v v = W Z ( y v + t y f v D v ) f v V Z ( x v + t x f v D v ) ( y v + t y f v D v ) f v β + f v ( 1 + ( y v + t y f v D v ) 2 f v 2 ) α γ ( x v + t x f v D v ) .
x v = H x v ,
H = K v ( R + 1 D c t N T ) K v 1 .
K v = [ f v 0 0 0 f v 0 0 0 1 ] .
R = [ cos ϕ 0 sin ϕ 0 1 0 sin ϕ 0 cos ϕ ] ,
t = [ t x 0 t z ] T .
H = K v [ cos ϕ 0 sin ϕ t x D c 0 1 0 sin ϕ 0 cos ϕ t z D c ] K v 1 .
x v = H x v = H K v [ I 0 ] X ,
H = H K v = [ f v cos ϕ 0 f v sin ϕ t x D c f v 0 f v 0 sin ϕ 0 cos ϕ t z D c ] .
cos ϕ t z D c D c t z D c = D v D c .
H = H K v [ f v D c D v cos θ 0 t x D v f v 0 f v D c D v 0 D c D v sin θ 0 1 ]
x v = f v D c D v X cos θ t x D v f v Z Z + D c D v X sin θ ,
y v = f v Y Z + D c D v X sin θ .
Z ̂ = Z ( ( x x ̂ 0 , y y ̂ 0 ) n ( x x 0 , y y 0 ) n + Z ( u rot e , v rot e ) n ) .
D = ( x x ̂ 0 ) 2 + ( y y ̂ 0 ) 2 ( x x 0 , y y 0 ) ( x x ̂ 0 , y y ̂ 0 ) + Z ( u rot e , v rot e ) ( x x ̂ 0 , y y ̂ 0 ) .
[ X ̂ , Y ̂ , Z ̂ , W ̂ ] = [ ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ] ,
D = f ̂ U ̂ f U + ( β f β ̂ f ̂ ) Z ,
D = x 2 + y 2 [ ( x o x e ) x + ( y o y e ) y ] + [ ( β f β ̂ f ̂ ) x + ( α f α ̂ f ̂ ) y ] Z .
u v = f v U Z + ( x v o x ) ( y v o y ) f v α f v ( 1 + ( x v o x ) 2 f v 2 ) β + γ ( y v o y ) ,
v v = f v V Z ( x v o x ) ( y v o y ) f v β + f v ( 1 + ( y v o y ) 2 f v 2 ) α γ ( x v o x ) ,
( o x , o y ) = ( t x f v D v , t y f v D v ) .
D = f ̂ v U ̂ f v U + ( β f v β ̂ f ̂ v ) Z ,
D = f ̂ v U ̂ f v U + [ ( β f v β ̂ f ̂ v ) + O 2 ( x v , y v ) ] Z ,
O 2 ( x v , y v ) = β ( x v o x ) 2 f v α ( x v o x ) ( y v o y ) f v β ̂ ( x v o ̂ x ) 2 f ̂ v + α ̂ ( x v o ̂ x ) 2 ( y v o ̂ y ) f ̂ v .
b = ( β f v β ̂ f ̂ v ) + O 2 ( x v , y v ) f ̂ v U ̂ ,
D = ( x v o ̂ x ) 2 + ( y v o ̂ y ) 2 ( x v o x ) ( x v o ̂ x ) + ( y v o ̂ y ) ( y v o ̂ y ) + O Z ,
O = ( β f v β ̂ f ̂ v ) ( x v o ̂ x ) + ( α f v α ̂ f ̂ v ) ( y v o ̂ y ) ;
D = ( x v o ̂ x ) 2 + ( y v o ̂ y ) 2 ( x v o x ) ( x v o ̂ x ) + ( y v o y ) ( y v o ̂ y ) + O Z ,
O = β ̑ f ( x v o ̂ x ) + α ̑ f ( y v o ̂ y ) ,
β ̑ f = β f v β ̂ f ̂ v + O x 2 ( x v , y v ) ,
α ̑ f = α f v α ̂ f ̂ v + O y 2 ( x v , y v ) ;
O x 2 ( x v , y v ) = β ( x v o x ) 2 f v + α ( x v o x ) ( y v o y ) f v + β ̂ ( x v o ̂ x ) 2 f ̂ v α ̂ ( x v o ̂ x ) ( y v o ̂ y ) f ̂ v ,
O y 2 ( x v , y v ) = α ( y v o y ) 2 f v + β ( x v o x ) ( y v o y ) f v + α ̂ ( y v o ̂ y ) 2 f ̂ v β ̂ ( x v o ̂ x ) ( y v o ̂ y ) f ̂ v .

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