Abstract

We explore the effect that an image-formation system’s point-spread function (PSF) has on measurement of the orientation and position of step edges. The theory of locales is used for numerical computation of bounds on precision as a function of PSF size and shape. It is shown that in the noise-free case the PSF should be 0.5–0.9 pixel in radius for optimal precision to be achieved. PSF shape is shown to have a much lesser effect on precision. A one-dimensional noisy-edge model is analyzed, and results similar to the no-noise case are obtained.

© 1994 Optical Society of America

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References

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  1. Y. C. Chu, “Resolution and accuracy in gray-level machine vision,” in VISION ’86 Conference Proceedings (Society of Manufacturing Engineers, Dearborn, Mich., 1986), pp. 6.1–6.13.
  2. R. A. Young, “Locating industrial parts with sub-pixel accuracies,” in Optics, Illumination, and Image Sensing for Machine Vision, D. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.728, 2–9 (1986).
    [CrossRef]
  3. S. H. Zisk, N. Wittels, “Camera edge response,” in Optics, Illumination, and Image Sensing for Machine Vision II, D. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.850, 9–16 (1987).
    [CrossRef]
  4. J. S. Chen, G. Medioni, “Detection, localization, and estimation of edges,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 191–198 (1989).
    [CrossRef]
  5. R. M. Haralick, “Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI–6, 58–68 (1984).
    [CrossRef]
  6. W. E. L. Grimson, E. C. Hildreth, “Comments on: Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-7, 121–127 (1985).
    [CrossRef]
  7. M. Hiraoglu, G. Nagy, “Digitizing-camera characteristics and edge locating accuracy in machine vision,” in VISION ’89 Conference Proceedings (Society of Manufacturing Engineers, Dearborn, Mich., 1989).
  8. J. W. Burnett, T. S. Huang, “Image mensuration by maximum a posterioriprobability estimation,” J. Opt. Soc. Am. 68, 157–166 (1978).
    [CrossRef]
  9. V. S. Nalwa, T. O. Binford, “On detecting edges,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-8, 699–714 (1986).
    [CrossRef]
  10. V. S. Nalwa, “Edge detector resolution improvement by image interpolation,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 446–451 (1987).
    [CrossRef]
  11. N. Haylo, R. W. Samms, “Combined optimization of image-gathering optics and image-processing algorithms for edge detection,” J. Opt. Soc. Am. A 3, 1522–1536 (1986).
    [CrossRef]
  12. A. J. Tabatabai, O. R. Mitchell, “Edge location to subpixel values in digital imagery,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 188–201 (1984).
    [CrossRef]
  13. E. M. Mikhail, M. L. Akey, O. R. Mitchell, “Detection and sub-pixel location of photogrammetric targets in digital images,” Photogrammetria 39, 63–83 (1984).
    [CrossRef]
  14. E. P. Lyvers, O. R. Mitchell, M. L. Akel, A. P. Reeves, “Subpixel measurements using a moment-based edge operator,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1293–1309 (1989).
    [CrossRef]
  15. L. J. Kitchen, J. A. Malin, “The effect of spatial discretization on the magnitude and direction response of simple differential edge operators on a step edge,” Comput. Vis. Graphics Image Process. 47, 243–258 (1989).
    [CrossRef]
  16. E. P. Lyvers, O. R. Mitchell, “Precision edge contrast and orientation estimation,” IEEE Trans. Patt. Anal. Mach. Intell. 10, 927–937 (1988).
    [CrossRef]
  17. Z. Kulpa “Area and perimeter measurement of blobs in discrete binary pictures,” Comput. Graphics Image Process. 6, 434–451 (1977).
    [CrossRef]
  18. C. S. Ho, “Precision of digital vision systems,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 593–601 (1983).
    [CrossRef]
  19. H. Klaasman, “Some aspects of the accuracy of the approximated position of a straight line of a square grid,” Comput. Graphics Image Process. 4, 225–235 (1975).
    [CrossRef]
  20. D. I. Havelock, “Geometric precision in noise-free digital images,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1065–1075 (1989).
    [CrossRef]
  21. D. I. Havelock, “Optimal estimation of planar position and orientation for straight edges,” in Proceedings of the Conference on Optical 3-D Measurement Techniques, (Herbert Wichmann Verlag GmbH, Karlsruhe, Ger., 1989), pp. 470–482.
  22. D. I. Havelock, “The topology of locales and its effects on position uncertainty,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 380–386 (1991).
    [CrossRef]
  23. R. G. White, R. A. Schowengerdt, “Effects of point-spread functions on the edge orientation precision of first derivative operators,” Opt. Eng. 31, 2239–2245 (1992).
    [CrossRef]

1992 (1)

R. G. White, R. A. Schowengerdt, “Effects of point-spread functions on the edge orientation precision of first derivative operators,” Opt. Eng. 31, 2239–2245 (1992).
[CrossRef]

1991 (1)

D. I. Havelock, “The topology of locales and its effects on position uncertainty,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 380–386 (1991).
[CrossRef]

1989 (4)

D. I. Havelock, “Geometric precision in noise-free digital images,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1065–1075 (1989).
[CrossRef]

J. S. Chen, G. Medioni, “Detection, localization, and estimation of edges,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 191–198 (1989).
[CrossRef]

E. P. Lyvers, O. R. Mitchell, M. L. Akel, A. P. Reeves, “Subpixel measurements using a moment-based edge operator,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1293–1309 (1989).
[CrossRef]

L. J. Kitchen, J. A. Malin, “The effect of spatial discretization on the magnitude and direction response of simple differential edge operators on a step edge,” Comput. Vis. Graphics Image Process. 47, 243–258 (1989).
[CrossRef]

1988 (1)

E. P. Lyvers, O. R. Mitchell, “Precision edge contrast and orientation estimation,” IEEE Trans. Patt. Anal. Mach. Intell. 10, 927–937 (1988).
[CrossRef]

1987 (1)

V. S. Nalwa, “Edge detector resolution improvement by image interpolation,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 446–451 (1987).
[CrossRef]

1986 (2)

1985 (1)

W. E. L. Grimson, E. C. Hildreth, “Comments on: Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-7, 121–127 (1985).
[CrossRef]

1984 (3)

R. M. Haralick, “Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI–6, 58–68 (1984).
[CrossRef]

A. J. Tabatabai, O. R. Mitchell, “Edge location to subpixel values in digital imagery,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 188–201 (1984).
[CrossRef]

E. M. Mikhail, M. L. Akey, O. R. Mitchell, “Detection and sub-pixel location of photogrammetric targets in digital images,” Photogrammetria 39, 63–83 (1984).
[CrossRef]

1983 (1)

C. S. Ho, “Precision of digital vision systems,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 593–601 (1983).
[CrossRef]

1978 (1)

1977 (1)

Z. Kulpa “Area and perimeter measurement of blobs in discrete binary pictures,” Comput. Graphics Image Process. 6, 434–451 (1977).
[CrossRef]

1975 (1)

H. Klaasman, “Some aspects of the accuracy of the approximated position of a straight line of a square grid,” Comput. Graphics Image Process. 4, 225–235 (1975).
[CrossRef]

Akel, M. L.

E. P. Lyvers, O. R. Mitchell, M. L. Akel, A. P. Reeves, “Subpixel measurements using a moment-based edge operator,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1293–1309 (1989).
[CrossRef]

Akey, M. L.

E. M. Mikhail, M. L. Akey, O. R. Mitchell, “Detection and sub-pixel location of photogrammetric targets in digital images,” Photogrammetria 39, 63–83 (1984).
[CrossRef]

Binford, T. O.

V. S. Nalwa, T. O. Binford, “On detecting edges,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-8, 699–714 (1986).
[CrossRef]

Burnett, J. W.

Chen, J. S.

J. S. Chen, G. Medioni, “Detection, localization, and estimation of edges,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 191–198 (1989).
[CrossRef]

Chu, Y. C.

Y. C. Chu, “Resolution and accuracy in gray-level machine vision,” in VISION ’86 Conference Proceedings (Society of Manufacturing Engineers, Dearborn, Mich., 1986), pp. 6.1–6.13.

Grimson, W. E. L.

W. E. L. Grimson, E. C. Hildreth, “Comments on: Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-7, 121–127 (1985).
[CrossRef]

Haralick, R. M.

R. M. Haralick, “Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI–6, 58–68 (1984).
[CrossRef]

Havelock, D. I.

D. I. Havelock, “The topology of locales and its effects on position uncertainty,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 380–386 (1991).
[CrossRef]

D. I. Havelock, “Geometric precision in noise-free digital images,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1065–1075 (1989).
[CrossRef]

D. I. Havelock, “Optimal estimation of planar position and orientation for straight edges,” in Proceedings of the Conference on Optical 3-D Measurement Techniques, (Herbert Wichmann Verlag GmbH, Karlsruhe, Ger., 1989), pp. 470–482.

Haylo, N.

Hildreth, E. C.

W. E. L. Grimson, E. C. Hildreth, “Comments on: Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-7, 121–127 (1985).
[CrossRef]

Hiraoglu, M.

M. Hiraoglu, G. Nagy, “Digitizing-camera characteristics and edge locating accuracy in machine vision,” in VISION ’89 Conference Proceedings (Society of Manufacturing Engineers, Dearborn, Mich., 1989).

Ho, C. S.

C. S. Ho, “Precision of digital vision systems,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 593–601 (1983).
[CrossRef]

Huang, T. S.

Kitchen, L. J.

L. J. Kitchen, J. A. Malin, “The effect of spatial discretization on the magnitude and direction response of simple differential edge operators on a step edge,” Comput. Vis. Graphics Image Process. 47, 243–258 (1989).
[CrossRef]

Klaasman, H.

H. Klaasman, “Some aspects of the accuracy of the approximated position of a straight line of a square grid,” Comput. Graphics Image Process. 4, 225–235 (1975).
[CrossRef]

Kulpa, Z.

Z. Kulpa “Area and perimeter measurement of blobs in discrete binary pictures,” Comput. Graphics Image Process. 6, 434–451 (1977).
[CrossRef]

Lyvers, E. P.

E. P. Lyvers, O. R. Mitchell, M. L. Akel, A. P. Reeves, “Subpixel measurements using a moment-based edge operator,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1293–1309 (1989).
[CrossRef]

E. P. Lyvers, O. R. Mitchell, “Precision edge contrast and orientation estimation,” IEEE Trans. Patt. Anal. Mach. Intell. 10, 927–937 (1988).
[CrossRef]

Malin, J. A.

L. J. Kitchen, J. A. Malin, “The effect of spatial discretization on the magnitude and direction response of simple differential edge operators on a step edge,” Comput. Vis. Graphics Image Process. 47, 243–258 (1989).
[CrossRef]

Medioni, G.

J. S. Chen, G. Medioni, “Detection, localization, and estimation of edges,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 191–198 (1989).
[CrossRef]

Mikhail, E. M.

E. M. Mikhail, M. L. Akey, O. R. Mitchell, “Detection and sub-pixel location of photogrammetric targets in digital images,” Photogrammetria 39, 63–83 (1984).
[CrossRef]

Mitchell, O. R.

E. P. Lyvers, O. R. Mitchell, M. L. Akel, A. P. Reeves, “Subpixel measurements using a moment-based edge operator,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1293–1309 (1989).
[CrossRef]

E. P. Lyvers, O. R. Mitchell, “Precision edge contrast and orientation estimation,” IEEE Trans. Patt. Anal. Mach. Intell. 10, 927–937 (1988).
[CrossRef]

E. M. Mikhail, M. L. Akey, O. R. Mitchell, “Detection and sub-pixel location of photogrammetric targets in digital images,” Photogrammetria 39, 63–83 (1984).
[CrossRef]

A. J. Tabatabai, O. R. Mitchell, “Edge location to subpixel values in digital imagery,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 188–201 (1984).
[CrossRef]

Nagy, G.

M. Hiraoglu, G. Nagy, “Digitizing-camera characteristics and edge locating accuracy in machine vision,” in VISION ’89 Conference Proceedings (Society of Manufacturing Engineers, Dearborn, Mich., 1989).

Nalwa, V. S.

V. S. Nalwa, “Edge detector resolution improvement by image interpolation,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 446–451 (1987).
[CrossRef]

V. S. Nalwa, T. O. Binford, “On detecting edges,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-8, 699–714 (1986).
[CrossRef]

Reeves, A. P.

E. P. Lyvers, O. R. Mitchell, M. L. Akel, A. P. Reeves, “Subpixel measurements using a moment-based edge operator,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1293–1309 (1989).
[CrossRef]

Samms, R. W.

Schowengerdt, R. A.

R. G. White, R. A. Schowengerdt, “Effects of point-spread functions on the edge orientation precision of first derivative operators,” Opt. Eng. 31, 2239–2245 (1992).
[CrossRef]

Tabatabai, A. J.

A. J. Tabatabai, O. R. Mitchell, “Edge location to subpixel values in digital imagery,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 188–201 (1984).
[CrossRef]

White, R. G.

R. G. White, R. A. Schowengerdt, “Effects of point-spread functions on the edge orientation precision of first derivative operators,” Opt. Eng. 31, 2239–2245 (1992).
[CrossRef]

Wittels, N.

S. H. Zisk, N. Wittels, “Camera edge response,” in Optics, Illumination, and Image Sensing for Machine Vision II, D. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.850, 9–16 (1987).
[CrossRef]

Young, R. A.

R. A. Young, “Locating industrial parts with sub-pixel accuracies,” in Optics, Illumination, and Image Sensing for Machine Vision, D. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.728, 2–9 (1986).
[CrossRef]

Zisk, S. H.

S. H. Zisk, N. Wittels, “Camera edge response,” in Optics, Illumination, and Image Sensing for Machine Vision II, D. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.850, 9–16 (1987).
[CrossRef]

Comput. Graphics Image Process. (2)

Z. Kulpa “Area and perimeter measurement of blobs in discrete binary pictures,” Comput. Graphics Image Process. 6, 434–451 (1977).
[CrossRef]

H. Klaasman, “Some aspects of the accuracy of the approximated position of a straight line of a square grid,” Comput. Graphics Image Process. 4, 225–235 (1975).
[CrossRef]

Comput. Vis. Graphics Image Process. (1)

L. J. Kitchen, J. A. Malin, “The effect of spatial discretization on the magnitude and direction response of simple differential edge operators on a step edge,” Comput. Vis. Graphics Image Process. 47, 243–258 (1989).
[CrossRef]

IEEE Trans. Patt. Anal. Mach. Intell. (11)

E. P. Lyvers, O. R. Mitchell, “Precision edge contrast and orientation estimation,” IEEE Trans. Patt. Anal. Mach. Intell. 10, 927–937 (1988).
[CrossRef]

C. S. Ho, “Precision of digital vision systems,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-5, 593–601 (1983).
[CrossRef]

D. I. Havelock, “Geometric precision in noise-free digital images,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1065–1075 (1989).
[CrossRef]

D. I. Havelock, “The topology of locales and its effects on position uncertainty,” IEEE Trans. Patt. Anal. Mach. Intell. 13, 380–386 (1991).
[CrossRef]

J. S. Chen, G. Medioni, “Detection, localization, and estimation of edges,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 191–198 (1989).
[CrossRef]

R. M. Haralick, “Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI–6, 58–68 (1984).
[CrossRef]

W. E. L. Grimson, E. C. Hildreth, “Comments on: Digital step edges from zero crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-7, 121–127 (1985).
[CrossRef]

A. J. Tabatabai, O. R. Mitchell, “Edge location to subpixel values in digital imagery,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 188–201 (1984).
[CrossRef]

V. S. Nalwa, T. O. Binford, “On detecting edges,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-8, 699–714 (1986).
[CrossRef]

V. S. Nalwa, “Edge detector resolution improvement by image interpolation,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9, 446–451 (1987).
[CrossRef]

E. P. Lyvers, O. R. Mitchell, M. L. Akel, A. P. Reeves, “Subpixel measurements using a moment-based edge operator,” IEEE Trans. Patt. Anal. Mach. Intell. 11, 1293–1309 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

R. G. White, R. A. Schowengerdt, “Effects of point-spread functions on the edge orientation precision of first derivative operators,” Opt. Eng. 31, 2239–2245 (1992).
[CrossRef]

Photogrammetria (1)

E. M. Mikhail, M. L. Akey, O. R. Mitchell, “Detection and sub-pixel location of photogrammetric targets in digital images,” Photogrammetria 39, 63–83 (1984).
[CrossRef]

Other (5)

Y. C. Chu, “Resolution and accuracy in gray-level machine vision,” in VISION ’86 Conference Proceedings (Society of Manufacturing Engineers, Dearborn, Mich., 1986), pp. 6.1–6.13.

R. A. Young, “Locating industrial parts with sub-pixel accuracies,” in Optics, Illumination, and Image Sensing for Machine Vision, D. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.728, 2–9 (1986).
[CrossRef]

S. H. Zisk, N. Wittels, “Camera edge response,” in Optics, Illumination, and Image Sensing for Machine Vision II, D. J. Svetkoff, ed., Proc. Soc. Photo-Opt. Instrum. Eng.850, 9–16 (1987).
[CrossRef]

M. Hiraoglu, G. Nagy, “Digitizing-camera characteristics and edge locating accuracy in machine vision,” in VISION ’89 Conference Proceedings (Society of Manufacturing Engineers, Dearborn, Mich., 1989).

D. I. Havelock, “Optimal estimation of planar position and orientation for straight edges,” in Proceedings of the Conference on Optical 3-D Measurement Techniques, (Herbert Wichmann Verlag GmbH, Karlsruhe, Ger., 1989), pp. 470–482.

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Figures (9)

Fig. 1
Fig. 1

Each curve in the set is the locus of points (ρ, θ) representing the line parameters that can produce the pixel value E for a rectangular PSF with Wx = Wy = 0.5. Note that each curve has two values of E associated with it.

Fig. 2
Fig. 2

Two distance functions are shown, centered on their respective pixels. The dashed curves represent the distance function of the second pixel transformed with respect to the origin of the first. Each of the four points at which the dashed curve intersects the distance function of the first pixel represents a valid set of line parameters. These points are indicated with arrows, and one of the points is used to illustrate how the line is defined by a point. The ambiguity is resolved by inspection of pixel values at either side of the lines.

Fig. 3
Fig. 3

Loci on feasible region boundaries for 16 quantization levels for different PSF’s: (a) rectangular PSF, Wx = 0.5, Wy = 1.0; (b) defocus PSF, R = 1.0; (c) Gaussian PSF, σx = σy = 0.5; (d) Airy-pattern PSF, B = 3.0.

Fig. 4
Fig. 4

Illustration of how feasible regions change as the number of pixels increases: (a) one pixel, (b) two pixels, (c) three pixels, (d) four pixels.

Fig. 5
Fig. 5

Example of how locales partition the parameter space. The locales are shown as a function of increasing defocus PSF size (N = 3, Q = 4). The shaded locales illustrate how locales can change size, shape, and position. (a) R = 0.1, (b) R = 0.2, (c) R = 0.3, (d) R = 0.4.

Fig. 6
Fig. 6

(a) Worst-case precision in ρ as a function of varying PSF size for Q = 4, N = 7; (b) worst-case precision in θ as a function of varying PSF size for Q = 4, N = 7.

Fig. 7
Fig. 7

(a) Average-case precision in ρ as a function of varying PSF size for Q = 8, N = 7; (b) average-case precision in θ as a function of varying PSF size for Q = 8, N = 7.

Fig. 8
Fig. 8

Average-case precision in ρ as a function of varying defocus PSF radius for Q = 4, N = 7, and two cases of rectangular sampling.

Fig. 9
Fig. 9

Rms error in ρ for an image of a vertical edge (M = 5, Q = 2) under varying defocus PSF size and noise standard deviation σ.

Tables (6)

Tables Icon

Table 1 Optimal PSF Sizes for Q = 4

Tables Icon

Table 2 Optimal PSF Sizes for Q = 8

Tables Icon

Table 3 Optimal PSF Sizes for Combined PSF’s (Q = 4, N = 7)

Tables Icon

Table 4 Optimal PSF Sizes for Combined PSF’s (Q = 8, N = 7)

Tables Icon

Table 5 Precision Values for Optimal Combined PSF’s (Q = 4, N = 7)a

Tables Icon

Table 6 Precision Values for Optimal Combined PSF’s (Q = 8, N = 7)a

Equations (23)

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

g ( x , y ) = f ( x , y ) h ( x , y )
h ( x , y ) = h d l ( x , y ) h def ( x , y ) h rect ( x , y )
h d l ( x , y ) = ( 2 J 1 ( B r ) B r ) 2 ,
h def ( x , y ) = { 1 / π R 2 if r 2 R 2 0 otherwise ,
h rect ( x , y ) = { 1 / 4 W x W y if - W x x W x and - W y y W y 0 otherwise ,
h Gauss ( x , y ) = K { 1 2 π σ x exp [ - ( x 2 / 2 σ x 2 ) ] } × { 1 2 π σ y exp [ - ( y 2 / 2 σ y 2 ) ] } .
f ( x , y ) = U ( ρ - x sin θ - y cos θ ) , U ( ξ ) = { 0 for ξ 0 1 for ξ > 0 .
- 1 / 2 ρ 1 / 2 ,             0 θ π / 4.
β θ = max R θ ( α θ ) ,
β ¯ θ = R θ α θ d θ ,
E i j = 1 4 W x W y × i - W x i + W x j - W y j + W y U ( ρ - x sin θ - y cos θ ) d y d x .
ρ 2 ( E i 2 j 2 ; θ 2 ) = r cos ( θ 2 - ψ 1 , 2 ) + ρ 2 ( E i 2 j 2 ; θ 2 ) .
L = { E Q + ½ Q for 0 E < 1 Q - ½ Q for E = 1 .
- ρ ^ P ρ ( ρ ) d ρ = ρ ^ P ρ ( ρ ) d ρ .
- π π 0 S PSF ( ρ , θ ) d ρ d θ = 0.8.
E i j = E i j + N i j .
P E i j [ E i j ( ρ , θ ) ] = P N [ N ; E i j ( ρ , θ ) , σ N ] .
P [ L i j ( ρ , θ ) ] = { - 1 / Q P [ E i j ( ρ , θ ) ] d E i j for L i j = 0.5 / Q L i j - 0.5 / Q L i j + 0.5 / Q P [ E i j ( ρ , θ ) ] d E i j for 1.5 / Q L i j ( Q - 1.5 ) / Q ( Q - 1 ) / Q P [ E i j ( ρ , θ ) ] d E i j for L i j = ( Q - 0.5 ) / Q .
P [ L ( ρ , θ ) ] = i = - n n j = - n n P [ L i j ( ρ , θ ) ] ;
ρ ^ ( L ) = - 0.5 0.5 0 π / 4 ρ P [ L ( ρ , θ ) ] d θ d ρ - 0.5 0.5 0 π / 4 P [ L ( ρ , θ ) ] d θ d ρ .
1 Q N 2 { - 0.5 0.5 0 π / 4 [ ρ - ρ ^ ( L ) ] 2 P [ L ( ρ , θ ) ] d θ d ρ - 0.5 0.5 0 π / 4 P [ L ( ρ , θ ) ] d θ d ρ } ,
Λ - = row = 1 M L - ( row ) ,
Λ + = row = 1 M L + ( row ) .

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