Abstract

The centroid method is a common procedure for subpixel location that is applied to a large number of optical sensors. In practice, it is always accompanied by thresholding algorithms used to eliminate undesirable background that may decrease precision. We present a full analytical description of the interaction between centroiding and thresholding applied over an intensity distribution corrupted by additive Gaussian noise. An in depth analysis of the most outstanding statistical properties of this relation (mean and variance) is also presented by means of simulated and experimental data. This work provides fundamental concepts to the designers of sensors that are based on centroid measurements to allow them to use thresholding correctly before centroid computation.

© 2004 Optical Society of America

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  2. R. C. Stone, “A comparison of digital centering algorithms,” Astron. J. 97, 1227–1237 (1989).
    [CrossRef]
  3. R. H. Garstang, “Hyperfine structure and the broadening of sunspot spectral lines,” J. Opt. Soc. Am. B 2, 311–313 (1984).
    [CrossRef]
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    [CrossRef]
  5. G. A. West, T. A. Clarke, “A survey and examination of subpixel measurement techniques,” in Close-Range Photogrammetry Meets Machine Vision, A. Gruen, E. P. Baltsavias, eds., Proc. SPIE1395, 456–463 (1990).
  6. M. R. Shortis, T. A. Clarke, T. Short, “A comparison of some techniques for the subpixel location of discrete target images,” in Videometrics II, S. F. El-Hakim, ed., Proc. SPIE2350, 239–250 (1994).
    [CrossRef]
  7. J. P. Fillard, “Subpixel accuracy location estimation from digital signals,” Opt. Eng. 31, 2465–2471 (1992).
    [CrossRef]
  8. W. Ruyten, “Subpixel localization of synthetic references in digital images by use of an augmented template,” Opt. Eng. 41, 601–607 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  23. J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.
  24. J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley–Interscience, New York, 1971).
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    [CrossRef]
  29. T. Metz, J. Walewski, C. F. Kaminski, “Maximum-likelihood curve fitting scheme for experiments with pulsed laser subject to intensity fluctuations,” Appl. Opt. 42, 1551–1563 (2003).
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  30. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]

2003 (1)

2002 (2)

J. Arines, J. Ares, “Minimum variance centroid thresholding,” Opt. Lett. 27, 497–499 (2002).
[CrossRef]

W. Ruyten, “Subpixel localization of synthetic references in digital images by use of an augmented template,” Opt. Eng. 41, 601–607 (2002).
[CrossRef]

2001 (2)

2000 (2)

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

J. Ares, T. Mancebo, S. Bará, “Position and displacement sensing with Shack-Hartmann wave-front sensors,” Appl. Opt. 39, 1511–1520 (2000).
[CrossRef]

1997 (1)

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of ray pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef] [PubMed]

1995 (2)

C. S. Fraser, M. R. Shortis, “Metric exploitation of still video imagery,” Photogrammet. Rec. 15, 107–122 (1995).
[CrossRef]

T. A. Clarke, “A frame grabber related error in subpixel target location,” Photogramet. Rec. 15, 315–322 (1995).
[CrossRef]

1994 (2)

G. Cao, X. Yu, “Accuracy analysis of Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994); (some small typographical errors in the formula appeared in the original paper and were corrected here).
[CrossRef]

R. G. Dorsch, G. Häusler, J. M. Herrmann, “Laser triangulation: fundamental uncertainty in distance measurement,” Appl. Opt. 33, 1306–1314 (1994).
[CrossRef] [PubMed]

1992 (1)

J. P. Fillard, “Subpixel accuracy location estimation from digital signals,” Opt. Eng. 31, 2465–2471 (1992).
[CrossRef]

1991 (2)

B. F. Alexander, K. Chew, “Elimination of systematic error in subpixel accuracy estimation,” Opt. Eng. 30, 1320–1331 (1991).
[CrossRef]

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

1990 (1)

1989 (2)

1985 (1)

R. Armstrong, D. Taley, “A survey of current solid state star tracker technology,” J. Astronaut. Sci. 33, 341–352 (1985).

1984 (1)

R. H. Garstang, “Hyperfine structure and the broadening of sunspot spectral lines,” J. Opt. Soc. Am. B 2, 311–313 (1984).
[CrossRef]

1982 (1)

G. A. Tyller, D. L. Fried, “Image-position error associated with a quadrant detector,” J. Opt. Soc. Am. A 72, 804–808 (1982).
[CrossRef]

1980 (1)

1971 (1)

B. Platt, R. V. Shack, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656–660 (1971).

Adrian, R. J.

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Alexander, B. F.

B. F. Alexander, K. Chew, “Elimination of systematic error in subpixel accuracy estimation,” Opt. Eng. 30, 1320–1331 (1991).
[CrossRef]

Ares, J.

Arines, J.

Armstrong, R.

R. Armstrong, D. Taley, “A survey of current solid state star tracker technology,” J. Astronaut. Sci. 33, 341–352 (1985).

Bará, S.

Bendat, J. S.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley–Interscience, New York, 1971).

Cao, G.

G. Cao, X. Yu, “Accuracy analysis of Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994); (some small typographical errors in the formula appeared in the original paper and were corrected here).
[CrossRef]

Chew, K.

B. F. Alexander, K. Chew, “Elimination of systematic error in subpixel accuracy estimation,” Opt. Eng. 30, 1320–1331 (1991).
[CrossRef]

Chutatape, O.

Clarke, T. A.

T. A. Clarke, “A frame grabber related error in subpixel target location,” Photogramet. Rec. 15, 315–322 (1995).
[CrossRef]

M. R. Shortis, T. A. Clarke, T. Short, “A comparison of some techniques for the subpixel location of discrete target images,” in Videometrics II, S. F. El-Hakim, ed., Proc. SPIE2350, 239–250 (1994).
[CrossRef]

G. A. West, T. A. Clarke, “A survey and examination of subpixel measurement techniques,” in Close-Range Photogrammetry Meets Machine Vision, A. Gruen, E. P. Baltsavias, eds., Proc. SPIE1395, 456–463 (1990).

Dorsch, R. G.

Fang, H.

Fillard, J. P.

J. P. Fillard, “Subpixel accuracy location estimation from digital signals,” Opt. Eng. 31, 2465–2471 (1992).
[CrossRef]

Fontanella, J. C.

Fraser, C. S.

C. S. Fraser, M. R. Shortis, “Metric exploitation of still video imagery,” Photogrammet. Rec. 15, 107–122 (1995).
[CrossRef]

Fried, D. L.

G. A. Tyller, D. L. Fried, “Image-position error associated with a quadrant detector,” J. Opt. Soc. Am. A 72, 804–808 (1982).
[CrossRef]

Garstang, R. H.

R. H. Garstang, “Hyperfine structure and the broadening of sunspot spectral lines,” J. Opt. Soc. Am. B 2, 311–313 (1984).
[CrossRef]

González, R. C.

R. C. González, R. E. Woods, “Image segmentation,” in Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), pp. 443–455.

Haralick, R. M.

R. M. Haralick, L. G. Shapiro, “Binary machine vision,” in Computer and Robot Vision (Addison-Wesley, Reading, Mass., 1992), Vol. I, pp. 14–28.

Häusler, G.

Herrmann, J. M.

Jenkins, E. B.

Kaminski, C. F.

Keeping, E. S.

J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.

Kenney, J. F.

J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.

Losada, M. A.

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of ray pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef] [PubMed]

Luo, G.

Mancebo, T.

Meng, H.

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

Metz, T.

Morgan, J.

Navarro, R.

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of ray pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Piersol, A. G.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley–Interscience, New York, 1971).

Platt, B.

B. Platt, R. V. Shack, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656–660 (1971).

Primot, J.

Pu, Y.

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

Rousset, G.

Ruyten, W.

W. Ruyten, “Subpixel localization of synthetic references in digital images by use of an augmented template,” Opt. Eng. 41, 601–607 (2002).
[CrossRef]

Shack, R. V.

B. Platt, R. V. Shack, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656–660 (1971).

Shapiro, L. G.

R. M. Haralick, L. G. Shapiro, “Binary machine vision,” in Computer and Robot Vision (Addison-Wesley, Reading, Mass., 1992), Vol. I, pp. 14–28.

Short, T.

M. R. Shortis, T. A. Clarke, T. Short, “A comparison of some techniques for the subpixel location of discrete target images,” in Videometrics II, S. F. El-Hakim, ed., Proc. SPIE2350, 239–250 (1994).
[CrossRef]

Shortis, M. R.

C. S. Fraser, M. R. Shortis, “Metric exploitation of still video imagery,” Photogrammet. Rec. 15, 107–122 (1995).
[CrossRef]

M. R. Shortis, T. A. Clarke, T. Short, “A comparison of some techniques for the subpixel location of discrete target images,” in Videometrics II, S. F. El-Hakim, ed., Proc. SPIE2350, 239–250 (1994).
[CrossRef]

Slater, D. C.

Southwell, W. H.

Stone, R. C.

R. C. Stone, “A comparison of digital centering algorithms,” Astron. J. 97, 1227–1237 (1989).
[CrossRef]

Taley, D.

R. Armstrong, D. Taley, “A survey of current solid state star tracker technology,” J. Astronaut. Sci. 33, 341–352 (1985).

Timothy, J. G.

Tyller, G. A.

G. A. Tyller, D. L. Fried, “Image-position error associated with a quadrant detector,” J. Opt. Soc. Am. A 72, 804–808 (1982).
[CrossRef]

Walewski, J.

West, G. A.

G. A. West, T. A. Clarke, “A survey and examination of subpixel measurement techniques,” in Close-Range Photogrammetry Meets Machine Vision, A. Gruen, E. P. Baltsavias, eds., Proc. SPIE1395, 456–463 (1990).

Woods, R. E.

R. C. González, R. E. Woods, “Image segmentation,” in Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), pp. 443–455.

Yu, X.

G. Cao, X. Yu, “Accuracy analysis of Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994); (some small typographical errors in the formula appeared in the original paper and were corrected here).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Appl. Opt. (5)

Astron. J. (1)

R. C. Stone, “A comparison of digital centering algorithms,” Astron. J. 97, 1227–1237 (1989).
[CrossRef]

Exp. Fluids (1)

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

J. Astronaut. Sci. (1)

R. Armstrong, D. Taley, “A survey of current solid state star tracker technology,” J. Astronaut. Sci. 33, 341–352 (1985).

J. Opt. Soc. Am. (2)

B. Platt, R. V. Shack, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656–660 (1971).

W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

R. H. Garstang, “Hyperfine structure and the broadening of sunspot spectral lines,” J. Opt. Soc. Am. B 2, 311–313 (1984).
[CrossRef]

Opt. Eng. (4)

J. P. Fillard, “Subpixel accuracy location estimation from digital signals,” Opt. Eng. 31, 2465–2471 (1992).
[CrossRef]

W. Ruyten, “Subpixel localization of synthetic references in digital images by use of an augmented template,” Opt. Eng. 41, 601–607 (2002).
[CrossRef]

B. F. Alexander, K. Chew, “Elimination of systematic error in subpixel accuracy estimation,” Opt. Eng. 30, 1320–1331 (1991).
[CrossRef]

G. Cao, X. Yu, “Accuracy analysis of Hartmann-Shack wavefront sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994); (some small typographical errors in the formula appeared in the original paper and were corrected here).
[CrossRef]

Opt. Lett. (2)

Optom. Vis. Sci. (1)

R. Navarro, M. A. Losada, “Aberrations and relative efficiency of ray pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef] [PubMed]

Photogramet. Rec. (1)

T. A. Clarke, “A frame grabber related error in subpixel target location,” Photogramet. Rec. 15, 315–322 (1995).
[CrossRef]

Photogrammet. Rec. (1)

C. S. Fraser, M. R. Shortis, “Metric exploitation of still video imagery,” Photogrammet. Rec. 15, 107–122 (1995).
[CrossRef]

Other (7)

G. A. West, T. A. Clarke, “A survey and examination of subpixel measurement techniques,” in Close-Range Photogrammetry Meets Machine Vision, A. Gruen, E. P. Baltsavias, eds., Proc. SPIE1395, 456–463 (1990).

M. R. Shortis, T. A. Clarke, T. Short, “A comparison of some techniques for the subpixel location of discrete target images,” in Videometrics II, S. F. El-Hakim, ed., Proc. SPIE2350, 239–250 (1994).
[CrossRef]

R. C. González, R. E. Woods, “Image segmentation,” in Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), pp. 443–455.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

J. F. Kenney, E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, Princeton, N.J., 1951), Part II.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley–Interscience, New York, 1971).

R. M. Haralick, L. G. Shapiro, “Binary machine vision,” in Computer and Robot Vision (Addison-Wesley, Reading, Mass., 1992), Vol. I, pp. 14–28.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup and simulated noiseless intensity distribution.

Fig. 2
Fig. 2

Grouped by pairs, the evolution of the mean and variance centroid values as a function of the threshold level for various Gaussian additive noise levels ση = {20, 10, 3, 10-6}. (a), (c), (e), (g), Evolution of the mean centroid: circles, simulated mean centroid estimation [Eqs. (20)]; solid curve, theoretically predicted mean centroid [relation (12)]; dashed–dotted curve, mean centroid value expected from the classical model of zero-mean noise. The dark gray regions show the theoretically predicted interval [relation (10)]. (b), (d), (f), (h) Similar evolution curves for the simulated [Eqs. (20)] and two theoretically predicted [relations (2) and (18)] centroid variances. Note that for the variance the dark gray region is delimited only by the theoretically predicted upper bound [relation (17)]. Nevertheless, in both cases we also show the 95% confidence intervals of the simulated mean and variance centroid estimations as light gray regions [relations (21, )]. Insets, enlarged view of the most representative part of each figure.

Fig. 3
Fig. 3

Evolution of the mean number of pixels that compose the centroided intensity distribution relative to the threshold value (left axis, darker curves); superimposed, the percentage standard deviation of the total intensity normalized to the mean total intensity (right axis, lighter curves). The curves correspond to the simulated noisy cases of Fig. 2 as follows: ση = 20, dashed; ση = 10, dotted-dashed; ση = 3, dotted: ση = 10-6, solid.

Fig. 4
Fig. 4

With the same format as Fig. 2, evolution of the mean and variance centroid values as a function of the threshold level for (a) simulated (N = 3000) and (b) theoretical estimates. The experimentally estimated values of the mean and variance centroid [Eqs. (20), N = 1500] were also plotted as crosses (×) with the 95% confidence intervals [expressions (21, )] represented as vertical error bars. The estimated value of the noise standard deviation was ση = 0.34 gray levels. (c) As in Fig. 3, solid curves, mean number of pixels that compose the centroided distribution and the normalized percentage standard deviation of the total intensity; crosses, experimentally estimated values of both magnitudes.

Equations (31)

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

Xc=i xiIi/i Ii.
σXc2=ση2IT2L3-L12.
Iui=Iiif IiU0if Ii<U.
Hi=1ηiU-Ii0ηi<U-Ii.
Xc= HiIixi HiIi=ZIT,
covA, B=AB-AB,
Xc=Z1IT+covZ, 1IT,
Xc=Z1IT±σZσ1/IT,
1IT1IT+σIT2IT3,
σ1ITσITIT2.
XcZIT+ZσIT2IT3±σZσITIT2.
Z= HiIixi+ Hiηixi,
IT= HiIi+ Hiηi,
σZ2= xi2Ii2Hi-Hi2+ xi2Hiηi2-Hiηi2+2  xi2IiHiηi1-Hi,
σIT2= Ii2Hi-Hi2+ Hiηi2-Hiηi2+2  IiHiηi1-Hi.
Xc HiIixi+ Hiηixi HiIi+ Hiηi.
Hi=U-Ii12πση exp-η22ση2dη,
Hiηi=U-Ii η 12πση exp-η22ση2dη,
Hiηi2=U-Ii η212πση exp-η22ση2dη,
σXc2=Xc2-Xc2=ZIT2-ZIT2.
σXc2=Z21IT2+covZ2, 1IT2-Z21IT2-covZ, 1IT2-2Z1ITcovZ, 1IT.
σXc2Z21IT2-Z21IT2+2Z1ITσZσ1/IT =σZ2σ1/IT2+σZ21IT2+Z2σ1/IT2+2Z×1ITσZσ1/IT.
σXc2σZ2σIT2IT4+σZ21IT+σIT2IT32+σIT2IT4 Z2+2 ZIT21IT+σIT2IT3σZσIT.
σXc2 xi2Ii2Hi-Hi2+ xi2ηi2Hi-ηiHi2+2  xi2IiηiHi1-Hi HiIi+ Hiηi2,
σXc2= ση2xi2Ii2=ση2Ii2L3-L12
Xˆc=N-1i=1NXci, σˆXc2=N-1-1i=1NXci-Xˆc2.
N-1i=1NXci-σˆXctN-1;0.025N, N-1i=1NXci+σˆXctN-1;0.025N,
N-1σˆXc2χN-1;0.0252, N-1σˆXc2χN-1;0.9752,
Hi=U-Ii12πση exp-η22ση2dη=121-erfU-Ii2ση,
Hiηi=U-Ii η 12πση exp-η22ση2dη=12π ση exp-U-Ii22ση2,
Hiηi2=U-Ii η212πση exp-η22ση2dη=12π σηU-Iiexp-U-Ii22ση2+12 ση21-erfU-Ii2ση.

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