Abstract

We present a technique for determining the contrast of an intensity distribution in the presence of additive noise and other effects, such as undesired local amplitude or offset variations. The method is based on the variogram function. It just requires the measurement of the variogram at only four points and, as a consequence, it is very fast. The proposed technique is compared with other standard techniques, showing a reduction in the error of the contrast measurement.

© 2007 Optical Society of America

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References

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  1. G. C. Holst, CCD Arrays, Cameras, and Displays (Society for Photo-Optical Instrumentation Engineers, 1996).
  2. P. Hariharan, Optical Interferometry (Academic, 1989).
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  4. W. K. Pratt, Digital Image Processing (Wiley, 1978).
  5. International Standardization Organization, Guide to the Expression of the Uncertainty in Measurement, Geneva (ISO, 1995).
  6. E. W. Weisstein, "Root-mean-square," From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Root-Mean-Square.html.
  7. S. Lai and G. Von Bally, "Fringe contrast evaluation by means of histograms," in OPTIKA '98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, and P. András, eds. Proc. SPIE 3573, 384-387 (1998).
  8. L. M. Sanchez-Brea, J. A. Quiroga, A. Garcia-Botella, and E. Bernabeu, "Histogram-based method for contrast measurement," Appl. Opt. 39, 4098-4106 (2000).
    [CrossRef]
  9. R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, 1985).
  10. N. A. Cressie, Statistics for Spatial Data (Wiley, 1991).
  11. L. M. Sanchez-Brea and E. Bernabeu, "On the standard deviation in CCD cameras: a variogram-based technique for non-uniform images," J. Electron. Imaging 11, 121-126 (2002).
    [CrossRef]
  12. L. M. Sanchez-Brea and E. Bernabeu, "Estimation of the standard deviation in three-dimensional microscopy by spatial statistics," J. Microsc. 218, 193-197 (2005).
    [CrossRef] [PubMed]
  13. P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1969).
  14. E. Keren and O. Kafri, "Diffraction effects in moire deflectometry," J. Opt. Soc. Am. A 2, 111-120 (1985).
    [CrossRef]

2005

L. M. Sanchez-Brea and E. Bernabeu, "Estimation of the standard deviation in three-dimensional microscopy by spatial statistics," J. Microsc. 218, 193-197 (2005).
[CrossRef] [PubMed]

2002

L. M. Sanchez-Brea and E. Bernabeu, "On the standard deviation in CCD cameras: a variogram-based technique for non-uniform images," J. Electron. Imaging 11, 121-126 (2002).
[CrossRef]

2000

1998

S. Lai and G. Von Bally, "Fringe contrast evaluation by means of histograms," in OPTIKA '98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, and P. András, eds. Proc. SPIE 3573, 384-387 (1998).

1996

G. C. Holst, CCD Arrays, Cameras, and Displays (Society for Photo-Optical Instrumentation Engineers, 1996).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

1991

N. A. Cressie, Statistics for Spatial Data (Wiley, 1991).

1989

P. Hariharan, Optical Interferometry (Academic, 1989).

1985

R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, 1985).

E. Keren and O. Kafri, "Diffraction effects in moire deflectometry," J. Opt. Soc. Am. A 2, 111-120 (1985).
[CrossRef]

1978

W. K. Pratt, Digital Image Processing (Wiley, 1978).

1969

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1969).

Bernabeu, E.

L. M. Sanchez-Brea and E. Bernabeu, "Estimation of the standard deviation in three-dimensional microscopy by spatial statistics," J. Microsc. 218, 193-197 (2005).
[CrossRef] [PubMed]

L. M. Sanchez-Brea and E. Bernabeu, "On the standard deviation in CCD cameras: a variogram-based technique for non-uniform images," J. Electron. Imaging 11, 121-126 (2002).
[CrossRef]

L. M. Sanchez-Brea, J. A. Quiroga, A. Garcia-Botella, and E. Bernabeu, "Histogram-based method for contrast measurement," Appl. Opt. 39, 4098-4106 (2000).
[CrossRef]

Bevington, P.

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1969).

Christiensen, R.

R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, 1985).

Cressie, N. A.

N. A. Cressie, Statistics for Spatial Data (Wiley, 1991).

Garcia-Botella, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic, 1989).

Holst, G. C.

G. C. Holst, CCD Arrays, Cameras, and Displays (Society for Photo-Optical Instrumentation Engineers, 1996).

Kafri, O.

Keren, E.

Lai, S.

S. Lai and G. Von Bally, "Fringe contrast evaluation by means of histograms," in OPTIKA '98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, and P. András, eds. Proc. SPIE 3573, 384-387 (1998).

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, 1978).

Quiroga, J. A.

Sanchez-Brea, L. M.

L. M. Sanchez-Brea and E. Bernabeu, "Estimation of the standard deviation in three-dimensional microscopy by spatial statistics," J. Microsc. 218, 193-197 (2005).
[CrossRef] [PubMed]

L. M. Sanchez-Brea and E. Bernabeu, "On the standard deviation in CCD cameras: a variogram-based technique for non-uniform images," J. Electron. Imaging 11, 121-126 (2002).
[CrossRef]

L. M. Sanchez-Brea, J. A. Quiroga, A. Garcia-Botella, and E. Bernabeu, "Histogram-based method for contrast measurement," Appl. Opt. 39, 4098-4106 (2000).
[CrossRef]

Von Bally, G.

S. Lai and G. Von Bally, "Fringe contrast evaluation by means of histograms," in OPTIKA '98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, and P. András, eds. Proc. SPIE 3573, 384-387 (1998).

Weisstein, E. W.

E. W. Weisstein, "Root-mean-square," From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Root-Mean-Square.html.

Appl. Opt.

J. Electron. Imaging

L. M. Sanchez-Brea and E. Bernabeu, "On the standard deviation in CCD cameras: a variogram-based technique for non-uniform images," J. Electron. Imaging 11, 121-126 (2002).
[CrossRef]

J. Microsc.

L. M. Sanchez-Brea and E. Bernabeu, "Estimation of the standard deviation in three-dimensional microscopy by spatial statistics," J. Microsc. 218, 193-197 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Other

P. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1969).

R. Christiensen, Linear Models for Multivariate, Time Series, and Spatial Data (Springer-Verlag, 1985).

N. A. Cressie, Statistics for Spatial Data (Wiley, 1991).

G. C. Holst, CCD Arrays, Cameras, and Displays (Society for Photo-Optical Instrumentation Engineers, 1996).

P. Hariharan, Optical Interferometry (Academic, 1989).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

W. K. Pratt, Digital Image Processing (Wiley, 1978).

International Standardization Organization, Guide to the Expression of the Uncertainty in Measurement, Geneva (ISO, 1995).

E. W. Weisstein, "Root-mean-square," From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Root-Mean-Square.html.

S. Lai and G. Von Bally, "Fringe contrast evaluation by means of histograms," in OPTIKA '98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, and P. András, eds. Proc. SPIE 3573, 384-387 (1998).

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Figures (6)

Fig. 1
Fig. 1

(a) Signal I ( x ) = 5 + sin ( 2 π x ) + r ( x ) , where r ( x ) represent an additive Gaussian noise with standard deviation σ = 0.5 . (b) Variogram for this signal when σ = 0 , dashed curve, and σ = 0.5 , solid curve.

Fig. 2
Fig. 2

(a) Contrast estimated with the variogram-based technique, Eq. (14), for the signal of Fig. 1, for different values of noise. (b) Relative error in the contrast estimation: error of the simulation, circles, error estimated with Eq. (15), solid line, and ± 2 e γ , s t , given in Eq. (17), dashed–dotted. (c) Comparison of the three techniques presented in the work (standard technique, rms technique, and variogram technique). In all the cases, circles represent the relative error using the technique; thick curves represent the average error given by Eqs. (5), (9), and (15), respectively. Dashed–dotted curves for the variogram-based technique represent ± e γ , s t .

Fig. 3
Fig. 3

(a) Signal I ( x ) = 5 + sin ( 8 π x ) + k sin ( π x ) sampled 1000 points between x ( 2 , 2 ) for k = 0.5 . (b) Relative error in the contrast estimation in terms of k: Direct definition of contrast, Eq. (1) dashed-dotted curve; rms technique, Eq. (7), dashed curve; and variogram technique, Eq. (14), solid curve.

Fig. 4
Fig. 4

(a) Signal I ( x ) = 10 + [ 1 + a cos ( π x / 2 ) ] sin ( 4 π x ) sampled 1000 points between x ( 2 , 2 ) for a = 0.5 . (b) Relative error in the contrast estimation in terms of a: Direct definition of contrast, Eq. (1) dashed–dotted; rms technique, Eq. (7) dash and variogram technique, Eq. (14) solid.

Fig. 5
Fig. 5

(a) Experimental intensity obtained after a diffraction grating (period 100   μm ) when it is illuminated with a monochromatic plane wave (wavelength 670   nm ). Z is the distance between the grating and a CMOS camera. Talbot planes are observed. (b) Fringes obtained for a position of high contrast and (c) fringes for a transition zone where contrast is low.

Fig. 6
Fig. 6

Contrast obtained with the different techniques presented; (a) direct definition of contrast, Eq. (1). (b) Rms technique, Eq. (7), and (c) variogram technique, Eq. (14).

Equations (17)

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C = I max I min I max + I min
I ( x ) = B + A sin ( 2 π x p ) + r ( x ) ,
C n o m = A B .
C s t = A + K σ B ,
e s t = | C s t C n o m C n o m | = K σ A ,
rms = ( [ I ( x ) I ] 2 d x d x ) 1 / 2 .
C r m s = 2 rms I .
C r m s = A 2 + 2 σ 2 B .
e r m s = | C r m s C n o m C n o m | = 1 + 2 ( σ A ) 2 1 .
γ ( h ) = 1 2 [ I ( x + h ) I ( x ) ] 2 ,
γ ( h ) = A 2 sin 2 ( π h p ) + σ 2 [ 1 δ ( h ) ] .
γ ̂ ( 0 ) 3 [ γ ( Δ x ) γ ( 2 Δ x ) ] + γ ( 3 Δ x ) .
A = γ ( p / 2 ) γ ̂ ( 0 ) ,
C γ = γ ( p / 2 ) γ ̂ ( 0 ) I .
e γ | C γ C n o m C γ | = 0 .
γ ( n Δ x ) = 1 2 ( N n ) i = 1 N n ( I i + n I i ) 2 ,
e γ , s t 1 N σ A + 2 N ( σ A ) 2 ,

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