Abstract

A histogram-based technique for robust contrast measurement is proposed. The method is based on fitting the histogram of the measured image to the histogram of a model function, and it can be used for contrast determination in fringe patterns. Simulated and experimental results are presented.

© 2000 Optical Society of America

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References

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  1. J. W. Coltman, “The specification of imaging properties by response to a sine wave input,” J. Opt. Soc. Am. 44, 468–471 (1954).
    [CrossRef]
  2. G. C. Holst, CCD Arrays, Cameras, and Displays (Society for Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1996).
  3. S. Lai, G. von Bally, “Fringe contrast evaluation by means of histograms,” in OPTIKA ’98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, P. András, eds., Proc. SPIE3573, 384–387 (1998).
  4. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  5. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  6. B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
    [CrossRef]
  7. T. Coleman, M. A. Branch, A. Grace, Optimization Toolbox for Use with MATLAB, users guide version 2 (MathWorks, Natick, Mass., 1996).
  8. A. Garcia-Botella, L. M. Sanchez-Brea, D. Vazquez-Molini, E. Bernabeu, “Modulation transfer function for translucent rough sheet,” Appl. Opt. 38, 5429–5432 (1999).
    [CrossRef]

1999 (1)

1954 (1)

Bernabeu, E.

Branch, M. A.

T. Coleman, M. A. Branch, A. Grace, Optimization Toolbox for Use with MATLAB, users guide version 2 (MathWorks, Natick, Mass., 1996).

Coleman, T.

T. Coleman, M. A. Branch, A. Grace, Optimization Toolbox for Use with MATLAB, users guide version 2 (MathWorks, Natick, Mass., 1996).

Coltman, J. W.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
[CrossRef]

Garcia-Botella, A.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Grace, A.

T. Coleman, M. A. Branch, A. Grace, Optimization Toolbox for Use with MATLAB, users guide version 2 (MathWorks, Natick, Mass., 1996).

Holst, G. C.

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

Lai, S.

S. Lai, G. von Bally, “Fringe contrast evaluation by means of histograms,” in OPTIKA ’98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, P. András, eds., Proc. SPIE3573, 384–387 (1998).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Sanchez-Brea, L. M.

Vazquez-Molini, D.

von Bally, G.

S. Lai, G. von Bally, “Fringe contrast evaluation by means of histograms,” in OPTIKA ’98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, P. András, eds., Proc. SPIE3573, 384–387 (1998).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (6)

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

S. Lai, G. von Bally, “Fringe contrast evaluation by means of histograms,” in OPTIKA ’98: 5th Congress on Modern Optics, G. Ákos, G. Lupkovics, P. András, eds., Proc. SPIE3573, 384–387 (1998).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

B. R. Frieden, Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1983).
[CrossRef]

T. Coleman, M. A. Branch, A. Grace, Optimization Toolbox for Use with MATLAB, users guide version 2 (MathWorks, Natick, Mass., 1996).

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Figures (8)

Fig. 1
Fig. 1

Relationship between h[ n] (circles) and h(n) (diamonds) for f(x) = a + b cos(wx), with b = 75, a = 125, x 1 = 0, x M = π/w, and M = 300. Only one of each of the four points is presented to improve the visibility of the figure.

Fig. 2
Fig. 2

Simulated one-dimensional fringe patterns with additive noise and different fringe profile: (a) for a sine fringe pattern, (b) quasi-sine fringe pattern, (c) quasi-square fringe pattern. For all cases a = 50, b = 200, σ = 15 g.l. (d), (e), (f): circles, histograms obtained from (a), (b), (c), respectively; curve, fits to sigmoidal histogram by means of minimization of Eq. (26).

Fig. 3
Fig. 3

Simulated (thin) and estimated (thick) fringe shape for the fringe patterns given in Fig. 1. The estimated fringe shape is obtained from results of Table 1. As we can see, there is a good agreement between the simulated and the estimated profiles.

Fig. 4
Fig. 4

Estimated uncertainty for contrast estimation in terms of Gaussian noise for several fringe shapes: (inverted triangles) λ = 0.263; (asterisks) λ = 0.160; (circles), λ = 0.0308. In this case the contrast is C = 0.6 (a = 50, b = 200).

Fig. 5
Fig. 5

(a) Estimation of λ in terms of the number of fringes per frame for three different shape profiles, (b) estimation of contrast for the same cases. The theoretical contrast is C = 0.6, and a noise of 10 g.l. has been added.

Fig. 6
Fig. 6

Sketch of the experimental setup for contrast measurement of bar tests when a translucent rough sheet is interposed between the bar test and the CCD camera.

Fig. 7
Fig. 7

Profiles from two-dimensional real fringe patterns and fits for their histograms: (a) and (d) sinusoidal low-contrast pattern, (b) and (e) sinusoidal high-contrast pattern, and (c) and (f) square high-contrast pattern. Only one of each of the four points is presented to improve the visibility of (d)–(f). Circles, experimental histogram; curve, the fit.

Fig. 8
Fig. 8

Measurements of contrast transfer function by means of histogram-based method, for a dielectric rough sheet with n = 1.523 and roughness parameter of τ/σ = 165 ± 4, at distances between bar test and sheet of d 1 = 5 cm (squares) and d 2 = 10 cm (circles), compared with the model proposed by Garcia-Botella et al.8

Tables (2)

Tables Icon

Table 1 Estimated Parameters for Fringe Profiles of Fig. 2 with a Histogram-Based Method

Tables Icon

Table 2 Estimated Parameters for Measurements of Real Fringe Patterns by Means of a Histogram-Based Method

Equations (30)

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MTFf=π4CTFf+CTF3f3-CTF5f5+CTF7f7+,
C=IMAX-IMINIMAX+IMIN,
hy=Ω δfx-ydx,
δgx=i1|g| δx-xi,
gx=fx-y,
hy=Ω1|f| δx-f-1ydx,
hy=1|ff-1y|.
hy=df-1ydy.
hy=1/|ωb2-y-a21/2|a-b<y<b+a0elsewhere.
Īr=Ir+nr,
h¯y=hy * py,
yn=n-1Δy,  n=1,, N,
xm=x1+m-1Δx,  m=1,, M,
y1fxm<y1+Δy/2,  n=1,yn-Δy/2fxm<yn+Δy/2,  1<n<N,yN-Δy/2fxm<yN,  n=N.
hn=|intf-1ynMIN+Δy/2/Δx-intx1/Δx|n=nMIN|intf-1yn+Δy/2/Δx-intf-1yn-Δy/2/Δx|nMIN<n<nMAX|intxM/Δx-intf-1ynMAX-Δy/2/Δx|+1n=nMAX0elsewhere,
nMAX=intfxM/Δy+1,nMIN=intfx1/Δy+1.
n=1N hn=M,
hn=f-1ynMIN+Δy/2-x1/Δxn=nMINf-1yn+Δy/2-f-1yn-Δy/2/ΔxnMIN<n<nMAXxM-f-1ynMAX-Δy/2/Δx+1n=nMAX0elsewhere,
n=1N hn=M,
fSx=a+b-a1+exp-x-x0/λ,
hSy=λb-a/b-yy-aya, b0elsewhere.
fS-1y=x0-λ lny-ba-y,
hSn=fS-1ynMIN+Δy/2-x1/Δxn=nMINλΔxln1+Δyb-n-1/2Δy1+Δyn-3/2Δy-anMIN<n<nMAXxM-fS-1ynMAX-Δy/2/Δx+1n=nMAX0elsewhere,
py, σ=12πσexp-y22σ2,
h¯Sn, a, b, λ, x0, σ=hSn * pn, σ,
E=n=1NhDn-h¯Sn, a, b, λ, x0, σ2,
C=fSxM-fSx1fSxM+fSx1.
h¯Sn, a, b, λ, x0, σ1, σ2=Λnhsn * pn, σ1+1-Λnhsn * pn, σ2,
E=n=1NhDn-h¯Sn, a, b, λ, x0, σ1, σ22.
CTFh, d, σ/τ=4πk=0-1k12k+1×exp-2k+12πhσdn-1τ2.

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