Abstract

We studied correlation between fractal dimensions and image contrast for metallic surfaces. The study has led to an interesting finding that the maximum fractal dimension of the object surface under imaging gives the best focal plane. The significant finding can be made use of to estimate the best focal plane or measure the focus error with high sensitivity of a few microns, which are well within depth of field of the microscopic imaging system.

© 2008 Optical Society of America

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References

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  1. B. B. Mandelbrot, The Fractal Geometry of Nature, (W. H. Freeman, San Francisco, New York, 1982).
  2. P. Kotowski, "Fractal dimension of metallic fracture surface," Int. J. Fract. 141, 269-286 (2006).
    [CrossRef]
  3. A. Helalizadeh, H. Muller-Steinhagen, and M. Jamialahmadi, "Application of fractal theory for characterisation of crystalline deposits," Chem. Eng. Sci. 61, 2069-2078 (2006).
    [CrossRef]
  4. D. K. Goswami and B. N. Dev, "Nanoscale self-affine surface smoothing by ion bombardment," Phys. Rev. B 68, 033401 (2003).
    [CrossRef]
  5. J. Henry, "Accuracy issues in chemical and dimensional metrology in the SEM and TEM," Meas. Sci. Technol. 18, 2755-2761 (2007).
    [CrossRef]
  6. G. V. Duinen, M. V Heel, and A. Patwardhan, "Magnification variations due to illumination curvature and object defocus in transmission electron microscopy," Opt. Express 13, 9085 (2005).
    [CrossRef] [PubMed]
  7. R. N. Bracewell, Fourier Analysis and Imaging, (Kluwer, New York, 2003).
  8. J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1996).
  9. G. Franceschetti and D. Riccio, Scattering, Natural Surfaces and Fractals, (Elsevier, 2007).
  10. S. S. Chen, J. M. Keller, and R. M. Crownover, "On the calculation of fractal features from images," IEEE T. Pattern Anal. 15, 1087-1090 (1993).
    [CrossRef]
  11. N. Sarkar and B. B. Chaudhuri, "An efficient approach to estimate fractal dimension of textural images," Pattern Recogn. 25, 1035-1044 (1992).
    [CrossRef]
  12. http://cse.naro.affrc.go.jp/sasaki/fractal/fractal-e.html.
  13. V. Krishnakumar and A. K. Asundi, "Defocus measurement using spackle correlation," J. Mod. Opt. 48, 935-940 (2001).

2007 (1)

J. Henry, "Accuracy issues in chemical and dimensional metrology in the SEM and TEM," Meas. Sci. Technol. 18, 2755-2761 (2007).
[CrossRef]

2006 (2)

P. Kotowski, "Fractal dimension of metallic fracture surface," Int. J. Fract. 141, 269-286 (2006).
[CrossRef]

A. Helalizadeh, H. Muller-Steinhagen, and M. Jamialahmadi, "Application of fractal theory for characterisation of crystalline deposits," Chem. Eng. Sci. 61, 2069-2078 (2006).
[CrossRef]

2005 (1)

2003 (1)

D. K. Goswami and B. N. Dev, "Nanoscale self-affine surface smoothing by ion bombardment," Phys. Rev. B 68, 033401 (2003).
[CrossRef]

2001 (1)

V. Krishnakumar and A. K. Asundi, "Defocus measurement using spackle correlation," J. Mod. Opt. 48, 935-940 (2001).

1993 (1)

S. S. Chen, J. M. Keller, and R. M. Crownover, "On the calculation of fractal features from images," IEEE T. Pattern Anal. 15, 1087-1090 (1993).
[CrossRef]

1992 (1)

N. Sarkar and B. B. Chaudhuri, "An efficient approach to estimate fractal dimension of textural images," Pattern Recogn. 25, 1035-1044 (1992).
[CrossRef]

Asundi, A. K.

V. Krishnakumar and A. K. Asundi, "Defocus measurement using spackle correlation," J. Mod. Opt. 48, 935-940 (2001).

Chaudhuri, B. B.

N. Sarkar and B. B. Chaudhuri, "An efficient approach to estimate fractal dimension of textural images," Pattern Recogn. 25, 1035-1044 (1992).
[CrossRef]

Chen, S. S.

S. S. Chen, J. M. Keller, and R. M. Crownover, "On the calculation of fractal features from images," IEEE T. Pattern Anal. 15, 1087-1090 (1993).
[CrossRef]

Crownover, R. M.

S. S. Chen, J. M. Keller, and R. M. Crownover, "On the calculation of fractal features from images," IEEE T. Pattern Anal. 15, 1087-1090 (1993).
[CrossRef]

Dev, B. N.

D. K. Goswami and B. N. Dev, "Nanoscale self-affine surface smoothing by ion bombardment," Phys. Rev. B 68, 033401 (2003).
[CrossRef]

Duinen, G. V.

Goswami, D. K.

D. K. Goswami and B. N. Dev, "Nanoscale self-affine surface smoothing by ion bombardment," Phys. Rev. B 68, 033401 (2003).
[CrossRef]

Heel, M. V

Helalizadeh, A.

A. Helalizadeh, H. Muller-Steinhagen, and M. Jamialahmadi, "Application of fractal theory for characterisation of crystalline deposits," Chem. Eng. Sci. 61, 2069-2078 (2006).
[CrossRef]

Henry, J.

J. Henry, "Accuracy issues in chemical and dimensional metrology in the SEM and TEM," Meas. Sci. Technol. 18, 2755-2761 (2007).
[CrossRef]

Jamialahmadi, M.

A. Helalizadeh, H. Muller-Steinhagen, and M. Jamialahmadi, "Application of fractal theory for characterisation of crystalline deposits," Chem. Eng. Sci. 61, 2069-2078 (2006).
[CrossRef]

Keller, J. M.

S. S. Chen, J. M. Keller, and R. M. Crownover, "On the calculation of fractal features from images," IEEE T. Pattern Anal. 15, 1087-1090 (1993).
[CrossRef]

Kotowski, P.

P. Kotowski, "Fractal dimension of metallic fracture surface," Int. J. Fract. 141, 269-286 (2006).
[CrossRef]

Krishnakumar, V.

V. Krishnakumar and A. K. Asundi, "Defocus measurement using spackle correlation," J. Mod. Opt. 48, 935-940 (2001).

Muller-Steinhagen, H.

A. Helalizadeh, H. Muller-Steinhagen, and M. Jamialahmadi, "Application of fractal theory for characterisation of crystalline deposits," Chem. Eng. Sci. 61, 2069-2078 (2006).
[CrossRef]

Patwardhan, A.

Sarkar, N.

N. Sarkar and B. B. Chaudhuri, "An efficient approach to estimate fractal dimension of textural images," Pattern Recogn. 25, 1035-1044 (1992).
[CrossRef]

Chem. Eng. Sci. (1)

A. Helalizadeh, H. Muller-Steinhagen, and M. Jamialahmadi, "Application of fractal theory for characterisation of crystalline deposits," Chem. Eng. Sci. 61, 2069-2078 (2006).
[CrossRef]

IEEE T. Pattern Anal. (1)

S. S. Chen, J. M. Keller, and R. M. Crownover, "On the calculation of fractal features from images," IEEE T. Pattern Anal. 15, 1087-1090 (1993).
[CrossRef]

Int. J. Fract. (1)

P. Kotowski, "Fractal dimension of metallic fracture surface," Int. J. Fract. 141, 269-286 (2006).
[CrossRef]

J. Mod. Opt. (1)

V. Krishnakumar and A. K. Asundi, "Defocus measurement using spackle correlation," J. Mod. Opt. 48, 935-940 (2001).

Meas. Sci. Technol. (1)

J. Henry, "Accuracy issues in chemical and dimensional metrology in the SEM and TEM," Meas. Sci. Technol. 18, 2755-2761 (2007).
[CrossRef]

Opt. Express (1)

Pattern Recogn. (1)

N. Sarkar and B. B. Chaudhuri, "An efficient approach to estimate fractal dimension of textural images," Pattern Recogn. 25, 1035-1044 (1992).
[CrossRef]

Phys. Rev. B (1)

D. K. Goswami and B. N. Dev, "Nanoscale self-affine surface smoothing by ion bombardment," Phys. Rev. B 68, 033401 (2003).
[CrossRef]

Other (5)

B. B. Mandelbrot, The Fractal Geometry of Nature, (W. H. Freeman, San Francisco, New York, 1982).

R. N. Bracewell, Fourier Analysis and Imaging, (Kluwer, New York, 2003).

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

G. Franceschetti and D. Riccio, Scattering, Natural Surfaces and Fractals, (Elsevier, 2007).

http://cse.naro.affrc.go.jp/sasaki/fractal/fractal-e.html.

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Figures (3)

Fig. 1.
Fig. 1.

(a). Intensity distribution scattered from a metallic fractal surface. (b) Fractal dimension, FD=2.6274, measured from the ln(N(ε))~ln(1/ε) curve. (c) Fractal dimension, D=(6+β)/2=2.687, measured from the log-log curve of power magnitude and frequency.

Fig. 2.
Fig. 2.

Fractal dimension measured from a sequence of images, which have a relative displacement of 50 µm. The maximum fractal dimension indicates the best focal plane. The result by the different methods – Fourier power spectrum (“triangle”) and box counting (“square”) agrees well with that by standard deviation (“circle”).

Fig. 3.
Fig. 3.

Fractal dimension measured from a sequence of images, which have a relative displacement of 10 µm. The peak having a maximum fractal dimension represents the best focal plane.

Equations (4)

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

I ( r ) = I 0 ( r ) k ( r r ) 2 dr 2 = I 0 ( r ) * k ( r ) 2 ,
σ 2 ( Δ r ) = I ( r + Δ r ) I ( r ) 2 Δ r 2 ( 3 D s ) .
FD = lim n log ( N ( A , ε ) ) log ( 1 ε ) ,
n = int ( Gray _ max ( i , j ) ε ) int ( Gray _ min ( i , j ) ε ) + 1 .

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