## Defocus measurement for random self-affine fractal surfaces

Optics Express, Vol. 16, Issue 5, pp. 2928-2932 (2008)

http://dx.doi.org/10.1364/OE.16.002928

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### 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

## 1. Introduction

2. P. Kotowski, “Fractal dimension of metallic fracture surface,” Int. J. Fract. **141**, 269–286 (2006). [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]

## 2. Theory

*k*(

*r*)|

^{2}of the imaging system with the intensity

*I*

_{0}(

*r*′) on the object plane [8],

*r*′ and

*r*are the position vector on the object plane and image plane, respectively. Considering incremental variance of the intensity distribution,

*σ*

^{2}(Δ

*r*)=〈|

*I*(

*r*+Δ

*r*)-

*I*(

*r*)|

^{2}〉, where Δ

*r*is an incremental distance over a fix direction. For an isotropic fractal surface, the incremental variance is expressed by the power function as a function of fractal dimension

*D*[9]

_{s}^{3}, e.g. a set of pixels in a grey-level image [10

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]

*FD*) using box counting of a set

*A*in a grey image is given as,

*N*

_{(A,ε)}denotes the number of boxes of side length,

*ε*=2

^{n},

*n*=1,2,3,4… An image of

*M×M*pixels can be divided into box of side length ε=(2)

^{n}. If the minimum grey level of the image in the grid (

*i*,

*j*) falls in box number

*k*, and the maximum grey level of the (

*i*,

*j*)

^{th}grid falls in box number

*1*, then n is counted as [11

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

*N*

_{(A,ε)}=∑

*n*(

*i*,

*j*) is counted for the different values of box dimension, ε. The FD can be estimated from the least-squares linear fit of the log(

*N*

_{(A,ε)})~ log(1/

*ε*) curve.

## 3. Experiment, results and discussion

*ε*=(1/2)

^{n}, n=2, 3, 4, 5, and 6), the value is FD=2.627. In addition, the fractal dimension

*D*was obtained in Fourier domain by least–square linear fit of the log-log curve of magnitude and frequency using the equation, D=(6+β)/2, where β is the slope of the log-log curve [9]. The fractal dimension is D=(6+β)/2=2.687, as shown in Fig. 1(c).

## 4. Conclusion

## Acknowledgments

## References and links

1. | B. B. Mandelbrot, |

2. | P. Kotowski, “Fractal dimension of metallic fracture surface,” Int. J. Fract. |

3. | A. Helalizadeh, H. Muller-Steinhagen, and M. Jamialahmadi, “Application of fractal theory for characterisation of crystalline deposits,” Chem. Eng. Sci. |

4. | D. K. Goswami and B. N. Dev, “Nanoscale self-affine surface smoothing by ion bombardment,” Phys. Rev. B |

5. | J. Henry, “Accuracy issues in chemical and dimensional metrology in the SEM and TEM,” Meas. Sci. Technol. |

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 |

7. | R. N. Bracewell, |

8. | J. W. Goodman, |

9. | G. Franceschetti and D. Riccio, |

10. | S. S. Chen, J. M. Keller, and R. M. Crownover, “On the calculation of fractal features from images,” IEEE T. Pattern Anal. |

11. | N. Sarkar and B. B. Chaudhuri, “An efficient approach to estimate fractal dimension of textural images,” Pattern Recogn. |

12. | |

13. | V. Krishnakumar and A. K. Asundi, “Defocus measurement using spackle correlation,” J. Mod. Opt. |

**OCIS Codes**

(080.1010) Geometric optics : Aberrations (global)

(110.0180) Imaging systems : Microscopy

(110.2960) Imaging systems : Image analysis

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 4, 2008

Revised Manuscript: February 8, 2008

Manuscript Accepted: February 14, 2008

Published: February 19, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Jun Wang, Wei Zhou, Lennie E. Lim, and Anand K. Asundi, "Defocus measurement for random self-affine
fractal surfaces," Opt. Express **16**, 2928-2932 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-2928

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### References

- B. B. Mandelbrot, The Fractal Geometry of Nature, (W. H. Freeman, San Francisco, New York, 1982).
- 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]
- D. K. Goswami and B. N. Dev, "Nanoscale self-affine surface smoothing by ion bombardment," Phys. Rev. B 68, 033401 (2003). [CrossRef]
- J. Henry, "Accuracy issues in chemical and dimensional metrology in the SEM and TEM," Meas. Sci. Technol. 18, 2755-2761 (2007). [CrossRef]
- 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]
- 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).
- 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]
- N. Sarkar and B. B. Chaudhuri, "An efficient approach to estimate fractal dimension of textural images," Pattern Recogn. 25, 1035-1044 (1992). [CrossRef]
- http://cse.naro.affrc.go.jp/sasaki/fractal/fractal-e.html.
- V. Krishnakumar and A. K. Asundi, "Defocus measurement using spackle correlation," J. Mod. Opt. 48, 935-940 (2001).

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