## Flexible focus function consisting of convex function and image enhancement filter |

Optics Express, Vol. 22, Issue 15, pp. 18668-18687 (2014)

http://dx.doi.org/10.1364/OE.22.018668

Acrobat PDF (5540 KB)

### Abstract

We propose a new focus function Λ that, like many of the existing focus functions, consists of a convex function and an image enhancement filter. Λ is rather flexible because for any convex function and image enhancement filter, it is a focus function. We proved that Λ is a focus function using a model and Jensen’s inequality. Furthermore, we generated random Λs and experimentally applied them to simulated and real blurred images, finding that 98% and 99% of the random Λs, respectively, have a maximum value at the best-focused image and most of them decrease as the defocus increases. We also applied random Λs to motion-blurred images, blurred images in different-sized windows, and blurred images with different types of noise. We found that Λ can be applied to motion blur and is robust to different-sized windows and different noise types.

© 2014 Optical Society of America

## 1. Introduction

1. W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu, “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express **15**(4), 1543–1552 (2007). [CrossRef] [PubMed]

3. Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing in computer microscopy: Selecting the optimal focus algorithm,” Microsc. Res. Tech. **65**(3), 139–149 (2004). [CrossRef] [PubMed]

5. J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. **24**(1), 100–111 (1976). [CrossRef] [PubMed]

6. F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry **6**(2), 81–91 (1985). [CrossRef] [PubMed]

7. E. Krotkov, “Focusing,” Int. J. Comput. Vis. **1**(3), 223–237 (1987). [CrossRef]

8. M. Subbarao, T. S. Choi, and A. Nikzad, “Focusing techniques,” Opt. Eng. **32**(11), 2824–2836 (1993). [CrossRef]

_{1}-norm of the gradient to construct the focus function [9]. Santos et al. presented several focus functions by summing the difference of the image higher than a threshold, summing the squares of the differences of the image higher than a threshold, and summing the squares of the image values higher than a threshold [10

10. A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. **188**(3), 264–272 (1997). [CrossRef] [PubMed]

11. J. Daugman, “How iris recognition works,” IEEE Trans. Circuits Syst. Video Technol. **14**(1), 21–30 (2004). [CrossRef]

12. B. J. Kang and K. R. Park, “A study on iris image restoration,” in *International Conference on Audio- and Video-Based Biometric Person Authentication* (2005), pp. 31–40. [CrossRef]

14. S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. **18**(9), 1237–1246 (2008). [CrossRef]

17. M. L. Mendelsohn and B. H. Mayall, “Computer-oriented analysis of human chromosomes. 3. Focus,” Comput. Biol. Med. **2**(2), 137–150 (1972). [CrossRef] [PubMed]

6. F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry **6**(2), 81–91 (1985). [CrossRef] [PubMed]

18. L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston Jr., “Comparison of autofocus methods for automated microscopy,” Cytometry **12**(3), 195–206 (1991). [CrossRef] [PubMed]

19. S. L. Brázdilová and M. Kozubek, “Information content analysis in automated microscopy imaging using an adaptive autofocus algorithm for multimodal functions,” J. Microsc. **236**(3), 194–202 (2009). [CrossRef] [PubMed]

20. H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol. **29**(3), 173–180 (2010). [CrossRef]

21. D. C. Tsai and H. H. Chen, “Effective autofocus decision using reciprocal focus profile,” in *18th IEEE International Conference on Image Processing (ICIP)* (IEEE, 2011). [CrossRef]

22. L. Xu, M. Mater, and J. Ni, “Focus detection criterion for refocusing in multi-wavelength digital holography,” Opt. Express **19**(16), 14779–14793 (2011). [CrossRef] [PubMed]

24. P. Gao, B. Yao, R. Rupp, J. Min, R. Guo, B. Ma, J. Zheng, M. Lei, S. Yan, D. Dan, and T. Ye, “Autofocusing based on wavelength dependence of diffraction in two-wavelength digital holographic microscopy,” Opt. Lett. **37**(7), 1172–1174 (2012). [CrossRef] [PubMed]

25. D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. **248**(3), 245–259 (2012). [CrossRef] [PubMed]

5. J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. **24**(1), 100–111 (1976). [CrossRef] [PubMed]

*I*denotes the image, (

*x,y*) denotes the coordinates of the image pixels,

*φ*is a convex function, and

*g*is an image enhancement filter.

*φ*can be a strictly convex function, such as the quadratic function

*x*< 0 and

*x*≥ 0). Further, the choice of the image enhancement filter

*g*is rather arbitrary; it can be a one-dimensional image enhancement filter such as

## 2. The proposed Λ

## 3. Modeling the defocus problem

*λ*with an

*f*/

*#*of

*F*focusing radiation, it iswhich can be approximated by the Gaussian function [26

26. G. V. Poropat, “Effect of system point spread function, apparent size, and detector instantaneous field of view on the infrared image contrast of small objects,” Opt. Eng. **32**(10), 2598–2607 (1993). [CrossRef]

27. F. F. Yin, M. L. Giger, and K. Doi, “Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique,” Med. Phys. **17**(6), 962–966 (1990). [CrossRef] [PubMed]

30. T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE **7498**, 74981H (2009). [CrossRef]

*I*

_{0}is the best focused image,

*I*

_{1}is the less blurred image,

*I*

_{2}is the more blurred image,

*PSF*

_{1}is the PSF of

*I*

_{1},

*PSF*

_{2}is the PSF of

*I*

_{2}, and

*G*is a Dispersion Function.

## 4. A brief proof

## Statements

## Continuous form of the problem

*Continuous form of the problem*: Consider

*I*

_{1}and

*I*

_{2}to be images of the same object plane, where

*I*

_{2}is more defocused than

*I*

_{1}. Let

*φ*be a convex function in

*R*

^{2}and

*g*be an arbitrary function in

*R*

^{2}; then we have

*R*

^{2}denotes two-dimensional vector space, and * is the convolution operator.

## Proof

*G*is a Dispersion Function.

*G*is a Dispersion Function.

### Discussion

*Ω*is the image window.

## 5. Experimental results

### 5.1 Preparation of experimental data

*p*represents an

_{m}*m*th-order polynomial, and

*p*represents an

_{n}*n*th-order polynomial.

*m*= 3,

*n*= 2; thus, the convex function was an eighth-order polynomial.

*g*were constructed. They were constructed as 5 × 5 matrices, and the values of each matrix were randomized in the range of [-0.5, 0.5]. There are two types of image filters in the known focus functions. One type is smoothing filters, such as Gaussian filters, in which the sum of all the elements is equal to 1; the other type is sharpening filters, such as a Laplacian filter, in which the sum of all the elements is equal to 0. Here, the random filters

*g*act as sharpening filters. Thus, the random filters

*g*were normalized such that the sum of all the elements is equal to 0. The normalization was carried out by moving

*g*vertically along the y-axis such that the sum of all the elements is equal to 0. Examples of the random image enhancement filters are displayed in Fig. 5.

32. M. J. Nasse and J. C. Woehl, “Realistic modeling of the illumination point spread function in confocal scanning optical microscopy,” J. Opt. Soc. Am. A **27**(2), 295–302 (2010). [CrossRef] [PubMed]

33. A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. **17**(10), 1737–1754 (2008). [CrossRef] [PubMed]

*I*

_{N}represents the image with noise,

*I*represents the original image, and

*n*represents the Poisson distributionwhere

_{P}*a*is the parameter of the Poisson distribution. The Gaussian distribution

*n*is given bywhere

_{G}*b*is the standard deviation of the noise.

*a*= 0.064,

*b*= 0;

*a*= 0,

*b*= 0.032; and

*a*= 0.064,

*b*= 0.032, three series of blurred images with noise were obtained, as shown in Fig. 14.

### 5.2 Experiments

*y*axis such that

*I*denotes the blurred image,

*i*denotes the index of the blurred image, and

*norm*indicates normalization.

*x*axis denotes the image index, and the

*y*axis denotes the Λ value. About 98% of the constructed Λs reach the maximum value at the best-focused image and most of them decrease as the defocus increases, which supports our proof in the previous section.

34. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. **12**(7), 629–639 (1990). [CrossRef]

## 6. Conclusions

## Appendix 1

### The Dispersion Inequality

**Statements**

*the Dispersion Function*and “

*disperse*”:

*The Dispersion Function*: Let

*σ*be an integrable function in

*R*

^{1}; then

*σ*is a

*Dispersion Function*if

*σ*satisfies the following two formulas:

*disperse*: Let

*f*be an integrable function in

*R*

^{1}; then we say that

*f*is

*disperse*when

*f*convolves by a Dispersion Function

*σ*.

**The Dispersion Inequality**

*The Dispersion Inequality:*Let

*φ*be a convex function on [

*α,β*],

*f*be an integrable function in

*R*

^{1},

*α*≤

*f*(

*x*) ≤

*β*,

*x*∈

*R*

^{1}, and

*σ*be a Dispersion Function. Then the Dispersion Inequality is stated as

*R*

^{1}is a one-dimensional vector space, and * is the convolution operator.

**Proof**

35. J. L. W. V. Jensen, “Sur les fonctions convexes et les inégalités entre les valeurs moyennes,” Acta Math. **30**(1), 175–193 (1906). [CrossRef]

*φ*is a convex function on [

*α,β*],

*f*and

*p*are integrable functions in

*R*

^{1},

*α*≤

*f*(

*x*) ≤

*β*,

*x*∈

*R*

^{1}, and

*p*satisfies the following two formulas:

### Discussion

## Appendix 2

### The finite form of Λ

**Statements**

**Lemma**

*The simplified finite form of Λ:*Consider

*I*and

_{1}*I*

_{2}to be images of the same object plane, where

*I*

_{2}is more defocused than

*I*

_{1},

*I*

_{1}and

*I*

_{2}to be

*m*×

*n*matrices, and

*φ*to be a convex function in

*R*

^{2}; then we have

**Proof**

*I*

_{2}is equal to

*I*

_{1}convolved by a Dispersion Function

*G*that is a

*p*×

*q*matrix, which indicates that from the left side of Eq. (26), we havewhere “mod” denotes modulus.

*G*is a Dispersion Function; thus, we have the following two formulas:

*φ*, real numbers

*x*

_{1}, …,

*x*, and positive weights

_{n}*a*, we have

_{j}**The finite form of Λ**

*The finite form of Λ:*Consider that

*I*

_{1}and

*I*

_{2}are images of the same object plane, where

*I*

_{2}is more defocused than

*I*

_{1};

*I*

_{1}and

*I*

_{2}are

*m*×

*n*matrices,

*φ*is a convex function in

*R*

^{2},

*g*is a

*p*×

*q*image enhancement filter, and

## Acknowledgments

## References and links

1. | W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu, “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express |

2. | P. Favaro, “Shape from focus, and, defocus: convexity, quasiconvexity and defocus-invariant textures,” in |

3. | Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing in computer microscopy: Selecting the optimal focus algorithm,” Microsc. Res. Tech. |

4. | Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing algorithm selection in computer microscopy,” in |

5. | J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. |

6. | F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry |

7. | E. Krotkov, “Focusing,” Int. J. Comput. Vis. |

8. | M. Subbarao, T. S. Choi, and A. Nikzad, “Focusing techniques,” Opt. Eng. |

9. | S. K. Nayar and Y. Nakagawa, ““Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell. |

10. | A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc. |

11. | J. Daugman, “How iris recognition works,” IEEE Trans. Circuits Syst. Video Technol. |

12. | B. J. Kang and K. R. Park, “A study on iris image restoration,” in |

13. | P. Langehanenberg, B. Kemper, and G. Bally, “Autofocus algorithms for digital-holographic microscopy,” in |

14. | S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol. |

15. | F. Quan, K. Han, and X. C. Zhu, “A new auto-focusing method based on the center blocking DCT,” in |

16. | W. Jian and H. B. Chen, “A novel auto-focus function,” in |

17. | M. L. Mendelsohn and B. H. Mayall, “Computer-oriented analysis of human chromosomes. 3. Focus,” Comput. Biol. Med. |

18. | L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston Jr., “Comparison of autofocus methods for automated microscopy,” Cytometry |

19. | S. L. Brázdilová and M. Kozubek, “Information content analysis in automated microscopy imaging using an adaptive autofocus algorithm for multimodal functions,” J. Microsc. |

20. | H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol. |

21. | D. C. Tsai and H. H. Chen, “Effective autofocus decision using reciprocal focus profile,” in |

22. | L. Xu, M. Mater, and J. Ni, “Focus detection criterion for refocusing in multi-wavelength digital holography,” Opt. Express |

23. | P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE |

24. | P. Gao, B. Yao, R. Rupp, J. Min, R. Guo, B. Ma, J. Zheng, M. Lei, S. Yan, D. Dan, and T. Ye, “Autofocusing based on wavelength dependence of diffraction in two-wavelength digital holographic microscopy,” Opt. Lett. |

25. | D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc. |

26. | G. V. Poropat, “Effect of system point spread function, apparent size, and detector instantaneous field of view on the infrared image contrast of small objects,” Opt. Eng. |

27. | F. F. Yin, M. L. Giger, and K. Doi, “Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique,” Med. Phys. |

28. | S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng. |

29. | A. P. Tzannes and J. M. Mooney, “Measurement of the modulation transfer function of infrared cameras,” Opt. Eng. |

30. | T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE |

31. | E. Artin, “Uber die Zerlegung definiter Funktionen in Quadrate,” Abh. Math. Seminar Univ. Hamburg |

32. | M. J. Nasse and J. C. Woehl, “Realistic modeling of the illumination point spread function in confocal scanning optical microscopy,” J. Opt. Soc. Am. A |

33. | A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process. |

34. | P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell. |

35. | J. L. W. V. Jensen, “Sur les fonctions convexes et les inégalités entre les valeurs moyennes,” Acta Math. |

**OCIS Codes**

(000.3870) General : Mathematics

(110.3000) Imaging systems : Image quality assessment

(180.0180) Microscopy : Microscopy

(260.5950) Physical optics : Self-focusing

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 2, 2014

Revised Manuscript: March 12, 2014

Manuscript Accepted: July 14, 2014

Published: July 25, 2014

**Virtual Issues**

Vol. 9, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Kai Wang, Yuntao Qian, Minchao Ye, and Zhijian Luo, "Flexible focus function consisting of convex function and image enhancement filter," Opt. Express **22**, 18668-18687 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18668

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### References

- W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu, “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express15(4), 1543–1552 (2007). [CrossRef] [PubMed]
- P. Favaro, “Shape from focus, and, defocus: convexity, quasiconvexity and defocus-invariant textures,” in ICCV (2007).
- Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing in computer microscopy: Selecting the optimal focus algorithm,” Microsc. Res. Tech.65(3), 139–149 (2004). [CrossRef] [PubMed]
- Y. Sun, S. Duthaler, and B. J. Nelson, “Autofocusing algorithm selection in computer microscopy,” in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2005) (IEEE, 2005).
- J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem.24(1), 100–111 (1976). [CrossRef] [PubMed]
- F. C. Groen, I. T. Young, and G. Ligthart, “A comparison of different focus functions for use in autofocus algorithms,” Cytometry6(2), 81–91 (1985). [CrossRef] [PubMed]
- E. Krotkov, “Focusing,” Int. J. Comput. Vis.1(3), 223–237 (1987). [CrossRef]
- M. Subbarao, T. S. Choi, and A. Nikzad, “Focusing techniques,” Opt. Eng.32(11), 2824–2836 (1993). [CrossRef]
- S. K. Nayar and Y. Nakagawa, ““Shape from focus,” IEEE Trans. Pattern Anal. Mach. Intell.16, 824–831 (1994).
- A. Santos, C. Ortiz de Solórzano, J. J. Vaquero, J. M. Peña, N. Malpica, and F. del Pozo, “Evaluation of autofocus functions in molecular cytogenetic analysis,” J. Microsc.188(3), 264–272 (1997). [CrossRef] [PubMed]
- J. Daugman, “How iris recognition works,” IEEE Trans. Circuits Syst. Video Technol.14(1), 21–30 (2004). [CrossRef]
- B. J. Kang and K. R. Park, “A study on iris image restoration,” in International Conference on Audio- and Video-Based Biometric Person Authentication (2005), pp. 31–40. [CrossRef]
- P. Langehanenberg, B. Kemper, and G. Bally, “Autofocus algorithms for digital-holographic microscopy,” in European Conference on Biomedical Optics, Optical Society of America (2007).
- S. Y. Lee, Y. Kumar, J. M. Cho, S. W. Lee, and S. W. Kim, “Enhanced autofocus algorithm using robust focus measure and fuzzy reasoning,” IEEE Trans. Circuits Syst. Video Technol.18(9), 1237–1246 (2008). [CrossRef]
- F. Quan, K. Han, and X. C. Zhu, “A new auto-focusing method based on the center blocking DCT,” in Fourth International Conference on Image and Graphics ( ICIG 2007) (2007).
- W. Jian and H. B. Chen, “A novel auto-focus function,” in 6th International Symposium on Advanced Optical Manufacturing and Testing Technologies (AOMATT 2012) (International Society for Optics and Photonics, 2012).
- M. L. Mendelsohn and B. H. Mayall, “Computer-oriented analysis of human chromosomes. 3. Focus,” Comput. Biol. Med.2(2), 137–150 (1972). [CrossRef] [PubMed]
- L. Firestone, K. Cook, K. Culp, N. Talsania, and K. Preston., “Comparison of autofocus methods for automated microscopy,” Cytometry12(3), 195–206 (1991). [CrossRef] [PubMed]
- S. L. Brázdilová and M. Kozubek, “Information content analysis in automated microscopy imaging using an adaptive autofocus algorithm for multimodal functions,” J. Microsc.236(3), 194–202 (2009). [CrossRef] [PubMed]
- H. Peter, J. Schulz, and K. H. Englmeier, “Content-based autofocusing in automated microscopy,” Image Anal. Stereol.29(3), 173–180 (2010). [CrossRef]
- D. C. Tsai and H. H. Chen, “Effective autofocus decision using reciprocal focus profile,” in 18th IEEE International Conference on Image Processing (ICIP) (IEEE, 2011). [CrossRef]
- L. Xu, M. Mater, and J. Ni, “Focus detection criterion for refocusing in multi-wavelength digital holography,” Opt. Express19(16), 14779–14793 (2011). [CrossRef] [PubMed]
- P. Ferraro, P. Memmolo, C. Distante, M. Paturzo, A. Finizio, and B. Javidi, “An autofocusing algorithm for digital holograms,” Proc. SPIE8384, 838408 (2012).
- P. Gao, B. Yao, R. Rupp, J. Min, R. Guo, B. Ma, J. Zheng, M. Lei, S. Yan, D. Dan, and T. Ye, “Autofocusing based on wavelength dependence of diffraction in two-wavelength digital holographic microscopy,” Opt. Lett.37(7), 1172–1174 (2012). [CrossRef] [PubMed]
- D. T. Elozory, K. A. Kramer, B. Chaudhuri, O. P. Bonam, D. B. Goldgof, L. O. Hall, and P. R. Mouton, “Automatic section thickness determination using an absolute gradient focus function,” J. Microsc.248(3), 245–259 (2012). [CrossRef] [PubMed]
- G. V. Poropat, “Effect of system point spread function, apparent size, and detector instantaneous field of view on the infrared image contrast of small objects,” Opt. Eng.32(10), 2598–2607 (1993). [CrossRef]
- F. F. Yin, M. L. Giger, and K. Doi, “Measurement of the presampling modulation transfer function of film digitizers using a curve fitting technique,” Med. Phys.17(6), 962–966 (1990). [CrossRef] [PubMed]
- S. E. Reichenbach, S. K. Park, and R. Narayanswamy, “Characterizing digital image acquisition devices,” Opt. Eng.30(2), 170–177 (1991). [CrossRef]
- A. P. Tzannes and J. M. Mooney, “Measurement of the modulation transfer function of infrared cameras,” Opt. Eng.34(6), 1808–1817 (1995). [CrossRef]
- T. Li, H. Feng, Z. Xu, X. Li, Z. Cen, and Q. Li, “Comparison of different analytical edge spread function models for MTF calculation using curve-fitting,” Proc. SPIE7498, 74981H (2009). [CrossRef]
- E. Artin, “Uber die Zerlegung definiter Funktionen in Quadrate,” Abh. Math. Seminar Univ. Hamburg5, 85–99 (1927).
- M. J. Nasse and J. C. Woehl, “Realistic modeling of the illumination point spread function in confocal scanning optical microscopy,” J. Opt. Soc. Am. A27(2), 295–302 (2010). [CrossRef] [PubMed]
- A. Foi, M. Trimeche, V. Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,” IEEE Trans. Image Process.17(10), 1737–1754 (2008). [CrossRef] [PubMed]
- P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell.12(7), 629–639 (1990). [CrossRef]
- J. L. W. V. Jensen, “Sur les fonctions convexes et les inégalités entre les valeurs moyennes,” Acta Math.30(1), 175–193 (1906). [CrossRef]

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