## Global approach for fitting 2D interferometric data

Optics Express, Vol. 15, Issue 8, pp. 4835-4847 (2007)

http://dx.doi.org/10.1364/OE.15.004835

Acrobat PDF (2346 KB)

### Abstract

The present paper describes a fitting procedure capable of providing a smooth approximation of experimental data distributed on a bi-dimensional domain, e.g. the typical output of an interferometric technique. The procedure is based on the optimization of an analytical model defined on the whole domain by the B-spline formulation. In the paper rectangular, circular and polygonal convex domains are considered in details, but, according to the need of the operating conditions, the procedure can be extended to domains of different shapes. The proposed procedure was initially calibrated by an analytical case study: a thin square plate simply supported along the edges and loaded by a uniform pressure. Subsequently, by the operative parameters defined by the analyses carried out on the analytic data, the fitting procedure was applied on experimental data obtained by phase shifting speckle interferometry.

© 2007 Optical Society of America

## 1. Introduction

8. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. **162**, 205–210, (1999). [CrossRef]

14. C. L. Bajaj and G. Xu, “Spline approximations of real algebraic surfaces,” J. Symbolic Comput. **23**, 315–333, (1997). [CrossRef]

16. F. Furgiuele, M. Muzzupappa, and L. Pagnotta, “A full-field procedure for evaluating the elastic properties of advanced ceramics,” Exp. Mech. **37**, 285–291, (1997). [CrossRef]

17. D. M. Bates and D. G. Watts, *Nonlinear regression analysis and its applications* (John Wiley & Sons, 1988). [CrossRef]

*MatLab*[19] or

*Mathematica*[20]. Moreover the application of the procedure to a non rectangular domain requires a space transformation consisting of further mathematical manipulations.

## 2. Mathematical formulation

*P*(

*x,y*) of a surface can be represented by the following equation:

*N*(

_{i,h}*x*) and

*N*(

_{j,k}*y*) are the mono-dimensional B-spline blending functions,

*W*are the weights to be evaluated by the optimization procedure, (

_{i,j}*n*+1) and (

*m*+1) are the number of the control points in

*x*- and

*y*- direction,

*h*and

*k*are the orders of the polynomial functions in

*x*-and

*y*- direction. The blending functions can be defined by the following recursive equation:

*ξ*is the

_{i}*i*-th element of the knot vector whose number of elements is equal to (

*n*+

*k*). By this approach a piecewise polynomial function of (

*k*-1) degree is obtained, that is

*C*

^{k-2}continuous, and it spreads throughout (

*k*+1) control points.

*k*times we obtain a curve starting and finishing from and to the ends knots. In the non-uniform formulation the intervals of the knot vector are not equispaced, and the internal knots can be even repeated.

*h*=

*k*=3) and 4x4 control points (

*n*+

*1*=4 and

*m*+

*1*=4) were used. It can be seen that each function affects only a limited portion of the whole domain.

### 2.1 The fitting procedure

*W*. The advantage of this approach consists in reducing the number of parameters to be optimized, and the low degree of the polynomial functions, that avoids unwanted fluctuations.

_{i,j}- Calibration of the spatial coordinates of the input data;
- Calculation of the blending functions;
- Construction of the model;
- Evaluation of the best fit parameters.

**M**is obtained with a very high number of rows (as many as the experimental data) and a reduced number of columns (as many as the parameters to be evaluated, the weights

*W*). The elements of this matrix are the values assumed by the blending functions at each experimental datum location. The known vector

_{i,j}**V**is simply the experimental data flattened in a column vector. So assuming the following form for the problem:

*l*-th datum, and

*p*is the total number of the experimental data. According to this approach the unknown vector

**W**is calculated as the pseudoinverse of

**M**, that can be denoted as

**M***, multiplied by the known vector

**V**. Another advantage of this formulation is that if more than one set of experimental data is carried out on the same object, the matrix

**M**must be calculated and inverted only the first time, whereas the unknown vectors for each set of data will be evaluated by a simple multiplication between

**M*** and

**V**.

### 2.2 Data distributed on a circular domain

*h*=

*k*=4), hence the knot vectors are of 10 and 20 elements in radial and circumferential directions, respectively. A non-periodic formulation was used for the radial coordinate while a periodic formulation is necessary for the circumferential coordinate. In Fig. 2 only the 6 blending functions for a fixed circumferential control point were reported, because in the periodic formulation these functions are simply translated (and eventually rotated in cylindrical coordinates) when moving from a control point to another. In the example reported in Fig. 2 the blending functions relative to the next or to the previous control points are obtained by a rotation of π/8 (=2π/16).

### 2.3 Data distributed on a domain of any shape

*and*

**f***that allow the transformation between the two coordinate systems by simple geometric considerations as those reported in Fig. 3(b).*

**f**^{−1}*x,y*)), while the first set of equations (for transforming the real coordinates (

*x,y*) into the auxiliary coordinates (ξ,η)) hides the geometric data used to generate them. This fact is due to the complexity introduced by the geometry of the problem, which however do not implies a complex algebraic formula, as it can be seen from eq. (5).

*N*triangles are obtained if

*N*is the number of the edges of the domain. Finally, according to the Fig. 4(b) each triangle can be transformed into a circular sector by the following general formula:

*x*

_{i},

*y*

_{i}) and (

*x,y*) are the coordinates of the triangle and the generic coordinates in the real domain respectively, (r,

*θ*) are the generic auxiliary coordinates that vary in the ranges [0,1] and [

*θ*

_{min},

*θ*

_{max}] respectively.

## 3. Application of the procedure

### 3.1 Application to analytical data

*w*(

*x,y*) is the out-of-plane displacement at the generic coordinate,

*p*is the pressure applied on the plate,

*E*and

*ν*are the Young’s modulus and Poisson’s ratio of the material,

*a*and

*b*are the length of the edges,

*t*is the thickness,

*n*and

*m*are two indices used to sweep the odd integer numbers between 0 and infinity. The loading configuration and a typical phase map of the out-of-plane displacements are reported in Fig. 6.

*h,k*) and the number of control points (

*n*+1,

*m*+1) necessary to fit accurately the simulated experimental data. By eq. (7) the displacements were evaluated on a grid of 300x300 elements, that is a proper number of information gathered by an experimental test. Then these data were fitted by varying the parameters (

*h,k*) and (

*n*+1,

*m*+1) and for each values assumed by the fitting parameters the maximum error (E

_{max}) and the standard deviation (σ) were evaluated. Because of the symmetry of the problem the same parameters were assumed along the

*x*- and

*y*- directions (

*h*=

*k*and

*n*=

*m*). Furthermore, as said in section 2, because of each blending function spreads throughout (

*k*+1) control points the order of polynomials fixes the minimum number of control points.

_{max}and σ and provides quantitative information to choose the fitting parameters. For example if a maximum error less than 0.1% is desired it is necessary to use 10 control points for

*h*=

*k*=3 or 7 control point for

*h*=

*k*=4, while more than 12 control points are necessary if a linear approximation (

*h*=

*k*=2) is adopted.

*k*=2) is not in general a good choice, especially when a displacement component must be fitted, because it provides very poor strain fields which must be evaluated by derivative operations.

_{1}; b) the difference between the fitted surface and the data with noise, σ

_{2}; c) the noise as is, σ

_{3}. In the simulations the level of noise was increased by increasing the standard deviation of the random data, which was varied between 0.02/3

^{0.5}and 0.10/3

^{0.5}by 5 steps. These values, calculated as

*r*/3

^{0.5}, are adopted in order to have a flat distribution in the range [-

*r*,

*r*] which results to have the same standard deviation of gaussian distribution. Obviously for the additive noise the mean value of both distributions is 0, while for the multiplicative noise the mean value is 1. The parameter

*r*is expressed as a fraction of the maximum displacement observed, in the simulations of the noise its value is varied until 10%, which represents a particularly severe noise.

_{1}is more than one order of magnitude less than σ

_{2}and σ

_{3}. For the additive noise the value σ

_{2}is practically equal to σ

_{3}, and it means that the noise is not able to bring to fail the procedure. For the multiplicative noise this fact does not happen because in this case it is necessary to take into account the distribution of the displacement; nevertheless the standard deviation σ

_{1}is still very small.

### 3.2 Application to experimental data

_{1}, S

_{2}and S

_{3}and loaded by a punctual force F. The experimental data are acquired only on a reduced area of the specimen because the loading fixture creates patches of shade. The specimen is a square with an edge of 50 mm, while the observed area is a square chosen in the middle of the specimen with an edge of about 35 mm, the experimental data consists of 400x400 pixels. The specimen is inserted along one arm of a Michelson interferometer capable of measuring out-of-plane displacements, according to the layout reported in Fig. 7(b). The reference beam is obtained by a rough surface fixed to a three degrees of freedom PZT actuator able to perform a uniform translation along its axis (necessary for phase shifting) and two tilting around an axis belonging to a plane parallel to the reference surface.

*h*=

*k*=4) B-spline surface with 7x7 control points were used, results are reported in Fig. 8 and Fig. 9. Figures 8(a) and 9(a) report the experimental data without any manipulation or fitting operation, while Fig. 8(b) and Fig. 9(b) are the fitted data represented as phase maps in order to be compared more easily with the experimental data. Figures 8(c) and 9(c) are the histograms of the error evaluated as the difference between the experimental and the fitted data for the known (continuous line) and unknown (dots) deformation field. In both cases the two error distributions, which are symmetric and have a mean equal to zero, are slightly different but they have the same standard deviation. In the upper-right corner of Fig. 8(c) and Fig. 9(c) there are the experimental phase maps obtained by tilting the reference surface which determine the same standard deviation of the fitted surface, while in the upper-left corner are reported the standard deviations of known (σ

_{k}) and unknown (σ

_{u}) deformation field. These quantities are expressed in radians because all the quantities are evaluated as phase variation, as also the errors used for plotting the histograms of Fig. 8(c) and Fig. 9(c); for the scope of the present paper it is not necessary the exact calibration in terms of the physical units of the measured quantity.

## 4. Conclusions

## References and links

1. | M. M. Frocht, |

2. | D. Post, B. Han, and P. Ifju, |

3. | R. K. Erf, |

4. | U. Schnars and W. Jueptner, |

5. | K. Creath, “Temporal phase measurement methods,” in |

6. | M. Kujawinska, “Spatial phase measurement methods,” in |

7. | D. C. Ghiglia and M. D. Pritt, |

8. | H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. |

9. | M. J. Huang, Z. N. He, and F. Z. Lee, “A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI,” Measurement |

10. | Q. Kemao, S. Hock Soon, and A. Asundi, “Smoothing filters in phase-shifting interferometry,” Opt. Laser Technol. |

11. | M. I. Younus and M. S. Alam, “Enhanced phase stepped interferometry via appropriate filtering,” J. Opt. Eng. |

12. | V. I. Vlad and D. Malacara, “Direct spatial reconstruction of optical phase from phase-modulated images,” Progress in Optics |

13. | J. Novak and A. Miks, “Least-squares fitting of wavefront using rational function,” Opt. Lasers Eng. |

14. | C. L. Bajaj and G. Xu, “Spline approximations of real algebraic surfaces,” J. Symbolic Comput. |

15. | B. Juttler and A. Felis, “Least-squares fitting of algebraic spline surfaces,” Adv. Comput. Math. |

16. | F. Furgiuele, M. Muzzupappa, and L. Pagnotta, “A full-field procedure for evaluating the elastic properties of advanced ceramics,” Exp. Mech. |

17. | D. M. Bates and D. G. Watts, |

18. | V. B. Anand, |

19. | A. Knight, |

20. | S. Wolfram, The Mathematica book, 5 |

21. | R. Szilard, |

**OCIS Codes**

(070.6110) Fourier optics and signal processing : Spatial filtering

(100.2000) Image processing : Digital image processing

(120.3180) Instrumentation, measurement, and metrology : Interferometry

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: January 16, 2007

Revised Manuscript: April 3, 2007

Manuscript Accepted: April 3, 2007

Published: April 5, 2007

**Citation**

Luigi Bruno, "Global approach for fitting 2D interferometric data," Opt. Express **15**, 4835-4847 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-4835

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### References

- M. M. Frocht, Photoelasticity, (John Wiley and Sons, 1941) Vol. I.
- D. Post, B. Han, and P. Ifju, High sensitivity moiré (Springer Verlag 1997).
- R. K. Erf, Speckle metrology (Academic Press 1978).
- U. Schnars and W. Jueptner, Digital holography (Springer 2005).
- K. Creath, "Temporal phase measurement methods," in Interferogram analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, 1993).
- M. Kujawinska, "Spatial phase measurement methods," in Interferogram analysis, D. W. Robinson and G. T. Reid, eds., (Institute of Physics Publishing, 1993).
- D. C. Ghiglia and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley & Sons, 1998).
- H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999). [CrossRef]
- M. J. Huang, Z. N. He and F. Z. Lee, "A novel methodology for enhancing the contrast of correlation fringes obtained by ESPI," Measurement 36, 93-100 (2004). [CrossRef]
- Q. Kemao, S. Hock Soon, and A. Asundi, "Smoothing filters in phase-shifting interferometry," Opt. Laser Technol. 35, 649-654 (2003). [CrossRef]
- M. I. Younus and M. S. Alam, "Enhanced phase stepped interferometry via appropriate filtering," J. Opt. Eng. 38, 1918-1923 (1999). [CrossRef]
- V. I. Vlad and D. Malacara, "Direct spatial reconstruction of optical phase from phase-modulated images," Prog. Opt. 33, 261-317 (1994). [CrossRef]
- J. Novak and A. Miks, "Least-squares fitting of wavefront using rational function," Opt. Lasers Eng. 44, 40-51 (2005).
- C. L. Bajaj and G. Xu, "Spline approximations of real algebraic surfaces," J. Symb. Comput. 23, 315-333 (1997). [CrossRef]
- B. Juttler and A. Felis, "Least-squares fitting of algebraic spline surfaces," Adv. Comput. Math. 17, 135-152 (2002). [CrossRef]
- F. Furgiuele, M. Muzzupappa and L. Pagnotta, "A full-field procedure for evaluating the elastic properties of advanced ceramics," Exp. Mech. 37, 285-291 (1997). [CrossRef]
- D. M. Bates and D. G. Watts, Nonlinear regression analysis and its applications (John Wiley & Sons, 1988). [CrossRef]
- V. B. Anand, Computer graphics and geometric modeling for engineers (John Wiley & Sons, 1993).
- A. Knight, Basics of Matlab and beyond (Chapman & Hall/CRC, 2000).
- S. Wolfram, Mathematica Book, 5th edition (Wolfram Media Inc, 2003).
- R. Szilard, Theory and analysis of plates: classical and numerical methods (Prentice-Hall, 1974).

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