## Simultaneous measurement of in-plane and out-of-plane displacements using pseudo-Wigner-Hough transform |

Optics Express, Vol. 22, Issue 7, pp. 8703-8711 (2014)

http://dx.doi.org/10.1364/OE.22.008703

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### Abstract

A new method based on pseudo-Wigner-Hough transform is proposed for the simultaneous measurement of the in-plane and out-of-plane displacements using digital holographic moiré. Multiple interference phases corresponding to the in-plane and out-of-plane displacement components are retrieved from a single moiré fringe pattern. The segmentation of the interference field allows us to approximate it with a multicomponent linear frequency modulated signal. The proposed method accurately and simultaneously estimates all the phase parameters of the signal components without the use of any signal separation techniques. Simulation and experimental results demonstrate the efficacy of the proposed method and its robustness against the variations in object beam intensity.

© 2014 Optical Society of America

## 1. Introduction

1. P. Picart, D. Mounier, and J. M. Desse, “High-resolution digital two-color holographic metrology,” Opt. Lett. **33**, 276–278 (2008). [CrossRef] [PubMed]

2. P. Picart, E. Moisson, and D. Mounier, “Twin-sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. **42**, 1947–1957 (2003). [CrossRef] [PubMed]

3. C. Kohler, M. R. Viotti, and A. G. Albertazzi Jr., “Measurement of three-dimensional deformations using digital holography with radial sensitivity” Appl. Opt. **49**, 4004–4009 (2010). [CrossRef] [PubMed]

4. S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. **3–4**, 223–228 (2005). [CrossRef]

5. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic moiré,” Appl. Opt. **50**, 4189–4197 (2011). [CrossRef] [PubMed]

6. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Estimation of multiple phases from a single fringe pattern in digital holographic interferometry,” Opt. Express **20**, 1281–1291 (2012). [CrossRef] [PubMed]

7. R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Laser Eng. **51**, 1168–1172 (2013). [CrossRef]

8. K. Pokorski and K. Patorski, “Separation of complex fringe patterns using two-dimensional continuous wavelet transform,” Appl. Opt. **51**, 8433–8439 (2012). [CrossRef] [PubMed]

5. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic moiré,” Appl. Opt. **50**, 4189–4197 (2011). [CrossRef] [PubMed]

6. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Estimation of multiple phases from a single fringe pattern in digital holographic interferometry,” Opt. Express **20**, 1281–1291 (2012). [CrossRef] [PubMed]

7. R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Laser Eng. **51**, 1168–1172 (2013). [CrossRef]

## 2. Theory

*x*,

*y*) is of the size

*N*×

*N*pixels. The pixels along the columns and rows are represented by

*x*and

*y*, respectively.

*A*

_{1}(

*x*,

*y*) and

*A*

_{2}(

*x*,

*y*) are the slowly varying or constant amplitudes; Δ

*φ*

_{1}(

*x*,

*y*) and Δ

*φ*

_{2}(

*x*,

*y*) represent the interference phases and

*η*(

*x*,

*y*) is the complex additive white Gaussian noise. In general, the interference phases are continuous functions of spatial coordinates

*x*and

*y*. Consequently, they can be approximated with the polynomials of appropriate order. However, if the interference phases vary rapidly, the required order for polynomial approximation of phases could be correspondingly high. The accuracy of phase parameter estimation decreases with the increase in the polynomial order especially in the case of multicomponent signals. Therefore, lower order polynomial approximation of interference phases is achieved by dividing the interference field into a number of non-overlapping segments

*L*in each column

*x*or in each row

*y*. Although, further analysis is carried out considering the signal segmentation in each column

*x*, it should be noted that the same analysis is true in case of signal segmentation in each row

*y*also. Over these segments, the interference phases are approximated with second order polynomial functions of

*y*with a multicomponent linear frequency-modulated signal representation of the interference field. Thus, for a given column

*x*, the interference field in the segment

*l*with

*l*∈ (1,

*L*) can be represented as, where, The phase parameters

*a*∈ {

*a*

_{l1},

*a*

_{l2}},

*b*∈ {

*b*

_{l1},

*b*

_{l2}},

*c*∈ {

*c*

_{l1},

*c*

_{l2}} correspond to

*initial phase*,

*mean spatial f requency*and

*sweep rate*, respectively. From the above equations, it can be understood that the phase parameters

*a*,

*b*and

*c*have to be accurately estimated for reliable estimation of the interference phases.

9. S. Barbarossa, “Analysis of multicomponent lfm signals by a combined wigner-hough transform,” IEEE T. Signal Process. **43**, 1511–1515 (1995). [CrossRef]

10. L. Cirillo, A. Zoubir, and M. Amin, “Parameter estimation for locally linear fm signals using a time-frequency hough transform,” IEEE T. Signal Process. **56**, 4162–4175 (2008). [CrossRef]

*(*

_{l}*y*) can be represented as, where,

*N*=

_{L}*N*/

*L*is the length of Γ

*(*

_{l}*y*);

*M*is the parameter defining window length,

*W*= 2

*M*+1;

*θ*represents the domain of phase parameters

*b*and

*c*;

*ω*=

*b*+

*cy*. The optimum value of

*M*= 0.1

*N*has been suggested [10

_{L}10. L. Cirillo, A. Zoubir, and M. Amin, “Parameter estimation for locally linear fm signals using a time-frequency hough transform,” IEEE T. Signal Process. **56**, 4162–4175 (2008). [CrossRef]

*b*and

*c*of all the signal components. PWHT is computationally more efficient as compared to WHT due to the windowing involved in PWVD. Additionally, the width of peak observed in PWHT parameter space is much larger than that observed for the WHT. This suggests that the PWHT offers improved numerical properties during optimization than that offered by the WHT. It is required to ensure that the initial estimates of phase parameters calculated using PWHT are accurate enough so that the optimization algorithm converges to true phase parameter values. In order to achieve this, peak detection in PWHT is performed on the grid of phase parameters

*b*and

*c*with appropriate grid spacings of Δ

*and Δ*

_{b}*, respectively. The values of Δ*

_{c}*and Δ*

_{b}*can be calculated as [10*

_{c}10. L. Cirillo, A. Zoubir, and M. Amin, “Parameter estimation for locally linear fm signals using a time-frequency hough transform,” IEEE T. Signal Process. **56**, 4162–4175 (2008). [CrossRef]

*x*one at a time. It is therefore required to group the estimated phases together with their respective counterparts from the other columns to obtain the complete 2D phase maps. To achieve this, a simple amplitude discrimination criteria is proposed. Different intensity levels are set up for each of the two object beams which result in different amplitudes of signal components in Eq. (1). It should be noted that the amplitude discrimination criteria is not used for the estimation of the phase parameters but only to discriminate between the estimated phases. On the contrary, the phase parameters of all signal components are estimated simultaneously. Consequently, the inherent error propagation effect of the sequential phase parameter estimation procedure based on amplitude discrimination criteria proposed in [7

7. R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Laser Eng. **51**, 1168–1172 (2013). [CrossRef]

*(*

_{l}*y*). Subsequently, the above explained steps (ii) and (iii) are used to re-estimate the phase parameters. These estimates are used as initial values for Nelder-Mead simplex optimization algorithm [11

11. D. S. Pham and A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE T. Signal Process. **55**, 56–65 (2007). [CrossRef]

## 3. Simulation and experimental results

*φ*

_{1}(

*x*,

*y*) and Δ

*φ*

_{2}(

*x*,

*y*) shown in Fig. 1(a) and Fig. 1(b). The ratio of signal amplitudes was set to 1.5 : 1. The moiré fringe pattern i.e. the real part of the interference field, Γ(

*x*,

*y*) given in Eq. (1), can be represented as, where,

*η*(

_{r}*x*,

*y*) is the real part of

*η*(

*x*,

*y*). The moiré fringe pattern and Fourier spectrum of the interference field are shown in Fig. 1(c) and Fig. 1(d), respectively. It can be observed that the spectrum of the signal components overlap each other. The proposed technique does not require any addition of carrier frequency for separation of signal components as proposed in [5

5. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic moiré,” Appl. Opt. **50**, 4189–4197 (2011). [CrossRef] [PubMed]

*L*= 4 segments in each column. The grid spacings of the parameter space for

*b*and

*c*were calculated using Eq. (6) and Eq. (7). Using these grid spacings, the phase parameters were estimated over a small grid of size 19 × 12.

*φ*

_{1}(

*x*,

*y*) and Δ

*φ*

_{2}(

*x*,

*y*) are plotted in Fig. 2(a) and Fig. 2(b), respectively. The error in the estimation of Δ

*φ*

_{1}(

*x*,

*y*) and Δ

*φ*

_{2}(

*x*,

*y*) are plotted in Fig. 2(c) and Fig. 2(d), respectively. The root-mean-square error (RMSE) in estimation of Δ

*φ*

_{1}(

*x*,

*y*) and Δ

*φ*

_{2}(

*x*,

*y*) were found to be 0.0558 and 0.0818 radians, respectively. The error in the phase estimation near the segment boundaries can be further reduced using overlapping segments [12

12. S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express **18**, 560–565 (2010). [CrossRef] [PubMed]

13. G. Rajshekhar and P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Laser Eng. **50**, iii–x (2012). [CrossRef]

**51**, 1168–1172 (2013). [CrossRef]

*φ*

_{1}(

*x*,

*y*) and Δ

*φ*

_{2}(

*x*,

*y*) were found to be 4.6797 and 13.0087 radians, respectively. This shows that the performance of PWHT based phase estimation is far superior as compared to that proposed in [7

**51**, 1168–1172 (2013). [CrossRef]

**56**, 4162–4175 (2008). [CrossRef]

6. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Estimation of multiple phases from a single fringe pattern in digital holographic interferometry,” Opt. Express **20**, 1281–1291 (2012). [CrossRef] [PubMed]

*A*

_{1}:

*A*

_{2}) using the simulated phases shown in Fig. 1(a) and Fig. 1(b). For each case, the calculated RMSE in the estimation of interference phases are given in Table 1. The RMSE values, found to be well below 0.1 radians, indicate that the accuracy of phase estimation is least affected by the variations in amplitude ratio. It should be noted that the variation in estimation error with varying amplitude ratio is mainly caused by the variation in the white Gaussian noise component added in the signal during each simulation run.

**20**, 1281–1291 (2012). [CrossRef] [PubMed]

**51**, 1168–1172 (2013). [CrossRef]

*nm*as shown in Fig. 3. Two distinct intensities were set for the two object beams using a beam intensity filter (IF) in the object beam OB2 arm to establish the amplitude discrimination criteria. The variable filter provides the flexibility of setting different object beam intensity ratios. Holograms were recorded with a CCD camera (XCL-U1000, Sony Corporation, Japan) of size 1600 × 1200 pixels. A circular membrane with clamped edges was used as a light diffusing object. The object was subjected to out-of-plane deformation with a point load and an in-plane rigid body rotation was superimposed on this deformation. The coordinate system used is also shown in the figure.

**50**, 4189–4197 (2011). [CrossRef] [PubMed]

## 4. Conclusion

## References and links

1. | P. Picart, D. Mounier, and J. M. Desse, “High-resolution digital two-color holographic metrology,” Opt. Lett. |

2. | P. Picart, E. Moisson, and D. Mounier, “Twin-sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. |

3. | C. Kohler, M. R. Viotti, and A. G. Albertazzi Jr., “Measurement of three-dimensional deformations using digital holography with radial sensitivity” Appl. Opt. |

4. | S. Okazawa, M. Fujigaki, Y. Morimoto, and T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. |

5. | G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic moiré,” Appl. Opt. |

6. | G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Estimation of multiple phases from a single fringe pattern in digital holographic interferometry,” Opt. Express |

7. | R. Kulkarni and P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Laser Eng. |

8. | K. Pokorski and K. Patorski, “Separation of complex fringe patterns using two-dimensional continuous wavelet transform,” Appl. Opt. |

9. | S. Barbarossa, “Analysis of multicomponent lfm signals by a combined wigner-hough transform,” IEEE T. Signal Process. |

10. | L. Cirillo, A. Zoubir, and M. Amin, “Parameter estimation for locally linear fm signals using a time-frequency hough transform,” IEEE T. Signal Process. |

11. | D. S. Pham and A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE T. Signal Process. |

12. | S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express |

13. | G. Rajshekhar and P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Laser Eng. |

**OCIS Codes**

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.2880) Instrumentation, measurement, and metrology : Holographic interferometry

(120.4290) Instrumentation, measurement, and metrology : Nondestructive testing

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: February 13, 2014

Revised Manuscript: March 18, 2014

Manuscript Accepted: March 26, 2014

Published: April 3, 2014

**Citation**

Rishikesh Kulkarni and Pramod Rastogi, "Simultaneous measurement of in-plane and out-of-plane displacements using pseudo-Wigner-Hough transform," Opt. Express **22**, 8703-8711 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8703

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### References

- P. Picart, D. Mounier, J. M. Desse, “High-resolution digital two-color holographic metrology,” Opt. Lett. 33, 276–278 (2008). [CrossRef] [PubMed]
- P. Picart, E. Moisson, D. Mounier, “Twin-sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. 42, 1947–1957 (2003). [CrossRef] [PubMed]
- C. Kohler, M. R. Viotti, A. G. Albertazzi, “Measurement of three-dimensional deformations using digital holography with radial sensitivity” Appl. Opt. 49, 4004–4009 (2010). [CrossRef] [PubMed]
- S. Okazawa, M. Fujigaki, Y. Morimoto, T. Matui, “Simultaneous measurement of out-of-plane and in-plane displacements by phase-shifting digital holographic interferometry,” Appl. Mech. Mater. 3–4, 223–228 (2005). [CrossRef]
- G. Rajshekhar, S. S. Gorthi, P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic moiré,” Appl. Opt. 50, 4189–4197 (2011). [CrossRef] [PubMed]
- G. Rajshekhar, S. S. Gorthi, P. Rastogi, “Estimation of multiple phases from a single fringe pattern in digital holographic interferometry,” Opt. Express 20, 1281–1291 (2012). [CrossRef] [PubMed]
- R. Kulkarni, P. Rastogi, “Multiple phase estimation in digital holographic interferometry using product cubic phase function,” Opt. Laser Eng. 51, 1168–1172 (2013). [CrossRef]
- K. Pokorski, K. Patorski, “Separation of complex fringe patterns using two-dimensional continuous wavelet transform,” Appl. Opt. 51, 8433–8439 (2012). [CrossRef] [PubMed]
- S. Barbarossa, “Analysis of multicomponent lfm signals by a combined wigner-hough transform,” IEEE T. Signal Process. 43, 1511–1515 (1995). [CrossRef]
- L. Cirillo, A. Zoubir, M. Amin, “Parameter estimation for locally linear fm signals using a time-frequency hough transform,” IEEE T. Signal Process. 56, 4162–4175 (2008). [CrossRef]
- D. S. Pham, A. M. Zoubir, “Analysis of multicomponent polynomial phase signals,” IEEE T. Signal Process. 55, 56–65 (2007). [CrossRef]
- S. S. Gorthi, G. Rajshekhar, P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010). [CrossRef] [PubMed]
- G. Rajshekhar, P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Laser Eng. 50, iii–x (2012). [CrossRef]

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