## Local frequency and envelope estimation by Teager-Kaiser energy operators in white-light scanning interferometry |

Optics Express, Vol. 22, Issue 15, pp. 18325-18334 (2014)

http://dx.doi.org/10.1364/OE.22.018325

Acrobat PDF (833 KB)

### Abstract

In this work, a new method for surface extraction in white light scanning interferometry (WLSI) is introduced. The proposed extraction scheme is based on the Teager-Kaiser energy operator and its extended versions. This non-linear class of operators is helpful to extract the local instantaneous envelope and frequency of any narrow band AM-FM signal. Namely, the combination of the envelope and frequency information, allows effective surface extraction by an iterative re-estimation of the phase in association with a new correlation technique, based on a recent TK cross-energy operator. Through the experiments, it is shown that the proposed method produces substantially effective results in term of surface extraction compared to the peak fringe scanning technique, the five step phase shifting algorithm and the continuous wavelet transform based method. In addition, the results obtained show the robustness of the proposed method to noise and to the fluctuations of the carrier frequency.

© 2014 Optical Society of America

## 1. Introduction

_{B}, has been introduced to track the interaction energy between two signals [16

16. J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy operator,” Elect. Lett. **40**, 778–780 (2004). [CrossRef]

17. A. Boudraa, “Relationships between Ψ_{B}-energy operator and some time-frequency representations,” IEEE Sig. Proc. Lett. **17**, 527–530 (2010). [CrossRef]

18. A. Boudraa, T. Chonavel, and J. Cexus, “Ψ_{B}-energy operator and cross-power spectral density,” Sig. Proc. **94**, 236–240 (2014). [CrossRef]

_{B}can be used as a correlator in both the time [19

19. A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-Ψ_{B}-energy operator-based signal detection,” JASA **123**, 4283–4289 (2008). [CrossRef]

18. A. Boudraa, T. Chonavel, and J. Cexus, “Ψ_{B}-energy operator and cross-power spectral density,” Sig. Proc. **94**, 236–240 (2014). [CrossRef]

20. F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. **17**, 2804–2815 (2005). [CrossRef]

_{B}correlator as local and non-linear methods. In this work, we show how this class of operators can be used in WLSI, computing the local phase by correlation, according to the estimated envelope and frequency to improve the surface measurement.

## 2. Experimental system and interferometric signal

*xyz*images. The fringe signals along the optical

*z*-axis at each pixel in an image are processed to determine the fringe envelope, with the peaks indicating the positions of the surface. For polychromatic interferometry, the total intensity is the sum of the interferences at each wavelength. A typical intensity signal obtained from a digital camera as the OPD is varied in the interferometer at a given point (

*x*,

*y*) on the material surface, can be approximated along the optical

*z*-axis by a modulated sinusoid as follows [2

2. K. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” JOSA A **13**, 832–843 (1996). [CrossRef]

*z*is a vertical scanning position along the optical axis,

*h*(

*x*,

*y*) represents the height of the surface,

*a*(

*x*,

*y*,

*z*) is an offset intensity containing low frequency components,

*b*(

*x*,

*y*) is a factor proportional to the reflected beam intensity, and

*α*(

*x*,

*y*) is an additional phase offset and

*C*(

*x*,

*y*,

*z*) is the envelope. The parameter

*lc*represents the coherence length of the light source and

*λ*

_{0}is the average wavelength of the light source. Generally the phase offset varies slowly from one point (

*x*,

*y*) to the next, and can be neglected, since only relative heights of the surface matter.

*λ*

_{0}is an averaged value, its possible fluctuations have been taken account using a flexible approach. Most often, the analysis of such signals is limited to the optical

*z*-axis. To overcome this problem and for more robust analysis, the signal could be processed according to slice (

*x*,

*z*) to take into account the neighborhood information.

## 3. Principles of the Teager-Kaiser energy operators

10. P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Sig. Proc. Lett. **2**, 152–154 (1995). [CrossRef]

*x*(

*t*) is given by where

*ẋ*(

*t*) and

*ẍ*(

*t*) denote the first and the second time derivative of

*x*(

*t*) respectively. Under realistic conditions [11

11. P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. **41**, 3024–3051 (1993). [CrossRef]

*s*(

*t*) =

*a*(

*t*) cos(

*ϕ*(

*t*)), the 1D TK energy operator yields as output Ψ[

*s*(

*t*)] ≈ [

*a*(

*t*)

*ϕ̇*(

*t*)]

^{2}. Thus the local envelope

*a*(

*t*) and the instantaneous frequency

*ϕ̇*(

*t*) can be estimated using the Energy separation algorithm (ESA) [11

11. P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. **41**, 3024–3051 (1993). [CrossRef]

*I*(

*x*

_{1},

*x*

_{2}) as a 2D TK operator In a similar manner to the 1D TK operator, for a narrowband image

*I*(

*x*

_{1},

*x*

_{2}) =

*a*(

*x*

_{1},

*x*

_{2}) cos(

*ϕ*(

*x*

_{1},

*x*

_{2})) the output of the 2D TK is given by Ψ[

*x*

_{1},

*x*

_{2}] ≈ [

*a*(

*x*

_{1},

*x*

_{2})

*ϕ̇*(

*x*

_{1},

*x*

_{2})]

^{2}. The spatially-varying amplitude

*a*(

*x*

_{1},

*x*

_{2}) can be interpreted as modeling local image contrast and the instantaneous frequency

*ω*(

*x*

_{1},

*x*

_{2}) = ∇

*ϕ*(

*x*

_{1},

*x*

_{2}) describes locally energy spatial frequencies. Applying Φ to

*∂I/∂x*

_{1},

*∂I/∂x*

_{2}and combining all energies, yields the 2D ESA [12

12. P. Maragos and A. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A **12**, 1867–1876 (1995). [CrossRef]

2. K. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” JOSA A **13**, 832–843 (1996). [CrossRef]

12. P. Maragos and A. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A **12**, 1867–1876 (1995). [CrossRef]

14. K. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express **13**, 8097–8121 (2005). [CrossRef] [PubMed]

15. F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc. **24**, 543–572 (2013). [CrossRef]

*a*(

**u**) and frequency vector

**w**= (

*w*

_{1},

*w*

_{2},...,

*w*)

_{n}*for any locally narrow band multidimensional signal such as:*

^{T}*s*(

**u**) ≃

*a*(

**u**) cos(

*ϕ*(

**u**)), where

**u**= (

*x*

_{1},

*x*

_{2},...,

*x*) and ∇

_{n}*ϕ*=

**w**(

**u**) ≃

**w**.

## 4. Ψ_{B} energy operator

## 5. The proposed method

_{B}correlator are combined for envelope extraction and phase retrieval of a WLSI signal. This algorithm, referred to as PETKB (Phase and Envelope estimated by the TK and the Ψ

*operators) is detailed as follows:*

_{B}- Select a slice
*s*(*z*) =*s*(*x*_{0},*y*_{0},*z*) corresponding to the site*x*=*x*_{0},*y*=*y*_{0}. - Estimate the envelope
*Ĉ*(*z*) =*Ĉ*(*x*_{0},*z*) and the instantaneous frequency*ν̂*(*z*) =*ν*of the signal using the TK energy operator along the optical_{z}*z*-axis. - Smooth the obtained envelope using an interpolating spline function.
- Identify the local maxima(s)
*z*of the envelope_{max}*Ĉ*(*z*) on the optical*z*-axis. - Around each of these local maxima on the optical
*z*-axis: where*ε*is a threshold value. Equation (9) gives the maximum of interaction between signals*s*(*z*) and*s*(_{θ}*z*) by the Ψ_{B}operator, located at*θ*=*θ̂*.

## 6. Results

25. P. Hariharan, B. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. **26**, 2504–2506 (1987). [CrossRef] [PubMed]

*ν̂*(

*z*) is estimated at the local maxima

*z*. The 1D TK algorithm extracts the height of the surface by computing the local maxima(s) of the 1D envelope

_{max}*C*

_{1}(

*z*) along the optical

*z*-axis. The 2D TK algorithm first computes the 2D local envelope

*Ĉ*(

*x*,

*z*) using the 2D TK energy operator, and then extracts the 1D envelope

*Ĉ*(

*x*

_{0},

*z*) by projection along the optical z-axis. The derivatives of the TK operators are computed using the Savitzky-Golay filter [27

27. R. Schafer, “What is a savitzky-golay filter?[lecture notes],” IEEE Sig. Proc. Mag. **28**, 111–117 (2011). [CrossRef]

*θ*(Eq. (8)) using five consecutive samples along the optical

*z*-axis. We used a generalization of this method, to any

*T*value, given by [28

_{e}28. J. Estrada, M. Servin, and J. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express **18**, 2632–2638 (2010). [CrossRef] [PubMed]

*ω*

_{0}=

*π*/2 corresponds to

*ω*

_{0}=

*π*/4 corresponds to

*λ*

_{0}= 640 nm corresponds to

*T*= 80 nm for

_{e}*ω*

_{0}=

*π*/2 and

*T*= 40 nm for

_{e}*ω*

_{0}=

*π*/4). The intensities

*I*

_{0},

*I*

_{−2},

*I*

_{−1},

*I*

_{1},

*I*

_{2}represent respectively the intensity value at

*z*=

*z*and their values around

_{max}*I*

_{0}spaced by

*T*. The intensity

_{e}*I*

_{0}has been initialized by identifying the local maximum along the optical

*z*-axis using the 1D TK method. Once

*θ*is estimated,

*z*is updated by

_{max}*ν̂*.

_{z}*ε*is set to 1 nm. The results in Figs. 1(b) and 1(c) show respectively the reference surface shape and the profile of a noisy signal along the optical

*z*-axis with an additive offset low frequency signal sampled at

*T*= 40 nm. Figures 1(d) and 1(e) show respectively a signal

_{e}*s*(

_{θ}*z*) superimposed on the original one (the offset is removed) and the estimated envelope

*Ĉ*(

*z*) along the optical axis, before and after the step 5 of the PETKB. This highlights the ability of the local phase retrieval to correct the position of the maximum first detected by the TK1D. Table 1 summarizes the results obtained for different Gaussian noise levels with

*T*= 80 and

_{e}*λ*

_{0}= 640 nm. We show the rate of relative errors compared with the reference surfaces for both surfaces: ”surface1” corresponds to the highest surface i.e, the higher level signal, and ”surface2” corresponds to the lower surface i.e, the weaker level optical signal. Clearly, in the noise free case the best results are provided by the PETKB, CWT and FSPS methods. In a noisy environment (10% or 20%) the PETKB performs slightly better than the CWT method and outperforms the other methods. Figure 2 shows that the profiles of surface1 and surface2 are well extracted by PETKB and FSPS methods [Figs. 2(a) and 2(b)], while 1D TK and 2D TK yield the same profiles but with noticeable fluctuations [Figs. 2(c) and 2(d)]. For surface2 the PETKB [Fig. 2(a)] performs slightly better than the FSPS [Fig. 2(b)]. The robustness of the methods has also been studied with respect to the fluctuations of the central carrier frequency value, which is also equivalent to assume that the

*T*measurement is not constant (for example a piezoelectric stepper leading to an inaccuracy in sampling). In this case randomized carrier frequencies around the mean value with a 5% margin interval vs additive Gaussian noise has been chosen. Results reported in Table 2 show that, on average, our method outperforms the other methods. Also the PFSM algorithm appears to be less robust than the PETKB and the CWT methods, specially for surface2 where it yields the highest error rate values. Furthermore, we have examined the impact of smaller sampling rates on the surface extraction. Table 3 shows the results for

_{e}*T*= 40 nm and with a fixed carrier frequency. The PETKB method consistently outperforms the other methods and particularly the CWT method for both the noiseless and noisy conditions. Table 4 shows the results for a non-constant carrier frequency (with a 5% margin). Again, as in Table 3 the same conclusions can be drawn. Even for a smaller sampling rate, we still show the effectiveness of the PETKB method in term of the error rate of surface extraction. We observe from the results of Tables 1 and 4 that, on average, the PETKB method performs better than the other methods. This performance can be attributed to the combined actions of the TK and the Ψ

_{e}_{B}operators. Indeed, the only action of the TK (1D TK or 2D TK) yields moderate performance in term of surface extraction and particularly in noisy conditions. The main advantages of the Ψ

_{B}operator are, first its efficient local ability as an instantaneous differential operator and second its ability as a correlator to accurately estimate the phase shift angle of the interferogram. In practice no more than 3 iterations are needed for the convergence of the PETKB algorithm, which is quite reasonable and helpful for the computing time performances. Simulations were carried out on a personal computer with a 2.13GHz Intel Core i3 and 4GB RAM. The computational time, for 256×256 interferometric image, of the CWT and the PETKB methods are compared. The PET KB takes 97 seconds while the CWT takes 95 seconds. We expect that if the scale depth of the CWT increases, then the associated CPU time will too. Even if in terms of computational time the two methods perform similarly, the PETKB provides better surface extraction results than the CWT. Finally, we show in Fig. 3 the demodulation results of real data with

*T*= 80 nm. These measurements were made on a system developed in our laboratory consisting of a Leitz-Linnik interference microscope with ×50 objectives (numerical aperture = 0.85) under white light illumination as described in [29

_{e}29. E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. **256**, 6144–6152 (2010). [CrossRef]

*μ*m PIFOC model (from PI) with integrated LVDT position sensor to give a linear response and a sensitivity of 5 nm. Figures 3(a) and 3(b) display respectively the original image and its 2D gradient. This derivative method allows us to eliminate the ”batwing effect” or envelope skewing, caused by interference from the top and bottom of the steps for heights less than the coherence length of the light used. Figures 3(c)–3(f) compare the robustness of the profile extraction using the PETKB, FSPS, 1D TK and 2D TK methods. Except for the few artefacts, the extracted surface profile by the PETKB is more accurate than the ones obtained by the other methods, and especially the FSPS method.

## 7. Conclusion

_{B}energy operators as instantaneous energy tracking tools. The TK envelope detection is combined with the TK frequency estimation to retrieve the envelope and instantaneous frequency simultaneously. Once the reference signal is extracted, a correlation technique based on the non-linear operator Ψ

*is used to identify the surface height information contained simultaneously in the phase and envelope of the signal along the optical*

_{B}*z*-axis. Computations performed on synthetic and real data with

*T*= 80 nm and

_{e}*T*= 40 nm along the optical

_{e}*z*-axis, show the interest of this TK class of operators, and highlight their robustness to the changes in carrier frequencies, compared to the CWT, FSPS and PFSM methods. In terms of execution time and error rate of surface extraction results, the PETKB method is more competitive than the CWT approach. The promising results obtained are a motivation to take into account more complex information and to apply the 2D and/or 3D detection (correlation) associated with the 2D and/or 3D envelope/frequency estimation. In future work we plan to extend this approach to multidimensional signals.

## Acknowledgments

## References and links

1. | C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. |

2. | K. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” JOSA A |

3. | P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt. |

4. | S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. |

5. | P. Sandoz, “Wavelet transform as a processing tool in white-light interferometry,” Opt. Lett. |

6. | M. Li, C. Quan, and C. Tay, “Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry,” Opt. Laser Technol. |

7. | Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” App. Opt. |

8. | H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. |

9. | J. Kaiser, “On a simple algorithm to calculate the energy of a signal,” in Proc. ICASSP, (1990), pp. 381–384. |

10. | P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Sig. Proc. Lett. |

11. | P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. |

12. | P. Maragos and A. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A |

13. | A. Boudraa, F. Salzenstein, and J. Cexus, “Two-dimensional continuous higher-order energy operators,” Opt. Eng. |

14. | K. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express |

15. | F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc. |

16. | J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy operator,” Elect. Lett. |

17. | A. Boudraa, “Relationships between Ψ |

18. | A. Boudraa, T. Chonavel, and J. Cexus, “Ψ |

19. | A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-Ψ |

20. | F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. |

21. | S. Petitgrand, “Méthodes de microscopie interférométrique 3D statiques et dynamiques pour la caractérisation de la technologie et du comportement des microsystèmes,” Ph.D. thesis (2005). |

22. | A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-Ψ |

23. | A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. |

24. | W. Zhang, C. Liu, and H. Yan, “Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,” Patt. Recong. |

25. | P. Hariharan, B. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. |

26. | P. Montgomery and J. Fillard, “Peak fringe scanning microscopy (pfsm): submicron 3d measurement of semiconductor components,” Interferometry: Techniques and Analysis pp. 12–23 (1755). |

27. | R. Schafer, “What is a savitzky-golay filter?[lecture notes],” IEEE Sig. Proc. Mag. |

28. | J. Estrada, M. Servin, and J. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express |

29. | E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.5070) Image processing : Phase retrieval

(100.3175) Image processing : Interferometric imaging

(100.4992) Image processing : Pattern, nonlinear correlators

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 12, 2014

Revised Manuscript: June 30, 2014

Manuscript Accepted: July 3, 2014

Published: July 22, 2014

**Citation**

F. Salzenstein, P. Montgomery, and A. O. Boudraa, "Local frequency and envelope estimation by Teager-Kaiser energy operators in white-light scanning interferometry," Opt. Express **22**, 18325-18334 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18325

Sort: Year | Journal | Reset

### References

- C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol.14, 1807 (2003). [CrossRef]
- K. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” JOSA A13, 832–843 (1996). [CrossRef]
- P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt.41, 4571–4578 (2002). [CrossRef]
- S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm.284, 2488–2493 (2011). [CrossRef]
- P. Sandoz, “Wavelet transform as a processing tool in white-light interferometry,” Opt. Lett.22, 1065–1067 (1997). [CrossRef] [PubMed]
- M. Li, C. Quan, and C. Tay, “Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry,” Opt. Laser Technol.40, 920–929 (2008). [CrossRef]
- Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” App. Opt.43, 2695–2702 (2004). [CrossRef]
- H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng.47, 1334–1339 (2009). [CrossRef]
- J. Kaiser, “On a simple algorithm to calculate the energy of a signal,” in Proc. ICASSP, (1990), pp. 381–384.
- P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Sig. Proc. Lett.2, 152–154 (1995). [CrossRef]
- P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc.41, 3024–3051 (1993). [CrossRef]
- P. Maragos and A. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A12, 1867–1876 (1995). [CrossRef]
- A. Boudraa, F. Salzenstein, and J. Cexus, “Two-dimensional continuous higher-order energy operators,” Opt. Eng.44, 7001–7010 (2005).
- K. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express13, 8097–8121 (2005). [CrossRef] [PubMed]
- F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc.24, 543–572 (2013). [CrossRef]
- J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy operator,” Elect. Lett.40, 778–780 (2004). [CrossRef]
- A. Boudraa, “Relationships between ΨB-energy operator and some time-frequency representations,” IEEE Sig. Proc. Lett.17, 527–530 (2010). [CrossRef]
- A. Boudraa, T. Chonavel, and J. Cexus, “ΨB-energy operator and cross-power spectral density,” Sig. Proc.94, 236–240 (2014). [CrossRef]
- A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-ΨB-energy operator-based signal detection,” JASA123, 4283–4289 (2008). [CrossRef]
- F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc.17, 2804–2815 (2005). [CrossRef]
- S. Petitgrand, “Méthodes de microscopie interférométrique 3D statiques et dynamiques pour la caractérisation de la technologie et du comportement des microsystèmes,” Ph.D. thesis (2005).
- A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-ΨB energy operator,” Int. J. Electron. Comm.63, 728–735 (2009). [CrossRef]
- A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc.135892, 1–8 (2008).
- W. Zhang, C. Liu, and H. Yan, “Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,” Patt. Recong.43, 3969–3976 (2010). [CrossRef]
- P. Hariharan, B. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.26, 2504–2506 (1987). [CrossRef] [PubMed]
- P. Montgomery and J. Fillard, “Peak fringe scanning microscopy (pfsm): submicron 3d measurement of semiconductor components,” Interferometry: Techniques and Analysis pp. 12–23 (1755).
- R. Schafer, “What is a savitzky-golay filter?[lecture notes],” IEEE Sig. Proc. Mag.28, 111–117 (2011). [CrossRef]
- J. Estrada, M. Servin, and J. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express18, 2632–2638 (2010). [CrossRef] [PubMed]
- E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci.256, 6144–6152 (2010). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.