## Continuous phase estimation from noisy fringe patterns based on the implicit smoothing splines |

Optics Express, Vol. 22, Issue 9, pp. 10775-10791 (2014)

http://dx.doi.org/10.1364/OE.22.010775

Acrobat PDF (2690 KB)

### Abstract

We introduce the algorithm for the direct phase estimation from the single noisy interferometric pattern. The method, named implicit smoothing spline (ISS), can be regarded as a formal generalization of the smoothing spline interpolation for the case when the interpolated data is given implicitly. We derive the necessary equations, discuss the properties of the method and address its application for the direct estimation of the continuous phase in both classical interferometry and digital speckle pattern interferometry (DSPI). The numerical illustrations of the algorithm performance are provided to corroborate the high quality of the results.

© 2014 Optical Society of America

## 1. Introduction

*n*is continuous and has first

*n*− 1 continuous derivatives. The smoothing spline [29, 30

30. C. Reinsch, “Smoothing by spline functions,” Numer. Math. **10**, 177–183 (1967). [CrossRef]

31. A. Federico and G. H. Kaufmann, “Local denoising of digital speckle pattern interferometry fringes by multiplicative correlation and weighted smoothing splines,” Appl. Opt. **44**(14), 2728–2735 (2005). [CrossRef] [PubMed]

32. M. J. Peyrovian and A. A. Sawchuk, “Restoration of noisy blurred images by a smoothing spline filter,” Appl. Opt. **16**(12), 3147–3153 (1977). [CrossRef] [PubMed]

*π*ambiguity or function symmetry ambiguity. Thus, we may avoid potentially difficult phase unwrapping at the cost of providing a reasonable initial approximation to the phase distribution. The common problem of numerous fringe processing algorithms (RPT, CWT, Fourier) in the low fringe frequency regions is not severe for the ISS, since it does not explicitly depend on the concept of the fringe frequency. Finally, it constitutes a new framework for the phase demodulation and with very small number of parameters that need to be specified, the method is not difficult to automatize.

## 2. About the method

30. C. Reinsch, “Smoothing by spline functions,” Numer. Math. **10**, 177–183 (1967). [CrossRef]

### 2.1. Cubic smoothing spline

*p*. On the opposite sides of the spectrum of solutions described by the smoothing spline are the direct spline interpolation (no restriction on smoothness,

*p*= 0) and straight line least squares fit (for

*p*→ ∞). Formally, the cubic smoothing spline is a spline function of degree

*n*= 3 minimizing the following penalized sum of squares (PSS) functional where

*S*=

_{i}*S*(

*x*), the spline function

_{i}*S*(

*x*) evaluated at the interpolation knots

*x*(one knot for each pixel in this case) and Ω represents the interpolation domain, i.e., in 1D it is the interval Ω = [

_{i}*x*

_{0},

*x*]. Data value at the

_{N}*i*-th interpolation knot is denoted by

*Y*. The stunning fact is that there exists an unique solution

_{i}*Ŝ*(

*x*), specified with the vector of values

*Ŝ*, minimizing such a PSS functional and that formula for calculating such a spline function can be given explicitly. There is no need for any special numerical procedure aimed at the PSS functional minimization, the solution is found just by the means of the elementary linear algebra. This is possible because the penalizing part of Eq. (1) can be written as a simple bilinear form [30

30. C. Reinsch, “Smoothing by spline functions,” Numer. Math. **10**, 177–183 (1967). [CrossRef]

*K*is an

*N*×

*N*matrix of constant coefficients that only depends on the spline degree and the knots distributions (see Appendix A for details in case of the cubic spline). Differentiating the PSS functional we find the system of equations Equaling derivatives to zero we find the solution

*Ŝ*(

*x*), given by with identity matrix

*I*. The numerical implementations of the smoothing spline typically take advantage of the

*K*matrix sparsity to solve the system of Eqs. (4), which is of particular importance if the large data set is considered.

### 2.2. Implicit smoothing spline

*Y*does not correspond directly to the sought distribution

_{i}*S*=

_{i}*S*(

*x*), but rather to the known scalar function

_{i}*f*of

*S*. Therefore we modify the first component (the error) of the PSS functional, leaving the second component (the smoothness) untouched. Following the derivation outlined in the former section, we find After differentiation and some straightforward algebra we find the following system of algebraic equations where we denoted elementwise multiplication by dot and differentiation with the apostrophe. Observe that if

_{i}*f*(

*S*) is an identity, i.e.,

*f*(

*S*) =

_{i}*S*, Eq. (6) reduces to Eq. (4) and we have a standard smoothing spline problem. We emphasize that this is not the differential equation, since we are looking for the vector

_{i}*Ŝ*and the function

*f*(and hence its derivatives) is known explicitly. For

*normalized*fringe pattern, which is of our interest,

*f*(

*S*) = cos(

_{i}*S*) and Eq. (6) becomes constituting a nonlinear system of equations with the unknown vector

_{i}*Ŝ*. This is unlike the case of a regular smoothing spline, where the system to be solved was linear, Eq. (4). However, as long as functions involved in Eq. (7) are continuous and smooth, the gradient-based nonlinear algebraic solvers are expected to work fine. In case of

*f*(

*S*) = cos(

_{i}*S*), the minimizer of the PSS given in Eq. (5) can not be unique even for the simple reason of the cosine function 2

_{i}*π*periodicity. This means that some attention has to be given to the choice of the initial condition for the nonlinear solver, which we discuss further.

*ϕ*) +

*N*(

*x*),

*ϕ*=

*x*

^{2}function spoiled with the additive white Gaussian noise

*N*(

*x*) of standard deviation

*σ*= 0.5. Figures 1(a) and 1(b) show the phase demodulation approach with the smoothing spline, i.e., denoise the intensity first and then decode phase with direct inversion of the cos(

*ϕ*) function. On the other hand, Figs. 1(c) and 1(d) show the ISS approach where it is phase distribution to be directly approximated by the smoothing spline. Constant function was used as the initial phase guess for the nonlinear solver. While the phase estimation quality obviously depends on the smoothing parameter

*p*, we have repeatedly found the direct phase estimation superior to intensity filtration and subsequent phase decoding.

## 3. Application for the fringe pattern phase decoding

*f*(

*S*) = cos(

_{i}*S*), some details of the method need to be further clarified. In Fig. 2 we show the flowchart of the general algorithm allowing to utilize ISS for the fringe pattern phase decoding. Subsequently, the algorithm is discussed in more details.

_{i}### 3.1. Nonlinear solver initial condition

*Ŝ*(

*x*) may converge to the solution branches mutually shifted by 2

*kπ*, integer

*k*. Initial phase guess must be provided. While the guess does not need to be very accurate and constant value was enough in the example shown in Fig. 1, it should indicate the closest solution branch to the nonlinear solver. In any situation, when the approximate phase distribution is known

*a priori*(e.g., when departure from the ideal object shape is measured), it can be used as the initial guess. In the subsequent demonstrations we address the more complicated case, when nothing is known about the expected phase distribution. Following [28

28. G. Wang, Y. J. Li, and H. Ch. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. **50**(19), 3110–3117 (2011). [CrossRef] [PubMed]

### 3.2. Choice of the smoothing parameter

*p*, controlling the smoothness of the phase estimate

*Ŝ*(

*x*). Wrong choice of

*p*may result either in preserving the unwanted noise component or oversmoothing the phase estimation. Many methods of selecting the smoothing parameter value were developed for the classic spline interpolation, among which the generalized cross-validation (GCV, see, e.g., [33

33. P. Craven and G. Wahba, “Smoothing noisy data with spline functions estimating the correct degree of smoothing by the method of generalized cross validation,” Numer. Math. **31**, 377–403 (1979). [CrossRef]

*very much*unpractical processing times. One of the reasons of the GCV method success is that effective, non-direct implementation is available. It can only be employed for the classic smoothing spline though, where linear equations are solved. While further investigations can reveal GCV methods (heuristic, most likely) available for the ISS algorithm, we are currently satisfied with the intuitive selection of the smoothness parameter

*p*. Larger values of

*p*provide better estimation quality for the data of lower signal to noise ratio. In all of the numerical tests presented further values of

*p*were kept around

*p*= 0.1.

### 3.3. Equation or functional?

*P*(

_{f}*S*) functional from Eq. (5). Not only these formulations are not strictly equivalent from the mathematical point of view, they also need numerical methods to be employed in a different manners. We investigated both cases, utilizing the powerful nonlinear solver based on the trust-region dogleg method [34, 35] (free Fortran and commercial Matlab implementations available). While for both problems solution should converge to the same vector

*Ŝ under proper initial conditions*, we observed different numerical performance. The method solving Eq. (7) repeatedly found the solution of higher quality, indicating better robustness of this formulation. This observation is consistent with the computational mathematic practice, see chapters IX-X of [36]. Since both approaches yielded similar time performance, we limit further demonstrations to the variant of the ISS implementation in which Eq. (7) is solved.

### 3.4. Extension to 2D

### 3.5. Data normalization

*b*(

*x*) and fringe modulation

*m*(

*x*), it is necessary to transform data into form Therefore, preprocessing is necessary to remove background illumination influence and the possible contrast variations from the input. However, the noise component does not need to be effectively treated at the preprocessing stage - the filtering properties of the implicit smoothing spline account for it. There are many methods available in the literature to perform the necessary preprocessing when the low quality fringe pattern is analyzed. One example is the fast and data-driven technique based on the fast and adaptive empirical mode decomposition [26

26. M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express **20**(21), 23463–23479 (2012). [CrossRef] [PubMed]

27. M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. **52**(1), 230–240 (2014). [CrossRef]

### 3.6. Relation to other phase estimation methods

- phase estimation is typically performed based on the local model, typically of a linear [13] or quadratic [14
13. L. Kai and Q. Kemao, “A generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express

**20**(11), 12579–12592 (2012). [CrossRef] [PubMed]] polynomial. WFT, CWT and ST can also be regarded as methods employing locally the linear phase model. Spline, on the other hand is a global function represented with a14. L. Kai and Q. Kemao, “Improved generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express

**21**(20), 24385–24397 (2013). [CrossRef] [PubMed]*different*polynomial between each adjacent knots. Since in our approach every single pixel constitutes a knot, we can represent*any*function defined on our discrete domain using splines, there is no model limitation whatsoever; - thanks to the smoothing splines algebraic properties, Eq. (2), we globally optimize just the phase values themselves rather than local sets of polynomial coefficients or other model parameters and since we are able to cast the optimization problem in the closed form of the system of algebraic equations depending only on the data values and unknown phase distribution, we may easily apply robust nonlinear algebraic solvers;
- global character means stronger dependence on the quality of the initial phase condition than in case of the local approaches. On the other hand this is the reason why the continuous phase distribution is found without the frequency integration;
- the smoothness term of the ISS is curvature-based, while methods such as RPT typically enforce smoothness based on the phase variability. This means that if smoothness is pushed to the limit, ISS is likely to find the best-fit constant frequency term while other methods will tend to reduce any phase variability. This implies that the choice of the smoothness parameter
*p*is less critical in the presented approach.

## 4. Numerical tests

### 4.1. General comparison between the compared methods

28. G. Wang, Y. J. Li, and H. Ch. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. **50**(19), 3110–3117 (2011). [CrossRef] [PubMed]

*Interpolation on the skeleton*is a direct phase estimation based on the set of interpolation knots being the positive and negative fringe ridges, i.e., the*skeleton*. Thin plate spline (TPS) interpolation is used, since it allows to perform calculations with the ungridded set of knots, results in smooth function and performs well even with low fidelity, disconnected skeleton [28]. The skeleton acts like a set of contour lines of phase distribution, separated by**50**(19), 3110–3117 (2011). [CrossRef] [PubMed]*π*radians (2*π*if only positive ridges are taken into account). Thus, the denser are the fringes and higher skeleton evaluation quality, the better method’s performance. Unlike ISS, this approach ignores the points outside of the skeleton. Also, the skeleton is fixed as the set of interpolation knots, while ISS allows, to certain extent, to correct the skeleton calculation errors, as it is only utilized for the nonlinear solver initial guess. Thin plate spline is expensive to evaluate, so this may not be the preferred technique when the skeleton is a very large set, e.g., for the spatial carrier fringe pattern. For some details regarding the skeleton calculation methods, see [2]. In the examples considered further, we assume that fringe ridges are calculated exactly and the adequate integer multiplicity of2. D. Malacara, M. Servin, and Z. Malacara,

*Interferogram Analysis for Optical Testing*(CRC, 2005). [CrossRef]*π*is assumed as a ridge value. While this is not the true value at the ridge pixel for the realistic (i.e., discretely sampled) image, it is still a high quality estimation - more than one could expect from the skeleton estimation on the real, noisy data. Note that such an approach renders results independent of the synthetic pattern noise level. We refer to this method further with the TPS acronym. Note that whenever the ISS algorithm utilizes the TPS method output as an initial phase, the final phase error can not become any larger in result of the ISS processing. Hence, to be precise, we rather observe how much more can ISS reduce the phase error than compare ISS to TPS.*Wavelet transform methods*constitute both popular and highly acknowledged phase decoding algorithms. In this comparison we utilized 2D complex continuous Morlet wavelet (CWT), that can be regarded as a windowed Fourier transform improved by utilizing additional relation between the frequency and the window width [17]. When it helped the phase calculation quality, we used unusually small value of the envelope wideness parameter (such an effect is not surprising for the low fringe densities). While theoretically this limits the mid-pass filter properties of the CWT, we found overall results to be much better - for typical width of the envelope Morlet CWT could not compete neither with TPS nor ISS in any of the considered examples other than the high frequency carrier image. Big advantage of the CWT is that there is no need for the additional preprocessing such as normalization or skeleton calculation. On the other hand, the resultant phase is mod 217. M. A. Gdeisat, D. R. Burton, and D. R. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt.

**45**(34), 8722–8732 (2006). [CrossRef] [PubMed]*π*wrapped. Also, even function ambiguity (cos(*ϕ*) = cos(−*ϕ*)) is present, leading to further unwrapping errors. One possible method to deal with this problem is described in [37]. In the error calculations given further we assumed that all wrapped phase ambiguities can be resolved with perfect accuracy. The quality was always worse when we performed the actual phase unwrapping. In the corresponding figures we show the phase decoding results in the wrapped form and error distributions of the perfectly unwrapped data.37. J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt.

**50**(16), 2425–2430 (2011). [CrossRef] [PubMed]

### 4.2. Synthetic interferometric patterns

#### 4.2.1. Test 1: small phase variation

*very*effective filtration algorithm). The comparison was performed for the phase distribution

*ϕ*

_{1}shown in Fig. 3(e) for the intensity images with additive white Gaussian noise of standard deviation

*σ*

_{1}= 0.1 and

*σ*

_{2}= 0.5. Image of the

*ϕ*

_{1}(

*x*,

*y*) function is confined to the [0, 3] set. Constant phase was utilized as the initial phase guess for the ISS nonlinear solver, since the skeleton interpolation was not feasible. We show phase estimation results for

*σ*

_{2}in Fig. 3, while phase errors for both noise levels are given in Table 1. While we do not indicate this in the results, it was also possible to perform the phase demodulation with the Morlet 2D CWT, with the strongly narrowed envelope. The error magnitude was similar as in case of the Gaussian filtration, far worse than for the ISS method.

#### 4.2.2. Test 2: continuous phase estimation

*ϕ*

_{2}= 5.25 ·

*ϕ*

_{1}. Phase variation is large enough to produce some fringes, see Fig. 4, so the wavelet method is more applicable than in the previous example. TPS skeleton interpolation and ISS are also viable. The Fourier method can not be applied because of the fringe orientation variation. Because of the large fringe curvature in relation to its frequency, envelope width parameter of the CWT was tuned for the higher quality (it is 0.35 of the typically used value). We show results for the noise standard deviation

*σ*

_{2}= 0.5 in Fig. 4 and give errors for

*σ*

_{1}= 0.1 and

*σ*

_{2}= 0.5 in Table 1.

#### 4.2.3. Test 3: fringes with spatial carrier

*ϕ*

_{3}= 2.625 ·

*ϕ*

_{1}distribution are present. This is the kind of input for which Fourier (classic method described in [3

3. M. Takeda, H. Ina, and S. Kobayashi, “Fourier transform methods of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. **72**(1), 156–160 (1982). [CrossRef]

*σ*

_{1}= 0.1 and

*σ*

_{2}= 0.5, respectively. As previously, figures are shown for the noise of

*σ*

_{2}= 0.5, see Fig. 5, while the errors for

*σ*

_{1}= 0.1 and

*σ*

_{2}= 0.5 are indicated in Table 2.

### 4.3. DSPI images processing

### 4.4. Processing experimental patterns

39. G. H. Kaufmann, “Nondestructive testing with thermal waves using phase shifted temporal speckle pattern interferometry,” Opt. Eng. **42**(7), 2010–2014 (2003). [CrossRef]

40. K. Patorski and L. Salbut, “Simple polarization phase-stepping scatterplate interferometry,” Opt. Eng. **43**(2), 393–397 (2004). [CrossRef]

### 4.5. Remarks on the numerical performance

## 5. Conclusions

- smoothing splines were formulated for arbitrary odd polynomial degree [30], in principle one could imagine extension of this algorithm to the quintic (or even higher order) spline interpolation.
30. C. Reinsch, “Smoothing by spline functions,” Numer. Math.

**10**, 177–183 (1967). [CrossRef] - weighted errors or spatially varying smoothness parameter (see, e.g., [41]) should work within the ISS framework in the same manner as with the classic smoothing spline. It could be a very useful feature in case when the pattern quality and/or signal to noise ratio vary throughout the domain.
41. C. de Boor, “Calculation of the smoothing spline with weighted roughness measure,” this paper can be downloaded at http://www.cs.wisc.edu.

- a reasonable method to automatically choose the smoothness parameter
*p*is necessary. - employing (meta)heuristic procedures, such as simulated annealing or genetic algorithms, to solve Eq. (5) would relax the initial phase guess restrictions by enabling the global extremum localization.
- more sophisticated ISS implementation will help with the computation time reduction. In particular, since the initial stage of the algorithm works as the set of independent 1D interpolation problems, parallelization should be feasible.

## 6. Appendix A

*x*. The differences vector

_{i}*h*=

_{i}*x*

_{i+1}−

*x*. Note that for our needs, when knots correspond simply to subsequent image pixels,

_{i}*h*is likely to be constant. Nevertheless, we give a more general equations here. The matrix

*K*, introduced in Eq. (2) and utilized by both smoothing splines and ISS methods, is in the form of with cubic spline basis Gram matrix of size

*N*− 2 ×

*N*− 2 and matrix

*Q*of size

*N*− 2 ×

*N*While Eq. (4) is correct, from the numerical point of view it is not a preferred way to calculate the smoothing spline. Typically, some equivalent linear equation is solved [29]. For the ISS algorithm implementation, we are currently directly solving numerically Eq. (7).

## 7. Appendix B

## Acknowledgments

## References and links

1. | D. Robinson and G. Reid, eds., |

2. | D. Malacara, M. Servin, and Z. Malacara, |

3. | M. Takeda, H. Ina, and S. Kobayashi, “Fourier transform methods of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. |

4. | C. Rodier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. |

5. | Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principle, applications and implementations,” Opt. Lasers Eng. |

6. | S. Fernandez, M. A. Gdeisat, J. Salvi, and D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. |

7. | Y. Ichioka and N. Inuiya, “Direct phase detecting system,” Appl. Opt. |

8. | O. Y. Kwon and D. M. Sough, “Multichannel grating phase-shift interferometers,” Proc. SPIE |

9. | K. Creath and J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. |

10. | M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. |

11. | H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express |

12. | C. Tian, Y. Y. Yang, D. Liu, Y. J. Luo, and Y. M. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. |

13. | L. Kai and Q. Kemao, “A generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express |

14. | L. Kai and Q. Kemao, “Improved generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express |

15. | L. Watkins, S. Tan, and T. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. |

16. | Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. |

17. | M. A. Gdeisat, D. R. Burton, and D. R. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. |

18. | L. R. Watkins, “Review of fringe pattern phase recovery using 1-D and 2-D continuous wavelet transforms,” Opt. Lasers Eng. |

19. | K. Pokorski and K. Patorski, “Processing and phase analysis of fringe patterns with contrast reversals,” Opt. Express |

20. | P. Etchepareborda, A. L. Vadnjal, A. Federico, and G. H. Kaufmann, “Phase-recovery improvement using analytic wavelet transform analysis of a noisy interferogram cepstrum,” Opt. Lett. |

21. | A. Dursun, Z. Sarac, H. S. Topkara, S. Ozder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement |

22. | M. Zhong, W. Chen, T. Wang, and X. Su, “Application of two-dimensional S-Transform in fringe pattern analysis,” Opt. Lasers Eng. |

23. | N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London A |

24. | N. E. Huang and Z. H. Wu, “A review on Hilbert-Huang transform method and its applications to geophysical studies,” Rev. Geophys. |

25. | Y. Lei, J. Lin, Z. He, and M. Zuo, “A review on empirical mode decomposition in fault diagnosis of rotating machinery,” Mech. Syst. Signal Process. |

26. | M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express |

27. | M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. |

28. | G. Wang, Y. J. Li, and H. Ch. Zhou, “Application of the radial basis function interpolation to phase extraction from a single electronic speckle pattern interferometric fringe,” Appl. Opt. |

29. | C. de Boor, |

30. | C. Reinsch, “Smoothing by spline functions,” Numer. Math. |

31. | A. Federico and G. H. Kaufmann, “Local denoising of digital speckle pattern interferometry fringes by multiplicative correlation and weighted smoothing splines,” Appl. Opt. |

32. | M. J. Peyrovian and A. A. Sawchuk, “Restoration of noisy blurred images by a smoothing spline filter,” Appl. Opt. |

33. | P. Craven and G. Wahba, “Smoothing noisy data with spline functions estimating the correct degree of smoothing by the method of generalized cross validation,” Numer. Math. |

34. | M. J. D. Powell, “A hybrid method for nonlinear equations,” in |

35. | M. J. D. Powell, “A Fortran subroutine for solving systems of nonlinear algebraic equations,” in |

36. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

37. | J. Ma, Z. Wang, B. Pan, T. Hoang, M. Vo, and L. Luu, “Two-dimensional continuous wavelet transform for phase determination of complex interferograms,” Appl. Opt. |

38. | J. W. Goodman, |

39. | G. H. Kaufmann, “Nondestructive testing with thermal waves using phase shifted temporal speckle pattern interferometry,” Opt. Eng. |

40. | K. Patorski and L. Salbut, “Simple polarization phase-stepping scatterplate interferometry,” Opt. Eng. |

41. | C. de Boor, “Calculation of the smoothing spline with weighted roughness measure,” this paper can be downloaded at http://www.cs.wisc.edu. |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

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

(120.3940) Instrumentation, measurement, and metrology : Metrology

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(120.6160) Instrumentation, measurement, and metrology : Speckle interferometry

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: January 23, 2014

Revised Manuscript: April 8, 2014

Manuscript Accepted: April 10, 2014

Published: April 28, 2014

**Citation**

Maciek Wielgus, Krzysztof Patorski, Pablo Etchepareborda, and Alejandro Federico, "Continuous phase estimation from noisy fringe patterns based on the implicit smoothing splines," Opt. Express **22**, 10775-10791 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10775

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### References

- D. Robinson, G. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurements (Institute of Physics, 1993).
- D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (CRC, 2005). [CrossRef]
- M. Takeda, H. Ina, S. Kobayashi, “Fourier transform methods of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]
- C. Rodier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987). [CrossRef]
- Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principle, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007). [CrossRef]
- S. Fernandez, M. A. Gdeisat, J. Salvi, D. Burton, “Automatic window size selection in Windowed Fourier Transform for 3D reconstruction using adapted mother wavelets,” Opt. Commun. 284(12), 2797–2807 (2011). [CrossRef]
- Y. Ichioka, N. Inuiya, “Direct phase detecting system,” Appl. Opt. 11(7), 1507–1514 (1972). [CrossRef] [PubMed]
- O. Y. Kwon, D. M. Sough, “Multichannel grating phase-shift interferometers,” Proc. SPIE 599, 273–279 (1985). [CrossRef]
- K. Creath, J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24, 365–379 (1996). [CrossRef]
- M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997). [CrossRef] [PubMed]
- H. Wang, Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009). [CrossRef] [PubMed]
- C. Tian, Y. Y. Yang, D. Liu, Y. J. Luo, Y. M. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49(2), 170–179 (2010). [CrossRef] [PubMed]
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