## Half-quadratic cost function for computing arbitrary phase shifts and phase: Adaptive out of step phase shifting

Optics Express, Vol. 14, Issue 8, pp. 3204-3213 (2006)

http://dx.doi.org/10.1364/OE.14.003204

Acrobat PDF (288 KB)

### Abstract

We present a phase shifting robust method for irregular and unknown phase steps. The method is formulated as the minimization of a half-quadratic (robust) regularized cost function for simultaneously computing phase maps and arbitrary phase shifts. The convergence to, at least, a local minimum is guaranteed. The algorithm can be understood as a phase refinement strategy that uses as initial guess a coarsely computed phase and coarsely estimated phase shifts. Such a coarse phase is assumed to be corrupted with artifacts produced by the use of a phase shifting algorithm but with imprecise phase steps. The refinement is achieved by iterating alternated minimization of the cost function for computing the phase map correction, an outliers rejection map and the phase shifts correction, respectively. The method performance is demonstrated by comparison with standard filtering and arbitrary phase steps detecting algorithms.

© 2006 Optical Society of America

## 1. Introduction

*I*= {

*I*

_{1},

*I*

_{2},… ,

*I*

_{K}} a set of

*K*phase shifted fringe patterns, then the

*k*

^{th}fringe pattern,

*I*

_{k}, is modelled by

*r*= [

*x,y*]

^{T}denotes a pixel position in the image lattice

*L*,

*a*

_{k}is the background illumination component,

*b*

_{k}is the fringe contrast,

*f*is the unknown phase, Δ = {Δ

_{1}, Δ

_{2},…, Δ

_{K}} is the phase shift vector and η

_{k}represents additive independent noise. We assume that a normalization process is performed on the fringe pattern so that the illumination components are for filtering out: thus we have that

*a*

_{k}≈ 0 and

*b*

_{k}is estimated. Therefore the normalized fringe pattern,

*Î*, is approximated by

*b*in the model because we do not want to perform division by zero

*b*values (or noise). However, without lost of generality, in the rest of the paper we assume

*b*≈ 1. and drop it from the rest of the derivation.

1. J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. **30**, 3018–3020 (2005). [CrossRef] [PubMed]

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A **22**, 1170–1175 (2005). [CrossRef]

*a*and

*b*, is eliminated) and the phase shifts, Δ, can be introduced with high accuracy. In such a case the wrapped phase

*f̂*, can be recovered with very simple algorithms; where

*f̂*=

*W*(

*f*) ≝

*f*+ 2

*πn*for an integer

*n*such that

*f̂*∈(-

*π,π*] (where

*W*denotes the wrapping operator). For instance, for a low level noise,

*f̂*can be computed from 4 fringe patterns (by assuming Δ

_{k}=

*kπ*/2) with:

*f*=

*W*

^{-1}(

*f̂*) the unwrapped phase computed with the unwrapping operator

*W*

^{-1}. However the phase unwrapping process is an

*ill posed*problem because a wrapped phase may correspond to multiple unwrapped phases, i.e. there exist many

*f̂*≠

*f*such that

*f̂*=

*W*(

*f̂*). For this reason, the unwrapping operator

*W*

^{-1}is, in general, implemented as the minimization of a regularized cost function, see for instance [4

4. M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. **29**, 504–506 (2004). [CrossRef] [PubMed]

## 2. Out of step phase shifting

5. J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White, and D.J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**, 2693–2703 (1974). [CrossRef] [PubMed]

8. C.J. Morgan, “Least squates estimation in phase-measurement interferometry,” Opt. Lett. **7**368–370 (1982). [CrossRef] [PubMed]

*δ*the computed phase shifts vector. The development of methods for computing arbitrary phase shifts has been a prolific research [9

9. G, Lai and T. Yatagai, ”Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A **8**, 822-(1991) [CrossRef]

22.
14.
C.J. Tay, C. Quan, and L. Chen, “Phase retrieval with a three-frame phase-shifting algorithm with an unknown phase shift,” Appl. Opt. **44**, 1401–1409 (2005). [CrossRef] [PubMed]

*I*and

*J*by means of the arccos of their correlation [12

12. H. van Brug, “Phase-step calibration for phase-stepped interferometry,” Appl. Opt. **38**3549–3555 (1999). [CrossRef]

*I*and

*J*[18

18. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. **28**, 1808–1810 (2003). [CrossRef] [PubMed]

*b*is the illumination contrast component (assumed equal for both interferograms). An alternative method was reported by Larkin in [17

17. K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express **9**, 236–253 (2001). [CrossRef] [PubMed]

*ill-posed*. To understood this claim, one can consider the case of 3 fringe patterns with unknown phase shifts, in such a case one have more unknowns (illumination components, phase and shifts) than data (that are, besides, corrupted with noise).

*ill-posed*problem could be solved by a minimum last squares procedure that implies to increase the number of computed fringe patterns with the subsequent increment in the complexity of the experimental procedure; or by using regularization techniques that implies to incorporate prior knowledge about the solution.

23. C. Rathjen, Statistical properties of phase-shift algorithms, J. Opt. Soc. Am. A **12**, 1997–2008 (1995). [CrossRef]

26. B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. **36**, 2070–2075 (1997). [CrossRef] [PubMed]

*π, π*], phase shifts.

## 3. Adaptive out of step phase shifting

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A **22**, 1170–1175 (2005). [CrossRef]

4. M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. **29**, 504–506 (2004). [CrossRef] [PubMed]

33. M. Rivera and J.L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image and Vision Computing **21**, 345–357 (2003). [CrossRef]

*δ*= {

*δ*

_{1},

*δ*

_{2},…,

*δ*

_{K}} of the unknown phase steps, Δ is available. Therefore, we have

*α*= {

*α*

_{1},

*α*

_{2},…,

*α*

_{K}} are the unknown phase step residuals. If we expect small residual phases, then a coarse estimation of the real phase can be computed by means of the minimization of (4) and an unwrapping method [4

4. M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. **29**, 504–506 (2004). [CrossRef] [PubMed]

*f*, and its coarse estimation,

*ψ*:

*ϕ*is an unknown residual phase field that corrects artifacts produced by noise and the residual steps,

*α*. Thus by substituting (8) and (9) in the normalized model (2), we obtain

*b*

_{k}≈ 1 for simplifying the presentation, but the fringe contrast term can and should be used in the procedure.

*ϕ*

_{r}, and therefore to compute the real phase,

*f*. For such a purpose we need to estimate the phase steps residuals,

*α*.

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A **22**, 1170–1175 (2005). [CrossRef]

*ψ*+

*ϕ*, and the true phase steps,

*δ*+

*α*, from a phase shifted pattern sets. Following [2

**22**, 1170–1175 (2005). [CrossRef]

*ϕ*

_{r}+

*α*

_{k}| is relatively small such that the first order Taylor series can be used to define the residual error:

*x*+

*y*) = cos

*x*cos

*y*- sin

*x*sin

*y*[as in the Grievenkamp’s residual error, Eq. (5)] and then by assuming

*y*small enough such that cos

*y*≈ 1 and sin

*y*≈

*y*; thus cos(

*x*+

*y*) ≈ cos

*x*-

*y*sin

*x*. Precisely, the advantage of formulating our problem as the computation of the small unknown corrections (instead of the whole value) is that the residual error can be approximated by the linear Taylor expansion

*E*

_{kr}. It is precautious to expect that there could be some places (pixels) where, actually, the residual error were large. Large residual error can be the result of a corrupted initial phase (for instance, phase corrupted with fringe harmonics), an unprecise initial phase shifts or data corruption (for instance, occlusions in the illumination). Then we incorporate a mechanism for weighting the contribution of the data in the estimation process. Such outliers rejection mechanism is implemented by a half-quadratic potential controlled by a parameter that avoids an over–detection or under-detection of atypical data.

*ill posed*problem of, then prior knowledge should be. We assume that unknown real phase

*f*is smooth in the sense of have small second derivatives. Thus we penalize the correction phase

*ϕ*to produces a smooth

*f*. Therefore, we propose to compute the phase correction field,

*ϕ*, the phase step corrections,

*α*, and an outliers detection field,

*ω*, by an alternating quadratic minimization of the cost function:

**22**, 1170–1175 (2005). [CrossRef]

*γ*) that enforce small values for the step correction vector,

*α*, and the residual phase field,

*ϕ*. Such terms improved significantly the convergence and stability of the minimization process. The details of the phase refinement procedure are formalized in Algorithm 1. It is important to note in Algorithm 1 that once a residual (

*ϕ*or

*α*) is computed, then the corresponding base variable (

*ψ*or

*δ*) is updated. Such a strategy reduces iteratively the value of the unknown residual and the fitness of the first order Taylor approach, and consequently the algorithm performance.

**Algorithm 1** Adaptive out of step phase shifts.

*g*= {

*g*

_{1},

*g*

_{2},… ,

*g*

_{K}} a fringe pattern set with expected phase steps equal to

*δ*and

*ψ*an initial coarse phase.

- Set, initially
*ϕ*= 0,*ω*= 1 and given ε > 0; - For all the pixels
*r*∈*R*: **while**||*g*- cos*ψ*|| > ε**do**- Compute
*ψ*_{r}=*ψ*_{r}+*ϕ*_{r}and then set*ϕ*_{r}= 0; - Compute
*α*= argmin_{α}*U*(*ϕ*= 0,*ω*,*α*); {use (13)} - Update
*δ*=*δ*+*α*and then set*α*= 0; - Compute
*ω*= argmin_{ω}*U*(*ϕ*= 0,*ω*,*α*= 0); {use (14)} - Compute
*ϕ*= argmin_{ϕ}*U*(*ϕ*,*ω*,*α*= 0); - {see Appendix A in [2
**22**, 1170–1175 (2005). [CrossRef] **end while**- Finish with results
*ψ*and*δ*;

*ψ*, can be computed with standard algorithms by assuming correct phase steps, i.e. by neglecting the residual steps,

*α*. Such a wrapped phase is corrupted with artifacts introduced by the residual steps. Then it recommended to use a robust algorithm for unwrapping the coarse wrapped phase. In particular, we use the half-quadratic convex unwrapping algorithm reported in [4

**29**, 504–506 (2004). [CrossRef] [PubMed]

*ψ*, may have a constant residual step,

*δ*

_{dc}, that can be approximated by a reduced search, i.e:

*D*= {

*d*

_{i}= 2

*πi*/

*N*}, for

*i*= 0,1,2,…,

*N*- 1, is a

*N*steps set (we use

*N*= 20 in our experiments). Alternatively, the method in Eqs. (6) or (7) can be used for the same purpose. Taking into account that the accuracy of these methods are reduced if the fringe patterns are corrupted by additive independent noise, see experiments in Fig. 4. Nevertheless how the

*δ*

_{dc}is estimated, we initialize the coarse phase by assigning:

*ψ*←

*ψ*+

*δ*

_{dc}. On the other hand, the steps

*δ*

_{k}’s can be initialized with the ideal values or, to reduce the risk of large

*α*residuals, can be estimated (as

*δ*

_{dc}).

*ϕ*= 0 and

*ω*fixed, cost function (12) can be written as:

*ψ̂*

_{kr}≝

*ψ*

_{r}+

*δ*

_{k}and the potential

*Q*(∙) contains the

*α*–independent terms. Equating to zero the partial gradient with respect to (w.r.t.)

*α*and solving for

*α*

_{k}, we obtain a closed formula for computing the

*α*’s optimum coefficients:

*ϕ*= 0 and

*α*= 0, we obtain a closed formula for computing the

*ω*field:

*ϕ*is obtained by solving the linear system that results of equaling to zero the partial gradient w.r.t.

*ϕ*of

*U*(

*ϕ*,

*α*= 0,

*ω*), keeping

*α*= 0 and keeping fixed

*ω*. In particular, we use a Gauss-Seidel scheme similar to the one proposed in the Appendix A of [2

**22**, 1170–1175 (2005). [CrossRef]

## 4. Experiments

## 5. Conclusion

## References and links

1. | J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. |

2. | M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A |

3. | M. Rivera, J.L. Marroquin, S. Botello, and M. Servin, “A robust spatio-temporal quadrature filter for multi-phase stepping,” Appl. Opt. |

4. | M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. |

5. | J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White, and D.J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. |

6. | J.E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. |

7. | J.E. Grievenkamp and J.H. Bruning, “Phase shifting interferometry,” in |

8. | C.J. Morgan, “Least squates estimation in phase-measurement interferometry,” Opt. Lett. |

9. | G, Lai and T. Yatagai, ”Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A |

10. | C. S. Vikram, W. K. Witherow, and J. D. Trolinger, ”Algorithm for phase-difference measurement in phase-shifting interferometry,” Appl. Opt. |

11. | C. Wei, M. Chen, and Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. |

12. | H. van Brug, “Phase-step calibration for phase-stepped interferometry,” Appl. Opt. |

13. | M. Chen, H. Guo, and C. Wei,“Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. |

14. | W. Li and X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Commun. |

15. | O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express |

16. | K.A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. |

17. | K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express |

18. | L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. |

19. | A. Patil, B. Raphael, and P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. |

20. | A. Patil, R. Langoju, and P. Rastogi, “An integral approach to phase shifting interferometry using a super-resolution frequency estimation method,” Opt. Express |

21. | A. Patil and P. Rastogi, “Nonlinear regression technique applied to generalized phase-shifting interferometry,” J. Mod. Opt. |

22. |
14.
C.J. Tay, C. Quan, and L. Chen, “Phase retrieval with a three-frame phase-shifting algorithm with an unknown phase shift,” Appl. Opt. |

23. | C. Rathjen, Statistical properties of phase-shift algorithms, J. Opt. Soc. Am. A |

24. | K. Hibino, B.F. Oreb, D.I. Farrant, and K.G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A |

25. | Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. |

26. | B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. |

27. | K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. |

28. | J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters for multi-phase stepping images,” Opt. Lett. |

29. | D. Geman and G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. |

30. | M.J. Black and A. Rangarajan, “Unification of line process, outlier rejection, and robust statistics with application in early vision,” Int. J. Comput. Vis. |

31. | P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. |

32. | M. Rivera and J.L. Marroquin, “Adaptive rest condition potentials: Second order edge-preserving regularization,” Comput. Vision Image Understand. |

33. | M. Rivera and J.L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image and Vision Computing |

**OCIS Codes**

(100.2650) Image processing : Fringe analysis

(100.3020) Image processing : Image reconstruction-restoration

(100.3190) Image processing : Inverse problems

(120.3180) Instrumentation, measurement, and metrology : Interferometry

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

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 22, 2006

Revised Manuscript: April 4, 2006

Manuscript Accepted: April 11, 2006

Published: April 17, 2006

**Citation**

Mariano Rivera, Rocky Bizuet, Amalia Martinez, and Juan A. Rayas, "Half-quadratic cost function for computing arbitrary phase shifts and phase: Adaptive out of step phase shifting," Opt. Express **14**, 3204-3213 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3204

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### References

- J. A. Guerrero, J. L. Marroquin, M. Rivera, J. A. Quiroga, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett. 30, 3018-3020 (2005). [CrossRef] [PubMed]
- M. Rivera, "Robust phase demodulation of interferograms with open or closed fringes," J. Opt. Soc. Am. A 22, 1170-1175 (2005). [CrossRef]
- M. Rivera, J.L. Marroquin, S. Botello and M. Servin, "A robust spatio-temporal quadrature filter for multi-phase stepping," Appl. Opt. 39, 284-292 (2000). [CrossRef]
- M. Rivera and J.L. Marroquin, "Half-quadratic cost functions for phase unwrapping," Opt. Lett. 29, 504-506 (2004). [CrossRef] [PubMed]
- J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White and D.J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974). [CrossRef] [PubMed]
- J.E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).
- J.E. Grievenkamp and J.H. Bruning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara ed. (John Wiley & Sons, Inc. New York, 1992) pp. 501-598.
- C.J. Morgan, "Least squates estimation in phase-measurement interferometry," Opt. Lett. 7368-370 (1982). [CrossRef] [PubMed]
- G, Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822- (1991) [CrossRef]
- C. S. Vikram, W. K. Witherow, and J. D. Trolinger, "Algorithm for phase-difference measurement in phaseshifting interferometry," Appl. Opt. 32, 6250-6252 (1993). [CrossRef] [PubMed]
- C. Wei, M. Chen, Z. Wang, "General phase-stepping algorithm with automatic calibration of phase steps," Opt. Eng. 38, 1357-1360 (1999). [CrossRef]
- H. van Brug, "Phase-step calibration for phase-stepped interferometry," Appl. Opt. 383549-3555 (1999). [CrossRef]
- M. Chen, H. Guo, and C. Wei,"Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000). [CrossRef]
- W. Li and X. Su, "Real-time calibration algorithm for phase shifting in phase-measuring profilometry," Opt. Commun. 40, 761-766 (2001).
- O. Soloviev and G. Vdovin, "Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Express 13, 3743-3753 (2005) [CrossRef] [PubMed]
- K.A. Goldberg and J. Bokor, "Fourier-transform method of phase-shift determination," Appl. Opt. 402886-1894 (2001) [CrossRef]
- K. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9, 236-253 (2001). [CrossRef] [PubMed]
- L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003). [CrossRef] [PubMed]
- A. Patil, B. Raphael and P. Rastogi, "Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search," Opt. Lett. 29, 1381-1383 (2004). [CrossRef] [PubMed]
- A. Patil, R. Langoju, and P. Rastogi, "An integral approach to phase shifting interferometry using a superresolution frequency estimation method," Opt. Express 12, 4681-4697 (2004) [CrossRef] [PubMed]
- A. Patil and P. Rastogi, "Nonlinear regression technique applied to generalized phase-shifting interferometry," J. Mod. Opt. 52, 573 - 582 (2005). [CrossRef]
- 14 C.J. Tay, C. Quan, L. Chen, "Phase retrieval with a three-frame phase-shifting algorithm with an unknown phase shift," Appl. Opt. 44, 1401-1409 (2005). [CrossRef] [PubMed]
- C. Rathjen, Statistical properties of phase-shift algorithms, J. Opt. Soc. Am. A 12, 1997-2008 (1995). [CrossRef]
- K. Hibino, B.F. Oreb, D.I. Farrant, and K.G. Larkin, "Phase shifting for nonsinusoidal waveforms with phaseshift errors," J. Opt. Soc. Am. A 12, 761-768 (1995). [CrossRef]
- Y. Surrel, "Design of algorithms for phase measurements by the use of phase stepping," Appl. Opt. 35, 51-60 (1996). [CrossRef] [PubMed]
- B. Zhao and Y. Surrel, "Effect of quantization error on the computed phase of phase-shifting measurements," Appl. Opt. 36, 2070-2075 (1997). [CrossRef] [PubMed]
- K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991). [CrossRef]
- J. L. Marroquin, M. Servin and R. Rodriguez-Vera, "Adaptive quadrature filters for multi-phase stepping images," Opt. Lett. 23, 238-240 (1998). [CrossRef]
- D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Image Process. 14, 367-383 (1992).
- M.J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996). [CrossRef]
- P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997). [CrossRef] [PubMed]
- M. Rivera and J.L. Marroquin, "Adaptive rest condition potentials: Second order edge-preserving regularization," Comput. Vision Image Understand. 88, 76-93 (2002). [CrossRef]
- M. Rivera, and J.L. Marroquin, "Efficient half-quadratic regularization with granularity control," Image and Vision Computing 21, 345—357 (2003). [CrossRef]

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