## Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes.

Optics Express, Vol. 15, Issue 5, pp. 2288-2298 (2007)

http://dx.doi.org/10.1364/OE.15.002288

Acrobat PDF (1595 KB)

### Abstract

We propose a new approach to demodulate a single fringe pattern with closed fringes by using Local Adaptable Quadrature Filters (LAQF). Quadrature filters have been widely used to demodulate complete image interferograms with carrier frequency. However, in this paper, we propose the use of quadrature filters locally, assuming that the phase is locally quasimonochromatic, since quadrature filters are not capable to demodulate image interferograms with closed fringes. The idea, in this paper, is to demodulate the fringe pattern with closed fringes sequentially, using a fringe following scanning strategy. In particular we use linear *robust quadrature filters* to obtain a fast and robust demodulation method for single fringe pattern images with closed fringes. The proposed LAQF method does not require a previous fringe pattern normalization. Some tests with experimental interferograms are shown to see the performance of the method along with comparisons to its closest competitor, which is the Regularized Phase Tracker (RPT), and we will see that this method is tolerant to higher levels of noise.

© 2007 Optical Society of America

## 1. Introduction

*moiré*interferometry, as well as in other areas of optical metrology, when one is working with transient events, one can have situations where it is necessary to deal with a single fringe pattern with closed fringes. As the information of interest is phase modulated by the fringe pattern, it is necessary to apply a demodulation method able to demodulate a single fringe pattern with closed fringes.

*a*(

*x,y*) is the background illumination and

*ϕ*(

*x,y*) the modulation term or contrast. The phase to be demodulated is f(x,y). If the fringe pattern has a carrier frequency, then it can be modeled in the following way:

*ϕ*(

*x,y*) is the modulating phase, and

*ω*is the carrier frequency. In this case, we can demodulate the fringe pattern by using quadrature filters like those used with the Fourier transform method [1

1. 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**156–160 (1982). [CrossRef]

2. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A , **3**847–855, 1986. [CrossRef]

*ϕ*(

*x,y*). However, this kind of methods fails to obtain the modulating phase when we have a fringe pattern like the one shown in (1), or in other words, when we have a single fringe pattern with closed fringes.

**LAQF**). In general, a quadrature filter is a band-pass filter that is zero in one half of the Fourier domain, and maps its input to a complex space. For example, if the input is a fringe pattern like the one shown in Eq. (1), the output is a complex signal whose real part is the fringe pattern itself, and the imaginary part is its quadrature. However, when we have closed fringes, the input signal quadrature, obtained with a quadrature filter, has abrupt sign changes when applied to the complete fringe pattern [2

2. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A , **3**847–855, 1986. [CrossRef]

3. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A , **18**1862–1870 (2001). [CrossRef]

4. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform.,” J. Opt. Soc. Am. A , **18**1871–1881 (2001). [CrossRef]

5. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A , **20**925–934 (2003). [CrossRef]

**RPT**) [6

6. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A , **18**689–695 (2001). [CrossRef]

6. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A , **18**689–695 (2001). [CrossRef]

### 1.1. RPT method

*ϕ*̂(

*x,y*) and frequencies

*u*(

*x,y*) and

*v*(

*x,y*):

*p*(η,ξ) =

*ϕ*̂(

*x,y*) +

*u*(

*x,y*)(

*x*- η) +

*v*(

*x,y*)(

*y*- ξ) is a phase plane and

*ϕ*̂(

*x,y*) is the estimated phase at site (

*x,y*) after the minimization. The closed region Γ is a neighborhood around site (

*x,y*), and

*λ*is a regularization parameter to strengthen the method against noise.

*I*́(

*x,y*) is the fringe pattern shown in Eq. (1), but normalized in the following way:

*a*(

*x,y*) is removed using a high-pass filter, and the modulation term

*b*(

*x,y*) is spatially normalized to the constant value 1, using normalization techniques for fringe patterns [7

7. J. A. Quiroga, J. A Gãmez-Pedrero, and A. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. , **197**43–51 (2001). [CrossRef]

8. J. A. Quiroga and M. Servin, “Isotropic *n*-dimensional fringe pattern normalization,” Opt. Commun. , **224**221–227 (2003). [CrossRef]

9. J. A. Guerrero, J. L. Marroquin, and M. Rivera, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. , **30**318–320 (2005). [CrossRef]

10. B. Strobel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. , **35**2192–2198 (1996). [CrossRef] [PubMed]

**FFS**) throughout this paper.

- 1. Apply a band-pass filter to remove the background illumination and attenuate the noise.
- 2. Normalize the fringe pattern in order to make the contrast component,
*b*(*x,y*), spatially constant. - 3. Demodulate the fringe pattern using the RPT.

*I*(

*x,y*) is the image interferogram modeled in (1), and σ

_{L}« σ

_{H}, i.e, σ

_{L}= 2.4 and σ

_{H}= 80, for 256 × 256 image interferograms or biggers. This process can be implemented in a fast way by using the fast Fourier transform [11].

7. J. A. Quiroga, J. A Gãmez-Pedrero, and A. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. , **197**43–51 (2001). [CrossRef]

8. J. A. Quiroga and M. Servin, “Isotropic *n*-dimensional fringe pattern normalization,” Opt. Commun. , **224**221–227 (2003). [CrossRef]

9. J. A. Guerrero, J. L. Marroquin, and M. Rivera, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. , **30**318–320 (2005). [CrossRef]

1. 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**156–160 (1982). [CrossRef]

3. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A , **18**1862–1870 (2001). [CrossRef]

4. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform.,” J. Opt. Soc. Am. A , **18**1871–1881 (2001). [CrossRef]

5. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A , **20**925–934 (2003). [CrossRef]

12. R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns,” Appl. Opt. , **41**5519–5526 (2002). [CrossRef] [PubMed]

12. R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns,” Appl. Opt. , **41**5519–5526 (2002). [CrossRef] [PubMed]

*b*̂ is the modulation estimation, and

*b*and

_{x}*b*its partial derivatives in

_{y}*x*and

*y*respectively. We have removed the (

*x,y*) dependence of each variable for more clarity. The term

*β*(η, ξ) is a modulation plane defined as

*β*(η,ξ) =

*b*̂(

*x,y*)+

*b*(

_{x}*x,y*)(

*x*- η)+

*b*(

_{y}*x,y*)(

*y*- ξ). If we see Eq. (3) and Eq. (7), we can see that the modified RPT method reported in [12

12. R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns,” Appl. Opt. , **41**5519–5526 (2002). [CrossRef] [PubMed]

**41**5519–5526 (2002). [CrossRef] [PubMed]

6. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A , **18**689–695 (2001). [CrossRef]

13. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A , **22** no. 6, 1170–1175 (2005). [CrossRef]

**41**5519–5526 (2002). [CrossRef] [PubMed]

14. R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A , **23**2724–2731 (2006). [CrossRef]

14. R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A , **23**2724–2731 (2006). [CrossRef]

3. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A , **18**1862–1870 (2001). [CrossRef]

**41**5519–5526 (2002). [CrossRef] [PubMed]

13. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A , **22** no. 6, 1170–1175 (2005). [CrossRef]

14. R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A , **23**2724–2731 (2006). [CrossRef]

**23**2724–2731 (2006). [CrossRef]

15. J. L. Marroquin, J. E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A , **14**779–791 (1997). [CrossRef]

## 2. Proposed LAQF method

*x,y*) is a site in

*L*, being

*L*the lattice where the interferogram is recorded. Now, assume that in a small region Γ around site (

*x,y*), the fringe pattern locally is quasimonochromatic. Then, locally the fringe pattern looks like:

*u*

_{0},

*v*

_{0}) are the local frequencies in

*x*and

*y*respectively. As one can see in Eq. (3), the RPT method also uses this assumption and fits the observed data in Γ around (

*x,y*) with a phase plane to estimate the phase at site (

*x,y*) (see Eq. 3). Although this technique works properly, we have found a better and more robust way using local adaptable quadrature filters to estimate the phase sequentially.

*x,y*), we use the Robust Quadrature Filters developed in [15

15. J. L. Marroquin, J. E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A , **14**779–791 (1997). [CrossRef]

15. J. L. Marroquin, J. E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A , **14**779–791 (1997). [CrossRef]

*x,y*), and we adapt its tuning frequency as we move through the image interferogram’s sites using a fringe following scanning or FFS. Hence, we call this method local adaptable quadrature filters or LAQF along this paper.

*x,y*):

*f*is complex and can be expressed as

*f*(η, ξ) = φ(η, ξ) +

*iψ*(η, ξ),for(η, ξ)∈Γ.

*R*

_{Γ}[

*f,I*] in (10) is commonly known as the data term, and it depends on the difference between the observed data / (in this case the interferogram) and the estimation model

*f*, in such a way that

*R*

_{Γ}[

*f,I*] is minimal when

*f*is close to

*f*. Here, we define

*R*

_{Γ}[

*f,I*] as the residual between the finite differences of the observed data and the finite differences of the filter estimation model in the following way:

*f*and

_{x}*I*, are the finite differences in

_{x}*x*and

*f*, and

_{y}*I*are the finite differences in

_{y}*y*. For example, the finite differences for

*f*in

*x*may be given as

*f*(

_{x}*x,y*) =

*f*(

*x,y*) -

*f*(

*x*- 1,

*y*), and so on. We define the data term in this way because if we take its Fourier transform, this term has a zero in the origin, which is a desired feature in the use of quadrature filters to demodulate fringe patterns (see [2

2. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A , **3**847–855, 1986. [CrossRef]

*V*

_{Γ}[

*f*] is usually called the regularization term. This term adds restrictions to the estimation model

*f*. For example, a commonly used regularization term to restrict the filter estimation model

*f*from being smooth (to obtain a low-pass filter), is the quadratic norm of the

*Laplacian*’s operator known as the

*membrane model*. In our case, since we want a quadrature filter, we define the regularization term in the following way, according to Ref. [15

**14**779–791 (1997). [CrossRef]

*i*= √-1,

*u*

_{0}is the tuning frequency in

*x*, and

*v*

_{0}the tuning frequency in

*y*. In the Fourier domain, this term, along with

*R*

_{Γ}[

*f,I*], looks like a band-pass filter that is zero in one half of the Fourier domain, and maximum in frequency (

*u*

_{0},

*v*

_{0}) (see [15

**14**779–791 (1997). [CrossRef]

*λ*in (10) is known as the regularization parameter that controls the strength of the quadrature filter (or its bandwidth).

*f*and equal it to zero. Then, we can solve the resulting linear equation system using fast straightforward algorithms such as the

*Gauss-Seidel*algorithm, although we can use more generic algorithms, like the

*Steepest-descent*used by the RPT method [16, 6

**18**689–695 (2001). [CrossRef]

**14**779–791 (1997). [CrossRef]

*x,y*), we minimize Eq. (10) with respect to

*f*using the tuning frequency (

*u*

_{0},

*v*

_{0}). Once given the quadrature filter

*f*by minimizing (10), we obtain the phase (modulus 2

*π*) in region Γ as:

*x,y*) by using (13), we move to the next site using the fringe following scanning or FFS as with the RPT. Just before we move to the next site, however, we need to update the LAQF tuning frequency to estimate the phase correctly in region Γ around the next site. The local frequencies for current site (

*x,y*) may be used as the tuning frequency to estimate the phase in the next site, and are obtained as the finite differences along

*x*and

*y*of the estimated phase given by (13). But, as we see from Eq. (13), the estimated phase in region Γ around site (

*x,y*) is wrapped into the [-π,π] interval. For this reason, we take the local frequencies for the current site (

*x,y*) in the following way:

*x*

_{+}and

*y*

_{+}are the coordinates of the previous already phase estimated site, given by the FFS, and they may take the following values:

*u*

_{0},

*v*

_{0}) = (0.25,0.24) with

*λ*= 50 to obtain the phase in Γ. We must remark that in this case, for illustration purposes we have chosen a region size too big compared with the sizes of

*n*×

*n*, with typically,

*n*∈ 5,6,7,8.

*x,y*) as the initial seed to demodulate the fringe pattern. Then, we visit each site from the fringe pattern using the FFS. For each visited site, we estimate its local phase by minimizing Eq. (10) and using Eq. (13). After this, we obtain the tuning frequency for the next site to visit using (14) and (15).

*x,y*), we must know previously the tuning frequency to use. Here, we propose a simple way to choose the initial seed (

*x,y*) and its tuning frequency by using a Gabor filter. A Gabor filter is a quadrature filter defined in the following way:

*u*

_{0},

*v*

_{0}) are its tuning frequencies. Thus, we tune the Gabor filter onto a frequency (

*u*

_{0},

*v*

_{0}) that can be in the fringe pattern. Then, we filter the fringe pattern with the tuned Gabor filter. Finally, we take as initial seed the site (

*x,y*) for which the Gabor filter magnitude response is maximal. For example, suppose that

*I*͂ is the filtered fringe pattern with the Gabor filter tuned to the frequency (

*u*

_{0},

*v*

_{0}), then we take as initial seed the site:

*u*

_{0},

*v*

_{0}) to start the demodulation process.

## 3. Tests and results

- 1. Apply a band-pass filter to attenuate the noise and remove the background contribution.
- 2. Demodulate the fringe pattern using the LAQF as described in previous section.

_{L}= 2.5 and σ

_{H}= 80 using a fast Fourier transform algorithm [11]. The computational time spent in this process, for a 256 × 256 image insterferogram, was of 0.543 seconds. For our tests we used a 64bit personal computer architecture in a Linux like operating system, and a C-language 64bit optimized compiler.

*λ*= 20 and neighborhood size 6×6. The parameters used with the LAQF are

*λ*= 5, neighborhood Γ size 6×6.

*λ*= 5 and neighborhood Γ size 8×8.

## 4. Discussion and conclusions

*n*×

*n*for

*n*∈ {5,6,7,8} are suitable to demodulate correctly an image interferogram with closed fringes. Actually there is no rule to say what is the optimum neighborhood size to use, nither for the other sequential methods like the RPT.

13. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A , **22** no. 6, 1170–1175 (2005). [CrossRef]

**41**5519–5526 (2002). [CrossRef] [PubMed]

**23**2724–2731 (2006). [CrossRef]

## References and links

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

2. | T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A , |

3. | K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A , |

4. | K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform.,” J. Opt. Soc. Am. A , |

5. | M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A , |

6. | M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A , |

7. | J. A. Quiroga, J. A Gãmez-Pedrero, and A. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. , |

8. | J. A. Quiroga and M. Servin, “Isotropic |

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

10. | B. Strobel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. , |

11. | E. O. Brigham, |

12. | R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns,” Appl. Opt. , |

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

14. | R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A , |

15. | J. L. Marroquin, J. E. Figueroa, and M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A , |

16. | Jorge Nocedal and Stephen J. Wright, Numerical Optimization. Springer (1999). |

**OCIS Codes**

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

(120.3940) Instrumentation, measurement, and metrology : Metrology

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 29, 2006

Revised Manuscript: January 25, 2007

Manuscript Accepted: January 29, 2007

Published: March 5, 2007

**Citation**

J. C. Estrada, M. Servin, and J. L. Marroquín, "Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes," Opt. Express **15**, 2288-2298 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2288

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### References

- 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, 156-160 (1982). [CrossRef]
- T. Kreis, "Digital holographic interference-phase measurement using the Fourier-transform method," J. Opt. Soc. Am. A 3, 847-855, 1986. [CrossRef]
- K. G. Larkin, D. J. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1862-1870 (2001). [CrossRef]
- K. G. Larkin, "Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1871-1881 (2001). [CrossRef]
- M. Servin, J. A. Quiroga, and J. L. Marroquin, "General n-dimensional quadrature transform and its application to interferogram demodulation," J. Opt. Soc. Am. A 20, 925-934 (2003). [CrossRef]
- M. Servin and J. L. Marroquin and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A 18, 689-695 (2001). [CrossRef]
- J. A. Quiroga, J. A G’omez-Pedrero, and A. Garc’ıa-Botella, "Algorithm for fringe pattern normalization," Opt. Commun. 19, 743-51 (2001). [CrossRef]
- J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt. Commun. 22, 4221- 227 (2003). [CrossRef]
- J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett. 30, 318-320 (2005). [CrossRef]
- B. Strobel, "Processing of interferometric phase maps as complex-valued phasor images," Appl. Opt. 35, 2192- 2198 (1996). [CrossRef] [PubMed]
- E. O. Brigham, The fast fourier tranform. (Prentice-Hall, 1974).
- R. Legarda-S’aenz and W. Osten and W. J¨uptner, "Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns," Appl. Opt. 41, 5519-5526 (2002).Q1 [CrossRef] [PubMed]
- M. Rivera, "Robust phase demodulation of interferograms with open or closed fringes," J. Opt. Soc. Am. A 22, 1170-1175 (2005). [CrossRef]
- R. Legarda-Saenz and M. Rivera, "Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns," J. Opt. Soc. Am. A 23, 2724-2731 (2006). [CrossRef]
- J. L. Marroquin and J. E. Figueroa and M. Servin, "Robust quadrature filters," J. Opt. Soc. Am. A 14, 779-791 (1997). [CrossRef]
- Jorge Nocedal and Stephen J. Wright, Numerical Optimization. Springer (1999).

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