Abstract

In optical measurement, spatial carrier fringe pattern analysis is suitable for measuring dynamic events in real-time. This paper presents a novel technique for analyzing a spatial carrier fringe pattern. It estimates the local phase gradients at a pixel from its neighborhood, by use of statistics of the intensity gradients. Using the estimated phase gradients, the phase map of the fringe pattern is recovered by solving numerical partial derivative equations or using an adaptive spatial carrier phase shifting (SCPS) algorithm. Simulation and experimental results demonstrate this algorithm to be valid.

© 2014 Optical Society of America

1. Introduction

In optical measurement, retrieving the phase map from a single fringe pattern enables measuring dynamic phenomena in real-time [13], or making measurement results immune from the influences of circumstantial disturbances and vibrations. In practice, however, analysis of a single fringe pattern usually suffers from a difficulty in isolating fringes from background intensities. Thanks to that, Takeda et al. [4] proposed in the early 1980s introducing a linear carrier into a fringe pattern so that its fringes may have much higher frequencies, gradients, or variations than the background intensities, thus providing a possibility of separating the fringes from background intensities mathematically. Since then, techniques of spatial carrier fringe pattern analysis have been extensively studied.

The mainstay technique for analyzing a spatial carrier fringe pattern is the Fourier transform (FT) method [410]. In the Fourier spectrum of the pattern, the fringe component corresponds to a pair of conjugately symmetric lobes, which locate at carrier frequency positions, and are disjointed from that of the background. Isolating one of the two lobes using a filter, taking its inverse FT (IFT), and then implementing arctangent operation yield the phase map of the pattern. Because the FT method is a global method, it has a limitation that the large deformation in the phase map increases the bandwidth of fringe component and thus may make the filtering process unreliable in separating the component of interest [7]. To solve this problem, the space-frequency analysis methods (i.e. the time-frequency analysis methods for temporal signals), which can provide spatial information and spectral information simultaneously, are usually employed. Among them, the windowed Fourier transform (WFT) method [11] determines spatial localization by sliding a window with constant size over the fringe pattern, and therefore it flaws in having a fixed resolution. By using scaled wavelets, the wavelet transform (WT) method [1215] possesses the ability of multi-resolution analysis. It gives high spatial resolution for high-frequency components, and high frequency resolution for low-frequency components. As an extension of WFT or the continuous WT, the S-transform (ST) method [16] uses a varying Gaussian window with its width and height depending on frequency, and hence enables yielding a better signal clarity. Although some efficient algorithms such as the fast Fourier transform (FFT) and the fast wavelet transform (FWT) have been invented, the transform-based methods generally have relatively high computational complexities.

Compared with processing in transform domains, a more straightforward solution is to process the fringe pattern in the spatial domain directly. According to the theory about the linear time-invariant (LTI) system, the filtering process in FT method can be equivalently implemented in the spatial domain by convoluting the fringe pattern with a complex kernel. Based on this principle, typically the spatial synchronous detection methods [1719] and some classical spatial carrier phase-shifting (SCPS) methods [2028] have been developed. With them, convolutions of the fringe pattern with the real and imaginary parts of the kernels are implemented individually, so that their formulas are generally similar to the standard temporal phase shifting formulas with known and fixed phase shifts. These spatial LTI methods usually use convoluting kernels with very small sizes in order to improve computational efficiencies or to take advantage of hardware implementations, at the expense of low frequency resolutions. With them, detuning errors induced by frequency mismatching may occur, especially when the phase map has a large deformation. Applying non-LTI systems to the fringe pattern is helpful for solving this problem. For example, assuming the local phase variation to be linear within a small region allows deducing SCPS formulas insensitive to linear phase errors [29, 30]; Employing a least-squares algorithm enables us estimating the local spatial frequencies of a pixel from its neighborhood and further calculating its phase by use of a two-dimensional (2D) SCPS algorithm [31]; Using an iterative strategy can improve the measurement accuracy of phase map by updating the phase shifts [32]; and in [33], the principal component analysis (PCA) as an effective tool for revealing the internal structure of data is applied for calculating the phase of a point from its neighborhood. In addition, some algorithms combine the spatial operation and space-frequency analysis. In [34], the local frequencies are estimated using a space-frequency analysis method, and then the phases are calculated using a SCPS formula. From these analyses, we know that the challenging issue for analyzing a spatial carrier fringe pattern with large phase deformation is the estimation of local phase gradients (i.e. local spatial frequencies).

In this paper we present, to the best of our knowledge, a novel technique for analyzing a spatial carrier fringe pattern. First, this technique estimates the local phase gradients at a pixel from its neighborhood, simply by use of statistics of the intensity gradients. Second, it recovers the phase map of the fringe pattern from the phase gradients just estimated, by using numerical integration or by using a 2D adaptive SCPS algorithm. Both numerical simulation and experiment are carried out to verify the validity of this proposed technique.

2. Estimating phase gradients from intensity gradients

2.1. Phase gradients

Consider a spatial carrier fringe pattern of the form

I(x,y)=a(x,y)+b(x,y)cos[uCx+υCy+ϕ(x,y)],
where a(x, y) and b(x, y) denote the background intensity and the modulation at the pixel (x, y), respectively; uC and vC denote the carrier frequencies along x and y directions, respectively, and ϕ(x, y) is the phase without carrier. Along these two directions, the gradients of phases, i.e. the local frequencies, are
u(x,y)=[uCx+υCy+ϕ(x,y)]x=uC+ϕ(x,y)x
and
υ(x,y)=[uCx+υCy+ϕ(x,y)]y=υC+ϕ(x,y)y,
respectively, which deviate from the carrier frequencies uC and vC, depending on how steeply ϕ(x, y) varies.

As introduced in Section 1, calculating phase gradients is central to developing an adaptive technique for analyzing a spatial carrier fringe pattern with large phase variations. If the phase gradients are estimated, we can recover the phase map by implementing a numerical integration or by employing a SCPS algorithm with adaptively varying phase shifts. In the following subsections, we will deduce a method for estimating the phase gradients from the intensity gradients.

2.2. Principle of phase gradient estimation

Intensity gradients describe how steeply a fringe pattern varies. They depend mainly on the phase gradients, as well as on the background intensities and modulations. For a pixel (x, y), we can calculate its intensity gradient along the horizontal direction using the forward difference formula

h(+)I(x,y)=I(x+1,y)I(x,y),
the backward difference formula
h()I(x,y)=I(x,y)I(x1,y),
and the central difference formula
HI(x,y)=I(x+1,y)I(x1,y)2.
If the intensities vary slowly across the fringe pattern, these three formulas give almost the same results. Whereas in the presence of spatial carrier, the intensities vary rapidly and hence using the three formulas may result in very different values, depending on phase gradients. This phenomenon implies that we can estimate the phase gradients via these intensity gradients.

We gain insight into the principle of estimating phase gradients by recalling the temporal phase shift estimation algorithm presented in [35]. With it, the intensity differences between three temporal phase-shifting interferograms are computed first, and then the standard deviations (SDs) of these differences are calculated. The phase shifts are estimated, by using the law of cosines, from a triangle whose lengths of sides are the SDs of the intensity differences. Emulating this temporal method in the spatial domain, the intensity differences between consecutive three pixels are ∇h( + )I(x, y), ∇h(-)I(x, y), and 2∇HI(x, y). Because the background intensities, modulations, and phase gradients vary slowly across the fringe pattern, we can estimate the SDs of ∇h( + )I(x, y), ∇h(-)I(x, y), and 2∇HI(x, y) within a sufficiently small neighborhood centered at the pixel (x, y). Assuming the neighborhood size to be (2K + 1) × (2K + 1), their SDs are calculated with

σh(+)(x,y)=1(2K+1)×(2K+1)s=KKt=KK[h(+)I(x+s,y+t)]2,
σh()(x,y)=1(2K+1)×(2K+1)s=KKt=KK[h()I(x+s,y+t)]2,
and
σH(x,y)=1(2K+1)×(2K+1)s=KKt=KK[HI(x+s,y+t)]2,
respectively. By use of the formula in [35], the phase gradient along the horizontal direction is estimated with
ωh(x,y)=2arccos[σh()(x,y)]2+[2σH(x,y)]2[σh(+)(x,y)]22σh()(x,y)[2σH(x,y)].
By noting that, although ∇h( + )I(x, y) is generally not equal to ∇h(-)I(x, y) for each single pixel, σh( + )(x, y) and σh(-)(x, y) as statistics have almost the same values, we simplify Eq. (10) as
ωh(x,y)=2arccosσH(x,y)σh(x,y),
where the superscript of σh(x, y) is removed, it can be calculated using either the forward difference formula in Eq. (4) or the backward difference formula in Eq. (5).

Based on the same principle, the phase gradient along the vertical direction can also be calculated. Using the forward and central difference formulas, we have

vI(x,y)=I(x,y+1)I(x,y)
and
VI(x,y)=I(x,y+1)I(x,y1)2.
Their SDs are
σv(x,y)=1(2K+1)×(2K+1)s=KKt=KK[vI(x+s,y+t)]2
and
σV(x,y)=1(2K+1)×(2K+1)s=KKt=KK[VI(x+s,y+t)]2.
The phase gradient along the vertical direction is

ωv(x,y)=2arccosσV(x,y)σv(x,y).

The phase gradients calculated using Eqs. (11) and (16) have a range from 0 to π radians, so the sign ambiguity arises. We have to calculate the phase gradients along the diagonal direction for determining the signs of phase gradients. The intensity gradients along the diagonal direction are

dI(x,y)=I(x+1,y+1)I(x,y)
and
DI(x,y)=I(x+1,y+1)I(x1,y1)2.
Their SDs are
σd(x,y)=1(2K+1)×(2K+1)s=KKt=KK[dI(x+s,y+t)]2
and
σD(x,y)=1(2K+1)×(2K+1)s=KKt=KK[DI(x+s,y+t)]2.
The phase gradient along the diagonal direction is

ωd(x,y)=2arccosσD(x,y)σd(x,y).

2.3. Two-dimensional implementation

The phase gradients calculated in the previous subsection, i.e. ωh(x, y), ωv(x, y), and ωd(x, y) are not practically useful. First, the practical phase gradients along x and y directions are not always positive, but may have the same or opposite signs depending on the fringe direction. Second, it is possible that the fringes are parallel to the horizontal or vertical direction, in which case the phase gradients, calculated in the previous subsection, along this direction may have very large errors. For solving these problems, we calculate the averaging values of σh(x, y), σv(x, y), and σd(x, y) over the whole fringe pattern, i.e. σh¯, σv¯, and σd¯, and then compare them for roughly determining the fringe direction. Among these three averaging values, the smallest one indicates a direction proximate to that of fringes. We should avoid directly calculating the phase gradients in this direction, and a switching mechanism as follows is used.

If σh¯σd¯ and σv¯σd¯,

{u(x,y)=ωh(x,y)v(x,y)=ωv(x,y)

If σh¯σv¯ and σd¯σv¯,

{u(x,y)=ωh(x,y)v(x,y)=ωd(x,y)ωh(x,y)

If σd¯σh¯ and σv¯σh¯,

{u(x,y)=ωd(x,y)ωv(x,y)v(x,y)=ωv(x,y)
In so doing, the phase gradients u(x, y) and v(x, y) are estimated, with their signs having been determined.

3. Phase recovering algorithms

3.1 Phase recovering using numerical integration

The phase map can be recovered by numerically integrating the phase gradients just calculated. This task can be implemented by use of least squares techniques [36, 37]. In this subsection, we will present a method based on the 2D discrete cosine transform (DCT) for doing it. Assuming the calculated phase gradients to be M × N matrices, we denote the recovered phase at pixel (x, y) as φ(x, y), the phase gradient at a position between two consecutive pixels can be estimated using the central difference formula u(x + 1/2, y)≈φ(x + 1, y)-φ(x, y). This gradient can also be estimated by averaging u(x, y) and u(x + 1, y), so we have

φ(x+1,y)φ(x,y)u(x+1,y)+u(x,y)2,
and similarly
φ(x,y+1)φ(x,y)v(x+1,y)+v(x,y)2.
The sum of squares of their residuals is
E=xy[φ(x+1,y)φ(x,y)u(x+1,y)+u(x,y)2]2+xy[φ(x,y+1)φ(x,y)v(x,y+1)+v(x,y)2]2.
In the least squares sense, minimizing E allows us recovering the phase map. Equating to zeros the derivatives of E with respect to φ(x, y), we get a system of linear equations
[φ(x+1,y)2φ(x,y)+φ(x1,y)]+[φ(x,y+1)2φ(x,y)+φ(x,y1)]=ρ(x,y)
with
ρ(x,y)=u(x+1,y)u(x1,y)2+v(x,y+1)v(x,y1)2.
Using Eq. (29), we compute ρ(x, y) for each pixel from the phase gradients of its neighboring pixels, and therefore the values of ρ(x, y) at boundaries cannot be available. As a result, the system based on Eq. (28) is underdetermined, i.e. its number of equations is smaller than that of unknowns. An additional boundary condition is required for solving it.

By padding zeros outside the ranges of the calculated phase gradients and then using Eq. (29), we calculate a matrix of ρ(x, y) with its size becoming (M + 2) × (N + 2), in which case we can regard Eq. (28) as the discretization of a Poisson equation

[2x2+2y2]ϕ(x,y)=ρ(x,y)
meeting the Neumann boundary conditions. We find its solution by use of 2D DCT [38]. First, we pad zeros to the phase gradients and then calculate ρ(x, y) using Eq. (29); second, we calculate 2D DCT of ρ(x, y), i.e. P(i, j) which is also a (M + 2) × (N + 2) matrix; third, we implement the inverse filtering procedure, viz.
Φ(i,j)=P(i,j)2(cosπiM+2+cosπjN+22),
where i = 0,1,…, M + 1 and j = 0,1,…, N + 1; and finally, calculating the inverse DCT of Φ(i, j) yields φ(x, y).

In the above steps, we can subtract from the phase gradients their mean values, and then calculate the matrix of ρ(x, y) using the zero-mean normalized phase gradients. Doing so will result in a phase map with its carrier and tilt component having been removed.

In fact, the procedure of this phase recovering method is very much similar to that of phase unwrapping technique proposed by Ghiglia and Romero [38], with a difference that the matrix of ρ(x, y) on the right-hand side of Eq. (28) is calculated from the estimated phase gradients. This phase recovering method, which is based on 2D DCT, is suitable for dealing with a rectangular aperture. If the aperture is circular or has other different shapes, we can refer to [36, 37] or some literature regarding weighted phase unwrapping techniques, in order to find a solution for the system of equations in Eq. (28) with (x, y) having non-rectangular ranges.

3.2 Phase recovering using 2D least squares SCPS algorithm

As proposed in [31], estimating the phase gradients allows us recovering the phase map of a spatial carrier fringe pattern using a 2D SCPS algorithm. In the neighborhood of a pixel, the estimated phase gradients are taken as the relative phase steps, so that its phase can be calculated in the least squares sense.

We specify the size of neighborhood to be (2L + 1) × (2L + 1) pixels. By noting that the background intensities and modulations are nearly constant within this neighborhood, and defining c0(x, y) = a(x, y), c1(x, y) = b(x, y)cos[uC x + vC y + ϕ(x,y)], and c2(x, y) = -b(x, y)sin[uC x + vC y + ϕ(x,y)], we have a system of linear equations

i=LLj=LL[1cos[u(x,y)i+v(x,y)j]0cos[u(x,y)i+v(x,y)j]cos2[u(x,y)i+v(x,y)j]000sin2[u(x,y)i+v(x,y)j]][c0(x,y)c1(x,y)c2(x,y)]=i=LLj=LL[I(x+i,y+j)I(x+i,y+j)cos[u(x,y)i+v(x,y)j]I(x+i,y+j)sin[u(x,y)i+v(x,y)j]].
Solving it for the unknowns, c0(x, y), c1(x, y), and c2(x, y), the phase is calculated with
φ(x,y)=arctanc2(x,y)c1(x,y)
in which the carrier component is not removed.

4. Numerical simulations

4.1. Phase gradient estimation

Here we perform numerical simulations for investigating the performances of our proposed approach. Figure 1(a) simulates a spatial carrier fringe pattern of 512 × 512 pixels, with its phase map shown in Fig. 1(b) and its carrier frequency being 0.5π radians/pixels along the direction of 40° from horizontal axis. We assume the background and modulation to be distributed as Gaussian functions with a 50% and 80% decrease in magnitude at the corners, respectively. Zero-mean Gaussian noise with the standard deviation (SD) of 0.02 is added into the intensities.

 figure: Fig. 1

Fig. 1 (a) is a simulated spatial carrier fringe pattern and (b) is its phase map.

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We use the proposed method in Section 2 for estimating the phase gradients from the spatial carrier fringe pattern in Fig. 1(a). In this procedure, a neighborhood of 11 × 11 pixels is specified for calculating the SDs of intensity gradients. In Fig. 2, the two panels of the left column show the theoretical phase gradients, calculated from the simulated phase map, along x and y directions, and the middle column gives the estimated ones using the proposed method. Comparing them, we find that the estimated phase gradients have almost the same distributions with the theoretical values. The right column illustrates their differences, in which the maximum errors are 0.0320 radians in u and 0.0192 radians in v.

 figure: Fig. 2

Fig. 2 (a) and (d) show the theoretical phase gradients in x and y directions, respectively. (b) and (e) are the estimated phase gradients along the two directions, respectively. (c) and (f) show the differences between the estimated and the theoretical phase gradients in x and y directions, respectively.

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Further simulations under different noise conditions are performed for investigating the accuracy of this method. As a result, the root-mean square (RMS) errors and the maximum errors in the estimated phase gradients are listed in Table 1, from which it is evident that the errors in the estimated phase gradients increase as the noise SDs increase. Even so, this algorithm still appears stable and robust. When relatively high noise with SD of 0.04 is added, the maximum error keeps being of the level of 0.01 radians, demonstrating that this method is less sensitive to random noise.

Tables Icon

Table 1. Simulation Results of Estimating Phase gradients in the Presence of Gaussian Noise

We also see from Table 1 that, in the absence of noise, the estimated phase gradients still contains some errors. They are caused by inaccuracies in estimating the variances of intensity gradients of pixels from their neighborhoods. Using a large neighborhood is helpful for improving the resolution of phase gradients, at the expense of computational time and spatial resolution. In these simulations, we use an 11 × 11 neighborhood. It covers at least two fringe widths, enabling giving satisfying results.

4.2. Phase map recovery

Continuing the simulations, we recover the phase map of the fringe pattern in Fig. 1(a), using the methods presented in Section 3, and compare their results with those of traditional methods. In Fig. 3, the panels of the left column, from top to bottom, show the phase maps recovered using the numerical integration algorithm presented in Section 3.1, the classical 1D SCPS algorithm with the five-step formula, the 2D SCPS algorithm presented in Section 3.2, and the FT method, with their carrier phase components having been removed. By subtracting from these results the predefined phase map in Fig. 1(b), the errors of these techniques are calculated and shown in the right column of Fig. 3. Table 2 summarizes the RMS phase errors of these algorithms.

 figure: Fig. 3

Fig. 3 Phase recovering results of the simulated fringe pattern. The left column, from top to bottom, shows the phase maps extracted using the numerical integration method, 1D SCPS method, 2D SCPS method, and FT method, respectively. The right column illustrates the phase errors corresponding to the above methods in turn.

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Tables Icon

Table 2. Simulation Results of Phase Recovery in the Presence of Gaussian Noise

From Fig. 3 and Table 2, we know that the used four algorithms all recover the phase map successfully, but they achieve different accuracies. With the numerical integration method, additional phase unwrapping step is not required, and the procedure of removing carrier and tilt phase component can be performed in advance by making the phase gradients zero mean normalized. However, because numerical integration involves an inverse high-pass filtering process, by which the signal component within low-band will be magnified significantly, its result may contain large errors of low frequencies, like those shown in Fig. 3(e). The classical 1D SCPS algorithm is fast in computation, but it uses a constant relative phase shift over the whole fringe pattern, thus it has detuning errors in its results, as shown with zigzag-type errors like those shown in Fig. 3(f). The 2D SCPS algorithm adaptively uses the estimated phase gradients as the local relative phase shift, and calculates the phase of each pixel in the least squares sense from a 11 × 11 neighborhood. It achieves very high accuracies, but has a somewhat higher computational complexity in contrast with the 1D algorithm. The FT method, as well known, has nonuniform accuracies over the fringe pattern, and usually the larger errors occur at aperture boundaries where the intensity signal is not continuous. By cutting off the data of aperture boundaries, the FT method has a small error as listed in Table 2.

5. Experiment and discussions

An experiment is carried out for verifying the feasibility of the proposed method in practical applications. Figure 4 shows an interferogram of a plane mirror, with its size being 391 × 391 pixels. In it, each fringe nearly contains five pixels in both the horizontal and vertical directions.

 figure: Fig. 4

Fig. 4 A practical interferogram with spatial carrier.

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By specifying a neighborhood of 11 × 11 pixels for calculating the variances of intensity gradients, the phase gradients along the horizontal and vertical directions are estimated using the method proposed in Section 2. Their distributions are shown in Fig. 5, from which we observe that the phase gradients vary across the fringe pattern, and have the largest absolute value near the corner with its coordinates being (391,1), corresponding to the top right corner of Fig. 4.

 figure: Fig. 5

Fig. 5 The phase gradients in (a) the horizontal and (b) vertical directions estimated from the interferogram in Fig. 4.

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Furthermore, we recover the phase map from Fig. 4 using the same methods as those in simulations. Their results are illustrated in Fig. 6, with (a) through (d) in turn being of the numerical integration, the 1D SCPS algorithm, the 2D SCPS algorithm, and the FT method. The phase maps in Fig. 6 are very much alike in their profiles, implying that all these algorithms can recover the phase map successfully. The numerical integration method presented in Section 3.1 avoids phase unwrapping implementation. Its limitation lies in containing errors of low frequency, which is observable when comparing the phase map in Fig. 6(a) with others. The 2D SCPS algorithm presented in Section 3.2 gives a more accurate phase map shown in Fig. 6(c), in contrast with the one shown in Fig. 6(b) which is obtained using the 1D algorithm and contains noticeable zigzag-type errors. The result of 2D SCPS algorithm is also comparable with that of FT method displayed in Fig. 6(d).

 figure: Fig. 6

Fig. 6 The phase maps of the interferogram in Fig. 4, recovered using (a) the numerical integration method, (b) 1D SCPS method, (c) 2D SCPS method, and (d) FT method.

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The FT method, as mentioned in Section 1, is the mainstay of techniques for spatial carrier fringe pattern analysis, and therefore it can be used as a benchmark for evaluating performances of other methods. In contrast with the FT method which involves a great number of complex multiplications despite the invention of FFT, spatial methods involve much fewer real computations thus being more efficient than the FT method, but generally they achieve lower accuracies. This paper proposes a method for estimating phase gradients from intensity gradients. Its significance for the spatial analysis is that, it can restrain detuning errors of the SCPS technique by taking advantage of its local adaptiveness. The 2D SCPS technique using local phase gradients, as demonstrated by both the simulation and experiment results, can achieve the same and even higher accuracy than the FT method. If using the numerical integration method, the same high accuracy cannot be achieved, but additional procedures for phase unwrapping and for carrier and tilt removing are cancelled.

Besides providing alternatives for the methods in the spatial domain, the proposed algorithm of phase gradient estimation is also potentially useful for enhancing the methods in the frequency domain. It can be combined with the FT method in order to improve its accuracy. An issue with the FT method is the design of filter. By using the filter, one of the lobes in the Fourier spectrum corresponding to fringes is isolated, and the position, size, and shape of the filter affect the measurement accuracies. Using the proposed algorithm allows us determine the average and range of the spatial frequencies of a fringe pattern, so it is helpful for exactly determining the carrier frequency and optimizing the filter parameters. Another issue is regarding boundary errors. The FT method has nonuniform accuracies over the fringe pattern, and usually the large errors appear at aperture boundaries where the intensity signal is not continuous, especially when the aperture is circular or of other shapes. In the situation that the aperture is not rectangular, we usually pad zeros outside the aperture to get a rectangular matrix for implementing 2D FT. Doing so may induce considerably large errors. For compressing the errors, multiplying the aperture with a 2-D Hanning window can partially soften the discontinuity at the aperture boundary [4], but a more effective solution is to perform extrapolation [8, 10, 39] instead of to pad zeros outside the aperture. Using the estimated phase gradients at boundaries enables predicting the intensities outside the aperture, thus extrapolating the aperture. In [31], aperture extrapolation using phase gradients has been demonstrated to be a very effective method for enhancing the accuracy of the FT method.

6. Conclusion

In this paper, we have proposed a novel spatial carrier fringe pattern analysis method. This method allows estimating the local phase gradients at a pixel from its neighborhood, by use of variances of the intensity gradients. Compared with the existing techniques for estimating phase gradients, e.g. those using least-square fitting, iterative strategy, or space-frequency analysis, the proposed algorithm has a simpler principle and is easier to implement. It is robust under noise conditions. In order to recover the phase map using the phase gradients just calculated, we presented a numerical integration method which recovers the phase map by solving a Poisson equation using 2D DCT. This phase recovering method does not require an additional phase unwrapping step, and can remove carrier and tilt phase component in advance by simply making the estimated phase gradients zero mean normalized. Its limitation lies in having low frequency errors in its result. We also introduced a 2D SCPS algorithm calculating the phase of each pixel in the least squares sense from its neighborhood. Because this 2D SCPS algorithm adaptively uses the estimated phase gradients as its local relative phase shift, high measurement accuracies are achieved. In addition, the proposed algorithm for estimating phase gradients also has potential uses in enhancing the methods in the frequency domain.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61178045), the Innovation Program (12ZZ098) from Shanghai Municipal Education Commission, and the National High Technology Research and Development Program (863 Program) of China (2012AA040507). Great thanks also to Miss Xing Wu for her kindly help in providing the interferogram.

References and links

1. X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010). [CrossRef]  

2. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010). [CrossRef]  

3. Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012). [CrossRef]  

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

5. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983). [CrossRef]   [PubMed]  

6. K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24(18), 3101–3105 (1985). [CrossRef]   [PubMed]  

7. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25(10), 1653–1660 (1986). [CrossRef]   [PubMed]  

8. C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987). [CrossRef]   [PubMed]  

9. J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995). [CrossRef]  

10. J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the fourier-transform method: theory and experimental tests,” Appl. Opt. 40(13), 2081–2088 (2001). [CrossRef]   [PubMed]  

11. K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004). [CrossRef]  

12. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24(13), 905–907 (1999). [CrossRef]   [PubMed]  

13. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005). [CrossRef]   [PubMed]  

14. A. Federico and G. H. Kaufmann, “Phase retrieval of singular scalar light fields using a two-dimensional directional wavelet transform and a spatial carrier,” Appl. Opt. 47(28), 5201–5207 (2008). [CrossRef]   [PubMed]  

15. S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48(36), 6893–6906 (2009). [CrossRef]   [PubMed]  

16. M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt. 51(5), 577–587 (2012). [CrossRef]   [PubMed]  

17. Y. Ichioka and M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11(7), 1507–1514 (1972). [CrossRef]   [PubMed]  

18. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984). [CrossRef]  

19. S. Tang and Y. Y. Hung, “Fast profilometer for the automatic measurement of 3-D object shapes,” Appl. Opt. 29(20), 3012–3018 (1990). [CrossRef]   [PubMed]  

20. L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22(10), 1535–1539 (1983). [CrossRef]   [PubMed]  

21. W. W. Macy Jr., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983). [CrossRef]   [PubMed]  

22. S. Toyooka and Y. Iwaasa, “Automatic profilometry of 3-D diffuse objects by spatial phase detection,” Appl. Opt. 25(10), 1630–1633 (1986). [CrossRef]   [PubMed]  

23. M. Kujawińska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61–67 (1991).

24. D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991). [CrossRef]  

25. R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992). [CrossRef]  

26. P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995). [CrossRef]  

27. Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997). [CrossRef]  

28. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006). [CrossRef]   [PubMed]  

29. P. L. Ransom and J. V. Kokal, “Interferogram analysis by a modified sinusoid fitting technique,” Appl. Opt. 25(22), 4199–4204 (1986). [CrossRef]   [PubMed]  

30. M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995). [CrossRef]  

31. H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. 46(7), 1057–1065 (2007). [CrossRef]   [PubMed]  

32. J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47(29), 5446–5453 (2008). [PubMed]  

33. Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012). [CrossRef]  

34. Z. Zhang and J. Zhong, “Spatial quasi-phase-shifting technique for single-frame dynamic fringe analysis,” Opt. Express 22(3), 2695–2705 (2014). [CrossRef]   [PubMed]  

35. H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013). [CrossRef]   [PubMed]  

36. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69(3), 393–399 (1979). [CrossRef]  

37. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980). [CrossRef]  

38. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11(1), 107–117 (1994). [CrossRef]  

39. R. J. Marks, “Gerchberg’s extrapolation algorithm in two dimensions,” Appl. Opt. 20(10), 1815–1820 (1981). [CrossRef]   [PubMed]  

References

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  • |

  1. X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
    [Crossref]
  2. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
    [Crossref]
  3. Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
    [Crossref]
  4. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [Crossref]
  5. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983).
    [Crossref] [PubMed]
  6. K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24(18), 3101–3105 (1985).
    [Crossref] [PubMed]
  7. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25(10), 1653–1660 (1986).
    [Crossref] [PubMed]
  8. C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987).
    [Crossref] [PubMed]
  9. J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
    [Crossref]
  10. J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the fourier-transform method: theory and experimental tests,” Appl. Opt. 40(13), 2081–2088 (2001).
    [Crossref] [PubMed]
  11. K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004).
    [Crossref]
  12. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24(13), 905–907 (1999).
    [Crossref] [PubMed]
  13. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005).
    [Crossref] [PubMed]
  14. A. Federico and G. H. Kaufmann, “Phase retrieval of singular scalar light fields using a two-dimensional directional wavelet transform and a spatial carrier,” Appl. Opt. 47(28), 5201–5207 (2008).
    [Crossref] [PubMed]
  15. S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48(36), 6893–6906 (2009).
    [Crossref] [PubMed]
  16. M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt. 51(5), 577–587 (2012).
    [Crossref] [PubMed]
  17. Y. Ichioka and M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11(7), 1507–1514 (1972).
    [Crossref] [PubMed]
  18. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984).
    [Crossref]
  19. S. Tang and Y. Y. Hung, “Fast profilometer for the automatic measurement of 3-D object shapes,” Appl. Opt. 29(20), 3012–3018 (1990).
    [Crossref] [PubMed]
  20. L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22(10), 1535–1539 (1983).
    [Crossref] [PubMed]
  21. W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983).
    [Crossref] [PubMed]
  22. S. Toyooka and Y. Iwaasa, “Automatic profilometry of 3-D diffuse objects by spatial phase detection,” Appl. Opt. 25(10), 1630–1633 (1986).
    [Crossref] [PubMed]
  23. M. Kujawińska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61–67 (1991).
  24. D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
    [Crossref]
  25. R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
    [Crossref]
  26. P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
    [Crossref]
  27. Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
    [Crossref]
  28. B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006).
    [Crossref] [PubMed]
  29. P. L. Ransom and J. V. Kokal, “Interferogram analysis by a modified sinusoid fitting technique,” Appl. Opt. 25(22), 4199–4204 (1986).
    [Crossref] [PubMed]
  30. M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
    [Crossref]
  31. H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. 46(7), 1057–1065 (2007).
    [Crossref] [PubMed]
  32. J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47(29), 5446–5453 (2008).
    [PubMed]
  33. Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012).
    [Crossref]
  34. Z. Zhang and J. Zhong, “Spatial quasi-phase-shifting technique for single-frame dynamic fringe analysis,” Opt. Express 22(3), 2695–2705 (2014).
    [Crossref] [PubMed]
  35. H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
    [Crossref] [PubMed]
  36. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69(3), 393–399 (1979).
    [Crossref]
  37. J. Herrmann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70(1), 28–35 (1980).
    [Crossref]
  38. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11(1), 107–117 (1994).
    [Crossref]
  39. R. J. Marks, “Gerchberg’s extrapolation algorithm in two dimensions,” Appl. Opt. 20(10), 1815–1820 (1981).
    [Crossref] [PubMed]

2014 (1)

2013 (1)

2012 (3)

2010 (2)

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

2009 (1)

2008 (2)

2007 (1)

2006 (1)

2005 (1)

2004 (1)

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004).
[Crossref]

2001 (1)

1999 (1)

1997 (1)

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

1995 (3)

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[Crossref]

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

1994 (1)

1992 (1)

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

1991 (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

1990 (1)

1987 (1)

1986 (3)

1985 (1)

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984).
[Crossref]

1983 (3)

1982 (1)

1981 (1)

1980 (1)

1979 (1)

1972 (1)

Arai, Y.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Bachor, H.-A.

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Barnes, T. H.

Bone, D. J.

Bryanston-Cross, P. J.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Chan, P. H.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Chen, M.

Chen, W.

Cuevas, F. J.

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[Crossref]

Du, Y.

Federico, A.

Feng, G.

Ghiglia, D. C.

Guo, H.

Heppner, J.

Herrmann, J.

Hung, Y. Y.

Hunt, B. R.

Ichioka, Y.

Ina, H.

Inuiya, M.

Iwaasa, Y.

Jiang, M.

Józwicki, R.

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

Kaufmann, G. H.

Kimbrough, B. T.

Kobayashi, S.

Kokal, J. V.

Kujawinska, M.

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

Li, H.

Li, S.

Lin, J.-F.

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Macy, W. W.

Marks, R. J.

Massig, J. H.

Mertz, L.

Mutoh, K.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Nugent, K. A.

Parker, S. C.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Peng, H.

Qian, K.

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004).
[Crossref]

Ransom, P. L.

Roddier, C.

Roddier, F.

Romero, L. A.

Salbut, L.

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

Sandeman, R. J.

Servin, M.

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[Crossref]

Shiraki, K.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Su, X.

Su, X.-Y.

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Takeda, M.

Tan, S. M.

Tang, S.

Toyooka, S.

Vargas, J.

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Watkins, L. R.

Weng, J.

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984).
[Crossref]

Xu, J.

Xu, Q.

Yamada, T.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Yang, Q.

Yokozeki, S.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

Zhang, Q.

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

Zhang, S.

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

Zhang, Z.

Zhang, Z. H.

Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

Zhong, J.

Zhong, M.

Zhou, S.

Appl. Opt. (19)

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22(24), 3977–3982 (1983).
[Crossref] [PubMed]

K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24(18), 3101–3105 (1985).
[Crossref] [PubMed]

D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25(10), 1653–1660 (1986).
[Crossref] [PubMed]

C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26(9), 1668–1673 (1987).
[Crossref] [PubMed]

J. H. Massig and J. Heppner, “Fringe-pattern analysis with high accuracy by use of the fourier-transform method: theory and experimental tests,” Appl. Opt. 40(13), 2081–2088 (2001).
[Crossref] [PubMed]

A. Federico and G. H. Kaufmann, “Phase retrieval of singular scalar light fields using a two-dimensional directional wavelet transform and a spatial carrier,” Appl. Opt. 47(28), 5201–5207 (2008).
[Crossref] [PubMed]

S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48(36), 6893–6906 (2009).
[Crossref] [PubMed]

M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt. 51(5), 577–587 (2012).
[Crossref] [PubMed]

Y. Ichioka and M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11(7), 1507–1514 (1972).
[Crossref] [PubMed]

S. Tang and Y. Y. Hung, “Fast profilometer for the automatic measurement of 3-D object shapes,” Appl. Opt. 29(20), 3012–3018 (1990).
[Crossref] [PubMed]

L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22(10), 1535–1539 (1983).
[Crossref] [PubMed]

W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983).
[Crossref] [PubMed]

S. Toyooka and Y. Iwaasa, “Automatic profilometry of 3-D diffuse objects by spatial phase detection,” Appl. Opt. 25(10), 1630–1633 (1986).
[Crossref] [PubMed]

B. T. Kimbrough, “Pixelated mask spatial carrier phase shifting interferometry algorithms and associated errors,” Appl. Opt. 45(19), 4554–4562 (2006).
[Crossref] [PubMed]

P. L. Ransom and J. V. Kokal, “Interferogram analysis by a modified sinusoid fitting technique,” Appl. Opt. 25(22), 4199–4204 (1986).
[Crossref] [PubMed]

H. Guo, Q. Yang, and M. Chen, “Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications,” Appl. Opt. 46(7), 1057–1065 (2007).
[Crossref] [PubMed]

J. Xu, Q. Xu, and H. Peng, “Spatial carrier phase-shifting algorithm based on least-squares iteration,” Appl. Opt. 47(29), 5446–5453 (2008).
[PubMed]

H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
[Crossref] [PubMed]

R. J. Marks, “Gerchberg’s extrapolation algorithm in two dimensions,” Appl. Opt. 20(10), 1815–1820 (1981).
[Crossref] [PubMed]

J. Mod. Opt. (2)

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44(4), 739–751 (1997).
[Crossref]

M. Servin and F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42(9), 1853–1862 (1995).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (4)

R. Józwicki, M. Kujawińska, and L. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–433 (1992).
[Crossref]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23(4), 391–395 (1984).
[Crossref]

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43(7), 1472–1473 (2004).
[Crossref]

J.-F. Lin and X.-Y. Su, “Two-dimensional Fourier transform profilometry for the automatic measurement of three dimensional object shapes,” Opt. Eng. 34(11), 3297–3302 (1995).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, “Digital phase-step interferometry: a simplified approach,” Opt. Laser Technol. 23(3), 147–150 (1991).
[Crossref]

Opt. Lasers Eng. (4)

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[Crossref]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[Crossref]

Z. H. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23(5), 343–354 (1995).
[Crossref]

Opt. Lett. (2)

Other (1)

M. Kujawińska and J. Wójciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. SPIE 1508, 61–67 (1991).

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Figures (6)

Fig. 1
Fig. 1 (a) is a simulated spatial carrier fringe pattern and (b) is its phase map.
Fig. 2
Fig. 2 (a) and (d) show the theoretical phase gradients in x and y directions, respectively. (b) and (e) are the estimated phase gradients along the two directions, respectively. (c) and (f) show the differences between the estimated and the theoretical phase gradients in x and y directions, respectively.
Fig. 3
Fig. 3 Phase recovering results of the simulated fringe pattern. The left column, from top to bottom, shows the phase maps extracted using the numerical integration method, 1D SCPS method, 2D SCPS method, and FT method, respectively. The right column illustrates the phase errors corresponding to the above methods in turn.
Fig. 4
Fig. 4 A practical interferogram with spatial carrier.
Fig. 5
Fig. 5 The phase gradients in (a) the horizontal and (b) vertical directions estimated from the interferogram in Fig. 4.
Fig. 6
Fig. 6 The phase maps of the interferogram in Fig. 4, recovered using (a) the numerical integration method, (b) 1D SCPS method, (c) 2D SCPS method, and (d) FT method.

Tables (2)

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Table 1 Simulation Results of Estimating Phase gradients in the Presence of Gaussian Noise

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Table 2 Simulation Results of Phase Recovery in the Presence of Gaussian Noise

Equations (33)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ u C x + υ C y + ϕ ( x , y ) ] ,
u ( x , y ) = [ u C x + υ C y + ϕ ( x , y ) ] x = u C + ϕ ( x , y ) x
υ ( x , y ) = [ u C x + υ C y + ϕ ( x , y ) ] y = υ C + ϕ ( x , y ) y ,
h ( + ) I ( x , y ) = I ( x + 1 , y ) I ( x , y ) ,
h ( ) I ( x , y ) = I ( x , y ) I ( x 1 , y ) ,
H I ( x , y ) = I ( x + 1 , y ) I ( x 1 , y ) 2 .
σ h ( + ) ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ h ( + ) I ( x + s , y + t ) ] 2 ,
σ h ( ) ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ h ( ) I ( x + s , y + t ) ] 2 ,
σ H ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ H I ( x + s , y + t ) ] 2 ,
ω h ( x , y ) = 2 arc cos [ σ h ( ) ( x , y ) ] 2 + [ 2 σ H ( x , y ) ] 2 [ σ h ( + ) ( x , y ) ] 2 2 σ h ( ) ( x , y ) [ 2 σ H ( x , y ) ] .
ω h ( x , y ) = 2 arc cos σ H ( x , y ) σ h ( x , y ) ,
v I ( x , y ) = I ( x , y + 1 ) I ( x , y )
V I ( x , y ) = I ( x , y + 1 ) I ( x , y 1 ) 2 .
σ v ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ v I ( x + s , y + t ) ] 2
σ V ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ V I ( x + s , y + t ) ] 2 .
ω v ( x , y ) = 2 arc cos σ V ( x , y ) σ v ( x , y ) .
d I ( x , y ) = I ( x + 1 , y + 1 ) I ( x , y )
D I ( x , y ) = I ( x + 1 , y + 1 ) I ( x 1 , y 1 ) 2 .
σ d ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ d I ( x + s , y + t ) ] 2
σ D ( x , y ) = 1 ( 2 K + 1 ) × ( 2 K + 1 ) s = K K t = K K [ D I ( x + s , y + t ) ] 2 .
ω d ( x , y ) = 2 arc cos σ D ( x , y ) σ d ( x , y ) .
{ u ( x , y ) = ω h ( x , y ) v ( x , y ) = ω v ( x , y )
{ u ( x , y ) = ω h ( x , y ) v ( x , y ) = ω d ( x , y ) ω h ( x , y )
{ u ( x , y ) = ω d ( x , y ) ω v ( x , y ) v ( x , y ) = ω v ( x , y )
φ ( x + 1 , y ) φ ( x , y ) u ( x + 1 , y ) + u ( x , y ) 2 ,
φ ( x , y + 1 ) φ ( x , y ) v ( x + 1 , y ) + v ( x , y ) 2 .
E = x y [ φ ( x + 1 , y ) φ ( x , y ) u ( x + 1 , y ) + u ( x , y ) 2 ] 2 + x y [ φ ( x , y + 1 ) φ ( x , y ) v ( x , y + 1 ) + v ( x , y ) 2 ] 2 .
[ φ ( x + 1 , y ) 2 φ ( x , y ) + φ ( x 1 , y ) ] + [ φ ( x , y + 1 ) 2 φ ( x , y ) + φ ( x , y 1 ) ] = ρ ( x , y )
ρ ( x , y ) = u ( x + 1 , y ) u ( x 1 , y ) 2 + v ( x , y + 1 ) v ( x , y 1 ) 2 .
[ 2 x 2 + 2 y 2 ] ϕ ( x , y ) = ρ ( x , y )
Φ ( i , j ) = P ( i , j ) 2 ( cos π i M + 2 + cos π j N + 2 2 ) ,
i = L L j = L L [ 1 cos [ u ( x , y ) i + v ( x , y ) j ] 0 cos [ u ( x , y ) i + v ( x , y ) j ] cos 2 [ u ( x , y ) i + v ( x , y ) j ] 0 0 0 sin 2 [ u ( x , y ) i + v ( x , y ) j ] ] [ c 0 ( x , y ) c 1 ( x , y ) c 2 ( x , y ) ] = i = L L j = L L [ I ( x + i , y + j ) I ( x + i , y + j ) cos [ u ( x , y ) i + v ( x , y ) j ] I ( x + i , y + j ) sin [ u ( x , y ) i + v ( x , y ) j ] ] .
φ ( x , y ) = arc tan c 2 ( x , y ) c 1 ( x , y )

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