Estimation of multiple phases from a single fringe pattern in digital holographic interferometry

Gannavarpu Rajshekhar; Sai Siva Gorthi; Pramod Rastogi

doi:10.1364/OE.20.001281

1. Introduction

The measurement of in-plane and out-of-plane displacements of a deformed object is required for many applications. For such multidimensional measurements in digital holographic interferometry (DHI), the diffuse object is illuminated along different directions [1] resulting in multiple object beams. For each illumination direction, there is an associated sensitivity vector and interference phase. The in-plane and out-of-plane displacements are related to the multiple phases for different directions through the corresponding sensitivity vectors. Hence, estimation of these multiple interference phases is required for multidimensional deformation measurements. For a multi-beam illumination setup, methods [2–4] based on phase shifting technique have been proposed in DHI, but they are not suitable for simultaneous measurements because of sequential illumination and further require multiple frames.

For simultaneous multidimensional measurements, various methods [1, 5–7] have been developed in DHI. In these methods, the desired multiple phases are estimated using filtering operations based on the application of a spatial carrier. However, such approach requires careful control of the carrier in the experimental setup to ensure sufficient spectral separation, which might be difficult in practice. In addition, the phases obtained after filtering are of wrapped form and thus require additional complex unwrapping operations which are time-consuming and susceptible to noise.

In the paper, we propose an approach that is capable of estimation of multiple phases from a single fringe pattern in DHI without the requirement of an additional spatial carrier. The proposed method relies on piecewise multicomponent polynomial phase signal formulation. The theory of the proposed method is discussed in the next section. The analysis of the method is presented in section 3 followed by conclusions.

2. Theory

To explore the proposed method, the illumination of a diffuse object by two beams as shown in Fig. 1 is considered. In DHI, with a reference wave and two object waves resulting from the dual-beam illumination, the reconstructed interference field is obtained as [7]

Γ (x, y) = A_{1} (x, y) exp [j Δ ϕ_{1} (x, y)] + A_{2} (x, y) exp [j Δ ϕ_{2} (x, y)] + η (x, y)

where A₁(x, y) and A₂(x, y) represent the amplitudes assumed to be slowly varying, η(x,y) denotes the noise and Δϕ₁(x, y) and Δϕ₂(x, y) are the interference phases corresponding to the respective beams. Estimation of the individual phases Δϕ₁(x, y) and Δϕ₂(x, y) in Eq. (1) is a challenging problem since Γ(x,y) is the sum of two complex signals. The applicability of the popular spatial fringe analysis methods based on Fourier transform [8], windowed Fourier transform [9] and wavelet transform [10] is limited to extracting a single phase distribution from a fringe pattern and their accuracy would be strongly affected for the retrieval of dual phases due to the spectral overlap of the two complex signals in absence of a spatial carrier. Similarly, the recently developed methods based on polynomial phase approximation [11–13] are also affected when multiple phases are present.

Fig. 1 Dual-beam illumination of a diffuse object

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In the proposed method, the reconstructed interference field Γ(x,y) is modeled as a piecewise multicomponent polynomial phase signal to directly estimate the phases Δϕ₁(x, y) and Δϕ₂(x, y). Accordingly, a given column/row of Γ(x,y) is divided into say N_w segments and the phase of each component in every segment is approximated as a polynomial. Hence, for a given column x and segment i where i ∈ [1,N_w], we have

Δ ϕ_{1 i} (y) = \sum_{m = 0}^{M} a_{i m} y^{m}

Δ ϕ_{2 i} (y) = \sum_{m = 0}^{M} b_{i m} y^{m}

Γ_{i} (y) = A_{1 i} (y) exp [j (\sum_{m = 0}^{M} a_{i m} y^{m})] + A_{2 i} (y) exp [j (\sum_{m = 0}^{M} b_{i m} y^{m})] + η_{i} (y)

Equation (4) represents a multicomponent polynomial phase signal of order M and the phases Δϕ_1i(y) and Δϕ_2i(y) in the i^th segment can be obtained from the polynomial coefficients [a_i₀, ··· ,a_iM] and [b_i₀, ··· ,b_iM]. To estimate these coefficients, product high-order ambiguity function (PHAF) [14] is used. The main advantage of using PHAF lies in its ability to suppress the unwanted interference caused due to multiple components which leads to robust estimation of the desired coefficients.

The m^th order PHAF is computed through the following steps [14]:

Initially, the high-order instantaneous moment (HIM) of m^th order is calculated from Γ_i(y) using a recursive operation, $\begin{matrix} x_{1} (y) = Γ_{i} (y), \\ x_{2} (y; τ_{k, 1}) = x_{1} (y) x_{1}^{*} (y - τ_{k, 1}), \\ ⋮ \\ x_{m} (y; τ_{k, m - 1}) = x_{m - 1} (y; τ_{k, m - 2}) x_{m - 1}^{*} (y - τ_{k, m - 1}; τ_{k, m - 2}) \end{matrix}$ where ‘*’ denotes complex conjugate and τ_k,m−1 is the k^th lag parameter.
Subsequently, the high-order ambiguity function (HAF) is obtained from the discrete Fourier transform (DFT) of the HIM as $\begin{array}{l} X_{m} (ω; τ_{k, m - 1}) = DFT [x_{m} (y; τ_{k, m - 1})] \\ = \sum_{y = 0}^{N - 1} x_{m} (y; τ_{k, m - 1}) exp [- j ω y] \end{array}$ where N is the sample size. The obtained m^th order HAF i.e. X_m(ω;τ_k,m−1) corresponds to a particular lag parameter τ_k,m−1.
The above steps are repeated for K different lag parameters i.e. [τ_1,m−1, ··· ,τ_K,m−1] to obtain different HAFs corresponding to k ∈ [1,K].
Finally, the multiple HAFs are scaled in frequency domain by the scaling factor P_m(τ_k,m−1,τ_1,m−1) and subsequently multiplied to obtain the m^th order PHAF of Γ_i(y) as $X_{K, m} (ω) = \prod_{k = 1}^{K} X_{m} (P_{m} (τ_{k, m - 1}, τ_{1, m - 1}) ω, τ_{k, m - 1})$ where $P_{m} (τ_{k, m - 1}, τ_{1, m - 1}) = {[\frac{τ_{k, m - 1}}{τ_{1, m - 1}}]}^{m - 1}$

For the multicomponent case, the interesting property of m^th order PHAF is that its spectrum exhibits peaks corresponding to the m^th order polynomial coefficients. For the dual component case of Γ_i(y) in Eq. (4), if the maxima of the PHAF spectum |X_K,m(ω)| occur at ω_max, then the coefficients a_im and b_im are given as,

a_{i m}, b_{i m} = \frac{ω_{\max}}{m! {(τ_{1, m - 1})}^{m - 1}}

To comprehend the working of PHAF, consider a multicomponent signal with cubic phases i.e. M = 3,

\begin{array}{l} Γ (y) = A_{1} exp [j (a_{0} + a_{1} y + a_{2} y^{2} + a_{3} y^{3})] \\ + A_{2} exp [j (b_{0} + b_{1} y + b_{2} y^{2} + b_{3} y^{3})] \forall y \in [1, N], N = 256 \end{array}

where the coefficients are [a₃, a₂, a₁, a₀] = [−0.00002, 0.008, 0.5, 0] for the first component and [b₃, b₂, b₁, b₀] = [0.00001, −0.006, −0.5, 0] for the second component. For the m^th order PHAF, the optimal lag parameter τ_k,m₋₁ ≈ N/m [14] and hence, we used the integer values of [τ_1,2,τ_2,2,τ_3,2] = [N/3, N/3 – 10,N/3+10] for K = 3, m = 3 and [τ_1,1,τ_2,1] = [N/2 – 10,N/2+ 10] for K = 2, m = 2 in the analysis.

To obtain the highest order coefficients i.e. a₃ and b₃, the third order (m = 3) PHAF is computed and its spectrum |X_K,m(ω)| is shown in Fig. 2(a) for the case of equal amplitudes i.e. A₁ = A₂. As the m^th order polynomial coefficient is related to the frequency through Eq. (9), all PHAF plots in the paper are shown on a frequency scaled axis, i.e. $ω / (m! τ_{1, m - 1}^{m - 1})$ instead of ω so that the coefficients a_m and b_m directly correspond to the peaks in the PHAF spectrum. From Fig. 2(a), it is clear that two distinct peaks are visible corresponding to $ω / (6 τ_{1, 2}^{2}) = - 0.00002$ and 0.00001 i.e. the third order coefficients.

Fig. 2 (a) Third order (m = 3) PHAF spectrum for (a) A₁ = A₂, (b) A₁ < A₂ and (c) A₁ > A₂. (d) Second order (m = 2) PHAF spectrum. (e) First order (m = 1) PHAF spectrum. (f) Original vs estimated first phase in radians.

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However, for the equal amplitude case, the main difficulty lies in matching the estimated coefficients with their respective components i.e. deciding whether the peak corresponds to a₃ or b₃. To resolve this, unequal amplitudes could be used so that the PHAF peak corresponding to the stronger component is dominant. This is shown in Fig. 2(b) and Fig. 2(c) for the cases A₁ = 1, A₂ = 1.5 and vice-versa. Consequently, m^th order PHAF peak detection would yield the m^th order polynomial coefficient for the stronger component.

For the case A₁ = 1.5, A₂ = 1 i.e. A₁ > A₂, tracing the peak of the third order (m = 3) PHAF spectrum as shown in Fig. 2(c) provides the estimate â₃ for the stronger component. Subsequently, to obtain the second order coefficient a₂, a peeling operation is carried out to decrement the polynomial order of the stronger component,

\begin{array}{l} Γ_{2} (y) = Γ (y) exp [- j {\hat{a}}_{3} y^{3}] \\ \approx A_{1} exp [j (a_{0} + a_{1} y + a_{2} y^{2})] + A_{2} exp [j (b_{0} + b_{1} y + b_{2} y^{2} + (b_{3} - {\hat{a}}_{3}) y^{3})] \end{array}

The stronger component in Γ₂(y) is reduced to a second order polynomial. Consequently, peak detection of the second order (m = 2) PHAF yields the estimate of a₂. The corresponding PHAF spectrum for Γ₂(y) is shown in Fig. 2(d) where the peak corresponds to â₂. Subsequently, to estimate the first order coefficient a₁, an additional peeling operation is carried out,

\begin{array}{l} Γ_{1} (y) = Γ_{2} (y) exp [- j {\hat{a}}_{2} y^{2}] \\ \approx A_{1} exp [j (a_{0} + a_{1} y)] + A_{2} exp [j (b_{0} + b_{1, y} + (b_{2} - {\hat{a}}_{2}) y^{2} + (b_{3} - {\hat{a}}_{3}) y^{3})] \end{array}

Since the stronger component in Γ₁(y) is now a first order polynomial, the peak of the first order (m = 1) PHAF provides estimate of a₁. The corresponding PHAF spectrum for Γ₁(y) is shown in Fig. 2(e) where the peak corresponds to â₁. Finally, the contribution of first order coefficient is removed from Γ₁(y) to obtain,

\begin{array}{l} Γ_{0} (y) = Γ_{1} (y) exp [- j {\hat{a}}_{1} y] \\ \approx A_{1} exp [j a_{0}] + A_{2} exp [j (b_{0} + (b_{1} - {\hat{a}}_{1}) y + (b_{2} - {\hat{a}}_{2}) y^{2} + (b_{3} - {\hat{a}}_{3}) y^{3})] \end{array}

The term A₁ exp[ja₀] in the above equation is a constant and does not vary with y. It can be separated by taking the mean of Γ₀(y) and hence the zeroth order coefficient a₀ and the amplitude A₁ can be approximately estimated as,

{\hat{a}}_{0} = angle [\frac{1}{N} \sum_{y = 1}^{N} Γ_{0} (y)]

{\hat{A}}_{1} = | \frac{1}{N} \sum_{y = 1}^{N} Γ_{0} (y) |

The above procedure estimates the first or stronger component’s polynomial coefficients iteratively from the highest order to the lowest. The estimates are [â₃, â₂, â₁, â₀, Â₁] = [−0.00002,0.008,0.5001,−0.0650,1.4956]. Effectively, all the parameters i.e. the polynomial coefficients and the amplitude of the first component are now known. Subsequently, the contribution of the first component is removed from Γ(y) to obtain the second component,

\begin{array}{l} Γ^{'} (y) = Γ (y) - {\hat{A}}_{1} exp [j ({\hat{a}}_{0} + {\hat{a}}_{1} y + {\hat{a}}_{2} y^{2} + {\hat{a}}_{3} y^{3})] \\ \approx A_{2} exp [j (b_{0} + b_{1 y} + b_{2} y^{2} + b_{3} y^{3})] \end{array}

Γ′(y) behaves as a third order polynomial phase signal. Accordingly, its third order PHAF spectrum, as shown in Fig. 3(a), exhibits a dominant peak at

ω / (6 τ_{1, 2}^{2}) = 0.00001

which corresponds to the third order coefficient b₃. The remaining coefficients for Γ′(y) are obtained in the same way as for the stronger component using successive polynomial order reduction by peeling and corresponding PHAF peak detection. The PHAF spectra for m = 2 and m = 1 are shown in Fig. 3(b) and Fig. 3(c). The estimated coefficients for the second component are [b̂₃, b̂₂, b̂₁, b̂₀] = [0.00001, −0.0060,−0.5001,−0.0562].

Fig. 3 (a) Third order (m = 3) PHAF spectrum. (b) Second order (m = 2) PHAF spectrum. (c) First order (m = 1) PHAF spectrum. (d) Original vs estimated second phase in radians. (e) Estimation error in radians for first phase. (f) Estimation error in radians for second phase.

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The phases for the first and second components are constructed from the polynomial coefficients as in Eq. (2) and Eq. (3). The original vs estimated first phase in radians is shown in Fig. 2(f). Similarly, the original vs estimated second phase in radians is shown in Fig. 3(d). The corresponding estimation errors in radians for the first and second phases are shown in Fig. 3(e) Fig. 3(f).

The above procedure, though shown for a third order multicomponent signal, can be easily generalized for other polynomial orders. For the M^th order multicomponent signal Γ_i(y) of Eq. (4), the two sets of polynomial coefficients i.e. [a_i0, ··· ,a_iM] and [b_i0, ··· ,b_iM] in the i^th segment are computed using the PHAF based estimation scheme discussed above. To minimize the errors due to noise and cross-component influence, these coefficient estimates are further refined using a multicomponent optimization routine based on Nelder-Mead simplex algorithm [15]. Using these coefficients, the multiple phases are constructed for the i^th segment using Eq. (2) and Eq. (3). Though the estimates obtained along a column are unwrapped, a phase-stitching procedure [11] between adjacent columns is used to obtain continuous phase distributions.

It needs to be emphasized that the amplitude based separability criterion ensures proper matching of the estimated coefficients to the respective components in each segment. From an experimental perspective, the amplitude inequality between the two components could be achieved in DHI by controlling the relative intensities of the individual object beams.

Finally, the proposed method to retrieve multiple phase distributions can be summarized as

Each column of the reconstructed interference field Γ(x,y) is divided into N_w segments to obtain Γ_i(y).
In each segment, Γ_i(y) is modeled as a multicomponent polynomial phase signal as in Eq. (4).
The coefficients [a_i0, ··· ,a_iM] of the stronger component are estimated by successive application of the PHAF peak detection and subsequent peeling operation as shown in Eq. (9–15).
The second component is obtained by removing the contribution of the first component from Γ_i(y) using Eq. (16).
The coefficients [b_i0, ··· ,b_iM] of the second component are estimated in a similar manner as step 3.
The phases Δϕ_1i(y) and Δϕ_2i(y) for the i^th segment are obtained using Eq. (2) and Eq. (3) from the refined coefficients.
The above steps are repeated for all segments and subsequently for all columns to obtain the overall multiple phase distributions.

3. Analysis

To analyze the proposed method, a reconstructed interference field with multiple phases was simulated in MATLAB,

Γ (x, y) = A_{1} exp [j Δ ϕ_{1} (x, y)] + A_{2} exp [j Δ ϕ_{2} (x, y)]

with A₁ = 2 and A₂ = 1 i.e. amplitude ratio of 2:1. Noise was added at signal-to-noise ratio (SNR) of 25 dB using MATLAB’s ‘awgn’ function. The original phases Δϕ₁(x, y) and Δϕ₂(x, y) in radians are shown in Fig. 4(a) and Fig. 4(b). The real part of Γ(x,y) which constitutes the fringe pattern (256 × 256 pixels) is shown in Fig. 4(c).

Fig. 4 (a) Original Δϕ₁(x, y) in radians. (b) Original Δϕ₂(x, y) in radians. (c) Fringe pattern. (d) Fourier spectrum |γ(ω_x,ω_y)|

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To highlight the problem of spectral overlap in the absence of a spatial carrier, the Fourier transform of Γ(x,y) was calculated as,

γ (ω_{x}, ω_{y}) = DFT [Γ (x, y)]

The Fourier spectrum i.e. |γ(ω_x,ω_y)| is shown in Fig. 4(d). From the figure, it is clear that the spectra of the two components are overlapped and not well separated. Due to the spectral overlap, the individual components with the phases Δϕ₁(x, y) and Δϕ₂(x, y) cannot be efficiently filtered in the frequency domain using the Fourier transform. Hence, in the absence of a spatial carrier, retrieval of multiple phases is difficult using current phase estimation methods.

The proposed method was applied to estimate the phases Δϕ₁(x, y) and Δϕ₂(x, y) from Γ(x,y). For the analysis, we used N_w = 4 and M = 2. The estimated first phase in radians is shown in Fig. 5(a). The corresponding phase estimation error in radians is shown in Fig. 5(b). Similarly, the estimated second phase in radians is shown in Fig. 5(c). The corresponding phase estimation error in radians is shown in Fig. 5(d). The root mean square errors (RMSE) for the estimation of Δϕ₁(x, y) and Δϕ₂(x, y) were 0.0256 and 0.0333 radians. From the analysis, it is clear that the proposed method is capable of reliable estimation of multiple phases from a single fringe pattern.

Fig. 5 (a) Estimated Δϕ₁(x, y) in radians. (b) Estimation error for Δϕ₁(x, y) in radians. (c) Estimated Δϕ₂(x, y) in radians. (d) Estimation error for Δϕ₂(x, y) in radians.

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4. Conclusions

The paper presents a piecewise multicomponent polynomial phase approximation based method for multiple phase retrieval which is desired for the simultaneous in-plane and out-of-plane deformation measurements in DHI. The major advantage of the proposed method lies in its ability to extract information about multiple phases from a single fringe pattern without the need of spatial carrier based spectral filtering, which is hitherto not possible with the current state-of-the-art spatial fringe analysis methods. In addition, the method provides continuous phase distributions and does not require complex unwrapping algorithms. We believe that the proposed method would enhance the applicability of DHI for simultaneous multidimensional measurements.

References and links

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Estimation of multiple phases from a single fringe pattern in digital holographic interferometry

Abstract

1. Introduction

2. Theory

3. Analysis

4. Conclusions

References and links

Cited By

Figures (5)

Equations (18)

Optics Express