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We propose a new method for signal separation from a multicomponent interference field recorded in a digital holographic interferometry setup. The setup consisting of multiple object illuminating beams results in an interference field containing multiple signal components. The proposed method utilizes an amplitude discrimination criteria established by setting different intensities to the object illuminating beams in order to separate the signal components iteratively. The signal separation is performed in a small block of the interference field at a time. The augmentation of the block matrix with its own rows and columns is performed which has an effect of noise subspace inflation. This operation offers an improved noise robustness to the signal separation capability of the proposed method. The simulation and experimental results are provided to substantiate the applicability of the proposed method in multidimensional deformation measurement.
© 2015 Optical Society of America
1. Introduction
Simultaneous measurement of multidimensional displacement components has generated a lot of interest in recent years because such measurements have the ability to provide a complete characterization of the deformation process. Optical techniques employed for such measurements are preferred over other techniques due to their capability of providing non-destructive, full-field measurements. Digital holographic interferometry (DHI) is one of such optical setups used for multidimensional deformation measurement. One of the important advantages associated with a DHI setup is that no phase-shifting or introduction of carrier fringes are required to obtain the interference phase estimate. For multidimensional deformation measurement, different experimental setups have been proposed in the context of DHI [1–4]. These optical set-ups basically involve the illumination of the object with multiple light beams from different directions. However, difficulties involved in grouping together pairs of such light beams with their corresponding reference beams, and in arranging them to obtain selective reconstruction of object waves associated to each of the light beams by either the use of spatial multiplexing or multiple laser sources have engendered overall complexity that hampers the realization of measurement operations.
Another approach [5–9] that has been suggested for the multidimensional deformation measurement consists of using multiple object illuminating beams allied to a single reference beam. The composite interference field recorded in such a setup is made up of a number of interference signals depending on the number of beams illuminating the object surface. The techniques based on this approach have been shown to derive the multidimensional deformation measurements from a single frame. The important limitation of these techniques lies in the inherent assumption of the low order polynomial form of the interference phases over a segment of the interference field, as there are situations where modeling the phase variation as a polynomial of not sufficiently higher order may not be justified. An alternative and significantly important approach has emerged recently in which the multidimensional deformation measurement is obtained by first separating the signal components present in the composite interference field, and then treating each of the separated signal components separately. The signal separation technique proposed in [10] is based on the introduction of a known carrier frequency along one of the two object beam paths. Two signal components present in the interference field are then separated in the space-frequency domain. Very recently, a technique based on the windowed Fourier transform (WFT) has also been proposed [11] for the signal separation from an interference field. In this paper, we present a new technique for signal separation which is based on the iterative separation and reconstruction of the signal components. The proposed method provides the signal separation from a single recording of the interference field and without any need for the introduction of the carrier fringes. The proposed method is also particularly well suited for applications with high density fringes without any special concern as there might be while using the WFT based method described in [11].
2. Theory
A schematic diagram of a digital holographic interferometry setup for multidimensional deformation measurement is shown in Fig. 1(a). A single laser light source is used to illuminate the object symmetrically to its surface normal. The same laser source is used to derive a reference beam which interferes with the light scattered by the object at the surface of hologram recording medium. The angles of the object illumination beams are set to be α1 = α2. The holograms are recorded using a CCD camera. The recording plane of the CCD camera is placed perpendicular to the object surface normal. The coordinate system is such that the the observation direction is along the z axis. The object illumination beams are placed in the x − z plane. This setup is thus sensitive to the object deformation along the x and z directions. Let us represent the complex interferogram recorded in a multi-wave DHI setup with two object illuminating beams and a single reference beam as
where, ; k ∈ [0, K − 1] and l ∈ [0, L − 1] represent the pixels along the rows and columns of the interferogram I[k, l], respectively; K and L represent the number of pixels along the rows and columns of the interferogram, respectively; φ1[k, l] and φ2[k, l] represent the interference phases associated with the beams 1 and 2, respectively; A1[k, l] and A2[k, l] represent the amplitudes associated with the beams 1 and 2, respectively, which usually have low spatial variations; the interferogram is assumed to be corrupted by a complex additive white Gaussian noise, ε[k, l]. The interference phases are proportional to the amount of deformation the object has gone through. Therefore, the space-frequency distributions of the signal components depend upon the object surface deformation. The amplitudes, on the other hand, depend on the intensities of the object illuminating beams. Consequently, the signal amplitudes can be controlled by adjusting the beam intensities. In order to accurately and reliably separate the signal components overlapping in space-frequency domain, we utilize an amplitude discrimination criteria. Accordingly, two distinct beam intensities are set for beam 1 and beam 2. As a result, the amplitudes of the signal components in Eq. (1) are set differently.
Fig. 1 (a) A schematic diagram of the digital holographic interferometry setup with two object illuminating beams (b) N × N sized two dimensional block representation of an interference field of size K × L.
In the proposed method, we first proceed by separating the higher amplitude signal in a block-wise manner. The interference field is divided into a number of small blocks each of size N × N as shown in Fig. 1(b). The block size is chosen such that in each block the interference phases have a simple form, such as a linear phase. Let us represent the interference field in a given block as
The singular value decomposition (SVD) of the matrix Ib decomposes it into a number of components based on their respective strengths. Let us represent the SVD of Ib as follows, where, * represents complex conjugate operation; Ub and Vb are the matrices containing the left and right singular vectors, ub and vb, respectively; Λb is the rectangular diagonal matrix with non-negative real singular values on its diagonal arranged in a decreasing order. In the present case, these singular values depend on the amplitudes of the signal components. Considering the case of A1 > A2, the first singular value and its associated singular vectors are related to the first signal component. Likewise, the second singular value and its associated singular vectors are related to the second signal component. In the absence of noise, the remaining singular values are zero. In the presence of noise, however, the remaining singular values are non-zero. As the interference phases are considered to have a simple structure of linear phase and the noise generally has a uniform space-frequency distribution, the singular values corresponding to the noise are always small as compared to the signal component. This fact is the key feature of the proposed method in separating the signal components from the interference field. Now, we note two important facts:- The singular values associated with the signal components are always greater than those associated with the noise components.
- The first and the second singular values correspond to the higher and the lower amplitude signal components, respectively.
We first illustrate the ability of the proposed method in separating the signal from noise. Let us consider a single component two dimensional complex signal example with a simulated linear phase of size 50 × 50 pixels. The signal was generated with a signal-to-noise ratio (SNR) of 0 dB. The wrapped phase of the noisy signal is shown in Fig. 2(a). The proposed technique of signal separation was applied to this signal. As only a single component is present in the signal, only the first, i.e., the highest singular value and its associated singular vectors are used for the signal reconstruction. The wrapped phase of the reconstructed signal is shown in Fig. 2(b). It is evident that the SVD based approach effectively removes the noise and reconstructs the signal.
Fig. 2 (a) Noisy phase fringe pattern. Filtered phase fringe pattern using the SVD of (b) Ib and (c) Ia.
A more robust method is adapted for the signal reconstruction in this paper. This method involves an augmentation of data matrix Ib with its own partitioned rows and columns. This operation basically increases the noise subspace dimension and allows the noise subspace to encapsulate maximum amount of noise present in the signal [12]. Accordingly, an augmented block matrix is generated with Ib as
where, for n = [0, N − 1]. Subsequently, SVD of the augmented matrix is computed as Similar to the signal reconstruction using the SVD of Ib, the signal is reconstructed using the SVD of Ia with the first singular value and its associated singular vectors. The inflated noise subspace dimension depends on the selection of the parameters P and Q. Depending on the number of signal components i present in the signal, the values of the parameters P and Q should be selected [12] as follows: The noisy phase fringe pattern shown in Fig. 2(a) is filtered and signal is reconstructed using the first singular value and its associated singular vectors computed in the SVD of Ia. In this case, i = 1. The wrapped phase of the reconstructed signal is shown in Fig. 2(c). It is quite clear that the augmented matrix based method provides more accurate signal reconstruction compared to the method based on the SVD of Ib.Now, let us consider the signal containing two components. The augmented matrix Ia is generated from this signal. As noted earlier, in the case of A1 > A2, the first and the second singular values correspond to the the first and the second signal components, respectively. At this stage, if we reconstruct a signal using the first singular value and its associated singular vectors, the first signal component is reconstructed. The second signal component is filtered out along with the noise component. The first component I(1) thus separated from the signal is stored and its contribution from the signal is removed as follows:
The signal I(2) primarily contains the second signal component along with the noise. This signal is filtered using the augmented matrix based method to remove the noise present in the signal. The reconstructed signal thus carries the information on the second signal component.To illustrate this point in more detail, a two dimensional signal example with two signal components with linear phases is considered. The signal was generated with A1 > A2 at an SNR of 0 dB. The wrapped phase of the noisy signal is shown in Fig. 3(a). The wrapped phases of the reconstructed first and second signal components are shown in Figs. 3(b) and 3(c), respectively. It can be deduced from these figures that the proposed method is capable of separating the signal components in the presence of severe noise.
Fig. 3 (a) Noisy phase fringe pattern. Phase fringe patterns corresponding to (b) first signal component and (c) second signal component.
An iterative method for the separation of signal components from the interference field I[k, l] is explained with the following steps:
- The interference field is divided into a number of blocks Ib’s each of size N × N.
- The first signal component I(1) is separated by applying the augmented matrix based signal reconstruction method to each Ib.
- A new signal I(2) containing the second signal component and the noise is obtained as I(2)= Ib − I(1).
- By applying the augmented matrix based signal reconstruction method to I(2), the second signal component is separated from noise.
- In the case of signal components with overlapping space-frequency distribution, above steps do not provide accurate signal separation. For a more robust signal reconstruction, the first signal component is further reconstructed by applying the signal reconstruction method to the signal I(1) = Ib − I(2).
- The above steps of extracting a single signal component at a time are executed iteratively to obtain the accurate reconstructions of both the signal components.
3. Numerical results
We have performed numerical experiments to evaluate the performance of the proposed method. Two simulated interference phases were used to generate a complex interferogram of size 256 × 256 pixels. The real part of this interferogram, i.e. the fringe pattern, is shown in Fig. 4(a). The signal amplitudes were selected such that A1 : A2 = 2 : 1. The interferogram was corrupted by the additive white Gaussian noise such that the signal-to-noise ratio (SNR) was 5 dB. The absolute value of the Fourier spectrum of the interferogram is shown in Fig. 4(b). It can be observed that the space-frequency distributions of the signal components overlap each other. The wrapped phases of the separated first and second signal components after the first iteration are shown in Figs. 4(c) and 4(d), respectively. It can be observed that although the signals are separated, small traces of the other component are still present in each of the signal components. However, after eight iterations, the signals are separated accurately and the wrapped phases of the first and second components are shown in Figs. 4(e) and 4(f), respectively. To evaluate the noise performance of the proposed method, the simulations were performed at SNRs varying between 2 and 20 dB using different block sizes. The RMSEs in the phase estimation were computed for each case and are plotted in Figs. 5(a) and 5(b), respectively. These RMSE values substantiate the fact that selection of a large block size containing more data is capable of filtering the noise effectively compared to the selection of small block sizes. In the case of small block sizes, it is not possible to obtain a reliable reconstruction of the second signal component. The signal separation is also found to be more reliable in the case of large block sizes. However, beyond a certain limit of the block size, the multiple phase estimation performance does not improve significantly. On the contrary, such a large block size selection may violate the assumption of the linear phase made in the proposed method. Additionally, the selection of large block sizes also results in increased computational burden. Based on these observations, we have selected the block size of 21 × 21 pixels in our simulation.
Fig. 4 (a) Noisy fringe pattern associated with an interferogram containing two signal components (b) Fourier spectrum of the interferogram. Phase fringe patterns of (c) first signal component and (d) second signal component after first iteration. Phase fringe patterns of (c) first signal component and (d) second signal component after eight iterations.
Fig. 5 RMSEs in the computation of phase of (a) first signal component and (b) second signal component in function of block size and SNR. RMSEs in the computation of phase of (a) first signal component and (b) second signal component in function of amplitude ratio and SNR.
The distinction between the signal amplitudes plays an important role in signal separation. We have studied the effect of the ratio of signal amplitudes on the signal separation and in turn on the multiple phase estimation accuracy. The simulations were performed to compute the RMSEs in the multiple phase estimation in function of the signal amplitude ratio and SNR. The computed RMSEs are plotted in Figs. 5(c) and 5(d), respectively. These plots suggest the range of amplitude ratios where the proposed method provides accurate phase estimation performance. The selection of the amplitude ratio between 1.5 and 2.5 seems to provide a consistent phase estimation accuracy. These amplitude ratios can be easily realized by appropriately setting the intensities of the object illuminating beams.
4. Experimental results
The performance of the proposed method is validated with a moiré fringe pattern recorded in a DHI experimental setup. The experimental setup consisted of a laser source of wavelength 532 nm. A single reference beam and two object illumination beams were derived from this laser source. The object illuminating beams were placed in the x − z plane symmetrical to the object surface normal. The observation plane, i.e., the CCD plane was placed perpendicular to the object surface normal set along the z direction. We have used a Sony XCL-U1000 monochrome CCD camera to record the holograms. The test object was a circular membrane of 6 cm diameter uniformly clamped along its edges. The deformation was applied by a point load and the object was rotated around the z-axis. A hologram was recorded in the object’s undeformed state and another was recorded in its deformed state. The numerical reconstruction of the two holograms allows for generating the complex interferogram corresponding to the object deformation.
The real part of this interferogram is shown in Fig. 6(a). The proposed method was applied for multiple phase estimation with the block size of N = 21. The wrapped phases of the separated signals are shown in Figs. 6(b) and 6(c). The unwrapped phases are obtained by applying a simple unwrapping algorithm and are shown in Figs. 6(d) and 6(e), respectively. The sum and difference of the estimated phases corresponding to the out-of-plane and in-plane components of displacement of the object surface are plotted in Figs. 7(a) and 7(b), respectively. For the sake of clarity, the wrapped form of the sum and difference of phases are given in Figs. 7(c) and 7(d), respectively.
Fig. 6 (a) Experimentally recorded fringe pattern. Phase fringe pattern of separated (b) first signal component and (c) second signal component. Unwrapped phases associated with (d) first signal component and (e) second signal component.
Fig. 7 Unwrapped form of (a) sum of phases (b) difference of phases. Wrapped form of (c) sum of phases (d) difference of phases.
5. Conclusion
In this paper, we have presented a new method of signal separation from a multicomponent interference field recorded in a digital holographic interferometry setup. The iterative method proposed for the signal separation consists of dividing the interference field into a number of small sized blocks and separating the signal components in each block based on an amplitude discrimination criteria. The noise subspace inflation is achieved by augmenting the block matrix by its own rows and columns which results in the improvement of noise robustness and signal separation. The performance of the proposed method is evaluated by carrying out error analysis in the interference phase estimation of the individual signal components as a function of signal to noise ratio, block size and amplitude ratio. The experimental verification is also performed using a DHI setup. The simulation and experimental results validate the applicability of the proposed method in multiple deformation measurement.
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