Automatic fringe enhancement with novel bidimensional sinusoids-assisted empirical mode decomposition

Chenxing Wang; Qian Kemao; Feipeng Da

doi:10.1364/OE.25.024299

1. Introduction

In optical measurement, fringe pattern analysis is important for accurate and robust phase extraction. In the cases of harsh environment or dynamic phenomena, capturing a single fringe pattern is more realistic and convenient, but analyzing it is more challenging. Spectral analysis methods, such as Fourier transform and wavelet transform [1–3], have shown their success in single fringe pattern analysis, but they encounter difficulties if the phase and/or the background are complicated. Empirical mode decomposition (EMD) was recently introduced into fringe analysis. EMD is an adaptive and data-driven technique, and decomposes a signal into intrinsic mode functions (IMFs). Its capability in removing noise and background in fringe analysis has been demonstrated [4–7]. However, their applications are still troubled by two difficulties. First, the mode mixing problem (MMP) often appears due to uneven distribution of extrema [8], making the decomposition less meaningful. Second, automatically recognizing noise and background to remove them is not easy if the fringe pattern is complicated, even without the MMP.

To solve the MMP, the ensemble EMD (EEMD) repeatedly adds white noise to homogenize the analyzed data [8], followed by averaging the ensemble of decomposition results. Although effective and widely used [9], EEMD is hindered in practical use due to the very heavy computation load. Inspired by the EEMD, regenerated phase-shifted sinusoids-assisted EMD (RPSEMD) was developed using a small number of adaptively designed sinusoids, which dramatically reduces the computation time with even better decomposition results [10]. The RPSEMD has been applied to process complicated signals [11] and analyze fringes line by line [12]. The attractive performance urges us to generalize the idea to 2D to solve the annoying MMP which persists in other bi-dimensional EMD methods including bi-dimensional EEMD (BEEMD) [13, 14] and fast and adaptive BEMD (FABEMD) [15, 16].

The mathematical decomposition results of EMD, in the application of fringe analysis, should be meaningfully grouped into noise, a sinusoidal fringe component and background. There are two directions to fulfill this task. One is to solve the MMP first and then group the resulted IMFs directly; the other is to analyze IMFs locally to reconstruct the fringe component, regardless of the MMP. For the first direction, the EEMD is too time-consuming [9, 13, 14, 16] and automatic grouping of IMFs has not yet been fully achieved [17, 18]. For the second direction, a selective reconstruction method is implemented by combining local regions selected from bi-dimensional IMFs (BIMFs) based on amplitude modulation analysis [19], and its efficiency and automation is further improved with the enhanced fast EMD (EFEMD) method [20]. Although these methods have shown good results and interesting applications [21, 22], sometimes human intervention is necessary to remove the background correctly and moreover, large errors will still be produced when the MMP appears.

Aiming at higher accuracy, efficiency and automaticity, we concentrated on both a better BEMD algorithm and an automatic fringe enhancement method. First, a novel BSEMD is proposed, which adopts the-state-of-art EFEMD as the base BEMD to achieve high efficiency, and adds our specially designed sinusoids to solve the MMP. Second, based on the results of BSEMD, a simple grouping strategy is proposed to separate the fringe component and a simple post-processing to further enhance it. Third, the characteristics of EMD are utilized for the grouping and in turn the grouping decides the automatic stop of the decomposition. This integration smartly avoids redundant and meaningless decomposition, and contributes to the automaticity, efficiency and accuracy of the whole method. Experiments show that the proposed method is able to handle various fringe patterns including complexly distorted and low-quality ones.

2. The proposed bi-dimensional sinusoids-assisted EMD (BSEMD)

BEMD has been proposed a few years ago [23]. With a similar procedure to EMD, BEMD detects the extrema of an image, constructs upper and lower envelopes from the extrema, and removes the mean of envelops from the image; this process is iterated (as inner iterations) a few times until a qualified BIMF is resulted. By repeating these operations (as outer iterations), a series of BIMFs are obtained from high frequency to low frequency so that

I (x, y) = \sum_{k = 1}^{K} {BIMF}_{k} (x, y) + r (x, y),

where K is the number of BIMFs and r(x, y) is a residual.

Been driven by data is an impressive advantage for the BEMD method. However, an accompanying problem comes in the meantime, i.e., uneven extrema distribution of the data will result in the mixture of components with different scales in a BIMF. In fact, the extrema of an image is hardly evenly distributed in practice, leading to the MMP at a high probability.

BEEMD methods solve the MMP by repeatedly performing BEMD on white noise added images and then averaging the ensemble of corresponding BIMFs as a final result. The base BEMD selection and the ensemble construction are two key factors determining the effectiveness and the efficiency of BEEMD methods [13, 14, 16]. Inspired by BEEMD, our proposed BSEMD inherits the main procedure of BEEMD, but with a better base BEMD algorithm and a smarter ensemble construction.

2.1 The selection of base BEMD

Among the existing BEMD methods, a noticeable one is the fast and adaptive BEMD (FABEMD) [15]. FABEMD omits the inner iteration and uses order statistics filters to construct the envelops of data, which not only maintains the decomposition quality but also reduces the computation cost from minutes to seconds.

Recently, FABEMD has been further improved into EFEMD which can process a fringe pattern (512 × 512 pixels) within 1 second in MATLAB [20, 21] and in the meantime ensure the decomposition quality similar to FABEMD. The time cost of EFEMD is reduced due to two aspects: (i) the envelopes are constructed by a smoothing filter using morphological operations, replacing the order statistics filters used in the FABEMD; (ii) the filter size is estimated in a simple way to avoid the verbose statistical sorting, i.e.:

T = \sqrt{\frac{M N}{P}}

where M × N is the image size and P is the average number of the detected maxima and minima. In view of the above advantages, EFEMD is selected as the base BEMD in this paper.

2.2 The ensemble construction

The ensemble method used in EEMD can also be applied to EFEMD, but the computation cost is unbearable. In RPSEMD [10–12], the ensemble size is greatly reduced from large (usually 200) to small (usually 4), by adding well-designed sinusoids. Consequently this sinusoids-assisted ensemble construction is generalized to the proposed BSEMD.

The structure of the algorithm: As shown in Fig. 1, BSEMD has a few outer iterations to decompose an image I into different scales. Each of the outer iteration generates a BIMF through the following three key steps: (i) For the i^th outer iteration, the input signal I_i is partially decomposed by EFEMD to obtain only the first BIMF, which is a temporary result denoted as ${BIMF}_{1}^{t}$ (I₁ = I for initialization). The properties of ${BIMF}_{1}^{t}$ are analyzed to design a 2D sinusoid s_i as an auxiliary signal that will be used to homogenize the distribution of peaks and valleys of the smallest significant scale of component in ${BIMF}_{1}^{t}$ . (ii) With ${(BIMF}_{1}^{t} + s_{i})$ being partially decomposed again, a new BIMF₁ is obtained and denoted as ${BIMF}_{1}^{+}$ , which contains the scale similar to s_i. Similarly, ${(BIMF}_{1}^{t} - s_{i})$ is partially decomposed to obtain ${BIMF}_{1}^{-}$ . Averaging the ensemble of ${BIMF}_{1}^{+}$ and ${BIMF}_{1}^{-}$ gives BIMF_i, where the add-ins, ± s_i, will be cancelled through the averaging. (iii) BIMF_i is then removed from the input I_i to obtain I_i₊₁ as the input of the next iteration.

Fig. 1 The structure of BSEMD.

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Note that in each iteration, (i) EFEMD is always partially executed and stops after the ${BIMF}_{1}^{t}$ , ${BIMF}_{1}^{+}$ or ${BIMF}_{1}^{-}$ is obtained; (ii) as mentioned above, there is no inner iteration in EFEMD in contrast to the traditional EMD; (iii) the ensemble size is only 2, in contrast to a big number in EEMD. These factors make the proposed BSEMD very efficient. In the meantime, it is also effective in solving the MMP, as will be demonstrated later.

The design of the sinusoid: Designing the 2D sinusoid is a crucial step for BSEMD. To make the extrema of the data distribute evenly, the auxiliary 2D sinusoid is designed as

s_{i} (x, y) = a_{i} \cos (2 π f_{i} x) \cos (2 π f_{i} y),

where i is the index of the iterations as in Fig. 1; a_i and f_i denote the amplitude and frequency, controlling the intensity and density of peaks and valleys of the 2D sinusoid, respectively.

The determination of a_i and f_i is related to the highest-frequency mode in the ${BIMF}_{1}^{t}$ as in Fig. 1. For simplicity, we denote two dominant modes in ${BIMF}_{1}^{t}$ as m1 and m2, with their amplitudes and frequencies as $(a_{m 1}, f_{m 1})$ and $(a_{m 2}, f_{m 2})$ , respectively. Suppose that $f_{m 1} > f_{m 2}$ so the mode m1 should be separated and extracted first. According to [24], successful modes separation requires:

f_{m 1} \geq 1.5 f_{m 2},

a_{m 1} f_{m 1} > a_{m 2} f_{m 2} .

The filter size T_i in Eq. (2) reflects the statistical distance between two peaks (or two valleys) of ${BIMF}_{1}^{t}$ , so the average frequency is defined as $f_{a v} = 1 / T_{i}$ and $f_{m 1} \geq f_{a v} \geq f_{m 2}$ can be deduced. Then f_i is determined as

f_{i} = {\begin{array}{l} 1 / 2, & i = 1 \\ 1.5 f_{a v} = 1.5 / T_{i}, & i > 1 \end{array}

which guarantees that the inequality (4) is satisfied so that the first dominant component can be separated from the others. It is worth mentioning that f_i should be restricted to be no more than 1/2, the Nyquist sampling rate. To be safe, we initialize

f_{1} = 1 / 2

to cope with the possible high-frequency noisy cases in practice.

As for the estimate of a_i, setting $a_{m 1} \geq a_{m 2}$ make the inequality (5) being satisfied since $f_{m 1} > f_{m 2}$ , which can be realized by taking a_m₁ as the largest amplitude of ${BIMF}_{1}^{t}$ , namely $m a x (| B I M F_{1}^{t} |)$ . In practice, we enlarge it to ensure the added $s_{i} (x, y)$ is always effective:

a_{i} = 4 \times \max (| {BIMF}_{1}^{t} |) .

3. The fringe grouping based on BSEMD

The proposed BSEMD can be applied to all general images. We now turn to the application of fringe enhancement. A fringe pattern $I (x, y)$ can be written as

I (x, y) = a (x, y) + b (x, y) \cos [φ (x, y)] + n (x, y),

where

a (x, y)

is the background intensity;

b (x, y) \cos [φ (x, y)]

is the useful term for phase calculation and called a phase-modulated (PM) signal;

n (x, y)

is noise. To extract the PM signal, we groupe all the obtained BIMFs in Eq. (9) in order to map them to Eq. (8), i.e.,

I (x, y) = \sum_{k = 1}^{k_{1}} {BIMF}_{k} (x, y) + \sum_{k = k_{1} +1}^{k_{2}} {BIMF}_{k} (x, y) + (\sum_{k = k_{2} + 1}^{K} {BIMF}_{k} (x, y) + r (x, y)),

where the three groups correspond to noise, the PM signal and background intensity successively, divided by the critical indexes k₁ and k₂. In our previous work [12], a fringe pattern can only be processed line by line, and each line is processed independently, thus many properties of 1D autocorrelation for each IMF are analyzed to ensure the correctness and robustness of the grouping. These properties are also useful for the BIMFs but can be much simplified and improved to cope with the 2D case.

3.1 The determinition of k₁

The following properties of noise and the PM signal are useful for their separation

(i) It is shown in [25] that the energy of the noise IMF is inversely proportional to the average period of the IMF, which is also true for BIMFs. Thus for BIMFs within the noise group we have $E (1) > E (2) > \dots > E (k) > \dots > E (k_{1})$ where E(k) represents the energy of a BIMF and can be calculated as $E (k) = A_{k} (0, 0)$ from the autocorrelation of BIMF_k below $A_{k} (τ_{1}, τ_{2}) = \sum_{(x, y)} [{BIMF}_{k} (x, y) {BIMF}_{k} (x - τ_{1}, y - τ_{2})] .$
(ii) E(k₁ + 1) will become large abruptly because it is the starting of the PM signal group.
These properties make E(k₁) a local minimum of E(k), which can be easily detected. In the special case of $E (1) < E (2)$ , $k_{1} = 1$ is set as BIMF₁ is always a noise term according to Eq. (6).

3.2The determinition of k₂

The next step is to determine k₂ to distinguish the PM signal from the background. In [12], grouping is applied to each line of a fringe pattern by using four properties, the global frequency (PB1), the energy (PB2), the amplitude-frequency parameter (PB3), and the orthogonality (PB4). While PB1 and PB3 are necessary for determining k₂, PB2 and PB4 are optional but they help to ensure the grouping correct and robust in coping with complicated fringe patterns. Different from the 1D case, in this paper, the BIMFs from BSEMD are generally MMP-free and mono-component, making them more meaningful and expressive. Consequently, PB1 and PB3 are sufficient to distinguish the two groups, described as follows:

(i) The ratio of global frequencies of two adjacent BIMFs (PB1).
The global frequency of BIMF_k, denoted as $f_{g} (k)$ , usually decreases when k increases, in the PM signal group. However, $f_{g}$ is reduced more drastically at the transition between two groups, as the background is assumed to have slow spatial variations thus a very low global frequency. This fact is well reflected by the ratio of global frequencies defined as $R_{f} (k) = f_{g} (k) / f_{g} (k + 1)$ . We consider $R_{f} (k) > 2$ as a drastic change and set this k as the initial estimation of k₂. $f_{g}$ can be estimated from $A_{k} (τ_{1}, τ_{2})$ in Eq. (10). Letting the first maximum on the positive axis of τ₁ be $A_{k} (τ_{1 g}, 0)$ and the one on the positive axis of τ₂ be $A_{k} (0, τ_{2 g})$ , we then calculate the frequencies in horizontal and vertical directions as $f_{1 g} = 1 / τ_{1 g}$ and $f_{2 g} = 1 / τ_{2 g}$ , respectively. Thus the global frequency is estimated as
$f_{g} = \sqrt{f_{1 g}^{2} + f_{2 g}^{2}} .$
(ii) The amplitude-frequency parameter (PB3).
A BIMF in the background group often has slow varying brightness, meaning that most of the points in ${BIMF}_{k_{2} + 1}$ have non-ignorable amplitudes but very small frequencies. An amplitude-frequency ratio is thus defined as
$R_{a f} (k) = a_{g} (k) / f_{g} (k),$
where a_g can be estimated from the global energy as $a_{g} (k) = \sqrt{E (k)}$ and $f_{g}$ has been given in Eq. (11). This R_af either has a local maximum value at k₂ or the largest increase from k₂ to k₂ + 1.

The implementation steps of determining k₂ are summarized as follows:

(i) Initial estimation: starting from $k = k_{1} + 1$ , compute $R_{f} (k)$ ; continuously increase k by 1 and repeat the computation until $R_{f} (k_{f}) > 2$ is found; set $k_{2 c} = k_{f}$ as an initial estimation of k₂. If no such k can be found, $k_{2} = K - 1$ is set.
(ii) Fine estimation: starting from $k = k_{2 c}$ , compute $R_{a f} (k)$ ; continuously increase k by 1 and repeat the computation until $R_{a f} (k_{R})$ is an extremum; set $k_{2} = k_{R} - 1$ if $R_{a f} (k_{R})$ is a maximum or $k_{2} = k_{R}$ if a minimum, as the fine and final estimation. If no such index k can be found, we set $k_{2} = k_{1} + 1$ if $R_{a f} (k)$ increases monotonically, and $k_{2} = K - 1$ if $R_{a f} (k)$ decreases monotonically, where K is the total number of BIMFs as shown in Eq. (1).

3.3 The integration of the decomposition and the grouping

As said, our proposed BSEMD can be applied to general images, where the decomposition stops when T in Eq. (2) is larger than $\sqrt{M N / 3}$ [20]. Such decomposition is called complete. In fringe enhancement, naturally a complete decomposition can be carried out first, followed by fringe grouping. However, in our implementation, the grouping information can be feedback to guide the BSEMD so that only partial decomposing is sufficient to achieve the same fringe enhancement result but with reduced computation cost: the decomposition stops once k₂ is determined. Thus each new separated BIMF will be grouped first and checked whether $k_{2}$ has been reached, in order to decide whether both the decomposition and grouping should be stopped.

4. Post-denoising for the PM signal

By now, the desired PM signal has been extracted. However, if noise is heavy, parasite noise may still be observed because noise spreads in all BIMFs including the PM signal group [25]. Post-denoising is proposed by utilizing the feature of a fringe pattern in the context of BSEMD. As a BIMF has mono frequency, a local part of the PM signal in a local region of a BIMF will rarely appear in other BIMFs, and correspondingly, this local region in other BIMFs should only present noise. Equivalently, in one BIMF, the PM signal and noise may appear in different isolated local regions. This unique and interesting feature enables us to develop the following novel denoising method, i.e., in a BIMF, we first identify and segment the signal and noise regions and then remove the noise regions.

The above segmentation can be easily achieved by applying Otsu’s n-thresholding to the instantaneous amplitude map of each BIMF, where $n = k$ is set for ${BIMF}_{k}$ by conservatively considering the worst case that all k modes from ${BIMF}_{1}$ to ${BIMF}_{k}$ may appear. The instantaneous amplitude of a BIMF can be extracted through a Hilbert spiral transform (HST) [26]. Open and Close morphological operations are followed to rectify some isolated small regions, and then the morphological dilation is used to moderately expand the determined local regions of fringe signals in order to make sure that no PM signal is removed mistakenly. The size of the structuring element in these morphological operations is set based on the global period of BIMF_k, namely, $2 / f_{g} (k)$ . Finally, ${BIMF}_{k}$ is segmented into n classes represented by the cluster indexes from 1 to n, so the regions with values of 1 are taken as the noise regions and set to be 0 for denoising.

To achieve better efficiency and robustness, we limit the segmentation and denoising to ${BIMF}_{k} (k_{1} + 1 \leq k < k_{3})$ where k₃ is obtained from ${BIMF}_{k_{3}}$ whose energy $E (k_{3})$ is the highest among the PM signal group $(k_{1} + 1 \leq k \leq k_{2})$ . Post-denoising is not applied to ${BIMF}_{k} (k_{3} \leq k \leq k_{2})$ as the noise therein is insignificant compared to the signals.

Note that similar denoising work can be found in [19] but it critically depends on the human intervention. In contrast, the post-denoising in this paper is simple and fully automatic.

5. Experiments

Experiments are carried out to show the effectiveness and efficiency of BSEMD. The ASR-EFEMD [20] and fast BEEMD (FBEEMD) [16] are selected for comparison, which represent the state-of-the-art EMD techniques for fringe analysis. The parameters for ASR-EFEMD are set according to [20], while those for FBEEMD are selected according to [8, 14, 16]. Manual work is applied to first remove the significant background for ASR-EFEMD and to remove both noise and background for FBEEMD to yield optimal results.

5.1 A closed fringe pattern

The closed fringe pattern shown in Fig. 2 is simple in structure yet challenging for processing. Figure 2(a) is a simulated ideal fringe pattern, which is modified into Fig. 2(b) with non-uniform modulation and background as well as random noise.

Fig. 2 A simulation of a closed fringe pattern.

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First, the decomposition results of Fig. 2(b) by the three methods are shown in Fig. 3. For the EFEMD result, significant noise are mixed with the fringe signal in Figs. 3(a1)-3(b1), and the fringe signal (center) and background (surrounding) are mixed in Fig. 3(f1). The similar MMP also appears in the FBEEMD result although the results have been improved. As for BSEMD, most of the noise has been separated into Figs. 3(a3)-3(b3) and little background is mixed with the fringe signal in Fig. 3(g3). For quantitative evaluation, the orthogonality of successive BIMFs (OSB) [27] is computed as

OSB = \frac{1}{M \times N} \sum_{k = 1}^{K - 1} [\sum_{(x, y)} B I M F_{k} (x, y) B I M F_{k + 1} (x, y)] .

The OSB values for the results of EFEMD, FBEEMD and BSEMD are 0.12, 0.13 and 0.09, respectively, showing that the proposed BSEMD algorithm presents the best orthogonality.

Fig. 3 The BIMFs by the: (a1)-(h1) EFEMD; (a2)-(i2): FBEEMD; (a3)-(i3): BSEMD.

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Second, the proposed grouping method is tested. The energy E(k) is computed and plotted in Fig. 4(a), from which k₁ = 3 is identified, meaning that ${BIMF}_{1}$ - ${BIMF}_{3}$ in Figs. 3(a3)-3(c3) belong to the unwanted noise group. Starting from $k = 4$ , $R_{f} (5) > 2$ is found from the $R_{f}$ curve in Fig. 4(b), so that $k_{2 c} = k_{f} = 5$ is determined as the initial estimation of k₂. Then, starting from ${BIMF}_{5}$ , the fine estimation is performed from the $R_{a f}$ curve in Fig. 4(c), where $k_{2} = k_{R} = 7$ is finally determined, meaning that ${BIMF}_{4}$ - ${BIMF}_{7}$ in Figs. 3(d3)-3(g3) are the desired PM signal group. The global frequencies of BIMF₁-BIMF₈ in Figs. 3(a3)-3(h3) estimated using Eq. (11) are about 0.471, 0.320, 0.202, 0.062, 0.052, 0.026, 0.014 and 0.006, the ratios of two adjacent global frequencies are about 1.47, 1.60, 3.23, 1.19, 2.00, 1.86 and 2.33, respectively. It can be seen that (i) the BIMFs are pretty like dyadic filteing results and (ii) there are two sudden changes between BIMF₃ and BIMF₄ as well as BIMF₇ and BIMF₈, which group the BIMFs into three components.

Fig. 4 Grouping parameters for all BIMFs of BSEMD: (a) the global energy E(k); (b) ratio values of global frequencies of two adjacent BIMFs; (c) the amplitude-frequency ratio R_af (k).

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Third, the post-denoising for the PM signal group is illustrated. According to Fig. 4(a), since BIMF₅ has the largest energy within the PM signal group $(4 \leq k \leq 7)$ , i.e., k₃ = 5, thus only BIMF₄ (Fig. 5(a)) needs post-denoising. Figures 5(b)-5(f) show the step-by-step results of the post-denoising and Fig. 5(g) shows the denoised BIMF₄. The final reconstructed PM signal is obtained by summing the results of Fig. 5(g) and Figs. 3(e3)-3(g3), which is given in Fig. 6(c).

Fig. 5 (a) The BIMF₄ of BSEMD, namely Fig. 3(d3); (b) the instantaneous amplitude map by HST; (c) the segmentation performed on the amplitude map by Otsu’s n-thresholding (n = 4); the results after: (d) the morphological open operation, (e) the morphological close operation, and (f) the dilation operation; (g) the final result after removing the noise region.

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Fig. 6 The reconstructed PM signal of: (a) ASR-EFEMD, (b) FBEEMD, (c) our method; the normalized PM signal of: (d) ASR-EFEMD; (e) FBEEMD; (f) our method.

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We also reconstruct the PM signal for the other two methods for comparison. For ASR-EFEMD method, general background still needs to be removed firstly. For the FFEMD results in Figs. 3(a1)-3(h1), the background is most significantly presented in the residual (Fig. 3(h1)), but it is also visibly leaked into Fig. 3(g1) and invisibly leaked into Fig. 3(f1). The residual in Fig. 3(h1) can be set by default belonging to the background, but it is difficult to decide whether Figs. 3(f1) and 3(g1) belong to the background because some of the PM signal also exists in these two BIMFs. Finally, all the Figs. 3(f1)-3(h1) are considered as the background to minimize the errors in Table 1. This manual intervention is also used in FBEEMD, where Figs. 3(c2)-3(g2) are decided as the PM signal group. The reconstructed fringes of the three methods are displayed in Figs. 6(a)-6(c), showing that ASR-EFEMD loses some information in the central part of fringes and both ASR-EFEMD and FBEEMD retain heavy residual noise. All these results are normalized by dividing the amplitude obtained using HST and the normialziation results are shown in Figs. 6(d)-6(f), where BSEMD is visually most satisfactory.

Table 1. The error statistics of the three methods.

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Furthermore, the above results are compared with the ground truth for quantitative evaluation, as shown in Table 1, where μ_err, σ_err and γ_err are the mean error, standard deviation of the errors and the largest error, respectively. For ASR-EFEMD, μ_err, σ_err and γ_err are all large as it selects only partial components of the PM signal in most cases. The normarlization greatly reduces σ_err but unfortunately amplifies γ_err. For FBEEMD, as a complete PM signal is extracted, much better results are generated. For our method, as a more reasonable PM is extracted, its result is the best among the three. We further record the time cost of the three methods in Table 1, where the human interventions for both ASR-EFEMD and FBEEMD are not taken into account. Our method has satisfactory efficiency even though it is about 2 times longer than ASR-EFEMD due to the ensemble size of only 2.

We also tested another closed fringe pattern shown in Fig. 7(a), which is similar to Fig. 2(b) but with severer non-uniformity in modulation and background, and was downloaded from the link provided in [28]. The above methods are repeated and the normalized results are shown in Fig. 7. Figures 7(b) and 7(c) are the two best results from ASR-EFEMD, with a difference of including or excluding a certain low-frequency BIMF, leaving us a difficulty to make decision. Figure 7(d) is from FBEEMD. Our result in Fig. 7(e) is shown to be most satisfactory.

Fig. 7 (a) A close fringe pattern with severe noise and background; the normalized PM signal of: (b) ASR-EFEMD with the central information reserving complete; (c) ASR-EFEMD with background removed thoroughly; (d) FBEEMD; (e) our method.

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5.2 Complexly distorted fringe patterns

A complex fringe pattern is simulated according to [20] and shown in Fig. 8(a). Due to the MMP of the decomposition, the OSB values of EFEMD and FBEEMD are both larger than that of our method, as shown in Table 2.

Fig. 8 (a) A complex fringe pattern; (b)-(d): The PM signal by ASR-EFEMD, FBEEMD and our method respectively; (e)-(g): the normalized PM signal of ASR-EFEMD, FBEEMD and our method, respectively.

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Table 2. The error statistics of the three methods.

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With the same operations as in the previous example, the final reconstructed PM signals are resulted as shown in Figs. 8(b)-8(g). All the methods are seen effective in enhancing this complicated fringe pattern. However, Fig. 8(b) has imperfections including sudden intensity changes (red arrow) and signal loss in the dense fringe area (blue arrow), and Fig. 8(c) has imperfections including noise (red arrow) and also signal loss in the dense fringe area (blue arrow). Those imperfections are not obviously obverved in Fig. 8(d). Error statistics are also computed and listed in Table 2, where our result is seen to be the best one. The time cost is similar to the previous example.

5.3 A low-quality fringe pattern from a fringe projection system

A low-quality fringe pattern (1236 × 1268) is captured from a fringe projection system for measuring a steel key with black plastic bar and shown in Fig. 9. Saturation, shadow, heavy noise and dark regions are observed. All three methods are attempted and their OSB values of the decomposition results are listed in Table 3, where BSEMD still outperforms the others. The reconstructed PM signals, both non-normalized and normalized, are shown in Figs. 10(a)-10(f). All the three methods are able to enhance the fringe pattern impressively, while our result looks most satisfactory, regarding either noise suppression or fringe structure preservation. It is interesting to note from Table 3 that the time costs of ASR-EFEMD and our method are almost the same in this experiment. This is because in our method, the decomposition is automatically stopped when k₂ is determined, which helps to save time.

Fig. 9 A real fringe pattern.

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Table 3. The OSB values and the processing time for reconstructing PM fringe signals.

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Fig. 10 (a)-(c): the PM signal by ASR-EFEMD, FBEEMD and our method respectively; (d)-(f): the corresponding normalizeded PM signal.

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5.4 Discussions

In this paper, BSEMD is proposed for better image decomposition, based on which a fringe pattern is grouped into noise, a PM signal and background, then the extracted PM signal is further enhanced. Possible extentions are discussed below.

(i) The grouping method, i.e., the determination of k₁ and k₂ based on the analysis of BIMF properites, is successful in all the above examples. However, grouping by optimization algorithms [29] may be a usefl alternative and worth exploring.
(ii) In practical measurement, over-exposure, shadows and discontinuities are often unavoidable. From Fig. 10, it is noticeable that our method works robustly regardless of these negative factors, which is interesting and valuable for further investigation.
(iii) The major purpose of fringe enhancement is to improve the phase retrival quality. Our initial results are positive, with one example shown in Fig. 11 below. However, careful investigation is necessary to take into accout different phase retrieval methods and their paramters.

Fig. 11 Retrieved phases by FFSD [30] from the PM signals by using (a) ASR-EFEMD, (b) FBEEMD and (c) our method respectively, which are presented in Figs. 6(d)-6(f) correspondingly

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6. Conclusions

In practical measurement, unpredictable noise and background are unavoidable, which easily cause the mode mixing problem (MMP) in EMD-based fringe analysis. In this paper, a bidimentional sinusoids-assisted EMD (BSEMD) algorithm is proposed, based on which noise and background can be removed automatically. In the BSEMD, our specially designed 2D sinusoids can help to make the extrema of each scale distribute evenly thus greatly reduce the chance of MMP. The phase-modulated signal can then be easily reconstructed through a simple grouping strategy and enhanced to be high-quality by further post-denoising. Besides the effectiveness, the proposed method is automatic with no parameter tuning, and is efficient not only because the ensemble size is only 2 but also as the decomposition is organically integrated with the grouping. The automaticity, effectiveness and efficiency are experimentally validated, showing its potential in practical use. The removal of background and noise will enhance the fringe quality and subsequently improves th accuracy of phase retrieval,which may help to increase the measurement range. For example, the measurable object slope can be increased up to three times for Fourier transform profiloemty if the backround is successfully removed [31]. Future works on an alternative grouping technique, processing of shadow and evaluation of phase retreival are discussed.

Funding and Acknowledgments

This work is supported by the National Natural Science Foundation of P. R. China (61405034, 51475092), National Research Foundation, Prime Minister’s Office, Singapore under its IDM Futures Funding Initiative, and Singapore AcRF Tier 1 (RG28/15).

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Method	Non-normalization			Normalization			Time (s)
Method	μ_err	σ_err	γ_err	μ_err	σ_err	γ_err	Time (s)
ASR-EFEMD	−0.006	0.32	0.90	−0.006	0.11	1.96	1.7
FBEEMD	−0.007	0.12	0.60	−0.019	0.14	1.88	74
Our method	−0.003	0.09	0.60	−0.002	0.07	0.74	4.0

Method	OSB values	Non-normalization			Normalization			Time (s)
Method	OSB values	μ_err	σ_err	γ_err	μ_err	σ_err	γ_err	Time (s)
ASR-EFEMD	0.88	−0.017	0.89	2.72	−0.005	0.08	0.98	1.9
FBEEMD	0.88	−0.011	0.40	2.41	−0.0001	0.07	0.98	76.5
Our method	0.52	−0.017	0.22	1.53	−0.004	0.06	0.94	3.7

Method	ASR-EFEMD	FBEEMD + our fringe enhancement	Our method
OSB values	571.1	376.8	278.5
Time (s)	38.1	629.6	40.4

Method	Non-normalization			Normalization			Time (s)
Method	μ_err	σ_err	γ_err	μ_err	σ_err	γ_err	Time (s)
ASR-EFEMD	−0.006	0.32	0.90	−0.006	0.11	1.96	1.7
FBEEMD	−0.007	0.12	0.60	−0.019	0.14	1.88	74
Our method	−0.003	0.09	0.60	−0.002	0.07	0.74	4.0

Method	OSB values	Non-normalization			Normalization			Time (s)
Method	OSB values	μ_err	σ_err	γ_err	μ_err	σ_err	γ_err	Time (s)
ASR-EFEMD	0.88	−0.017	0.89	2.72	−0.005	0.08	0.98	1.9
FBEEMD	0.88	−0.011	0.40	2.41	−0.0001	0.07	0.98	76.5
Our method	0.52	−0.017	0.22	1.53	−0.004	0.06	0.94	3.7

Automatic fringe enhancement with novel bidimensional sinusoids-assisted empirical mode decomposition

Abstract

1. Introduction