Presented method for fringe pattern enhancement has been designed for processing and analyzing low quality fringe patterns. It uses a modified fast and adaptive bidimensional empirical mode decomposition (FABEMD) for the extraction of bidimensional intrinsic mode functions (BIMFs) from an interferogram. Fringe pattern is then selectively reconstructed (SR) taking the regions of selected BIMFs with high modulation values only. Amplitude demodulation and normalization of the reconstructed image is conducted using the spiral phase Hilbert transform (HS). It has been tested using computer generated interferograms and real data. The performance of the presented SR-FABEMD-HS method is compared with other normalization techniques. Its superiority, potential and robustness to high fringe density variations and the presence of noise, modulation and background illumination defects in analyzed fringe patterns has been corroborated.
© 2012 OSA
Optical interferometry enables accurate quantitative studies in diverse physical and engineering fields. In majority of applications desired information is encoded in the fringe pattern phase distribution, i.e., shape and position of fringes. Quantitative and fast analysis of fringe patterns is performed using computer-aided automatic analysis methods, see for example [1–3]. Under out-of-laboratory data acquisition conditions it is difficult to make use of the temporal phase shifting (TPS) method providing the highest accuracy. Stringent requirements on the recording of component interferograms complicate the process of data acquisition and analysis. Single fringe pattern acquisition, processing and analysis approaches are much more immune to environmental disturbances and require much simpler experimental setups. Additionally they enable investigating transient events. Their accuracy depends mainly on the algorithmic solutions applied.
In some experiments, mainly in experimental mechanics, material testing, 3D shape determination and production engineering applications, a single fringe pattern contains noise, nonuniform background and intensity modulation variations. The departures from uniform sinusoidal intensity distributions, additionally, cause significant errors in the fringe pattern phase demodulation. The usually utilized solution is to normalize the fringe pattern prior to applying the phase extraction algorithm, e.g., the regularized phase tracking (RPT) in its original [4,5] and modified versions [6,7]. Several normalization solutions have been proposed over the last decade. They include, to mention some of them, interferogram clipping [4,5], 2-D envelope transform coupled with spin filtering , two orthogonal bandpass filtering , the quadrature transform isotropic operator (Hilbert transform) algorithm , adaptive filtering as a linear combination of isotropic bandpass filtering , continuous wavelet transform (CWT) processing coupled with synthesized interferogram clipping [12,13], the directional derivative approach , and the bidimensional empirical mode decomposition aided by the partial Hilbert transformation .
In this studies we present an algorithm utilizing 2D extension of the empirical mode decomposition (EMD) method and the spiral phase Hilbert transform as image processing tools. The one-dimensional version of EMD was firstly developed in . It is adaptive and data-driven approach. Unlike in the Fourier or wavelet methods, no predefined decomposition basis is used; it is rather derived from the signal itself. EMD is designed to deal with nonlinear and nonstationary data. The bidimensional EMD (BEMD) is a fully two-dimensional method, which interpolates envelopes of the corresponding data extrema with functions as bidimensional cubic splines [17,18] or radial based functions . Some comparison of different interpolation techniques is given in . Reported applications of BEMD in the fringe pattern processing and analysis deal with noise reduction in digital speckle interferometry , fringe pattern normalization , phase measurement in temporal speckle interferometry  and evaluation of amplitude encoded fringe patterns . The practical impact of BEMD is limited by the calculation time – spline interpolation on the irregular grid is the most expensive part of the algorithm.
To overcome this limitation FABEMD (Fast Adaptive BEMD) approach was recently proposed [24,25] in which the envelope determination of conventional BEMD is modified by replacing the 2D surface interpolation by an order-statistics-based filtering followed by a smoothing operation. Variable window size selection for extrema and smoothing filters enable image different decompositions; it results in the approach adaptivity. Beside significantly shortening the computation time, more accurate estimation of the bidimensional intrinsic mode functions (BIMFs), representing image features at various spatial scales is obtained in many cases. In this paper we use the modified FABEMD method proposed recently  to provide a continuous decrease of the number of extrema at subsequent decompositions stages, without artificially increasing the filter mask window width.
To conduct the amplitude demodulation of extracted BIMFs (required for subsequent pattern normalization) we apply in our studies the analytic signal – a well-established 1D signal processing tool [see Eq. (1)]. Real part of the complex analytic signal (sA) is the signal under test with removed bias term (s). The imaginary part is created using Hilbert transform (sH):
In our approach we use the 2D Hilbert transform extension by means of the spiral phase Hilbert transform [see Eq. (2) and Eq. (3)] proposed in [27,28]. We denote the spiral phase method with HS (Hilbert spiral) . The spiral phase function in Fourier domain is defined as:Eq. (4)], as
Once we have determined the amplitude distribution of the fringe pattern we divide the FABEMD filtered (noise and bias components removed) interferogram under study by the calculated amplitude distribution map. This step completes normalization process. As has been shown in  the spiral phase transform method outperforms the partial Hilbert transform method (PHT, used in ) in terms of the fringe pattern amplitude demodulation quality. In particular, HS behaves better when dealing with high frequency dense fringes and is insensitive to fringe orientation angle.
The fringe pattern enhancement technique proposed in this paper conducts the fringe pattern bias term removal, noise suppression and amplitude demodulation using a novel selective reconstruction (SR) approach. Only regions with high amplitude function values |A(x,y)| of each BIMF are assigned to the reconstruction process. High modulation (contrast) distribution values describe regions with sharply extracted denoised fringes. By neglecting spurious noisy parts of BIMFs and focusing only on regions carrying important information we increase the normalization quality. In the numerical and experimental sections of the paper we will demonstrate that the proposed SR-FABEMD-HS method (selective reconstruction - FABEMD - spiral phase Hilbert transform; FABEMD-HS and SR are novel contributions of the paper) provides normalization of better quality than other reported techniques. Proposed algorithm is especially designed for dealing with noisy interferograms containing high fringe density variation together with slow (low spatial frequency) amplitude and background distribution intensity changes. Naturally when analyzing less complex fringe patterns our approach also gives excellent results.
2. Proposed algorithm
In this section we present the modified FABEMD algorithm used for the interferogram BIMFs extraction. Moreover we outline a novel selective reconstruction method alongside with discussion of its properties.
2.1 The FABEMD method
The undertaken task of denoising and normalization of complex low-quality fringe patterns is partially carried out using the Fast Adaptive BEMD method. In FABEMD the envelope surfaces fitting by means of the corresponding data extrema interpolation (regular BEMD) is replaced by an order-statistic-based filtering followed by a smoothing operation. The filter size selection plays dominant role in the whole decomposition process. Order-statistic filter width (OSFW) is determined by its type and the Euclidean (the lowest and the highest) distances between adjacent minima/maxima points (for details, see [24,25]). OSFW type 1 and type 2 define filter mask window width as one of the lowest distances between adjacent extrema. Using small filter width (the same for order-statistic and smoothing operations) at each decomposition stage results in high number of extracted BIMFs. Every BIMF has a relatively narrow Fourier spectrum of low amplitude. OSFW type 3 and type 4 select the filter mask window width as one of the highest distances between adjacent extrema. Large filters shorten decomposition process into a few extracted components and a residue. Each BIMF has relatively wide Fourier spectrum of high amplitude. The set of BIMFs (and smoothed modulation distributions) for the selective reconstruction algorithm is calculated using the FABEMD OSFW type 1 technique because of its high spatial resolution. We choose to have a large set of BIMFs to increase the spatial frequency range of sharply extracted fringes.
To ensure the continuous decrease of the spatial frequency of components stored in consecutive BIMFs two important conditions need to be satisfied. Correct decomposition can be characterized by: (1) a decreasing number of extrema points, and (2) increasing the order-statistic and smoothing filter mask window width obtained at subsequent decomposition steps. Analyzing fringe patterns (matrices of 512x512 pixels) with FABEMD OSFW type 1 and type 2 methods one faces a serious limitation. Using constant extrema detector size for whole decomposition process the order statistic filter size does not increase naturally over the set of BIMFs. One way to cope with this problem is to artificially augment filter width multiplicating it by a predefined factor (e.g. 1,5 - approach proposed in ). When dealing with fringe patterns the need to enlarge the filter width occurs very often for all the BIMFs and residue beside BIMF1 (especially when using 3x3 and 5x5 extrema detectors – detectors of that type are mainly used to obtain good spatial resolution of BIMFs). This is due to dense extrema maps detected at each decomposition stage by the constant small size sliding window. The adaptivity of the FABEMD method is therefore greatly reduced – one uses the same set of filter widths for every image (the method is not fully data-driven for 3x3 extrema detector). In many cases the whole set of filters depends on the size of extrema detector (it determines the size of the filters used for BIMF1 extraction) and the value of predefined factor.
To increase adaptivity of the FABEMD method utilized for interferogram decomposition we proposed in  a simple modification. The adjustment step for the extrema detector window width is added at the beginning of calculating each BIMF. We find a proper sliding window size, which provides detection of extrema maps meeting conditions (1) and (2). Having proper minima and maxima maps we continue decomposition as described in  and . In empirical decompositon methods the larger the number of BIMF is, the lower spatial frequency component of the original image it contains. It seems natural to augment the extrema detector size with the increase of the BIMF number. Proposed modification extends the computation time but it is specially tailored to yield adaptivity of FABEMD. The adaptivity is the unique property of empirical mode decomposition based algorithms.
2.2 Selective reconstruction approach
Once we have the fringe pattern decomposed into a set of BIMFs we perform a band-pass filtering by neglecting first few BIMFs (high frequency noise part) and several last ones including residue (low frequency noise and bias term). No extra denoising nor detrending in preprocessing is needed. Different BIMFs contain good quality fringes of different spatial frequencies – the larger the BIMF number the lower the spatial frequency of sharply extracted fringes. When we deal with interferogram with fringes of constant period (frequency) we choose only one BIMF to perform normalization – the greater the fringe spacing the bigger BIMF number will be chosen. We can also reconstruct the interferogram by simply adding several BIMFs and perform amplitude demodulation in order to normalize the fringe pattern (dividing reconstructed interferogram by its modulation distribution). As we will show in the numerical part of the paper this straightforward approach gives good results when analyzing intensity distributions containing fringe density variations, modulation and background illumination defects but not suffering from the presence of noise.
In case of complicated noisy fringe patterns with high fringe density variations we propose to calculate the modulation distribution of each selected BIMF with HS, smooth it using the FABEMD method (choosing manually, basing on our experience and visual evaluation, one of the last BIMFs as an appropriately smoothed modulation distribution) and take to the reconstruction process only the regions of each BIMF with high modulation values. Regions with high modulation (contrast) values are those with sharply extracted fringes. We achieve this isolation by setting a threshold for modulation distribution which helps to distinguish “valuable fringes”. First we set threshold value to 0 – it is a mean value of each intrinsic mode function and for that reason it is a proper start for BIMF thresholding. Then we slightly change the threshold value around zero and control the Q value of reconstructed and normalized optical fringe pattern. When the Q value control is not possible, like in the case of experimental data analysis, we use visual judgment to slightly change threshold value around 0 in order to neglect the noisy parts of each reconstructing BIMF. This so-called selective reconstruction approach prevents low-contrast noisy parts of BIMFs from corrupting the reconstructed interferograms. The quality of the whole normalization process is increased. Robustness to noise and high fringe density variation is the unique feature of the SR-FABEMD-HS method. However the operator needs to ensure proper threshold selection (and BIMF selection, but this is rather unambiguous). Too low threshold level will not result in any significant gain – the noisy regions will still corrupt the outcome. On the other hand, too high threshold may start dismiss the good regions resulting in the normalization quality drop.
Computation time of the proposed fringe pattern enhancement method strongly depends on the image under study. However, the processing time of a single 512x512 image does not exceed 1 minute with Matlab implementation, on a medium class PC, which is satisfactory in comparison with BEMD-PHT performance reported in . Results obtained using the presented enhancement algorithm depend on how carefully one selects the information carrying BIMFs and sets the threshold level for smoothed modulation maps to obtain good quality BIMFs region isolation.
3. Numerical studies
Proposed SR-FABEMD-HS algorithm is compared with previously developed methods, i.e., two orthogonal bandpass filtering (TOBF) , the directional derivative approach DD  (parameters of both methods are given in Table 1 ) and bidimensional empirical mode decomposition with partial Hilbert transformation (BEMD-PHT) . To minimize computational time and boundary errors connected with the BEMD-PHT method we introduce Delauney triangulation  of extended extrema maps and piecewise cubic interpolation on triangles for envelope generation. Numerical evaluations were conducted using Matlab environment.
3.1 Constant period circular fringe pattern analysis
We have simulated four interferograms as 512x512 matrices in grayscale. Simulated interferogram No. 1 (SI1) contains constant period circular fringes with modulation and background variations. In case of SI2 we additionally spoiled it with white Gaussian noise (variance 0.03). Both simple fringe patterns and their cosine term are presented in Fig. 1 . Computer generation of fringe patterns enables one to carefully design the cosine term and the background and modulation distortions. Numerical simulations also allow one to quantitatively measure the performance of the normalization technique by comparing the normalized fringe pattern with the design defect-free cosine term of simulated interferogram. As a comparison method we use well-established quality index Q calculation . It is defined as:Eq. (5)] using 8x8 sliding window and averaged to obtain a single value from range [-1,1], with 1 corresponding to the perfect match.
To normalize SI1 we used the sum of BIMF3 to BIMF6. There is no need to utilize selective reconstruction approach when analyzing constant period fringes. Very good quality of the normalized fringe pattern can be observed in Fig. 2 . Beside qualitative evaluation we measured the similarity between simulated cosine term and normalized interferogram using quality index Q, very high value Q = 0.9348 was obtained. Similar result was obtained for SI2 (Fig. 2) – it expresses good denoising abilities of the FABEMD method. To normalize SI2 we take into account the sum of BIMF5 and BIMF6. In Table 1 we present quantitative assessment of various normalization methods applied to different images. As we can see the SR-FABEMD-HS algorithm (frankly yet without selective reconstruction part) shows great outcome. In case of constant spacing circular fringes the proposed approach demonstrates robustness to noise, modulation and background defects. The DD method  is also immune to noise but exhibits lower normalization quality (problems with coefficient calculations in the center region of closed fringes and slight misalignment introduced between the fringes).
Figure 3 shows decent performance of the BEMD-PHT normalization method. Two problems can be noted, however, i.e., errors at the boundaries and poor denoising. Low quality normalization at interferogram boundaries can be attributed to very low modulation distribution values, interpolation defects and partial Hilbert transform disadvantages. Denoising is performed by subtracting first two noisy BIMFs. Inferior denoising abilities of BEMD-PHT as compared with the results of SR-FABEMD-HS, see Fig. 2, indicate that one is able to extract smoother intrinsic mode functions utilizing FABEMD than classical BEMD (see also ).
3.2 Complex interferogram studies
Next step of our numerical studies has been devoted to enhancing the quality of complex noisy interferograms with high fringe density (and orientation) variations together with modulation and background illumination distortions. For the purpose of examining the robustness to noise we generated and normalized two interferograms (Fig. 4 ): one noise-free (SI3) and another spoiled with white Gaussian noise (SI4, variance 0.03).
In case of the complex fringe pattern SI3 the DD method  fails to normalize regions with high spatial frequency difference resulting in Q = 0.7717 (e.g., dense fringes around the center and sparse fringes on the left hand side of the interferogram). For noisy pattern SI4 similar Q value is obtained – it shows the method robustness to noise which is a unique property of the DD technique. Results obtained with the BEMD-PHT method are presented in Fig. 5 . This algorithm exhibits good performance when analyzing noise-free synthetic pattern SI3 (Q = 0.8523). However some problems appear when we spoil our synthetic pattern with noise (SI4). Denoising cannot consist in neglecting BIMF1 because it contains dense fringes near the interferogram center. In case of noisy synthetic pattern SI2 the absence of dense fringes enabled us to erase first two BIMFs to denoise the fringes. To suppress the noise in SI4 we used simple 3x3 sliding window averaging. In result the value of Q has decreased below the value obtained by the DD method.
Finally our numerical studies have been focused on processing complex fringe patterns utilizing the proposed SR-FABEMD-HS method. As previously noted selective reconstruction helps to normalize noisy, complex interferograms with modulation and background illumination variations. Therefore we apply it only to normalize the pattern SI4. For the normalization of SI3 we filtered out the bias term understood as a residual part (two last BIMFs and actual residue) of the decomposed interferogram, Fig. 6(a) . Next we performed amplitude demodulation and intensity normalization using spiral Hilbert transform, Fig. 6(b), obtaining excellent result Q = 0.9228.
In Fig. 7 we present the set of eleven BIMFs and residue extracted from SI4 with the FABEMD OSFW type 1 method. First we focus on evaluating the performance of FABEMD-HS method without resorting to selective reconstruction. For this purpose we conduct interferogram band-pass filtering by neglecting BIMF1 (high frequency noise) and residual part (last three BIMFs and residue - bias term and low frequency noise). We perform amplitude demodulation of the filtered fringe pattern using the spiral Hilbert transform and we normalize it (Fig. 8 ). The value of Q obtained, Q = 0.8228, is higher than for any other method [9,14,15], see the quantitative evaluation of each method using Q value in Table 1. To further increase the normalization quality we propose to use the selective reconstruction approach. We calculate the modulation distribution (again using HS) of BIMFs Nos. 2 to 8 containing regions with denoised and sharply extracted fringes. We smooth each modulation distribution using FABEMD decomposition (taking into account only one or several last BIMFs). By setting the threshold value for the smoothed modulation distribution (always around 0) we isolate regions with high modulation values corresponding to sectors with sharply extracted denoised fringes. Two examples of the described isolation process are outlined in Fig. 9 (for BIMF4 and BIMF7). In Table 2 . we outlined threshold values of smoothed modulation distribution BIMFs utilized for the selective reconstruction process (regions with sharp fringe recognition).
Once we have processed all of the information carrying BIMFs we proceed with adding them, calculating modulation distribution of this selective sum (using HS) and dividing selectively reconstructed fringe pattern by its modulation distribution to obtain normalized interferogram (Fig. 10 ). Using described approach we achieved significantly increased Q value (Q = 0.8631).
3.3 Phase demodulation accuracy evaluation
The final goal of the optical fringe pattern analysis is its phase distribution calculation in order to obtain information about element/phenomena under study. The accuracy of phase decoding determines the quality of the metrological setup. To increase the measurement accuracy we perform fringe pattern enhancing. In this section we simulate fringe patterns, corrupt them with noise and low frequency modulation and background intensity changes, and normalize them using analyzed methods. Both simulated images are π/2 shifted so we can use the analytic signal tool for the phase distribution determination as an arctangent of imaginary part (first fringe pattern) divided by real part (second fringe pattern). Computer generated ideal phase shift does not contribute to the phase recovery error budget. The imperfections are entirely connected with utilized normalization method which is the main reason of the proposed analytic-signal-based phase recovery evaluation scheme.
We simulate the interferogram with vertical fringes with spatial frequency increasing toward the right hand side of the image (SI5). We corrupt it with white Gaussian noise (variation 0.15) and slow background and modulation variations (low spatial frequency components). It can be considered somehow as a simulation of the cantilever bending. This pattern is enhanced using analyzed methods. For the BEMD-PHT approach we use BIMFs Nos. 2 to 9 concerning BIMF1 as a noise carrying component. For the FABEMD-HS we consider BIMFs Nos. 3 to 8. Taking advantage of the proposed selective reconstruction algorithm we use set of thresholds values and smoothed modulation distribution BIMFs outlined in Table 3 . The same set of parameters was employed to enhance both π/2 phase-shifted fringe patterns. Normalized interferograms alongside with the simulated fringe pattern are shown in Fig. 11 . Then to calculate phase distribution we use the analytic signal concept. We enhance the π/2 phase-shifted imaginary part fringe pattern (SI6) using analyzed methods (results are similar to those already shown in Fig. 11, therefore we do not present them) and decode phase distribution as an arctangent of Im/Re. The phase maps are presented in Fig. 12 . After simulations we get three sets of images enabling evaluation of normalization and phase demodulation quality exhibited by analyzed methods. Computing Q values (with ideal cosine terms and true phase as a reference) we can quantitatively confirm the superiority of the SR-FABEMD-HS algorithm over other methods (see Table 4 ). The difference between SR and no SR can be readily assessed (it is due to considerable fringe noise reduction provided by SR).
For clarity of the method discussion the above presented simulation work concerned the example of a monotonically changing phase distribution, Fig. 11(b). In case of complex patterns with closed fringes the phase ambiguity (two possible values of phase, θ and θ + π) is encountered. Several solutions are available to resolve the phase ambiguity issue such as, for example, combining windowed Fourier ridges with regularized phase tracking [4–6, 30], frequency-guided algorithm [31,32] and the 2D continuous wavelet transform method . For the present paper conciseness the complex pattern decoding issue is not included, please see cited references.
4. Experimental data processing
To corroborate the potential of the proposed SR-FABEMD-HS method for enhancing complex, low quality experimental data we have processed real DSPI fringes obtained in the studies of in-plane displacements of a stiff epoxy polymer cantilever beam, see Fig. 13 . Figure 14 presents results obtained using the DD method  and the BEMD-PHT method . Both techniques exhibit low quality normalization performance because of considerable fringe spatial frequency variations over the whole correlogram and noise susceptibility.
Proposed SR-FABEMD-HS method is robust to high fringe density variations and presence of noise (also speckles, in simulation studies we used white Gaussian noise) which makes it a good tool for processing fringes shown in Fig. 13. Selectively reconstructed and denoised DSPI fringe pattern using BIMFs Nos. 5 to 9 is presented in Fig. 15(a) . Normalized correlogram of relatively good quality is shown in Fig. 15(b). We can visually assess the advantages of proposed selective reconstruction approach by observing results obtained using the FABEMD-HS method presented in Fig. 15(c) and 15(d). The FABEMD method speckle pattern denoising results resemble, to some extent, those exhibited by the bidimensional ensemble empirical mode decomposition (BEEMD) method proposed in . This topic stays out of the scope of this paper, however, and will be presented in a separate paper.
One of potential applications of the presented normalization method could be proposed for archiving cultural heritage objects using the structured light projection technique. Exemplary results are presented in Fig. 16 . Image of the statue obtained using structured illumination for 3D shape determination is normalized using first two BIMFs (we take into account BIMF1 because of lack of noise in the analyzed pattern and the presence of high spatial frequency fringes). Cross sections outlined in Fig. 17 show the analyzed intensity distributions compared with the sinusoidal one. The upper plinth fragment of the statue (with quasi-constant fringe period) is studied. Prior to normalization the root-mean square error between the actual cross section of the upper plinth fragment of Fig. 16(a) and ideal sinusoid (simulated in Matlab, period of 12 pixels) was equal to 70,6. Proposed normalization technique increased the sinusoidal resemblance decreasing the RMS value to 46,9. Detailed quantitative studies of the SR-FABEMD-HS method for the 3D shape measurement will be given in a separate paper.
Please note that the discussion of the method presented in the paper is confined to random noise (speckling, photon noise, detector electronics noise). In case of structural illumination applications for shape recovery just shown in Fig. 16 a variety of global illumination (interreflections, subsurface scattering, etc.) and illumination defocus effects are present  and references therein. Denoising captured images in the presence of these effects is challenging but left for separate studies of the structured illumination method to keep this paper compact enough.
We presented an adaptive method for fringe pattern enhancement – its denoising and normalization. Proposed algorithm uses a modified fast and adaptive bidimensional empirical mode decomposition (FABEMD) for the extraction of bidimensional intrinsic mode functions (BIMFs) from an interferogram. In comparison with regular BEMD it needs less time to calculate more accurate BIMFs – the improvement is made both in terms of speed and accuracy of the decomposition. Fringe pattern band-pass filtering is performed by neglecting several first BIMFs (high frequency noise part) and few last ones including the residue (low-frequency noise part and the bias term). No extra denoising nor detrending in preprocessing is needed. Fringe pattern is then selectively reconstructed (SR) taking the regions of selected BIMFs with high modulation values only. This novel approach prevents low-contrast noisy parts of BIMFs from corrupting the reconstructed interferograms. Amplitude demodulation for intensity normalization of the reconstructed fringe pattern is conducted using the Hilbert spiral phase transform (HS). Proposed method has been tested using computer generated interferograms and real data; satisfactory results have been obtained in both cases. The performance of the presented SR-FABEMD-HS technique has been compared with other normalization schemes; its superiority has been shown. The method robustness to high fringe density variations and the presence of noise, modulation and background illumination defects in processed low quality fringe patterns has been corroborated. The results of the proposed algorithm strongly depend, however, on the operator choices. The presented low quality fringe pattern enhancement technique could gain from the automatic BIMFs selection and modulation map thresholding. This issue will be the topic of our further studies.
This work was supported by Ministry of Science and Higher Education budget funds for science in the years 2012-2015 as a research project in the “Diamond Grant” programme and partly by statutory funds. The authors would like to thank OGX group http://ogx.mchtr.pw.edu.pl for the access to experimental image of the statue presented in Fig. 16. The authors also thank Zofia Sunderland for providing Matlab codes. The authors thank editors and anonymous reviewers for their help and comments.
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