A Novel and Optimized Sine–Cosine Transform Wavelet Threshold Denoising Method Based on the sym4 Basis Function and Adaptive Threshold Related to Noise Intensity

Abstract

In digital shearography, the speckle noise of the phase fringe pattern has a negative effect on the accuracy and reliability of the phase unwrapping procedure. A novel and optimized sine–cosine transform wavelet threshold denoising method is proposed to suppress speckle noise. Fast phase denoising can be achieved by using the proposed method while preserving the phase reversal information. The details of the selected wavelet basis function, the optimal decomposition level, the threshold function, and the denoising threshold are also provided in this manuscript. In particular, the decomposition level is analyzed and optimized through simulation analysis according to the speckle suppression index and the adaptive denoising method. The experimental results show that the proposed method has more adaptive ability in practical application than the sine–cosine transform average denoising method with the selected mask and iterative procedure, which speeds the denoising process up and takes better-unwrapped phase patterns.

1. Introduction

Digital shearography has the following advantages: full-field, non-contact, high-precision, non-destructive, and real-time measurement [1,2,3,4,5,6]. Therefore, it is widely used in aerospace, civil engineering, transportation, machinery, materials, and other non-destructive testing fields [7,8,9,10]. In digital shearography, a beam of coherent light is used to illuminate a rough object. The light beam scattered on the surface of an irregular object is divided into two transversely misplaced beams through the shear element and they interfere with each other in the overlapping area of the detection plane [11,12,13]. The wrapped phase pattern with a large amount of noise can be obtained by processing the interferograms recorded before and after the deformation of the object, respectively. Several phase denoising and unwrapping methods are used to obtain the phase information derived from the deformation with high sensitivity. Thus, defect detection and condition monitoring can be realized with the correlation model between the phase information and the physical quantity to be measured.

In order to improve the accuracy of phase unwrapping, speckle noise needs to be filtered [14]. Several methods have been proposed to filter the speckle noise of the wrapped phase pattern, which mainly focuses on the spatial and transform domains. Spatial domain denoising uses various smoothing methods to process the value of each pixel to achieve the purpose of filtering. In terms of the spatial domains method, Aebischer, Hubert A et al. [15]. proposed an average denoising method based on sine–cosine transform, transforming the discontinuous phase of the wrapped phase pattern into a continuous phase. Qiyang Xiao et al. [16] proposed a sine–cosine transform average denoising method based on wavelet packet decomposition to calculate noise energy and determine the cycle threshold, limiting algorithm cycles. Transform domain denoising mainly transforms the noisy wrapped pattern through a particular transformation. It processes it with a specific method in the transform domain and inverse transform to obtain a denoised pattern. Yonghong Wang et al. [17] proposed a method combining sine–cosine transform with ideal low-pass filtering. Ying Xing et al. [18] proposed an improved adaptive total variation (TV) denoising method based on the block-matching and 3D filtering (BM3D) method for ESPI fringes. Jian Zhang et al. [19] proposed that traditional wavelet filtering for shearography fringe patterns of composite materials can reduce noise and preserve details. Xueling Ning et al. [20] proposed a sine–cosine transform and stationary wavelet transform method based on the db4 wavelet basis function and high penalty threshold, and it can improve the filtering efficiency. The phase reversal information may be destroyed in several of the above methods and affect the accuracy of defect detection. Several methods are too complex and inefficient for real-time and fast denoising. To the best of our knowledge, the sine–cosine transform average denoising method has been widely used for shearography interferometric patterns and speckle interferometric patterns. Still, the mask and iterations need to be selected according to the direction and fringe density in practical application.

This paper proposes a novel and optimized sine–cosine transform wavelet threshold denoising method. In this method, the sym4 wavelet basis function is used according to the characteristics of the laser speckle. Meanwhile, the optimal wavelet decomposition level is ascertained according to the denoising effect. Combined with the speckle characteristics and noise intensity, a soft threshold global denoising method is also proposed to process the wavelet coefficients adaptively and fast. Thus, the complication of the threshold calculation process is decreased, therefore improving the calculation efficiency. This paper provides the filtering differences caused by different decomposition levels, and the optimal decomposition level is obtained through simulation. The proposed method is applied to process several shearography–interferometric phase fringe patterns and speckle–interferometric phase fringe patterns. Compared to the sine–cosine transform average denoising method, the better denoising effect is achieved adaptively and efficiently in the proposed method.

2. The Principle of the Sine–Cosine Transform Wavelet Threshold Denoising Method

The noise in shearography wrapping phase patterns is mainly additive Gaussian white noise. Suppose a noiseless pattern is f. After adding Gaussian white noise z with the noise intensity $\epsilon$, the noiseless pattern degenerates the noisy pattern $\tilde{f}$. The relationship between the noiseless pattern f and the noisy degraded pattern $\tilde{f}$ can be expressed as [19]

$$\tilde{f}=f+\epsilon z$$

(1)

In the denoising process, the wavelet basis function, decomposition levels, threshold, and threshold function directly affect the processing effect. The flowchart of the proposed algorithm is shown in Figure 1.

(1)

Sine and cosine transform of the wrapped phase pattern.

The wrapped phase pattern is transformed by sine and cosine transform, respectively.

$$\left\{\begin{array}{c}\hfill Spw=sin\phi \hfill \\ \hfill Cpw=cos\phi \hfill \end{array}\right.$$

(2)

where $\phi$ is the wrapping phase pattern and represents the pattern after the sine–cosine transform, respectively.

(2)

The sine and cosine patterns are decomposed into N levels by the sym4 wavelet basis function.

The 2-D discrete wavelet transform (DWT) algorithm [21,22] for patterns is shown in Figure 2. LOD means wavelet decomposition low-pass filter and HiD means wavelet decomposition high-pass filter. The 2-D DWT algorithm decomposes the low-frequency approximation coefficients $c{\tilde{f}}_j$ into four parts at scale $j+1$: (1) Low-frequency coefficients $c{\tilde{f}}_{j+1}$; (2) horizontal detail coefficients $c{D}^{\left(h\right)}{}_{j+1}$; (3) vertical detail coefficients $c{D}^{\left(v\right)}{}_{j+1}$; (4) diagonal detail coefficients $c{D}^{\left(d\right)}{}_{j+1}$.

The wavelet decomposition transform can be expressed as

$$c\tilde{f}=cf+c\epsilon z$$

(3)

where c represents the wavelet decomposition transform, $c\tilde{f}$ is the wavelet coefficients of the noisy degraded pattern $\tilde{f}$, $cf$ is the wavelet coefficients of the noiseless pattern f, and $c\epsilon z$ is the wavelet coefficients of the noise z.

The selection of the wavelet basis function should consider its support length, symmetry, vanishing moment, regularity, and similarity. Symlets wavelet is usually expressed as symM where M is $2,3,\dots ,8$. Compared with other wavelets, the sym4 wavelet where M = 4 in symM has better symmetry to reduce phase distortion in signal analysis and reconstruction. Therefore, the sym4 wavelet is selected to decompose and reconstruct the wrapping phase pattern. Perform N-level decomposition on the sine and cosine patterns in step (1):

$$\left\{\begin{array}{c}\hfill cSpw=c{f}_1+c{\epsilon }_1{z}_1\hfill \\ \hfill cCpw=c{f}_2+c{\epsilon }_2{z}_2\hfill \end{array}\right.$$

(4)

where $f_1,f_2$ are the ideal sine and cosine patterns, respectively, ${\epsilon }_1,{\epsilon }_2$ are the noise intensity of sine and cosine noisy patterns, respectively. And $z_1,z_2$ are random Gaussian noise in sine and cosine noisy patterns, respectively.

(3)

Threshold denoising for wavelet coefficients.

After the wavelet basis function decomposes the noisy pattern, it is necessary to select the appropriate threshold and denoise the coefficients of each level with the threshold function [23]. Therefore, it is essential to determine the threshold and threshold processing function.

The selection of a threshold directly affects the effect of denoising. Currently, the main threshold selection methods are fixed threshold estimation, extreme threshold estimation, unbiased likelihood estimation, and heuristic estimation. But the above thresholds have nothing to do with the noise intensity. According to the global threshold form proposed by Donoho et al. [20], the fixed threshold form and noise intensity can filter out Gaussian white noise. Combined with the characteristics of speckle Gaussian white noise, the threshold form of the combination of fixed threshold form and noise level is selected for denoising.

$$\left\{\begin{array}{c}\hfill T_{hr}=\sigma \sqrt{2ln\mathrm{N}}\hfill \\ \hfill \sigma =\frac{median\left(\mid k_N\right)}{0.6745}\hfill \end{array}\right.$$

(5)

where $T_{hr}$ is the denoising threshold, N is the decomposition level, $\sigma$ is the noise intensity, and $k_N$ is the original wavelet coefficients in level N.

The wavelet coefficient should be processed for denoising according to the threshold function. The threshold function mainly includes the hard threshold function and soft threshold function. A hard threshold denoising function can easily cause local jitter and remove useful signal information. There is no sudden change in the wavelet coefficients when using the soft threshold denoising function. Therefore, the soft threshold processing function is adopted for the wrapped phase patterns.

When the absolute value of the wavelet coefficient is more than the given threshold, the wavelet coefficient subtracts the threshold for the soft threshold processing function. The wavelet coefficient is set to zero when it is less than the threshold.

$${\eta }_H\left(k,T_{hr}\right)=\left\{\begin{array}{c}k-T_{hr}, k\ge T_{hr}\hfill \\ 0, \left|k\right|<T_{hr}\hfill \\ k+T_{hr}, k\le -T_{hr}\hfill \end{array}\right.$$

(6)

where k is the original wavelet coefficients, and ${\eta }_H\left(k,T_{hr}\right)$ is wavelet coefficients processed by the threshold.

The soft threshold global denoising of wavelet coefficients can be expressed as

$$\left\{\begin{array}{c}\hfill Sp{w}^{\prime }={\eta }_H\left(cSpw,T_{hr}\right)\hfill \\ \hfill Cp{w}^{\prime }={\eta }_H\left(cCpw,T_{hr}\right)\hfill \end{array}\right.$$

(7)

where $Sp{w}^{\prime },Cp{w}^{\prime }$ are the denoised wavelet coefficients of the sine and cosine patterns, respectively.

(4)

Wavelet reconstruction of sine and cosine patterns after threshold processing.

The 2-D inverse discrete wavelet transform (IDWT) algorithm for patterns is shown in Figure 3. The 2-D IDWT algorithm reconstructs the approximation coefficients $c{\tilde{f}}_{j+1}$, $c{D}^{\left(h\right)}{}_{j+1}$, $c{D}^{\left(v\right)}{}_{j+1}$ and $c{D}^{\left(d\right)}{}_{j+1}$ into $c{\tilde{f}}_j$ at scale j.

The wavelet reconstruction transform can be expressed as

$$\left\{\begin{array}{c}\hfill Sp{w}^{*}=Re\left(Sp{w}^{\prime }\right)\hfill \\ \hfill Cp{w}^{*}=Re\left(Cp{w}^{\prime }\right)\hfill \end{array}\right.$$

(8)

where “Re” means wavelet reconstruction, and $Sp{w}^{*}$, $Cp{w}^{*}$ are the reconstructed sine and cosine patterns, respectively.

(5)

Calculate the inverse tangent of sine and cosine after denoising.

$$pw=arctan\left(\frac{Sp{w}^{*}}{Cp{w}^{*}}\right)$$

(9)

where $pw$ is the denoised shearography–interferometric phase fringe patterns.

(6)

Calculating speckle suppression index to determine the optimal decomposition level. The speckle suppression index (SSI) reflects the ability of noise suppression [24]. And the smaller the speckle suppression index is, the better the denoising effect is.

$$I_{\mathrm{S}\mathrm{S}\mathrm{I}}\left(\mathrm{N}\right)=\frac{\sqrt{{\sigma }_I}}{{\mu }_I}/\frac{\sqrt{{\sigma }_F}}{{\mu }_F}$$

(10)

where ${\sigma }_I,{\mu }_I$ are the variance and mean of a denoised pattern, and ${\sigma }_F,{\mu }_F$ are the variance and mean of a noisy pattern.

Calculate the speckle suppression index when the decomposition level is $3,\cdots ,N$, $\left(N\subseteq N^{*}\right)$. The optimal decomposition level can be regarded as

$${\mathrm{N}}_0=argmin\left\{{\mathrm{I}}_{\mathrm{SSI}}\left(\mathrm{N}\right)\right\}$$

(11)

where $N_0$ is the optimal decomposition level, and $argmin\left\{{\mathrm{I}}_{\mathrm{SSI}}\left(\mathrm{N}\right)\right\}$ represents the decomposition level among $3,\cdots ,N$ when the SSI is minimum.

3. Simulation and Comparative Analysis

Shearography–interferometric phase fringe patterns of 600 × 600 pixels with different noise and fringe densities are provided. Assume that the out-of-plane displacement $W(x,y)$ satisfies the following:

$$W(x,y)=w·exp\left(-\frac{x^2+y^2}{2}\right)\dots w\subseteq [10,40]\&w\subseteq N^{*}$$

(12)

where $x,y\in [-3,3]$ are the normalized distance units.

Set the shearing direction along x. Then, the relationship between the unwrapping phase $\phi (x,y)$ and the out-of-plane displacement $W(x,y)$ satisfies the following:

$$\phi (x,y)=\frac{4\pi }{\lambda }\frac{\delta W(x,y)}{\delta x}∆x$$

(13)

where $\lambda$ is the wavelength of light in the system, $∆x$ is the amount of shear in the x direction, and $\delta W(x,y)/\delta x$ is $W(x,y)$ the difference with respect to x. Calculate the wrapping phase of ${\phi }^{\prime }(x,y)$.

$${\phi }^{\prime }(x,y)=mod[\phi (x,y),2\pi ]$$

(14)

We then generate four phase-shifted fringe patterns according to the following formula [25]:

$$I_i(x,y)=1+cos\left[{\phi }^{\prime }(x,y)+{\delta }_i\right]+\mathrm{noise}(x,y)$$

(15)

where i = 1, 2, 3, 4 is the phase-shifting step, ${\delta }_i=(i-1)\pi /2$ is the phase-shift amounts, and $noise(x,y)$ is the Gaussian white noise for each pattern. Then, we can use the standard four-step phase shift method to restore the phase. The noise of the power intensity I = 0.2, 0.5, 0.7, 1 dBw, in which dBw represents the absolute noise intensity, is added to the noiseless shearography–interferometric phase fringe patterns obtained by simulation with w = 10, 20, 30. Then, the sine–cosine transform wavelet threshold denoising method with decomposition levels of 3, 4, 5, and 6 is used to denoise the pattern. According to the four-step phase shift method, the wrapped patterns of Gaussian white noise with different noise intensities and fringe densities are processed with different wavelet decomposition levels, as shown in Figure 4, Figure 5 and Figure 6.

The speckle suppression indices of the above fringe patterns are calculated and compared, as shown in Figure 7.

The three-group shearography–interferometric phase fringe patterns under different noises are processed by the sine–cosine transform wavelet threshold denoising method with varying decomposition levels. As shown in Figure 7a–c, the distribution of SSI with low fringe density is concentrated, while it is just the opposite with high fringe density. The SSI analysis results show that the decomposition level has little effect on the filtering effect of low-density fringes. On the contrary, the decomposition level will significantly affect the filtering effect for high-density fringes, so it is particularly important to determine the decomposition level. The SSI with five-level decomposition is always the smallest regardless of fringe density and noise intensity. The denoising effect of five-level decomposition in the sine–cosine transform wavelet threshold denoising method is the best for shearography–interferometric phase fringe patterns.

The four kinds of noisy wrapped patterns with different fringe densities and noise intensity are processed by the sine–cosine transform wavelet threshold denoising method. The axial phases of the noiseless and denoised speckle pattern are extracted, as shown in Figure 8. As we can see, the noise is well removed, and the changing trend is consistent. The denoised phase is preserved well, with the details maintained.

4. Experimental Analysis and Discussion

4.1. Construction of the Experimental System

As shown in Figure 9, we have built a common optical path to obtain shearography–interferometric phase fringe patterns and speckle–interferometric phase fringe patterns based on digital speckle pattern interferometry (DSPI) and digital shearography (DS). The object to be measured is an aluminum plate with a diameter of 250 mm and eight screws around it. There is a spiral structure in the center to load the out-of-plane displacement of the aluminum plate. The laser in the system is a single-mode fiber laser with 532 nm wavelength. The focal length of the imaging lens is 35 mm. The 4f lens comprises two double-glued lenses with a focal length of 75 mm for image transmission. M1, M2, and M3 are the reflecting mirrors with a diameter of 25 mm. Where M1 is the shearing mirror, and M2 is the phase-shifting mirror. BS1, BS2, and BS3 are splitter prisms with 50 mm sides.

The laser is split into two beams by BS1. One of the beams is expanded to irradiate the surface of the aluminum plate. An imaging lens with a focal length of 35 mm is used to collect scattered light derived from the surface of an aluminum plate as the object light. When optical switch one is on and optical switch two is off, the DS optical path with a 4f system is formed by M1 and M2. When optical switch two is on and optical switch one is off, the DSPI optical path is formed by M1 and M3.

4.2. Analysis of Experimental Results

For comparison, the sine–cosine transform average denoising method is also used to process these results and analyze the effect of the mask and iterations. The phase fringe patterns with different densities are processed by the sine–cosine transform average denoising method with mask sizes of 3 × 3 and 27 × 27 for five and fifteen iterations.

The sparse fringes processing results using the sine–cosine transform average denoising method with different masks and iterations are shown in Figure 10. The denoising effect of sparse fringes becomes better by using the 3 × 3 mask with increasing iterations, as shown in Figure 10b,c. A satisfactory denoising effect can also be achieved by using the 27 × 27 mask with five iterations, as shown in Figure 10d. But more iterations will over-smooth the phase reversal information and lead to loss of details, as shown in Figure 10e. The execution time increases as the number of iterations increases regardless of the mask size, as shown in Figure 11a.

The dense fringes processing results using the sine–cosine transform average denoising method with different masks and iterations are shown in Figure 12. A large amount of speckle noise still exists in the wrapped patterns with the 3 × 3 mask and fifteen iterations, as shown in Figure 12b,c. The denoising result with the 27 × 27 mask and five iterations is better than that with the 3 × 3 mask and fifteen iterations, as shown in Figure 12d. However, the large-sized mask may deteriorate the quality of the wrapped patterns with increasing iterations, as shown in Figure 12e. The relationship between the calculation time and the number of iterations is shown in Figure 11b. Increasing the number of iterations with this small mask will lead to too much time and lower calculation efficiency.

Therefore, different masks and iterations must be considered for different shearography fringe phase patterns. Small-size masks are suitable for the high-density fringe pattern, and large-size masks are ideal for the low-density fringe pattern. The sine–cosine transform average denoising method is applied to process the fringe pattern with the consideration of the denoising efficiency caused by the iterations. The denoising effect does not operate well with too few iterations. If too many iterations exist, it is easy to cause over-filtering, adhesion, and low efficiency. In practical application, we need to select the size and shape of the mask and the number of iterations for a satisfactory denoising effect. The decomposition levels and the denoising threshold of the sine–cosine transform wavelet threshold denoising method in this paper are determined adaptively rather than through artificial arbitrariness.

As shown in Figure 13, shearography–interferometric phase fringe patterns and speckle–interferometric phase fringe patterns are processed by the sine–cosine transform average denoising method and the sine–cosine transform wavelet threshold denoising method, respectively. The appropriate mask and iterations are selected to achieve satisfactory filtering results for the sine–cosine transform average denoising method.

Each sine and cosine patterns for different noisy speckle fringes have different denoising thresholds. The sine–cosine transform wavelet threshold denoising method achieves denoising adaptively according to the different denoising thresholds.

To show more details of the sine–cosine transform (SCT) wavelet threshold denoising method, we calculate the denoising thresholds of each sine and cosine patterns after the sine–cosine transform, as shown in

Table 1. The denoising threshold of each sine and cosine patterns.

The Pattern	The Denoising Threshold
(a)—DS-sine pattern	3.544
(a)—DS-cosine pattern	3.885
(d)—DS-sine pattern	3.730
(d)—DS-cosine pattern	4.051
(g)—DS-sine pattern	2.952
(g)—DS-cosine pattern	3.120
(j)—DS-sine pattern	3.161
(j)—DS-cosine pattern	3.109

.

The denoising effect can be evaluated with the indices of code execution time, peak signal-to-noise ratio, and structure similarity [26]. Peak signal-to-noise ratio (PSNR) reflects the detail retention ability of a pattern, and the higher the PSNR, the better the detail retention of the pattern. Structural similarity (SSIM) is a full-reference pattern quality evaluation index that measures pattern similarity from brightness, contrast, and structure.

When the execution time is similar, the SSI of the sine–cosine transform wavelet threshold method is lower than that of the sine–cosine transform average method, as shown in

Table 2. Comparison of SSI after the sine–cosine transform wavelet threshold denoising method and sine–cosine transform average denoising method when execution time is similar.

The Pattern	Execution Time/s	SSI	PSNR	SSIM
(a)—DS-SCT-wavelet	∼1.850	0.064	41.370	0.884
(a)—DS-SCT-average	∼1.850	0.074	40.380	0.886
(d)—DS-SCT-wavelet	∼1.900	0.009	40.571	0.852
(d)—DS-SCT-average	∼1.900	0.024	40.618	0.853
(g)—DSPI-SCT-wavelet	∼1.950	0.155	42.031	0.944
(j)—DSPI-SCT-average	∼1.950	0.205	42.068	0.945
(j)—DSPI-SCT-wavelet	∼2.100	0.210	41.339	0.948
(j)—DSPI-SCT-average	∼2.100	0.233	42.437	0.953

. Meanwhile, the PSNR and SSIM for the two methods are almost identical. The experimental results show that the sine–cosine transform wavelet threshold denoising method with five-level decomposition is better than the sine–cosine transform average denoising method.

5. Conclusions

This paper focuses on the denoising method in digital shearography. A sine–cosine transform wavelet threshold denoising method based on the sym4 basis function and threshold related to noise intensity is proposed. According to the comparative analysis of the speckle suppression index, the decomposition level with the best denoising effect is determined in the proposed method. The experimental results show that the denoising effect of the proposed method is better than that of the sine–cosine transform average denoising method with the similar execution time. Furthermore, the proposed method can adaptively set the threshold to process the speckle–interferometric patterns without artificially setting the mask and iterations compared to the sine–cosine transform average denoising method. This method can realize fast denoising and reserve phase reversal information well. It is beneficial to unwrap the phase information accurately. Simulation analysis and experimental results verify the effectiveness of the proposed method. This method can play an important role in practical application scenarios such as defect detection, stress/strain monitoring, life prediction of the specimen, etc. In the next steps of the research, we will perfect the shearography–interferometric phase fringe patterns simulation algorithm to provide support for the optimization of the denoising algorithm, deeply explore the influence of different wavelet basis functions on the denoising effect, and use parallel computing to further improve the denoising efficiency of the algorithm.