Ultrasonic Phased Array Compressive Imaging in Time and Frequency ...

sensors Article

Ultrasonic Phased Array Compressive Imaging in Time and Frequency Domain: Simulation, Experimental Verification and Real Application Zhiliang Bai, Shili Chen *, Lecheng Jia and Zhoumo Zeng State Key Laboratory of Precision Measurement Technology and Instrument, Tianjin University, Tianjin 300072, China; [email protected] (Z.B.); [email protected] (L.J.); [email protected] (Z.Z.) * Correspondence: [email protected]; Tel.: +86-22-2740-2366 Received: 2 April 2018; Accepted: 1 May 2018; Published: 8 May 2018







Abstract: Embracing the fact that one can recover certain signals and images from far fewer measurements than traditional methods use, compressive sensing (CS) provides solutions to huge amounts of data collection in phased array-based material characterization. This article describes how a CS framework can be utilized to effectively compress ultrasonic phased array images in time and frequency domains. By projecting the image onto its Discrete Cosine transform domain, a novel scheme was implemented to verify the potentiality of CS for data reduction, as well as to explore its reconstruction accuracy. The results from CIVA simulations indicate that both time and frequency domain CS can accurately reconstruct array images using samples less than the minimum requirements of the Nyquist theorem. For experimental verification of three types of artificial flaws, although a considerable data reduction can be achieved with defects clearly preserved, it is currently impossible to break Nyquist limitation in the time domain. Fortunately, qualified recovery in the frequency domain makes it happen, meaning a real breakthrough for phased array image reconstruction. As a case study, the proposed CS procedure is applied to the inspection of an engine cylinder cavity containing different pit defects and the results show that orthogonal matching pursuit (OMP)-based CS guarantees the performance for real application. Keywords: ultrasonic phased array; compressive sensing; image reconstruction; time and frequency domain; engine cylinder cavity

1. Introduction Ultrasonic phased array is one of the most widely used imaging modalities in current industrial non-destructive evaluation (NDE) due to its increased flexibility, faster detection speed, higher inspection quality and radiation-free operation [1]. With programmable time delays, a single linear ultrasonic array can be used to undertake various inspections and produce real-time 2D images, which brings efficiency in the structural health monitoring (SHM) of some industrial components with complex geometries [2,3]. Performed digitally, it is required that the analog signals first be sampled in the inspection, which is confined to traditional Nyquist-Shannon sampling limitation. Although the Nyquist rate is defined as twice the highest frequency component in the signal, in practice, oversampling (generally 4 to 10 times that of the transducer central frequency) is implemented in order to improve resolution, reduce noise and avoid aliasing. Furthermore, as image techniques developing and the requirement of inspection accuracy improving, the number of elements involved in phased array typically rises. Consequently, the A-scan lines in an image increase, leading to a huge amount of data collection from the system front-end, eventually exerting pressures on data acquisition sensors. Therefore, it is imperative to develop compression methods to effectively reduce the sampling rate. Sensors 2018, 18, 1460; doi:10.3390/s18051460

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In recent years, Compressive Sensing (CS) has become a challenging field that has driven a lot of research interests in SHM applications, such as, the sparse recovery optimization in wireless sensor network [4], the mode separation in Lamb wave-based long-range damage detection [5], the reliable estimation of vehicular position in general traffic scenarios [6] and the recovery of the lost data in civil SHM [7]. Exploiting a priori knowledge that many natural measurement signals admit a sparse representation on proper basis or a redundant dictionary, CS provides one approach to achieve qualified signal reconstruction with fewer measurements compared with Nyquist sampling criteria [8]. Contrary to the traditional compressions, the CS encoder is based on a very simple and extremely low-power hardware, while most of the complexity and energy requirements are transferred to the decoding stage [9]. Up till now, CS has found increasing interests in the ultrasonic inspection community, which generally falls into two categories: medical ultrasound of diagnostic sonography [10–14] and guided wave-based NDE applications [15–20]. For the former, by treating ultrasound signals within the FRI framework, Wagner et al. generalized the concept of compressed beamforming following the spirit of Xampling [10]. Lorintiu et al. presented a CS-based reconstruction of 3D ultrasound data using dictionary learning and line-wise subsampling, which confirmed a better performance than conventional fixed transforms [11]. Foroozan et al. employed wave atom dictionary as a low dimension projection and the robust Capon beamformer was combined with CS, instead of using the delay-and-sum method [13]. In the case of ultrasonic NDE, López et al. analyzed the application of CS techniques in order to achieve faster scanning in the ultrasonic imaging of cargo containers [15]. Di et al. used a random sampling scheme based on CS to minimize the number of points at which the field is measured [16]. Mesnil et al. proposed a reconstruction technique to estimate the location of sources and structural features interacting with the waves from a set of sparse measurements [17]. Wang et al. applied a dictionary algorithm on sparse representation for Lamb-wave-based damage detection [18]. In all cases, some only apply CS in ultrasound single signals [11,12,17,18] while others extend it to ultrasound images. On the other hand, there are several reports concentrating on the use of CS in frequency domain recently. Chernyakova et al. [21] demonstrated on in vivo cardiac data that reductions up to 1/28 of the standard rates are possible, using only a portion of the beamformed signal’s bandwidth. Their study was also extended to the 3D beamforming in frequency domain and satisfactory results can be obtained [22]. Mishra et al. [23] demonstrated a significant superiority of frequency domain CS reconstruction but no further explanation was offered. Compared with a large number of literatures on medical ultrasound or guided wave-based NDE fields, the CS applications on ultrasonic phased array are still inadequate and deserve more attentions. In this work, we applied the CS framework to a general phased array image reconstruction in both time and frequency domain, aiming at verifying the feasibility and potentiality of CS in this field. Employing the peak signal to noise ratio (PSNR, in dB) and the structural similarity (SSIM) as the evaluation criteria, images of different sampling rate (SR) are recovered under various compression ratios (CR). The results comparisons between time and frequency domain are presented and the causes behind the performances are explained, especially the superiority in frequency domain. In order to investigate the influence of SR level on the recovery performance, we compare the PSNR of different SRs when keeping the measurement points the same. Besides, considering the presence of close defects in real-world components, the recoveries when the distance of defects is different are also included, accompanied by results analysis. Here, the Orthogonal Matching Pursuit (OMP) was utilized to reconstruct all images. Simulations based on CIVA platform, experiments on three kinds of artificial flaws (through-holes, electrical discharge machining (EDM) notches and flat-bottom holes) and real application of engine cylinder cavity inspection are exploited to demonstrate the performance of the proposed framework. Figure 1 illustrates the simplified procedure of the proposed CS-based scheme for ultrasonic phased array image reconstruction in time and frequency domain.

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Figure 1. The simplified architecture of the proposed compressive sensing (CS)-based scheme.

Figure 1. The simplified architecture of the proposed compressive sensing (CS)-based scheme.

The paper is structured as the following: Section 2 presents an introduction of the CS theory,

The papersparsity is structured as thecoding, following: Section 2with presents an introduction of the CSvia theory, including of transform measurement incoherence and reconstruction optimization. Section 3 shows the simulated study based on images obtained from Section 4 via including sparsity of transform coding, measurement with incoherence and CIVA. reconstruction contains the details3 shows of experiment setup and of the implementation process. Finally, Section 5 optimization. Section the simulated study based on images obtained from CIVA. Section 4 summarizes the main findings of this work and ends the paper. contains the details of experiment setup and of the implementation process. Finally, Section 5 summarizes the main findings of this work and ends the paper. 2. Overview of CS

CS, as novel sampling paradigm that goes against the common wisdom in data acquisition, 2. Overview of aCS successfully combines sampling and compression together. To make this possible, two fundamental

CS, as a novel sampling paradigm that goes against the common wisdom in data acquisition, prerequisites play a critical role: sparsity, which relates to the signal essence; and incoherence, which successfully combines sampling and compression together. To make this possible, two fundamental associates with sensing modality. prerequisites play a critical role: sparsity, which relates to the signal essence; and incoherence, which 2.1. Sparsity of Transform Coding associates with sensing modality. Although many natural signals are not sparse in their own domain, they admit a sparse representation when expressed in a proper basis. Mathematically speaking, given a basis or dictionary Ψ we represent vector as a linear combination of a few atoms: ∈  N ×N ,natural ∈  Nsparse Although many signals are fnot in their own domain, they admit a sparse

2.1. Sparsity of Transform Coding

representation when expressed in a proper basis. Mathematically speaking, given a basis or dictionary N f = Ψ x = x ψ N × N N (1)  i i Ψ∈R , we represent vector f ∈ R as a linear combination of a few atoms: i=1

xi are coefficients and xi =  f ,ψ i  =ψ iT f . If Nat most K components of xi are nonzero, we f = Ψx = ∑ xi ψi (1) possibility that these coefficients are close to 0, say that f is K-sparse. In practice, there is a strong i =1

where

rather than equal to 0, we regard it as compressible. At present, the commonly used sparse T f. If at most basesand include transform (DFT), Wavelet redundant wheretransformation xi are coefficients xi = hf,Discrete ψi i = ψiFourier K components of xi arebasis, nonzero, we say that dictionaries and adaptive decomposition. f is K-sparse. In[24] practice, there issparse a strong possibility that these coefficients are close to 0, rather than theregard following simulations and experiments, the most typical transformationbases equal to 0,Inwe it as compressible. At present,DFT, the commonly usedorthogonal sparse transformation is employed. It can be expressed as include Discrete Fourier transform (DFT), Wavelet basis, redundant dictionaries [24] and adaptive N −1 sparse decomposition. f =  x(n) e− j 2π kn/ N , 0 ≤ k ≤ N −1 (2) n =0 In the following simulations and experiments, DFT, the most typical orthogonal transformation is employed. It can be expressed as

2.2. Measurement with Incoherence

N −1

j2πkn/N f = this x ( n )e− , 0a≤measurement k ≤ N − 1 system that acquire M linear With consideration of noise, begins with ∑section n =0 measurements data:

2.2. Measurement with Incoherence

y = Φf + n

(2)

(3)

With consideration of noise, this section begins with a measurement system that acquire M linear measurements data: y = Φf + n (3)

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where n is an unknown error term and Φ ∈ R M× N can play a role of dimensionality reduction, for M is typically much smaller than N. For a sparse or compressible signal f, we can rewrite Equation (3) as y = Ax + n

(4)

where A = ΦΨ is denoted as the sensing matrix. To guarantee the robustness and validity of the K-sparse signal recovery, a sufficient condition is that matrix A satisfies the restricted isometry property (RIP) of order k if there exists a δk ∈ (0, 1) such that [8]

(1 − δk )kxk22 ≤ kAxk22 ≤ (1 + δk )kxk22

(5)

holds for all K-sparse vectors x. It has been proved that an equivalent condition of RIP is the incoherence between Φ and Ψ [25]. The coherence is defined as √

 µ(Φ, Ψ) = N · max ϕk , ψj (6) 1≤k,j≤ N

where ϕ and ψ represent any two elements of Φ and Ψ. The less correlated elements they have, the smaller coherence will be get. It is shown that random matrices with Gaussian or Bernoulli distributions can satisfy the RIP with high probability [26]. 2.3. Reconstruction via Optimization If the original signal f is rationally sparse and the RIP holds, it is possible to accurately recover x by convex programming: xˆ = argmink x k1 subject to kAx − yk2 ≤ ε

(7)

where kxk1 = ∑| xi | is the l1 norm, k · k2 represents the standard Euclidean norm and ε2 is a likely upper bound on the noise power knk22 . For the sake of computing efficiency, one of the most frequently used greedy algorithms, that is, OMP, will be adopted in our work. The mathematical description of OMP is formally defined in Algorithm 1 [27]. Algorithm 1 Orthogonal Matching Pursuit (OMP) Input: A signal y ∈ R N , a matrix A ∈ R M× N . Initialize: Set the support set Ω0 = ∅, the residual error r0 = y and put the counter k = 1. Identify: Find a column an from A that most correlates with the residual error and record the correlation coefficient: nk ∈ arg max |hrk−1 , an i|, Ωk = Ωk−1 ∪ {nk } n=1,2,...N

Estimate: compute the best approximating coefficients: xk = argminx ky − AΩk xk2 Iterate: Update the residual and counter: rk = y − AΩ k xk , k = k + 1 Until: Stopping criterion holds. Output: The vector xˆ with components xˆ (n) = xk (n)(n ∈ Ωk ) and xˆ (n) = 0 otherwise.

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3. Simulation Results and Discussion 3. Simulation Results and Discussion To justify the feasibility of CS framework and to see how it helps for data compression in To justify the feasibility of CS framework to see how it helps for data compression in ultrasonic ultrasonic phased array imaging, a simulationand procedure based on CIVA software (Version 9.0, Paris, phased imaging, a simulation France) array is provided in this section. procedure based on CIVA software (Version 9.0, Paris, France) is provided in this section. 3.1. Simulation Settings 3.1. Simulation Settings Developed by CEA (the French Atomic Energy Commission) since the early 90’s, the CIVA Developed by CEA (the French Atomic Energy modeling Commission) since 90’s,platform, the CIVA making gathers gathers most influential parameters and advanced tools intothe an early expertise most influential and modeling tools into an expertise platform, making it more it more and moreparameters widely used inadvanced the industrial NDE fields (Ultrasound, Eddy Current, Radiography, and more widely used in the industrial NDE fields (Ultrasound, Eddy Current, Radiography, ...) [28,29]. ...) [28,29]. As Figure 2a shows, we simulated an array aperture comprising a 64-element linear phased As Figure 2a shows, we simulated an array aperture comprising a 64-element linear phased array array transducer transducer of of 55 MHz MHz center center frequency. frequency. A A 20 20 mm mm thickness thickness flat flat wedge wedge was was used used to to eliminate eliminate the the potential near-field influence. More detailed parameters are shown in Table 1. The aluminum specimen potential near-field influence. More detailed parameters are shown in Table 1. The aluminum with threewith 1 mmthree diameter holes was inspected planeusing B-scan modality. horizontal specimen 1 mmthrough diameter through holes wasusing inspected plane B-scanThe modality. The and verticaland space werespace set aswere 10 mm, 6 mm, 4 mm and 42 mm mm,and respectively. In the following horizontal vertical set 8asmm, 10 mm, 8 mm, 6 mm, 2 mm, respectively. In the implementations, 16 element16chips are chips excited time, resulting 49 sequences contained in each following implementations, element areeach excited each time, resulting 49 sequences contained image. is the2b original image ofimage 10 mm-space throughthrough holes. holes. in each Figure image.2b Figure is the original of 10 mm-space

(b)

(a)

Figure 2. (a) The simulation setup using CIVA software; (b) the original image of 10 mm-space through Figure 2. (a) The simulation setup using CIVA software; (b) the original image of 10 mm-space holes. through holes. Table 1. Array parameters for simulation study. Table 1. Array parameters for simulation study.

Array Parameter Array Frequency Parameter Center Center Frequency Element Count Element Element Count Pitch Element Pitch Element Width Element Width Element Element Elevation Elevation Pulse Type Type Pulse −6dB dBBandwidth Bandwidth −6

Value Value 5 MHz 5 MHz 64 64 0.60 mm 0.60 mm 0.50 mm 0.50 mm 10.0 mm 10.0 mm Gaussian weighted Gaussian weighted 50% 50%

3.2. Time Domain Reconstructions Reconstructions In the employed thethe discrete Fourier basisbasis as sparse basis and theand Gaussian random thesimulation, simulation,wewe employed discrete Fourier as sparse basis the Gaussian matrix sensingas matrix, exploiting properties of calculation and implementation randomas matrix sensing matrix,their exploiting their properties efficiency of calculation efficiency and simplicity. Due to the restriction ofthe hardware implementation, in our work, various levels were implementation simplicity. Due to restriction of hardware implementation, in ourCR work, various

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obtained by removing some parts of the original samples, as many reports generally used [12,30]. Here, the CR is determined by the ratio: M (8) CR = (1 − ) × 100% N where M is the measurement points in the given domain and N represents the signal length of each sequence. Quantitative evaluation of the proposed CS method was performed with respect to the recovered two-dimensional image. To quantify the reconstruction performance, we mainly used the PSNR, given by: 2552 PSNR = 10lg( ) (9) MSE Given a m × n original image I and its reconstructed version R, the mean squared error (MSE) measures the average of the squares of the deviations between I and R: MSE =

1 m −1 n −1 ∑ [ I (i, j) − R(i, j)]2 mn i∑ =0 j =0

(10)

Although a higher PSNR generally indicates a better performance of CS algorithms, PSNR is just a kind of approximation to human perception of recovery quality. In our work, comparison between the original and reconstructed images is also performed by calculating the perception-based structural similarity (SSIM) index [31]: SSIM( P, Q) = [l ( P, Q)]α · [c( P, Q)] β · [s( P, Q)]γ

(11)

where l ( P, Q), c( P, Q) and s( P, Q) represent luminance comparison function, contrast comparison function and structure comparison function between images P and Q, respectively. α > 0, β > 0 and γ > 0 are parameters used to adjust the relative importance of the three components. The resultant SSIM index is a decimal value between 0 and 1, and value 1 is only reachable in the case of two identical sets of data. In any given detection area, different sampling rate (SR) means different signal length of the time traces. Therefore, the image compression potentiality was explored by varying the SR levels and 20 MHz, 25 MHz, 30 MHz, 40 MHz SR were considered in the simulation. The 40 MHz SR results in an overall number of 1168 real-valued samples (N = 1168), the data lengths of other SRs are proportional. To eliminate the negligible difference caused by the randomness of Gaussian matrix, each PSNR or SSIM data point averaged out the results of 100 runs. Figure 3 shows the comparative results as a function of CR. It is clear that the reconstruction quality generally decreases as CR rises. In the case of 65% CR or above, the reduction of SR leads to lower PSNR as well as SSIM. When SR 25 MHz, the impact of different SRs on the recovery performance is relatively small, with the PSNR > 43.99 dB and the SSIM > 0.8931 at 60% CR. Reconstructed images obtained by the proposed CS method, using 50%, 40%, 30% samples per image line in time domain, are shown in Figure 4. Here, the SR is 25 MHz. As can be seen from comparison with the original image, shown in Figure 2b, the recovery results when CR is 50% or 60% are in good agreement with our expectation: three defects are clearly preserved and almost no reconstruction errors exist. However, satisfactory performance cannot be realized if CR is 70%, although the defect echoes are roughly recognizable.

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(b) (b)

(a) (a)

Figure 3. 3. Comparative Comparative (a) (a) Peak Peak signal signal to to noise noise ratio ratio (PSNR), (PSNR), (b) (b) Structural Structural similarity similarity (SSIM) (SSIM) between between Figure Figure 3. Comparative (a) Peak signal to noise ratio (PSNR), (b) Structural similarity (SSIM) between different sampling rates (SRs) in the time domain, using simulated data. different sampling rates (SRs) in the time domain, using simulated data. different sampling rates (SRs) in the time domain, using simulated data.

(a) (a)

(b) (b)

(c) (c)

Figure 4. The recovery results of 25 MHz SR simulated image in time domain, (a) Compression ratio Figure 4. recovery results of MHz SR simulated image in domain, (a) ratio Figure 4. The ThePSNR recovery results of 25 25 SRPSNR simulated image in time time (a) Compression Compression (CR) = 50%, = 45.05 dB; (b) CRMHz = 60%, = 43.99 dB; (c) CR =domain, 70%, PSNR = 36.71 dB. ratio (CR) (CR) ==50%, 50%,PSNR PSNR==45.05 45.05dB; dB;(b) (b)CR CR==60%, 60%,PSNR PSNR==43.99 43.99dB; dB;(c) (c)CR CR==70%, 70%,PSNR PSNR= =36.71 36.71dB. dB.

Notably, in the case of 25 MHz SR and 60% CR, the measurement points used for gratifying Notably, in the case of 25 MHz SR and 60% CR, the measurement points used for gratifying ). As 5 MHzreconstruction ascase many of 10 SR original ( 25 ×（1 −points 60%） =10 Notably, inare the ofas 25that MHz SRMHz and 60% CR, thesample measurement used forthe gratifying ). As the 5 MHzreconstruction are as many as that of 10 MHz SR original sample ( 25 ×（1 − 60%） =10 center frequency areasemployed dB bandwidth is 50%, the minimum sampling rate reconstruction arepulses as many that of 10 and MHzits SR−6original sample (25 × (1 − 60% the 5 MHz) = 10). As center frequency pulses are employed and its −6 dB bandwidth is 50%, the minimum sampling rate is 15 MHz according to the Nyquist theory. The fact indicates that simulated images can be perfectly center frequency pulses are employed and its −6 dB bandwidth is 50%, the minimum sampling rate is 15 MHz according to the Nyquist theory. The fact indicates that simulated images can be perfectly recovered time domain fewer samples Nyquist sampling limitation. Likewise, for is 15 MHz in according to thefrom Nyquist theory. Thethan fact the indicates that simulated images can be perfectly recovered in time domain from fewer samples than the Nyquist sampling limitation. Likewise, for 20 MHz SR CR, the is thethan same, PSNR being 43.96 dB and SSIM being 0.9296. recovered inand time50% domain fromconclusion fewer samples thewith Nyquist sampling limitation. Likewise, for 20 MHz 20 MHz SR and 50% CR, the conclusion is the same, with PSNR being 43.96 dB and SSIM being 0.9296. As50% can CR, be readily seen, either lower or higher CR causes reconstructed SR and the conclusion is the the same, withSR PSNR being 43.96 dB andthe SSIM being 0.9296.images to As can be readily seen, either the lower SR or higher CR causes the reconstructed images to distort, because both seen, caseseither meanthe the lessSR measurement (M)reconstructed used for recovery. we As can be readily lower or higher CRpoints causes the images Next, to distort, distort, because both cases mean the less measurement points (M) used for recovery. Next, we concentrate on less finding out whether level impact on we the concentrate reconstruction because bothour casesinterest mean the measurement pointsthe (M)SR used for has recovery. Next, our concentrate our interest on finding out whether the SR level has impact on the reconstruction performance. In Table 2 we compare the PSNR of different SRs when keeping the measurement interest on finding out whether the SR level has impact on the reconstruction performance. In Table 2 performance. In Table 2 we compare the PSNR of different SRs when keeping the measurement points the same. Contrary to earlierSRs results, the measurement points up to 260, we compare the PSNR of different whenfor keeping the measurement points the PSNR same. decreases Contrary points the same. Contrary to earlier results, for the measurement points up to 260, PSNR decreases with the increase of SR. This observation illustrates that the reconstruction accuracy of OMP is of likely to earlier results, for the measurement points up to 260, PSNR decreases with the increase SR. with the increase of SR. This observation illustrates that the reconstruction accuracy of OMP is likely affected by the ratio of M/N,that especially when M is comparatively small. This can be explained by the This observation illustrates the reconstruction accuracy of OMP is likely affected by the ratio of affected by the ratio of M/N, especially when M is comparatively small. This can be explained by the fact that for low when M/N, M theis structure of sensing becomes ill-balanced, has adverse M/N, especially comparatively small.matrix This can be explained by the which fact that foran low M/N, fact that for low M/N, the structure of sensing matrix becomes ill-balanced, which has an adverse impact on theof recovery. increased 280, we can which see thathas there is no obvious PSNR distinctions the structure sensing After matrix becomestoill-balanced, an adverse impact on the recovery. impact on the recovery. After increased to 280, we can see that there is no obvious PSNR distinctions between different SRs. we can see that there is no obvious PSNR distinctions between different SRs. After increased to 280, between different SRs. In ultrasonic phased array NDE, there are always defects that are very close to each other. Therefore, it is indispensable for CS to distinguish these adjacent flaws in the reconstructed image. Using 25 MHz SR and 60% CR, Figure 5 presents the recovery results when the distance of defects (D) is different. As can be seen, the reconstructions are all quite qualified, which shows that the proposed

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Table 2. Comparative PSNR of different SRs when keeping the same sampling points, using Table 2. Comparative PSNR of different SRs when keeping the same sampling points, using simulated data. simulated data.

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8 of 21 Measurement Points 200 220 240 260 280 300 320 340 Measurement Points 200 220 240 260 280 300 320 340 20 MHz SR (584 points) 37.50 38.45 40.59 42.57 43.92 43.98 44.37 44.57 20 MHz SR (584 points) 37.50 38.45 40.59 42.57 43.92 43.98 44.37 44.57 25 MHz (730 points) 34.99 37.56 accuracy 37.65 41.06 44.19 44.23 44.44 smaller OMP-based CS SR guarantees the reconstruction in terms43.83 of close defects. Actually, 25 MHz SR (730 points) 34.99 37.56 37.65 41.06 43.83 44.19 44.23 44.44 MHz SR (876about points) 34.31 defects 30 distance brings better PSNR.35.60 37.14 39.81 42.22 44.19 44.23 44.39 30 MHz SR (876 points) 34.31 35.60 37.14 39.81 42.22 44.19 44.23 44.39 40 MHz SR (1168 points) 32.35 33.99 35.70 39.38 43.81 44.13 44.28 44.53 40 MHz SR (1168 points) 32.35 33.99 35.70 39.38 43.81 44.13 44.28 44.53

Table 2. Comparative PSNR of different SRs when keeping the same sampling points, using simulated data.

In ultrasonic phased array NDE, there are always defects that are very close to each other. In ultrasonic phased array NDE, there defects that very close other. Points 220are always 240 260 280 are 320 to each 340image. Therefore,Measurement it is indispensable for CS200 to distinguish these adjacent flaws in 300 the reconstructed Therefore, it is indispensable for CS to distinguish these adjacent flaws in the reconstructed image. Using 2520 MHz CR, Figure the recovery of defects MHzSR SRand (584 60% points) 37.50 5 presents 38.45 40.59 42.57 results 43.92 when 43.98the distance 44.37 44.57 Using 2525MHz SR and points) 60% CR, Figure 5 presents the recovery results when the 44.23 distance44.44 of defects MHz SR (730 34.99 37.56 37.65 41.06 43.83 44.19 (D) is different. As can be seen, the reconstructions are all quite qualified, which shows that the (D) is different. As(876 canpoints) be seen, the reconstructions all quite42.22 qualified, which shows that the MHz SR 34.31 35.60 37.14are 39.81 proposed30OMP-based CS guarantees the reconstruction accuracy in terms44.19 of close44.23 defects.44.39 Actually, 40 OMP-based MHz SR (1168CS points) 32.35 33.99 35.70 accuracy 39.38 43.81 44.53 proposed guarantees the reconstruction in terms44.13 of close44.28 defects. Actually, smaller defects distance brings about better PSNR. smaller defects distance brings about better PSNR.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure SRSR (CR = 60%) images in time domain when thethe distance of Figure5.5.The Therecovery recoveryresults resultsofof2525MHz MHz (CR = 60%) images in time domain when distance Figure 5. The recovery results of 25 MHz SR (CR = 60%) images in time domain when the distance of defects is different (a) D = 2 mm, PSNR = 45.62 dB; (b) D = 4 mm, PSNR = 45.33 dB; (c) D = 6 mm, PSNR of defects is different (a) D = 2 mm, PSNR = 45.62 dB; (b) D = 4 mm, PSNR = 45.33 dB; (c) D = 6 mm, defects is different (a) D = 2 mm, PSNR = 45.62 dB; (b) D = 4 mm, PSNR = 45.33 dB; (c) D = 6 mm, PSNR =PSNR 45.17 = dB; (d) D = 8(d) mm, 44.25 dB. 45.17 dB; D =PSNR 8 mm,=PSNR = 44.25 dB. = 45.17 dB; (d) D = 8 mm, PSNR = 44.25 dB.

3.3. Frequency Domain Reconstructions 3.3. Frequency Frequency Domain Domain Reconstructions Reconstructions 3.3. As a next step, we implemented reconstructions in frequency domain on simulated array As aa next implemented reconstructions in frequency domain on simulated array images. As nextstep, step,wewe implemented reconstructions in frequency domain on simulated array images. Here, the Discrete Cosine transform (DCT), rather than the DFT, is employed in our work. Here, the Discrete Cosine transform (DCT), rather than the DFT, is employed in our work. Thework. DCT images. Here, the Discrete Cosine transform (DCT), rather than the DFT, is employed in our The DCT only retains the cosine components of DFT, so the operation is more efficient and simplified. onlyDCT retains the cosine components of DFT, so the operation is more is efficient and simplified. In this The only retains the cosine components of DFT, so the operation more efficient and simplified. In this work, we use DCT coefficients to conduct CS recovery and the image in time is obtained by work, we use DCT coefficients to conduct CS recovery and the image in time is obtained by performing In this work, we use DCT coefficients to conduct CS recovery and the image in time is obtained by performing an inverse DCT transform. Figure 6 provides a sketch of the procedure used to an inverse DCT transform. 6 provides of the usedprocedure to reconstruct performing an inverse DCTFigure transform. Figurea 6sketch provides a procedure sketch of the used the to reconstruct the undersampled phased array images in frequency domain. The sparse basis and undersampled phased array images in frequency domain. The sparse basis and sensing matrix reconstruct the undersampled phased array images in frequency domain. The sparse basis and sensing matrix exploited in the reconstructions are same as the previous section. exploited in theexploited reconstructions are same as the are previous section. sensing matrix in the reconstructions same as the previous section.

Figure 6. The sketch of the procedure used to reconstruct phased array images in frequency domain. Figure 6. The sketch of the procedure used to reconstruct phased array images in frequency domain. Figure 6. The sketch of the procedure used to reconstruct phased array images in frequency domain.

In Figure 7, the PSNR values are drawn as a function of CR and for different SRs. As an initial In Figure 7, the PSNR values are drawn as a function of CR and for different SRs. As an initial observation, we7,can that the recovery performance in of frequency domain is highly better than In Figure thenote PSNR values are drawn as a function CR and for different SRs. As an initial observation, we can note that the recovery performance in frequency domain is highly better than that in time domain. Except for 20 MHz SR, the PSNR values at 80% CR are all higher than 47 dB, observation, we can note that in frequency domain highlythan better that in time domain. Except forthe 20recovery MHz SR,performance the PSNR values at 80% CR are allishigher 47than dB, which istime better than any resultfor in time domain. In addition, the SSIMs ofCR all the reconstructed images that in domain. Except 20 MHz SR, the PSNR values at 80% are all higher than 47 dB, which is better than any result in time domain. In addition, the SSIMs of all the reconstructed images in presence of frequency domain are very close In to addition, 1, so we would not of present here. Contrary images to the which is better than any result in time domain. the SSIMs all the reconstructed in presence of frequency domain are very close to 1, so we would not present here. Contrary to the in presence of frequency domain are very close to 1, so we would not present here. Contrary to the situation in time domain, lower SR lends to better PSNR when CR 70%, even if the differences are less

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situation in time domain, lower SR lends to better PSNR when CR 70%, even if the differences are situation timethe domain, lower SR lends toinbetter PSNR when CR that 70%, even no if the differences than 1 dB.1inFrom recovery images shown Figure 8, it can say error in are the less than dB. From the recovery images shown in Figure 8, itbecan be sayalmost that almost no occurs error occurs less than 1 dB. From the recovery images shown in Figure 8, it can be say that almost no error occurs reconstruction, using only 20% samples. in the reconstruction, using only 20% samples. in the reconstruction, using only 20% samples.

Figure 7. Comparative PSNR between different SRs in frequency domain, using simulated data. Figure 7. 7. Comparative Comparative PSNR PSNR between between different different SRs SRs in in frequency frequency domain, domain, using using simulated simulated data. Figure data.

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Figure 8. The recovery results of 20 MHz SR simulated image in frequency domain, (a) CR = 70%, Figure 8. The recovery results of 20 SR simulated Figure The dB; recovery 20 MHz MHz SR dB. simulated image image in in frequency frequency domain, domain, (a) (a) CR CR == 70%, 70%, PSNR =8.49.73 (b) CRresults = 80%,of PSNR = 44.83 PSNR = 49.73 dB; (b) CR = 80%, PSNR = 44.83 dB. PSNR = 49.73 dB; (b) CR = 80%, PSNR = 44.83 dB.

To explain the superiority of the frequency domain CS, we investigate the sparsity of the oneTo explain the superiority of the frequency domain CS, we investigate the sparsity of the onedimensional timethe trace as well as its coefficients, throughCS, expressing them in the the discrete To explain superiority of DCT the frequency domain we investigate sparsityFourier of the dimensional time trace as well as its DCT coefficients, through expressing them in the discrete Fourier basis. An arbitrary A-scan, 9a,coefficients, is extractedthrough from 25expressing MHz SR image and 9b one-dimensional time trace shown as wellin asFigure its DCT them in theFigure discrete basis. An arbitrary A-scan, shown in Figure 9a, is extracted from 25 MHz SR image and Figure 9b shows its corresponding Fourier spectrum. CS, the DCT of A-scan Fourier basis. An arbitrary A-scan, shown In in frequency Figure 9a, domain is extracted from 25 coefficient MHz SR image and shows its corresponding Fourier spectrum. In frequency domain CS, the DCT coefficient of A-scan signal plays the role of input signal Fourier and is shown in Figure 9c,d is the corresponding DFT coefficient spectrum. Figure 9b shows its corresponding spectrum. In frequency domain CS, the DCT signal plays the role of input signal and is shown in Figure 9c,d is the corresponding DFT spectrum. WeA-scan manually setplays a threshold out the main is, only values that are bigger of signal the roletoofpick input signal and iselements, shown inthat Figure 9c,d the is the corresponding DFT We manually set a threshold to pick out the main elements, that is, only the values that are bigger than 1% of the maximum can be designated as “Information Value (IV)”. The IV of raw A-scan signal spectrum. We manually set a threshold to pick out the main elements, that is, only the values that are than 1% of the maximum can be designated as “Information Value (IV)”. The IV of raw A-scan signal is 87, and it isbe60. The calculated results indicate that the is bigger thanfor 1%DCT of thecoefficients, maximum can designated as “Information Value (IV)”. TheDCT IV ofcoefficient raw A-scan is 87, and for DCT coefficients, it is 60. The calculated results indicate that the DCT coefficient is sparseristhan thefor original time domainit signal, thatcalculated is, the excellent in frequency domain signal 87, and DCT coefficients, is 60. The results performance indicate that the DCT coefficient is sparser than the original time domain signal, that is, the excellent performance in frequency domain benefits from its higher sparsity. sparser than the original time domain signal, that is, the excellent performance in frequency domain benefits from its higher sparsity. Figure different situations vary in defects interval distance and compares the benefits from10itsconsiders higher sparsity. Figure 10 considers different situations vary in defects interval distance and compares the PSNRs in two domains when keeping 70% vary CR. We note that the results in and frequency domain are at Figure 10 considers different situations in defects interval distance compares the PSNRs PSNRs in two domains when keeping 70% CR. We note that the results in frequency domain are at least 10.8 dB higher than that in time domain, which confirms the capability of frequency domain CS in two domains when keeping 70% CR. We note that the results in frequency domain are at least least 10.8 dB higher than that in time domain, which confirms the capability of frequency domain CS in various defects cases. 10.8 dB higher than that in time domain, which confirms the capability of frequency domain CS in in various defects cases. various defects cases.

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Figure Figure 9. 9. (a) (a) Extracted Extracted A-scan A-scan signal signal from from 25 25 MHz MHz SR SR image; image; (b) (b) the the Discrete Discrete Fourier Fourier Transform Transform (DFT) (DFT) Figure 9. (a) Extracted A-scan signal from 25 MHz SR image; (b) the Discrete Fourier Transform (DFT) spectrum of (a); (c) the Discrete Cosine Transform (DCT) coefficients of (a); (d) the DFT spectrum of spectrum of (a); (c) the Discrete Cosine Transform (DCT) coefficients of (a); (d) the DFT spectrum of spectrum of (a); (c) the Discrete Cosine Transform (DCT) coefficients of (a); (d) the DFT spectrum of (c). (c). (c).

Figure 10. Comparative in time Figure 10. 10. Comparative Comparative PSNRs PSNRs between between different different defects defects interval interval spaces, spaces, using using 25 25 MHz MHz SR SR in in time time Figure PSNRs between different defects interval spaces, using 25 MHz SR domain and 20 MHz SR in frequency domain, and CR is 70% for both domains. domain and 20 MHz SR in frequency domain, and CR is 70% for both domains. domain and 20 MHz SR in frequency domain, and CR is 70% for both domains.

To To conclude, conclude, using using samples samples less less than than the the minimum minimum requirements requirements of of the the Nyquist Nyquist theorem, theorem, both both To conclude, using samples less than the minimum requirements of the Nyquist theorem, both time domain and frequency domain CS verify their abilities to accurately reconstruct time domain and frequency domain CS verify their abilities to accurately reconstruct ultrasonic ultrasonic time domain and frequency domain CS verify their abilities to accurately reconstruct ultrasonic phased phased phased array array simulated simulated images. images. In In addition, addition, CS CS in in frequency frequency domain domain allows allows one one to to obtain obtain aa more more array simulated images. Indue addition, CS in frequency domain allows one to obtain a more satisfactory satisfactory performance to its inherent higher sparsity. satisfactory performance due to its inherent higher sparsity. performance due to its inherent higher sparsity. 4. 4. Experimental Experimental Results Results and and Discussion Discussion 4. Experimental Results and Discussion 4.1. Images and Algorithm Parameters 4.1. Apparatus, Apparatus, Real-Time Real-Time Images Images and and Algorithm Algorithm Parameters Parameters 4.1. Real-Time For experimental standard PC, commercial ultrasonic phased For experimental experimental testing, testing, the the lab lab setup setup includes includes aaa standard standard PC, PC, aaa commercial commercial ultrasonic ultrasonic phased phased For testing, setup includes array detector (Multi 2000, Paris, France), a 5 MHz linear array with 64 elements and specimen with array detector detector (Multi (Multi 2000, 2000, Paris, Paris, France), France), aa 55 MHz MHz linear linear array array with with 64 64 elements elements and and specimen specimen with with array artificial defects. We particularly chose devices that share the same parameters with simulation for artificial defects. defects. We particularly particularly chose chose devices devices that that share share the the same same parameters parameters with with simulation simulation for for artificial ease of performance comparison. The phased array was placed on aa 20 mm-depth Plexiglas wedge ease of performance comparison. The phased array was placed on 20 mm-depth Plexiglas wedge ease of performance comparison. The phased array was placed on a 20 mm-depth Plexiglas wedge to to the probe from wearing excessively. Linear scanning was exploited in inspection and to protect protect probe from wearing excessively. Linear scanning exploited in the the inspection protect thethe probe from wearing excessively. Linear scanning waswas exploited in the inspection andand all all the signals were sampled at 20 MHz, 25 MHz, 33 MHz and 50 MHz, respectively. all the signals were sampled at 20 MHz, 25 MHz, 33 MHz and 50 MHz, respectively. the signals were sampled at 20 MHz, 25 MHz, 33 MHz and 50 MHz, respectively. The test specimen was manufactured from with three kinds of The test testspecimen specimen manufactured from Aluminum Aluminum kinds of defects. defects. The was was manufactured from Aluminum with threewith kinds three of defects. Measurements Measurements of through-holes and electrical discharge machining (EDM) notches were performed Measurements of and through-holes and electrical discharge machining (EDM) were of through-holes electrical discharge machining (EDM) notches were notches performed asperformed shown in as in Figure 11a. Figure 11b shows flat-bottom holes inspected. The as shown shown Figure11b 11a. Figure 11b shows how how flat-bottom holes were were The specimen specimen Figure 11a.inFigure shows how flat-bottom holes were inspected. The inspected. specimen geometry and geometry and details of defects distribution are indicated in Figure 11c. geometry and details of defects indicated details of defects distribution aredistribution indicated inare Figure 11c. in Figure 11c.

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Figure (b) 11. (a) The inspection of through-holes and electrical discharge machining (EDM) notches; (b) Figure 11. (a) The inspection of through-holes and electrical(c) discharge machining (EDM) notches; The inspection of flat-bottom holes; (c) The specimen geometry and defects distribution. (b) The inspection of flat-bottom (c) The and specimen geometry and defects (EDM) distribution. Figure 11. (a) The inspection ofholes; through-holes electrical discharge machining notches; (b)

The inspection of flat-bottom (c) The geometry and12. defects distribution. it is clear that Part of real-time images ofholes; 50 MHz SR specimen are shown in Figure For through-holes,

Part of real-time images of 50 MHz SR are shown in Figure 12. For through-holes, is clear that the signal intensity decreases with the transmission distance. For flat-bottom holes and EDMitnotches, Part of real-time images of 50 MHz SR are shown in Figure 12. For through-holes, it is clear although a strong back wall echo or side reflection echo is quite obvious, the pre-machined flaws are the signal intensity decreases with the transmission distance. For flat-bottom holes and EDMthat notches, the signal intensity And, decreases with the transmission distance. For flat-bottom holes andwith EDMthe notches, all well detected. more remarkably, the measured depths coincide very well actual although a strong back wall echo or side reflection echo is quite obvious, the pre-machined flaws are although a strong back wall echorecorded or side reflection echoarray is quite obvious, the pre-machined flaws are positions. All And, kinds of defects by linear ensure an overall understanding all well detected. more remarkably, thethemeasured depths coincide very well with of thetheactual all well detected. remarkably, theofmeasured detection, locationAnd, and more quantitatively sizing defects. depths coincide very well with the actual positions. All kinds of defects recorded by the linear array ensure an overall understanding of the positions. All kinds of defects recorded by the linear array ensure an overall understanding of the detection, location andand quantitatively detection, location quantitativelysizing sizingof of defects. defects.

Figure 12. The real-time measurements of three kinds of defects.

4.2. Time Domain Reconstructions Figure 12. The real-time measurements of three kinds of defects. Figure 12. The real-time measurements of three kinds of defects. Similar as the simulation, the images sampled from 20 MHz, 25 MHz, 33 MHz and 50 MHz SR 4.2. Time Domain Reconstructions were considered in the experiment. In our work, 16 elements were activated each time, leading to 49 4.2. Time Domain Reconstructions sequences image (64 − 16 +the 1 = images 49). We sampled first take from the through-holes as an 33 example to perform Similarper as the simulation, 20 MHz, 25 MHz, MHz and 50 MHztime SR domain recovery. ease calculation and16 comparison, 512 points containing theleading main part of SR Similar as the simulation, theofimages from 20 MHz, 25 MHz, 33 time, MHz and 50 were considered inFor the the experiment. In oursampled work, elements were activated each toMHz 49 defects zone was intercepted MHz The average results were of 100 runs are illustrated in Figure perin image (64 − 16 +in 1 =5049). We first take theelements through-holes asactivated an example to perform time weresequences considered the experiment. In ourSR. work, 16 each time, leading to 13. Considering the complexity of real experiment, we first note that the overall performances are domain recovery. For the ease of calculation and comparison, 512 points containing the main part of 49 sequences per image (64 − 16 + 1 = 49). We first take the through-holes as an example to perform worse zone than with the simulated images. To exact, whenresults CR is 60%, PSNRs at least reduce 5 dB was intercepted 50 MHz SR.be The average of 100the runs are illustrated in Figure time defects domain recovery. For the in ease of calculation and comparison, 512 points containing the main except for 50 MHz SR. For SSIMs, the negative effects are more exposed. In addition, the 13. Considering the complexity of real experiment, we first note that the overall performances are part of defects zone was intercepted in 50 MHz SR. The average results of 100 runs are illustrated in performances are more liable to affected by the SR level. For 50 MHz SR, the results are relatively worse than with the simulated images. To be exact, when CR is 60%, the PSNRs at least reduce 5 dB Figuresatisfactory 13. Considering the complexity of real experiment, we first note that the overall performances not more By contrast, when the exposed. SR drops In to addition, 20 MHz, the the except for 50when MHzCR SR.is For SSIMs,than the 60%. negative effects are more are worse than withisthe simulated images. To be exact, when CR is 60%, the PSNRs at least reduce 5 dB reconstruction just barely good in 20% CR. performances are more liable to affected by the SR level. For 50 MHz SR, the results are relatively

except for 50 MHz SR. CR For is SSIMs, the negative effects are morewhen exposed. In addition, theMHz, performances satisfactory when not more than 60%. By contrast, the SR drops to 20 the are more liable to affected by the SR level. For 50 MHz SR, the results are relatively satisfactory when reconstruction is just barely good in 20% CR. CR is not more than 60%. By contrast, when the SR drops to 20 MHz, the reconstruction is just barely good in 20% CR.

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Figure 13. Comparative (a) PSNR, (b) SSIM between different SRs in time domain, using experimental

(b) Figure 13. Comparative(a) (a) PSNR, (b) SSIM between different SRs in time domain, using experimental through-holes data. through-holes data. Figure 13. Comparative (a) PSNR, (b) SSIM between different SRs in time domain, using experimental To clearly show through-holes data.the statistic results, the scattered point distribution of 100 releases testing (50 MHz SR, 60% CR)the is shown inresults, Figure 14, thedistribution 3σ (σ is the standard deviation) criterion To clearly show statistic thecombined scatteredwith point of 100 releases testing (50 MHz To clearly show the statistic results, the scattered point distribution of 100 releases testing (50 of reliability theory. It is clear that all the PSNR results lie within three standard deviations on either SR, 60% CR) is shown in Figure 14, combined with the 3σ (σ is the standard deviation) criterion of MHz SR, 60% CR) is value), shown in Figure 14, combined with theof3σOMP (σ is algorithm. the standard deviation) criterion side of μ (the mean which confirms the reliability By calculating all the reliability theory. It is clear that all the PSNR results lie within three standard deviations on either side of reliability theory.values It is clear that all the results within three3standard deviations either standard deviation of different SRsPSNR and CRs, welie show in Table that, except for someonfailures of µ (the mean value), which confirms the reliability of OMP algorithm. By calculating all the standard side μ is (the mean value), which the reliability of OMP algorithm. By3σcalculating whenofSR low or CR is high, mostconfirms of the situations can successfully fit with the criterion. all the deviation values of different and CRs, show that, except someforfailures when SR is standard deviation valuesSRs of different SRswe and CRs,in weTable show3in Table 3 that,for except some failures low or CR is high, most of the situations can successfully fit with the 3σ criterion. when SR is low or CR is high, most of the situations can successfully fit with the 3σ criterion.

Figure 14. Distribution of 100 OMP runs based on 3σ criterion of reliability theory in time domain, using 50 MHz SR experimental through-holes data when CR is 60%. Figure 14. Distribution of 100 OMP runs based on 3σ criterion of reliability theory in time domain, Figure 14. Distribution of 100 OMP runs based on 3σ criterion of reliability theory in time domain, Table 50 3. MHz The standard deviation values (σ) ofdata different and CRs in time domain (√ means the using SR experimental through-holes whenSRs CR is 60%. using 50 MHz SR experimental through-holes data when CR is 60%. situation fits with 3σ criterion, X means not). Table 3. The standard deviation values (σ) of different SRs and CRs in time domain (√ means the 20 30 40 50 60 70 √ 80 Tablesituation 3. CR The(%) standard fits with 3σdeviation criterion, Xvalues means (σ) not).of different SRs and CRs in time domain ( means the σ 0.4322 0.3878 0.4217 0.3930 0.4593 6.4072 9.3127 situation with 20 fits MHz SR 3σ criterion, X means not). Fit 3σ criterion √ √ √ √ √ X X CR (%) 20 30 40 50 60 70 80 0.3677 0.3878 0.4539 0.4217 0.3876 0.3930 0.4211 0.4593 0.4125 6.4072 4.5018 9.3127 5.3882 σ 0.4322 25 CR20 (%) 20 30 40 50 60 70 80 MHz SR Fit 3σ criterion √ √ √ √ √ XX XX σ σ 0.4322 0.3878 0.4217 0.3930 0.4593 6.40725.3882 0.4482 0.3975 0.4421 0.3903 0.3759 0.5225 3.56979.3127 0.3677 0.4539 0.3876 0.4211 √ √ √ √ 0.4125 √ 4.5018 20 MHz SR SR 33 MHz 25 Fit 3σ criterion X √ X Fit 3σ criterion √ √ √ √ √ X X σ σ 0.3677 0.4539 0.3876 0.4211 0.4125 4.50183.5697 0.3855 0.4203 0.3547 0.3971 0.3632 0.4125 3.43385.3882 0.4482 0.3975 0.4421 0.3903 √ √ √ √ 0.3759 √ 0.5225 25 MHz SR SR 50 MHz 33 Fit Fit 3σ 3σ criterion X criterion √ √ √ √ √ √ X X 0.3855 0.4203 0.3547 0.3971 0.3632 0.4125 3.43383.5697 σ σ 0.4482 0.3975 0.4421 0.3903 0.3759 0.5225 √ √ √ √ √ √ 50 MHz 33 MHz SR SR 3σoriginal criterion Fit 3σ criterion √ results √ of 50%√ CR, 60% X CRX and Figure 15a isFit the image of√ 50 MHz√ SR. The√ recovered

70% CR are shown in Figures 15b–d,0.3855 respectively. We can see that, for 50% CR, the 0.4125 recovery 3.4338 is quite σ 0.4203 0.3547 0.3971 0.3632 √ √ √ √ √ √ 50 MHz SR 15a isFit the original image of 50In MHz SR. The recovered 50% 60% CRX and 3σ criterion goodFigure and almost no obvious error occurs. the case of 60% CR, sixresults defectsofcan be CR, easily identified, 70% CR are shown in Figures We can see that, for 50% CR, the is quite although there are some slight15b–d, errorsrespectively. in the areas without defect echo. But when CRrecovery rise to 70%, the good and almost no obvious error occurs. In the case of 60% CR, six defects can be easily identified, reconstruction is so distorted that some defects are almost overwhelmed by errors. Figure 15a is the original image of 50 MHz SR. The recovered results of 50% CR, 60% CR and although there are some slight errors in the areas without defect echo. But when CR rise to 70%, the 70% CR are shownisin 15b–d, respectively. can overwhelmed see that, for 50% CR, the recovery is quite reconstruction soFigure distorted that some defects areWe almost by errors.

good and almost no obvious error occurs. In the case of 60% CR, six defects can be easily identified,

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Figure 15.15. (a)(a) TheThe original experimental through-holes image of 50 MHz SRMHz and the results Figure original experimental through-holes image of 50 SRreconstructed and the reconstructed (c)= 41.25 dB; (d) CR = 70%, PSNR (d) = 34.45 (a) (b) CR = 50%, PSNR =(b) in time domain; 45.36 dB; (c) CR = 60%, PSNR results in time domain; (b) CR = 50%, PSNR = 45.36 dB; (c) CR = 60%, PSNR = 41.25 dB; (d) CR = 70%, dB. Figure 15. (a) The original experimental through-holes image of 50 MHz SR and the reconstructed results PSNR = 34.45 dB. in time domain; (b) CR = 50%, PSNR = 45.36 dB; (c) CR = 60%, PSNR = 41.25 dB; (d) CR = 70%, PSNR = 34.45 We also perform the PSNR comparison at different SRs when keeping the sampling points dB.

We also perform the PSNR comparison at different SRs keepingpoints the sampling points consistent. Table 4 reports the results using experimental data. Forwhen measurement less than 220, consistent. Table 4 reports the results using experimental data. For measurement points less than theWe performance indeed becomes worse with improvement of SR, whichthe shows similarpoints change220, also perform the PSNR comparison atthe different SRs when keeping sampling the performance indeed becomes worse with the improvement of SR, which shows similar change rule compared thatthe in simulation. Besides, the gapdata. of PSNRs narrows withpoints measurement consistent. Table 4with reports results using experimental For measurement less thanpoints’ 220, rule compared with that becomes in simulation. Besides, gap ofare PSNRs narrows with measurement and the difference disappears if measurement points more than 240.shows It is therefore to therise performance indeed worse with the the improvement of SR, which similar natural changepoints’ confirm that lower M/N ratio affects the accuracy of OMP, while the reconstruction performance rise and the difference disappears if measurement points are more than 240. It is therefore natural rule compared with that in simulation. Besides, the gap of PSNRs narrows with measurement points’ is to independent withM/N SRdisappears when pointspoints increase some extent. confirm lower ratiomeasurement affects the accuracy of OMP, while the240. reconstruction is rise andthat the difference if measurement aretomore than It is therefore performance natural to independent when measurement points of increase to some confirm that with lowerSR M/N ratio affects the accuracy OMP, while theextent. reconstruction performance is Table 4. Comparative PSNR of different SRs when keeping the same sampling points, using

independent with SR when measurement points increase to some extent. experiment data. Table 4. Comparative PSNR of different SRs when keeping the same sampling points, using experiment data. TableMeasurement 4. Comparative PSNR different180 SRs when same sampling points, using Points of 160 200 keeping 220 the240 260 280 300 experiment data. 25 MHz SR (256 points) Measurement Points

39.44 42.47 44.47 44.52 44.63 N/A N/A N/A 160 180 200 220 240 260 280 300 33 MHz SR (341 points) 36.61 41.04 42.10 43.32 43.89 44.68 45.19 45.31 Measurement Points 160 180 200 220 240 260 280 300 25 MHz SR(512 (256points) points) 34.82 39.44 36.37 42.47 41.01 44.47 42.51 44.52 43.56 44.63 45.36 N/A N/A N/A MHz SR 45.66 45.87 25 50 MHz SR (256 points) 39.44 42.47 44.47 44.52 44.63 N/A N/A N/A 33 MHz SR (341 points) 36.61 41.04 42.10 43.32 43.89 44.68 45.19 45.31 33 MHz SR (341 points) 50 MHz SR (512 points) 36.61 34.8241.04 36.3742.10 41.0143.32 42.5143.89 43.5644.68 45.36 45.19 45.66 45.31 45.87 ToMHz experimentally demonstrate image quality in terms evaluate its 50 SR (512 points) 34.82 the 36.37 41.01 42.51 43.56of various 45.36 defects 45.66 and 45.87 impact on reconstruction accuracy, we calculated and compared the PSNRs of three typical defects To showed experimentally demonstrate the image that, qualityexpected, in terms of various defects and evaluate and the results in Figure 16.the It turns recoveries EDM notches and its To experimentally demonstrate imageout quality as in terms of the various defectsofand evaluate its impact on reconstruction accuracy, we calculated and compared the PSNRs of three typical defects flat-bottom holes reach sufficiently good PSNRs,and although sufferthe some reduction in typical lower CRs. Since impact on reconstruction accuracy, we calculated compared PSNRs of three defects andthe showed theimages resultsofinboth Figure 16. It turns out back that, wall as expected, the echo, recoveries of EDM notches and detected two defects contain echo or side the preliminary analysis and showed the results in Figure 16. It turns out that, as expected, the recoveries of EDM notches and flat-bottom holes reach sufficiently goodPSNRs, PSNRs, although suffer some reduction in lower CRs. Since is that these strong echoes have agood negative impact on thesuffer performances. flat-bottom holes reach sufficiently although some reduction in lower CRs. Since

the of both bothtwo twodefects defectscontain contain back wall echo or side echo, the preliminary analysis thedetected detectedimages images of back wall echo or side echo, the preliminary analysis isisthat haveaanegative negativeimpact impact performances. thatthese thesestrong strong echoes echoes have onon thethe performances.

Figure 16. Comparative PSNRs between three kinds of defects in time domain, using 50 MHz SR experimental data. Figure 16. Comparative PSNRs between three kinds of defects in time domain, using 50 MHz SR

Figure 16. Comparative PSNRs between three kinds of defects in time domain, using 50 MHz SR experimental data. experimental data.

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In Figure EDM notches and flat-bottom holes, as well as the In Figure 17, 17, we wepresent presentthe theoriginal originalimages imagesofof EDM notches and flat-bottom holes, as well as In Figure 17, we present the original images of EDM notches and flat-bottom holes, as well as reconstructed results when CR is 60%. Although the image qualities, corresponding to the proposed the reconstructed results when CR is 60%. Although the image qualities, corresponding to the the reconstructed results when CR is 60%. Although the image qualities, corresponding to the CS methodCS inmethod time domain with OMP-based reconstruction, are reducedare compared the original proposed in time domain with OMP-based reconstruction, reducedwith compared with proposed CS method in time domain with OMP-based reconstruction, are reduced compared with versions, important information, for example, the measured defects, essential for structural health the original versions, important information, for example, the measured defects, essential for the original versions, important information, for example, the measured defects, essential for assessment, are all clearly preserved. structural health assessment, are all clearly preserved. structural health assessment, are all clearly preserved.

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Figure 17. The original 5050MHz SR experimental image of (a) EDM notches, (c) flat-bottom holes and Figure 17. 17. The Theoriginal original50 MHz experimental image of EDM (a) EDM notches, (c) flat-bottom Figure MHz SRSR experimental image of (a) notches, (c) flat-bottom holesholes and the reconstructed results in time domain, (b) CR 60%, PSNR = 35.46 dB (d) CR = CR 60%,= 60%, PSNR = andreconstructed the reconstructed results in time domain, (b)==CR = 60%, = 35.46 the results in time domain, (b) CR 60%, PSNRPSNR = 35.46 dB (d)dB CR(d) = 60%, PSNR = 35.96 dB. 35.96 dB. PSNR = 35.96 dB.

4.3. Frequency Domain Reconstructions 4.3. Frequency Frequency Domain Domain Reconstructions Reconstructions 4.3. In section, the the performance performance ofthe the frequencydomain domain reconstruction is given and compared In this this section, and compared to In performanceof of thefrequency frequency domainreconstruction reconstructionis isgiven given and compared to the time domain results. Same as the simulation, the DCT transform is employed. Figure 18 shows thethe time domain results. shows to time domain results.Same Sameasasthe thesimulation, simulation,the theDCT DCTtransform transformisisemployed. employed. Figure 18 shows the image calculations of different SRs. We first note that the performances, both PSNRs and SSIMs the image image calculations calculations of of different different SRs. SRs. We We first both PSNRs PSNRs and SSIMs the note that the performances, both SSIMs dramatically outperform the time domain results. To be more specific, the lowest PSNR in 60% CR dramatically outperform the time domain results. To be more specific, the lowest PSNR in 60% CR is is dramatically outperform the time domain results. To nearly 9.5 dB higher than that in time domain, and for SSIM, that is 0.49. When CR is 70%, the PSNR nearly 9.5 dB dB higher than that in time domain, and for SSIM, that is 0.49. When When CR CR is is 70%, 70%, the the PSNR PSNR nearly of 50 MHz SR image can be 41.61 dB, meaning comparatively gratifying reconstruction. But for time of 50 50 MHz MHz SR SR image image can can be be 41.61 41.61 dB, dB,meaning meaning aaacomparatively comparatively gratifying gratifying reconstruction. reconstruction. But for time time of for domain, a 34.45 dB is obviously insufficient for defects recognition. Another observation is that the domain, a 34.45 dB is obviously insufficient for defects recognition. Another observation is that the SR domain, 34.45 dB is obviously insufficient for defects recognition. Another observation is that the SR level in frequency domain seems no longer as significant as that in time domain. The PSNRs of 20 level in frequency domain seems no longer as significant as that time domain. The PSNRs of 20 MHz SR level in frequency domain seems no longer as significant as in that in time domain. The PSNRs of 20 MHz SR are, maybe a little bit lower, not very far compared with 50 MHz SR, as for SSIMs, sometimes SR are, a littlea bit lower, not very compared with 50 MHz SR, asSR, forasSSIMs, sometimes even MHz SRmaybe are, maybe little bit lower, not far very far compared with 50 MHz for SSIMs, sometimes even better. Figure 19 shows the distributions of 100 OMP tests when 20 MHz SR experimental data better.better. FigureFigure 19 shows the distributions of 100 of OMP MHz experimental data is used even 19 shows the distributions 100 tests OMPwhen tests 20 when 20SR MHz SR experimental data is used and the50%, CR which is 50%,completely which completely reach the 3σ criterion of reliability theory. and theand CR the is reach the 3σ criterion of reliability theory. theory. is used CR is 50%, which completely reach the 3σ criterion of reliability

(a) (a)

(b) (b)

Figure 18. Comparative (a) PSNR, (b) SSIM between different SRs in frequency domain, using Figure 18. Comparative Comparative PSNR, (b) SSIM between different SRs in frequency domain, using Figure 18. (a)(a) PSNR, experimental through-holes data.(b) SSIM between different SRs in frequency domain, using experimental experimental data. through-holesthrough-holes data.

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Figure 19. Distribution of 100 orthogonal matching pursuit (OMP) runs based on 3σ criterion of Figure 19. Distribution 100 orthogonal matching pursuit (OMP) runs based on criterion of Figure 19. Distribution of of 100 orthogonal pursuit (OMP) runs based ondata 3σ 3σ criterion reliability theory in frequency domain, using matching 20 MHz SR experimental through-holes when CRof reliability theory in frequency domain, using 20 MHz SR experimental through-holes data when theory in frequency domain, using 20 MHz SR experimental through-holes data when CRCR isreliability 50%. is 50%. is 50%.

We have explained in simulation section that the better performance in frequency domain, in We have explained in simulation section that the better performance in frequency domain, in comparison with time domain, owes to its higher Now, according to the results We have explained in simulation section that thesparsity. better performance in frequency domain,inin comparison with time domain, owes to its higher sparsity. Now, according to the results in experimental case,time we offer an interpretation on the performance similarity different SRs. It comparison with domain, owes to its higher sparsity. Now, according to between the results in experimental experimental case, we offer an interpretation on the performance similarity between different SRs. It iscase, widely that different SR brings differentsimilarity samplingbetween points. Indifferent time domain, weknown offer an interpretation on the about performance SRs. Itthe is defect widely is widely known that different SR brings about different sampling points. In time domain, the defect information locates in all the sampling points. When CR is given, lower SR means that the known that different SR brings about different sampling points. In time domain, the defect information information locates in all the sampling points. When CR is given, lower SR means that the measurement used for CS reconstruction are less, hence to that poorthe results. Of course, we locates in all points the sampling points. When CR is given, lower SRleads means measurement points measurement points used for CS reconstruction are less, hence leads to poor results. Of course, we have confirmed through simulation and experiment that, if measurement points are not too few, used for CS reconstruction are less, hence leads to poor results. Of course, we have confirmed through have confirmed through simulation and experiment that, if measurement points are not too few, recovery performance has nothing do with SR. By contrast, in frequency domain,performance the ratio SR/N simulation and experiment that, iftomeasurement points are not too few, recovery has recovery performance has nothing to do with SR. By contrast, in frequency domain, the ratio SR/N defines the grid DFT and N is always proportional SR. So, the frequency grid is thein nothing tofrequency do with SR. By in contrast, in frequency domain, the ratiotoSR/N defines the frequency grid defines the frequency grid in DFT and N is always proportional to SR. So, the frequency grid is the same here. Therefore, the to points with information the useful are the same DFTfor andallNSRs is always proportional SR. So, therelevant frequency grid is theof same for all band SRs here. Therefore, same for all SRs here. Therefore, the points with relevant information of the useful band are the same for SRs. To illustrate N-point was A-scan signals. We can see the theallpoints with relevantintuitively, informationthe of the usefulDFT band areapplied the sametofor all SRs. To illustrate intuitively, for all SRs. To illustrate intuitively, the N-point DFT was applied to A-scan signals. We can see the point numberDFT of effective band to retains almost theWe same for point different SRs, as in Figure the N-point was applied A-scan signals. canlevel see the number of illustrated effective band retains point number of effective band retains almost the same level for different SRs, as illustrated in Figure 20. almost the same level for different SRs, as illustrated in Figure 20. 20.

Figure frequency spectrums Figure20. 20.The The frequency spectrumsofofdifferent differentSRs. SRs. Figure 20. The frequency spectrums of different SRs.

Before showing the frequency domain recovery performance of different kinds of defects, we Before showing the the frequency domain recovery performance of different kinds ofkinds defects, we showwe Before showing frequency domain recovery performance of different of defects, show in Figure 21, one extracted A-scan signal from 20 MHz SR image and its corresponding DFT in show Figurein21,Figure one extracted A-scan signal from 20 MHz SR20 image corresponding DFT spectrum. 21, one extracted A-scan signal from MHzand SR its image and its corresponding DFT spectrum. It can be inferred that, using traditional sampling method, at least 17 MHz SR is required It can be inferred that, using traditional sampling method, at least 17 MHz SR is required to avoid spectrum. It can be inferred that, using traditional sampling method, at least 17 MHz SR is required to avoid aliasing. Remarkably, it is possible to obtain satisfactory recovery using the equivalent of 10 aliasing. Remarkably, it is possibleitto satisfactory recovery using the equivalent 10 MHz (for to avoid aliasing. Remarkably, is obtain possible to obtain satisfactory recovery using theofequivalent of 10 MHz (for 50% CR), even 8 MHz (for 60% CR) samples, employing our proposed frequency domain 50% CR),(for even 8 MHz (for 60% CR) (for samples, employing our proposedour frequency domain CS scheme. MHz 50% CR), even 8 MHz 60% CR) samples, employing proposed frequency domain CS scheme. CS scheme.

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(a) (a)

(b) (b)

Figure 21. 21. (a) (a) Extracted Extracted A-scan A-scan signal signal from from 20 20 MHz MHz SR SR image; image; (b) (b) the the DFT DFT spectrum spectrum of of (a). (a). Figure Figure 21. (a) Extracted A-scan signal from 20 MHz SR image; (b) the DFT spectrum of (a).

Figure 22 presents the recovery results of through-holes, EDM notches and flat-bottom holes, Figure 22 presents the recovery results of EDM notches notches and and flat-bottom flat-bottom holes, Figure 22 SR presents of through-holes, through-holes, EDM using 20 MHz imagethe andrecovery 50% CR.results The performances are totally comparable with that of 50holes, MHz using 20 MHz SR and 50% CR. performances are totally using 20 MHz SR image image and CR. The The performances are totally comparable comparable with with that that of of 50 50 MHz MHz SR in time domain, shown in50% previous section. SR SR in in time time domain, domain, shown shown in in previous previous section. section.

(a) (a)

(b) (b)

(c) (c)

Figure 22. The recovery results of 20 MHz SR experimental image in frequency domain, (a) throughFigure 22. The recovery of 20 SR in domain, throughholes, PSNR = 42.91 dB; results (b) EDM = 41.50 dB;image (c) flat-bottom holes, PSNR =(a) 41.68 dB. Figure 22. The recovery results ofnotches, 20 MHz MHz PSNR SR experimental experimental image in frequency frequency domain, (a) throughholes, PSNR PSNR = = 42.91 dB; (b) (b) EDM EDM notches, notches, PSNR PSNR == 41.50 41.50 dB; dB; (c) (c) flat-bottom flat-bottomholes, holes,PSNR PSNR== 41.68 41.68dB. dB. holes, 42.91 dB;

Finally, in Figure 23 we show the comparative results of various defects. We can see that the Finally, in Figure 23 between we show the comparative results of various defects. We can see that the performance defects are smaller than that in time domain, which Finally, indisparities Figure 23 we showdifferent the comparative results of various defects. We can see that the performance disparities between different defects are smaller than that in time domain, which confirms thedisparities stabilization of the proposed DCT-based reconstruction. As a performance between different defects are smallerfrequency than that indomain time domain, which confirms confirms theour stabilization of framework the proposed DCT-based frequency domain reconstruction. As a conclusion, proposed CS is able to drastically reduce the data in ultrasonic phased the stabilization of the proposed DCT-based frequency domain reconstruction. As a conclusion, our conclusion, ourWhat’s proposed CSthe framework able to drastically reducestrengthen the data inthe ultrasonic phased array imaging. more, results inisfrequency further efficiency of CS, proposed CS framework is able to drastically reducedomain the data in ultrasonic phased array imaging. array imaging. What’s more, the results in frequency domain further strengthen the efficiency of CS, based on the fact that a breakthrough of the Nyquist sampling limitation is feasible. What’s more, the results in frequency domain further strengthen the efficiency of CS, based on the fact based on the fact that a breakthrough of the Nyquist sampling limitation is feasible. that a breakthrough of the Nyquist sampling limitation is feasible.

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Figure 23. Comparative PSNRs between three kinds of defects in frequency domain, using 20 MHz Figure 23. Comparative PSNRs between three kinds of defects in frequency domain, using 20 MHz SR Figure 23. Comparative SR experimental data.PSNRs between three kinds of defects in frequency domain, using 20 MHz experimental data. SR experimental data.

4.4. Real Application in Engine Cylinder Cavity Inspection 4.4. Real Real Application Application in in Engine Engine Cylinder Cylinder Cavity Cavity Inspection Inspection 4.4. As a case study, the proposed CS-based scheme was applied to the inspection of engine cylinder As aa case case study, the the proposed proposed CS-based CS-based scheme scheme was was applied applied to to the the inspection inspection of of engine engine cylinder cylinder As cavity in thisstudy, part. cavity in this part. cavity in thisengine part. cylinder cavity plays the role of flowing the cooling liquid, as shown in Figure 24a. The The engine engine cylinder cylinder cavity plays plays the role role of of flowing flowing the the cooling cooling liquid, liquid, as as shown shown in in Figure 24a. 24a. The It can be corroded overcavity time, whichthe may bring about serious hidden dangers and haveFigure catastrophic It can can be be corroded corroded over over time, time, which may about serious serious hidden hidden dangers dangers and and have have catastrophic It which may bring bring consequences. To solve this problem, most about of the researchers focused their studiescatastrophic on corrosion consequences. To solve this problem, most of the researchers focused their studies on corrosion corrosion consequences. To solve this problem, most of the researchers focused their studies on behavior or anticorrosion coolant. Despite the fact that these approaches can prevent the corrosions behavior or coolant. Despite the that approaches can prevent thethe corrosions to behavior or anticorrosion anticorrosion coolant. thefact fact thatthese these can prevent corrosions to an utmost degree, they are stillDespite inadequate in gaining theapproaches details of the flaws. Accordingly, routine an utmost degree, they are still inadequate in gaining the details of the flaws. Accordingly, routine to an utmost can degree, they are differences still inadequate in gaining the of the flaws. routine detection make some by identifying thedetails potential risks andAccordingly, the ultrasonic phased detection can make some differences by identifying the potential risks and the ultrasonic phased array detection can make some differences by identifying the potential risks and the ultrasonic phased array has been considered as an effective method for inspection of the cavity [32,33]. has been considered as an effective method for inspection of thewe cavity [32,33]. array has been considered as an effective method forthe inspection of the cavity Considering the concave cylinder surface of cavity, designed a[32,33]. convex wedge installing Considering the concave cylinder surface of the cavity, we designed a convex wedge installing on as the concave cylinder surface of specimen the cavity,containing we designed a convex wedge installing onConsidering the phased array transducer. An aluminous artificial pit defects, regarded thethe phased array transducer. aluminous specimen containing artificial pit defects, regarded as the on the phased array transducer. An aluminous specimen containing artificial pit defects, regarded as most common defectsAn caused by corrosion, was employed for our detection. The pits were most common defects caused by corrosion, was employed for our detection. The pits were machined themachined most common defects diameters caused byand corrosion, was employed our detection. pits were of with different taper angles in order to for preserve the main The characteristics with and taperthe angles in order tosystem preserve the main 24c characteristics real defects. machined withdiameters different diameters and taper angles in order to preserve the main characteristics realdifferent defects. Figure 24b shows experimental and Figure exhibits theofdetails of aofpit. Figure 24b shows the experimental system and Figure 24c exhibits the details of a pit. The specific real defects. Figure 24b shows experimental system Figure in 24cTable exhibits detailsthat of a all pit.the The specific dimensions andthe angles of pit defects areand tabulated 5. Itthe is noted dimensions and angles ofand pit angles defects inin Table 5. Itsection. is Table noted 5. that allnoted the phased array The specific dimensions pittabulated defects are tabulated in It is that all the phased array configurations remainofare same with that the last configurations remain same with thatsame in the lastthat section. phased array configurations remain with in the last section.

(b) (b)

(a) (a)

(c) (c)

Figure 24. (a) The structure of engine cylinder; (b) The experimental system; (c) The details of pit Figure 24. (b)(b) The experimental system; (c) (c) TheThe details of pit defects. Figure 24. (a) (a)The Thestructure structureofofengine enginecylinder; cylinder; The experimental system; details of defects. pit defects.

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Table 5. The diameters (R) and taper angles (θ) of pit defects. Table 5. The diameters (R) and taper angles (θ) of pit defects. Table 5. The diameters (R) and taper angles (θ) of pit defects. No. No. 1 1 2 2 3 3 4 4 5 5 6 6 77 8 8 9 9 10 10 11 No. 1 2 3 4 5 6 7 85 9 2 10 311 R (mm) 2 3 4 5 2 3 4 R (mm) 2 3 4 5 2 3 4 5 2 3 4 4 590 2 60 3 604 θ (◦ ) R (mm) 120 2120 3120 4120 5 90 2 90 3 90

θ (°) θ (°)

11 12 12 54 5 60 120 120 120 120 90 90 90 90 60 60 60 60 120 120 120 120 90 90 90 90 60 60 60 60

12 5 60

In Figure Figure 25, 25, the the original original image image of of No. No. 33pit pitand andits itsreconstructed reconstructedversions versionsin intime timeand andfrequency frequency In In Figure 25, the original image of No.domain, 3 pit and50itsMHz reconstructed versions in time and frequency domain are given and compared. For time SR image was used for the reconstruction domain are given and compared. For time domain, 50 MHz SR image was used for the reconstruction domain andFigure compared. For time domain, 50 real MHzapplication, SR image was for the reconstruction and the are CRgiven is 60%. 60%. 25b shows shows that, for this this theused performance is generally generally and the CR is Figure 25b that, for real application, the performance is and the CR is 60%. Figure 25b shows that, for this real application, the performance is generally satisfactory. Quite evidently, the recovery quality is much better in frequency domain, with the fact fact satisfactory. Quite evidently, the recovery quality is much better in frequency domain, with the satisfactory. Quite evidently, the recovery quality is much better in frequency domain, with the fact that lower lower measurement CR) was used. A more important observation is that, that measurementpoints points(20 (20MHz MHzSR, SR,50% 50% CR) was used. A more important observation is that lower measurementimage points (20 MHz SR, 50% was used. A more observation is from the reconstructed shown in Figure 25c,CR) frequency domain CS important can filterfilter part that, from the reconstructed image shown in Figure 25c, frequency domain CS inherently can inherently that, from the reconstructed image shown in Figure 25c, frequency domain CS can inherently filter background noises and enhance image quality. part background noises and enhance image quality. part background noises and enhance image quality.

(a) (a)

(b) (b)

(c) (c)

Figure 25. ofof No. 3 pit; (b)(b) the reconstructed image of Figure 25. (a) (a)The Theoriginal original50 50MHz MHzSR SRexperimental experimentalimage image No. 3 pit; reconstructed image Figure 25. (a) The original 50 MHz SR experimental image of No. 3 pit; (b) thethe reconstructed image of (a) in time domain, CR =CR 60%, PSNRPSNR = 36.86= dB; (c) dB; the recovery result inresult frequency domain, domain, SR = 20 of in time domain, = 60%, 36.86 the recovery in frequency (a) (a) in time domain, CR = 60%, PSNR = 36.86 dB; (c) the (c) recovery result in frequency domain, SR = 20 MHz, CRMHz, = 50%, = 40.84 dB. SR = 20 CRPSNR = 50%, = 40.84 dB. MHz, CR = 50%, PSNR = PSNR 40.84 dB.

We show in Figure 26 the comparative PSNRs between time and frequency domain for all pit We show show in in Figure Figure 26 26 the the comparative comparative PSNRs between time and frequency domain for all pit defects. Besides the obvious superiority of frequency domain reconstructions, the results indicate that defects. Besides Besidesthe theobvious obvioussuperiority superiority frequency domain reconstructions, the results indicate of of frequency domain reconstructions, the results indicate that the recovery performance is less likely to be affected by defects geometry, which validate the that the recovery performance is less likely to be affected by defects geometry, which validate the recovery performance is less likely to be affected by defects geometry, which validate the practicability of our proposed scenario in real application. practicability of our proposed proposed scenario scenario in in real real application. application.

Figure 26. Comparative PSNRs between time and frequency domain for all pit defects, using 50 MHz Figure 26. Comparative PSNRs between time and frequency domain for all pit defects, using 50 MHz SR, 60%26. CRComparative in time domain and between 20 MHz time SR, 50% in frequency domain. Figure PSNRs andCR frequency domain for all pit defects, using 50 MHz SR, 60% CR in time domain and 20 MHz SR, 50% CR in frequency domain. SR, 60% CR in time domain and 20 MHz SR, 50% CR in frequency domain.

5. Conclusions 5. Conclusions 5. Conclusions Ultrasonic phased array imaging has always been a promising method for NDE applications Ultrasonic phased array imaging has always been a promising method for NDE applications because of its unique inspection flexibility higher However, to theapplications increase of Ultrasonic phased array imaging has and always beensensitivity. a promising methoddue for NDE because of its unique inspection flexibility and higher sensitivity. However, due to the increase of element a significant amount of data acquired and needs to be processed. In increase this work, because numbers, of its unique inspection flexibility andishigher sensitivity. However, due to the of element numbers, a significant amount of data is acquired and needs to be processed. In this work, we proposed a CS-based data reduction framework and applied it into phased array image we proposed a CS-based data reduction framework and applied it into phased array image compression both in time field and frequency domain. compression both in time field and frequency domain.

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element numbers, a significant amount of data is acquired and needs to be processed. In this work, we proposed a CS-based data reduction framework and applied it into phased array image compression both in time field and frequency domain. Based on CIVA platform, simulated images of different SRs were reconstructed and evaluated using both PSNR and SSIM index. In time domain, for 40 MHz SR (70% CR) or 25 MHz (60% CR), the recovery results are so gratifying that three defects are clearly preserved and can be easily identified, which suggest a breakthrough of the Nyquist sampling limitation. In frequency domain, the reconstruction performance is considerably better than that in time domain, with at least 49 dB PSNR for 70% CR. We offer an explanation from a perspective of “Information Value” and attribute the excellence to its higher sparsity of DCT coefficients. In addition, the results of different defects interval distance show that the proposed OMP-based CS scheme guarantees the reconstruction quality for close defects. In experimental verification, images obtained from through-holes, EDM notches and flat-bottom holes are reconstructed when CR ranges from 20% to 80%. It is found that, in time domain, the recovery performances are relatively satisfactory when CR is 60% for 50 MHz SR, although not as good as that with simulated images. What’s more, the results are more liable to be affected by the SR level and we find it practically impossible to break Nyquist limitation. By contrast, both PSNRs and SSIMs in frequency domain appreciably outperform the time domain results with a 28.9% gain in PSNR and a 141% improvement in SSIM when SR is 20 MHz. For all three kinds of flaws, the coincident satisfactory recoveries confirm the stabilization of the proposed DCT-based frequency domain reconstruction. It is worth noting that, using the DCT coefficients to perform reconstruction, qualified recovery using the equivalent of 10 MHz samples can be obtained, which is below the sampling limitation and means a real breakthrough of the phased array data reduction. We also present that, in the real application of engine cylinder cavity inspection, the reconstruction performance keeps consistent for all pit defects with various diameters and taper angles, therefore verifies the enormous potential of our proposed scheme in the near future. Besides, both in simulation and experiment, we compare the PSNR of different SRs when keeping the measurement points the same to find out whether SR truly influence the reconstruction performance. We find that lower M/N indeed has an adverse impact on the accuracy, because it leads to an ill-balanced sensing matrix. When the measurement points increase to some extent, however, the recoveries are independent with SR. The results imply that, rather than signal bandwidth, the success of CS relates the number of required measurements that carrying real information. That is to say, the sampling does not depend on signal bandwidth but on signal sparsity, which is the fundamental difference between CS and traditional sampling approach. Therefore, the conventional concept of “sampling rate” is no longer applicable in CS. It is merely because of the limitation of physical implementation, removing some parts of the original sample is used in this work. Author Contributions: Z.B. and S.C. conceived and designed the experiments; Z.B. and L.J. performed the experiments; Z.B. analyzed the data; Z.B. and Z.Z. wrote the paper. Funding: This work was supported by the National Key Research and Development Program of China (No. 2016YFF0101800) and the National Natural Science Foundation of China (No. 61473205 and 51604192). Acknowledgments: We thank Yanbo Zhao for taking photographs of the experimental setups; Tianshu Xu, Xiaobo Rui and Zhou Sha for scientific discussions. Conflicts of Interest: The authors declare no conflict of interest.

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