The Maximum Entropy Method in Ultrasonic Non-Destructive ... - MDPI

entropy Article

The Maximum Entropy Method in Ultrasonic Non-Destructive Testing—Increasing the Resolution, Image Noise Reduction and Echo Acquisition Rate Evgeny Gennadievich Bazulin SPC ECHOPLUS, 123458 Moscow, Russia; [email protected]; Tel.: +7-495-7809248 Received: 26 June 2018; Accepted: 17 August 2018; Published: 20 August 2018







Abstract: The use of linear methods, for example, the Combined Synthetic Aperture Focusing Technique (C–SAFT), does not allow one to obtain images with high resolution and low noise, especially structural noise in all cases. Non-linear methods should improve the quality of the reconstructed image. Several examples of the application of the maximum entropy (ME) method for ultrasonic echo processing in order to reconstruct the image of reflectors with Rayleigh super-resolution and a high signal-to-noise ratio are considered in the article. The use of the complex phase-shifted Barker code signal as a probe pulse and the compression of measured echoes by the ME method made it possible to increase the signal-to-noise ratio by more than 20 dB for the image of a flat-bottom hole with a diameter of 1 mm in a model experiment. A modification of the ME method for restoring the reflector image by the time-of-flight diffraction (TOFD) method is considered, taking into account the change of the echo signal shape, depending on the depth of the reflector. Using the ME method, 2.5D-images of models of dangling cracks in a pipeline with a diameter of 800 mm were obtained, which make it possible to determine their dimensions. In the object with structural noise, using the ME method, it was possible to increase the signal-to-noise ratio of the reflector image by more than 12 dB. To accelerate the acquisition of echoes in the dual scan mode, it is proposed to use code division multiple access (CDMA) technology based on simultaneous emission by all elements of the array of pseudo-orthogonal signals. The model experiment showed the effectiveness of applying the ME method. Keywords: antenna array; full matrix capture (FMC); triple scanning; structural noise; maximum entropy (ME); C–SAFT; total focusing method (TFM); time-of-flight diffraction (TOFD); code division multiple access (CDMA)

1. Introduction At present, in the practice of ultrasonic testing (UT), flaw detectors using antenna arrays are widely used. Two technologies are used to visualize the internal volume of objects: phased antenna array (PAA) technology [1], which is by far the most widely used technology and digital focusing with an antenna (DFA) technology [2,3]. In the paper [4] devoted to the comparison of the capabilities of PAA and DFA flaw detectors, it was concluded that DFA technology is more promising from an algorithmic point of view, and therefore the paper is devoted to the processing of echoes measured by DFA flaw detectors. One can afford such an analogy: PAA technology can be compared with dough, from which one can quickly bake only bread, and DFA technology is flour, salt and other ingredients that will allow one to knead the dough for any bakery or confectionery, but in this case, one need more time to get the final result. With single-sided access, typical for ultrasonic inspection practice, it is effective to use two antenna arrays of Ne elements mounted on prisms and placed on the left (N–side) and on the right (P–side) from Entropy 2018, 20, 621; doi:10.3390/e20080621

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the welded joint to obtain a high-quality image of the reflectors. At the first stage of the application of application of the DFA-technology, signals are measured during emission reception by the DFA-technology, echo signals are echo measured during the emission andthe reception by and all combinations allpairs combinations of of pairs elements of [5]. the In antenna array[6], [5]. In mode the paper [6],recording this modeisofcalled echo of of elements the of antenna array the paper this of echo recording is called the dual scan mode, and in [7] the full matrix capture (FMC) mode. Echoes, the dual scan mode, and in [7] the full matrix capture (FMC) mode. Echoes, radiated by one element radiated by one element andareaccepted by allThe elements, are should called asupport shot. The equipment should and accepted by all elements, called a shot. equipment work on the N–(arrow support work on the N– (arrow with a blue fill in Figure 1), P– (arrow with a green fill) and NP– with a blue fill in Figure 1), P–(arrow with a green fill) and NP–(arrows with a yellow fill) acoustic (arrows with a yellow fill) acoustic channels. The acquisition of echoes by an antenna array operating channels. The acquisition of echoes by an antenna array operating in the dual scan mode when it the dual scan mode it is mechanically is called a triple scan mode or the migration isinmechanically movedwhen is called a triple scanmoved mode [6], or the migration of the[6], antenna array [8]. of the antenna array [8]. Joint coherent processing of echoes by the C–SAFT method for antenna different Joint coherent processing of echoes by the C–SAFT method for different positions of the positions of the antenna array allows obtaining a high-quality image of the reflectors, which array allows obtaining a high-quality image of the reflectors, which fundamentally distinguishes the fundamentally distinguishes the algorithmic capabilities of DFA–flaw detectors from PAA-flaw algorithmic capabilities of DFA–flaw detectors from PAA-flaw detectors [4]. For echoes measured in detectors [4]. For echoes measured in triple-scan is possible obtain focused triple-scan mode, it is possible to obtain focused mode, imagesitbeyond the to near-range of theimages antennabeyond array, the near-range of the antenna array, whichwith is especially important objects with a thickness of more which is especially important for objects a thickness of morefor than 70 mm and for objects with than 70 mm and for objects with high structural noise. high structural noise.

Figure Figure 1.1. Scheme Scheme of of measurements measurements of of echoes echoes by by two twoantenna antennaarrays arrayson onprisms. prisms. Black Black rectangles rectangles schematically schematicallyshow showthe theelements elementsof ofthe theantenna antennaarray. array.

(ri ) is reconstructed Inthe thesecond secondstage, stage, image of reflectors the reflectors by the Combined In thethe image of the ε(ri ) isεreconstructed by the Combined Synthetic Aperture (C–SAFT) [5,8], based[5,8], on the measured echoes, which can which be modified SyntheticFocusing ApertureTechnique Focusing Technique (C–SAFT) based on the measured echoes, can be to take into account the multipath propagation of ultrasound in the object with uneven boundaries modified to take into account the multipath propagation of ultrasound in the object with uneven and the presence with acoustic properties [9,10]. Sometimes the C–SAFT algorithm boundaries and of theregions presence ofdifferent regions with different acoustic properties [9,10]. Sometimes the C– isSAFT calledalgorithm the Total Focusing (TFM) [7]. In Method the Born (TFM) approximation, theBorn measured echo signalsthe p is called Method the Total Focusing [7]. In the approximation, can be represented as a system of linear Equations [7]: measured echo signals p can be represented as a system of linear Equations [7]: G+ ε +n,n , pp = =Gε

(1) (1)

n —the additive where GGisisthe thematrix matrixwhich whichdescribes describesthe the propagation propagation of of the the ultrasonic ultrasonic pulse, additive where pulse, n—the andε εcan canbebelexically lexicallyformed formedfrom fromthe the matrices, matrices, noise of of the the measurements. noise measurements. Note Notethat thatthe thevector vector pp and andvice viceversa. versa. and A fragment of the the image measurement of A fragment of image having having aa maximum maximumamplitude amplitudeininthe theregion regionofofa asharp sharp measurement the acoustic properties of the monitoring object, for example, the boundary of the discontinuity, is of the acoustic properties of the monitoring object, for example, the boundary of the discontinuity, to the the discontinuity discontinuity iscalled calledananindication. indication.AAfalse falseindication indicationisisan anindication indicationthat that does does not not correspond correspond to boundary and is formed due to the limitations of the acoustic path model. boundary and is formed due to the limitations of the acoustic path model. Thehigher higherthe theresolution resolutionofofthe thereflector reflectorimage image and lower noise level, thus number The and thethe lower thethe noise level, thus thethe number of of reflectors, their sizes be determined more accurately. increases the reliability of reflectors, their sizes andand typetype cancan be determined more accurately. ThisThis increases the reliability of the the UT. The maximum entropy method, which is a nonlinear method, should increase the resolution UT. The maximum entropy method, which is a nonlinear method, should increase the resolution and and reduce the image noise level. The paper considers several options of the ME method application for obtaining high-quality image of reflectors and for increasing the speed of echo signal acquisition.

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reduce the image noise level. The paper considers several options of the ME method application for obtaining high-quality image of reflectors and for increasing the speed of echo signal acquisition. Section 2 summarizes the principles of reflectors image reconstruction from measured echoes. Section 3 describes a model experiment in which the possibility of increasing the signal-to-noise ratio of reflectors in materials with high attenuation when compressing complex signals by the ME method is demonstrated. In Section 4 a modification of the ME method for the case of a variable waveform measured in the TOFD mode is considered. Section 5.1 describes the results of a model experiment for obtaining a super-resolution image for a set of point-type reflectors and for a pipeline fragment of 800 mm diameter with fracture models. Section 5.2 presents the results of a model experiment to improve the image quality of flat-bottomed holes with a diameter of 1 mm. Section 5.3 demonstrates the effectiveness of the ME method to produce an image of side drilled holes in a composite weld with a high level of structural noise. Section 6 is devoted to improving the speed of echo acquisition in FMC mode using CDMA technology and the ME method. 2. ME Method in Ultrasonic Nondestructive Testing 2.1. Restoring the Reflectors Image There are many ways to solve the classical problem by the Formula (1) from the point of view of solving the system of linear algebraic expressions. Because the matrix is typically ill-conditioned, in addition to a simple treatment of the matrix there are other solutions. One of them is to find an estimate as a solution of the unconditional optimization problem when the square of the solution residual is chosen as a criterion of the quality of the recovered image: χ2 (εˆ ) = kGˆε − pk = (Gˆε − p) T (Gˆε − p),

(2)

where the symbol T means the matrix transpose operation. The reconstructed estimate εˆ can be written as: εˆ = argmin (χ2 (εˆ )). (3) εˆ ∈ R

Ni,x · Ni,z

The solution of the inverse problem in the form (3) is called the least squares method. To search with the maximum speed of the minimum by Equation (3), one needs to use second order optimization methods [11]. Their application requires knowledge of gradient and Hessian, which for Equation (2) is calculated as follows: ∇χ2 (ε) = 2GT (Gε − p), ∇∇χ2 (ε) = 2GT G (4) In terminology [12], the solution of a degenerate system of linear algebraic Equation (1) with respect to providing the minimum of residual χ2 (ε) is called a pseudo-solution. There can be infinitely many pseudo-solutions and such parameters as resolution, speckle noise level, etc., may be far from ideal in general. Thus, in the work [13] the images of the reflectors based on the echo signals in the numerical experiment were found by the least square method. The image recovery algorithm was named extended synthetic aperture focusing technique (ESAFT). There is an explicit regularized solution to Equation (3), which is presented below: εˆ =

GT p, GGT + γE

(5)

where E is the diagonal matrix, and γ is the spectral power density of the interference n. This formula is equivalent in structure to the formula that describes the Wiener filter in the frequency domain [14]. With the help of (5), it is possible to obtain high-quality images, but in the case of a low level of interference n. Finally, the function εˆ estimate can be represented by analogy with the correlation formula in the matrix form:

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εˆ c = GT p,

(6)

To solve ill-conditioned problems a method of regularization was developed by Tikhonov [12], which justifies the replacement of the problem in the form of (1) by the problem of optimization of input data p resistant to small changes: εˆ = argmin



 χ2 (εˆ ) + αΩ(εˆ ) ,

(7)

N ·N εˆ ∈ R i,x i,z

where Ω(εˆ ) is the stabilizing functional. The purpose of using stabilizing functionals is to take into account some a priori information about the solution when solving an ill-conditioned problem and thereby narrow the search for solutions. This a priori information can range from the simplest requirement for non-negativity of the solution, the minimality of the solution energy to restrictions on a known auto-correlation function, a given spectrum structure, and so on. The entropy value of vector εˆ can be used as a stabilizing functional Ω(εˆ ). The ME method is used in spectral estimation problems. In 1967, Berg [15] derived a formula to estimate the spectral density of the signal power by known samples of the autocorrelation function based on the requirement of the maximum entropy corresponding to the time series in the form H (ε) = ∑ log ε i . Moreover, it was shown that the estimation of the spectral density of the signal power obtained by the autoregressive (AR) model and the spectral power density obtained by the maximum entropy method is identical in the case of the description of the signal as a Gaussian random process. Frieden in 1972 [16] showed the possibility of using Shannon entropy as a stabilizing functional in the context of the Tikhonov regularization method [12]. The possibility of achieving super-resolution in the image acquisition system (one-dimensional images, a system with diffraction limit) is demonstrated. Researches have shown the effectiveness of the practical application of the ME method in the reconstruction of images in tomography [17–19], radio astronomy [20–22], nuclear magnetic resonance [23], as well as in ultrasonic testing [24,25]. The paper [26] is devoted to the deconvolution of high-frequency (10–40 MHz) echoes by the maximum entropy method in the measurement of the thickness of layered structures. Thus, the method of maximum entropy as a solution of the optimization problem with the entropy of image estimation as a stabilizing functional: εˆ = argmin εˆ ∈ R



 χ2 (εˆ ) − αH (εˆ ) ,

(8)

Ni,x · Ni,z

where H is the entropy of a set of discrete independent random variables, which is defined for the case of real, non-negative ε i as: Ni,x Ni,z

H (ε) = −

∑

ε i ln ε i = −Ω(ε),

(9)

i =1

where Ni,x Ni,z is the number of pixels in the reconstructed image can be very useful for developing methods for obtaining image reflectors. The entropy calculated by this formula coincides with the entropy of a set of discrete independent random variables and is called Shannon entropy [27]. In practice, the so-called cross-entropy (other names: conditional entropy [27], relative entropy [20], the Kulbek-Labler distance is used [28]: Ni,x Ni,z

H (ε) = −

∑

i =1

ε i ln

εi , mi

(10)

where m is an a priori model or evaluation of the type of solution ε. As a simple model, one can use a constant value eµ, in which µ is understood as an estimate of the average value of the background intensity of the image. The calculation of cross-entropy allows one to bypass one of the problems associated with the use of the maximum entropy criterion. The fact is that when some of the evaluation

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values εˆ i approach zero, the logarithm in the expression for entropy (9) takes too large negative values, which makes it difficult to converge to images with zero background. For cross-entropy, its gradient elements will be close to zero for εˆ i which are close to µ. The logarithm in the maximum entropy criterion, as noted in [14], serves as an additional opportunity to prevent the appearance of elements with negative values in the recovered image. Since the recovered information ε about the reflector is considered as a real number, which can take both positive and negative values, Equation (10) cannot be used. Different approaches exist to adapt the maximum entropy criterion to data with negative values [29]. One can calculate the entropy of the module of vector ε in the form: p zi = | ε i | = ε i 2 , Ni,x Ni,z (11) zi . H|| (ε) = − ∑ zi ln eµ i =1

The gradient and the Hessian in the entropy calculation of Equation (11) will look as follows: ∂H|| (ε) ∂ε i

=−

ε i ln zµi zi

,

∂2 H|| (ε) ∂(ε i )

2

=−

εi . z3i

(12)

In the article [30] it is proposed to calculate the entropy of the alternating function by the following formula: p zi = ε i 2 − 4µ2 ,  , Ni,x Ni,z  (13) +ε i H± (ε) = − ∑ zi − 2µ − ε i ln zi2µ . i =1

In this case, the gradient and the Hessian of the entropy will look as follows:   z + εi ∂H± (ε) = − ln i , ∂ε i 2µ

∂2 H± (ε) ∂(ε i )

2

1 =− . zi

(14)

Considering that the entropy calculated by Equation (11) has two maxima, and the entropy calculated by Equation (13) has one maximum, Equations (13) and (14) were used in this paper. Note that in the Equations (12) and (14) the Hessian has the form of a diagonal matrix, which can significantly accelerate the work of optimizing algorithms and reduce the requirements for the sizes of the operative memory. In the paper [31] for work with alternating sign functions the method of generalized entropy is proposed, when the alternating sign function ε is represented as the difference of two positive definite functions: ( ε+ = ε+− , if ε+− ≥ 0 ε+− = ε+ − ε− , where , (15) ε− = −ε+− , if ε+− < 0 This approach has a technical drawback. Since the size of the vector ε+− is doubled, the memory required for matrix G calculations is quadrupled. 2.2. Processing of Echo Signals Presented in a Complex Form In the practice of ultrasonic testing, there are problems when it is necessary to increase the resolution of the pulses present in the measured echo signal p. In this case, we will mean by ε the type of the initial δ-pulses applied to the input of a linear system with a pulse response s. To solve the deconvolution problem, the matrix G columns will be formed by the response s with a cyclic shift, that is, the matrix G will be circulant.

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In general, the vectors s, p and the matrix G can be complex. Similar to how it was done in [29], we introduce the following agreement: complex vector and the matrices will be written in the following extended form: ! ! pRe GRe −GIm p≡ , G≡ . (16) pIm GIm GRe The gradient and Hessian of the residual criterion are calculated by Equation (4). The gradient and Hessian of the criterion H will be as follows: Nt

H (ε) = − ∑ zi ln ∂H (ε i ) ∂εRe i ∂2 H ( ε i ) ∂(

2 εRe i

)

=

=

zi eµ ,

i =1 εRe ln(z /µ) − i zi i ,

zi =

∂H (ε i ) ∂εIm i

Im −εRe i − ε i ln( zi /µ ) , z3i

∂2 H ( ε i ) Im ∂εRe i ∂ε i

=

∂2 H ( ε i ) Re ∂εIm i ∂ε i

q


2

=−


2 + εIm , i

εIm i ln( zi /µ ) , zi

Re −εIm i − ε i ln( zi /µ ) 2 Im z3i ∂ εi Re Im ε i ε i (ln(zi /µ)−1) z3i

∂2 H ( ε i )

(

=

εRe i

)

=

,

.

2.3. The Choice of the Lagrange Coefficient and Background Coefficient The main methodological problem of practical application of the ME method is the choice of algorithm parameters: the evaluation of the background amplitude µ and the Lagrange coefficient α. The role of α is to establish a consensus between obtaining an exact solution of the ill-posed problem and satisfying the restriction caused by the stabilizing functional. There are precise and empirical methods for choosing the regularization parameter: the residual method [12,32], the cross-validation method [33], the generalized minimization problem [34], the L-curve method [35] and others. However, some methods are iterative and therefore require significant computational resources, while others require additional information, such as the noise dispersion value σ (n), which may not always be estimated with sufficient accuracy. In many practical works devoted to the application of the ME method, the problem (7) is solved for the set of Lagrange coefficients α, and the best solution according to the given criterion is chosen from the set of obtained estimates εˆ α . In this paper, from the set of estimates εˆ α for the set {α1 , α2 , . . . , α Nα }, the estimate was chosen according to the L-curve criterion. The introduction of another parameter µ and the need for its evaluation creates the additional complexity of the entropy calculation by the Formula (10). Researches have shown that when model and experimental data are restored, sufficiently close results εˆ are observed for the values of the background coefficient µ differing by several orders. In [30], it is recommended to choose a value µ equal to one-hundredth of the mean square value of the reconstructed image εˆ . 3. Increasing the Signal-To-Noise Ratio of the Image Due to Deconvolution of Complex Signals To increase the signal/noise ratio in objects with high absorption, complex signals can be emitted [36] if the radiating electronics of the flaw detector support this mode. After the compression procedure, which can be performed using matched filtering, the longitudinal resolution of the image will be about the same as using simple signals of minimal duration, but the signal ratio increases many times over. The compression of the echo signals using a matched filter can result in “side lobes”. In the paper [37], it was proposed to use the ME method for the compression of complex signals to achieve the effect of superresolution. Practical application of the ME method for compression of signals is impossible without a demonstration of its stability to the influence of noise. In the model experiment, the piezoelectric transducer (resonance frequency of 4.0 MHz, input angle of 40 degrees, half the opening angle of 20 degrees) moved along the scanning line, which is shown by the red arrow at Figure 2. Echo signals reflected from a 1.0 mm diameter flat-bottomed hole

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achieve the effect of superresolution. Practical application of the ME method for compression of achieve the effect of superresolution. Practical application of the ME method for compression of signals is impossible without a demonstration of its stability to the influence of noise. signals is impossible without a demonstration of its stability to the influence of noise. In 2018, the model the piezoelectric transducer (resonance frequency of 4.0 MHz, input Entropy 621 experiment, 7 of 22 In the20,model experiment, the piezoelectric transducer (resonance frequency of 4.0 MHz, input angle of 40 degrees, half the opening angle of 20 degrees) moved along the scanning line, which angle of 40 degrees, half the opening angle of 20 degrees) moved along the scanning line, which is is shown bybythe reflectedfrom fromaa1.0 1.0mm mmdiameter diameter flat-bottomed shown thered redarrow arrowatatFigure Figure 2. 2. Echo Echo signals signals reflected flat-bottomed withwith a 45-degree slope at a depth of 40.0 mm weremm recorded (Figures 2(Figures and 3). The phase-manipulated hole a 45-degree slope at a depth of 40.0 were recorded 2 and 3). The phasehole with a 45-degree slope at a depth of 40.0 mm were recorded (Figures 2 and 3). The phasesignal according to the Barker code with a length of 13 was used as a probing signal. manipulated signal with aa length lengthofof13 13was wasused usedasas a probing signal. manipulated signalaccording accordingto tothe theBarker Barker code code with a probing signal.

Figure 2. Drawing of the reference block with planar holes. Figure Figure2.2.Drawing Drawingofofthe thereference referenceblock blockwith withplanar planarholes. holes.

Figure 3. View of the reference block from the side of the holes with a diameter of 1 mm. The depth of flat-bottomed holes with numbers from 1 to 4 following 2, 10, 20 and 40 mm. Figure 3. View of the reference block from the side of the holes with a diameter of 1 mm. The depth Figure 3. View of the reference block from the side of the holes with a diameter of 1 mm. The depth of of flat-bottomed holes with numbers from 1 to 4 following 2, 10, 20 and 40 mm. flat-bottomed holes the withresult numbers from 1 to 4 following 2, 10, of 20 aand 40 mm. Figure 4a shows of visualization of the image flat-bottomed hole when a simple

pulse is emitted. Visualization means finding the echo envelope and displaying it without spatial Figure 4a shows the result of visualization of the image of a flat-bottomed hole when a simple processing along the the central beam of the transducer all points of the spatial hole aperture. the 4a shows result of visualization ofecho thefor image of a and flat-bottomed whenOn aspatial simple pulse Figure is emitted. Visualization means finding the envelope displaying it without picture, the lines of black outline the boundaries of the hole and the sample. It can be seen that the pulse is emitted. Visualization means finding the echo displaying it withoutOn spatial processing along the central beam of the transducer forenvelope all pointsand of the spatial aperture. the indication from the hole, the center of which is indicated by a cross-shaped marker, is barely detected processing along theblack central beam the of the transduceroffor allhole points ofthe thesample. spatial aperture. On thethat picture, picture, the lines of outline boundaries the and It can be seen because ofblack the outline noise level comparableofwith its amplitude. FigureIt4b shows thethat result of thethe the lines of the boundaries the hole and the sample. can be seen the indication indication from the hole, the center of echo which is indicated by a cross-shaped marker, is barely reconstruction of the image by these signals by the method of projection in spectral spacedetected (PSS from theofhole, the center of which is indicated by aamplitude. cross-shaped marker, is barelythe detected because because the noise level comparable with its Figure 4b shows result ofinthe or FT–SAFT) [38]. The signal-to-noise image obtained by the PSS increased by more than 12 dB of the noise level comparable with its amplitude. Figure 4b shows the result of the reconstruction of reconstruction of the image by these echo signals by the method of projection in spectral space (PSS comparison with the visualization, and the front resolution decreased from 15 mm to 1 mm. Figure theFT–SAFT) image by[38]. theseThe echo signals by theimage method of projection in spectral space (PSS or FT–SAFT) or signal-to-noise obtained by the more thansignals, 12 dB[38]. in 4c shows the image of the bottom of the hole recovered byPSS the increased method ofbyPSS in the The signal-to-noise image obtained by the PSS increased by more than 12 dB in comparison with the comparison with and and the front decreased from by 15 the mmmethod to 1 mm. compressed by the visualization, matched filtering, Figureresolution 4d—signals, compressed of Figure ME visualization, and theoffront resolution decreased from 15 mm 1inmm. Figure 4c increase shows image 4c shows the the(8). bottom hole recovered by to the method of PSS in thethe signals, according to image the formula The useofofthe a complex signal resulted an additional in the ofSNR the bottom the12 hole by and the method of PSS in thesignal signals, compressed by the of more than dB.recovered After the compression of the complex by the ME the matched noise compressed byof the matched filtering, Figure 4d—signals, compressed by method, the method of ME filtering, and Figure 4d—signals, compressed by the method of ME according to the Formula (8). level decreased by at least 20The dB, and theabeam resolution from mm to 0.2 mm. Since according to the formula (8). use of complex signaldecreased resulted in an 0.7 additional increase in the The use of a complex signal resulted in an additional increase in the SNRbyofthe more 12 dB.the After the SNR of more than 12 dB. After the compression of the complex signal MEthan method, noise compression of the complex signal by the ME method, the noise level decreased by at least 20 dB, level decreased by at least 20 dB, and the beam resolution decreased from 0.7 mm to 0.2 mm. Since and the beam resolution decreased from 0.7 mm to 0.2 mm. Since the Barker code was used, the level of «side lobes» is insignificant after the compression of the echo signals by the matched filtration.

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of 22 used, the level of «side lobes» is insignificant after the compression of the8echo signals by the matched filtration.

Figure 4. Images of the flat-bottomed hole 4, recovered from the the original original echoes echoes (a), (a), by by the PSS method method Figure recovered from (b) and and after after their their compression compression by by the the matched matchedfiltering filtering(c) (c)and andby bythe theME MEmethod method(d). (d). (b)

Thus, the use of complex signals and their compression by the МЕ method can increase the Thus, the use of complex signals and their compression by the ME method can increase the signal-to-noise ratio by more than 40 dB. This is a very large margin of sensitivity for ultrasonic signal-to-noise ratio by more than 40 dB. This is a very large margin of sensitivity for ultrasonic testing testing in materials with high absorption. in materials with high absorption. 4. Reconstruction Reconstruction of of the the Image Image by by Tofd-Echoes Tofd-Echoes 4. In the the practice practice of of ultrasonic ultrasonic nondestructive nondestructive testing, testing, the the time-of-flight time-of-flight diffraction diffraction (TOFD) (TOFD) [39] [39] In method is is widely widely used, used, when when pulsing pulsing and and receiving receiving transducers transducers with with aa wide wide beam beam spread spread pattern pattern method move along the welded joint along an axis perpendicular to the plane of Figure 5. The measured y move along the welded joint along an axis y perpendicular to the plane of Figure 5. The measured echo echo signals allow to quite detecteffectively, cracks quite sinceecho the p ( informative t , y ) are informative signals p(t, y) are and allow toand detect cracks sinceeffectively, the diffraction signals fromecho the upper lower thelower crackedges differ of in phase by about degrees. small diffraction signalsand from the edges upper of and the crack differ 180 in phase by The about 180 size of acquired echo signals data setecho allows monitoring a verymonitoring high speed.at a very high speed. degrees. The small size of acquired signals data setat allows In Figure Figure 5, 5, the the green green arrows arrows schematically schematically show show the the trajectory trajectory of the the propagation propagation of the the longitudinal wave pulse from the fragment of the radiating radiating piezo crystal to the the fragment fragment of the the receiving The point-type point-typereflector reflectoratatthe thepoint point rriisisshown shownininred redεε((rr)i )= = i ). receiving piezo piezo crystal. crystal. The δ (rδ−(rri − ) . rTo i i To calculate the received echo signal, it is necessary to calculate the delay time and amplitude for all calculate the received echo signal, it is necessary to calculate the delay time and amplitude for all the the rays when the pulse is emitted when an electric pulse se (t) with a central frequency of 5 MHz rays when the pulse is emitted when an electric pulse se (t ) with a central frequency of 5 MHz is is applied to the piezo crystal and to integrate on the surface of the piezo crystal, which will allow applied to the piezo crystal and to integrate on the surface of piezo allow calculating the pulse strm (ri , t) at the point ri . A set of trajectoriesthe from the crystal, reflectorwhich to the will surface of strm (rcan ri. A setthe calculating thepiezo pulsecrystal at used the point of trajectories reflector to of the receiving to estimate echo signalfrom p(ri , the t) measured bythe thesurface receiver. i , t ) be The finite dimensions of the piezo crystal lead to the fact that the shape of the measured echo signal the receiving piezo crystal can be used to estimate the echo signal p(ri , t) measured by the receiver. will change. The pulse from the reflector (depth 25 mm) at the intersection of the acoustic axes of the The finite dimensions of the piezo crystal lead to the fact that the shape of the measured echo signal transducers practically the same as the probe pulse and is shown in Figure 5 by the balloon with the will change. The pulse from the reflector (depth 25 mm) at the intersection of the acoustic axes of the red border. The echo signal from the point-type reflector at a depth of 45 mm is shown in Figure 5 by transducers practically the same as the probe pulse and is shown in Figure 5 by the balloon with the the blue balloon. The scale of both graphs is the same along the time axis and is equal to 1 µs, and the red border. The echo signal from the point-type reflector at a depth of 45 mm is shown in Figure 5 by amplitudes are equalized. It can be seen that when the distance from the acoustic axis of the echo the blue balloon. The scale of both graphs is the same along the time axis and is equal to 1 µs, and the shape changes significantly. amplitudes are equalized. It can be seen that when the distance from the acoustic axis of the echo shape changes significantly.

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Figure 5. Echoes measurement in the TOFD mode.

Figure 5. 5. Echoes measurement inin the TOFD mode. Figure Echoes measurement the TOFD mode.

p (ri , t ) , it is possible to construct a matrix G and to reduce the Thus, after calculating ,along t()r,i it a matrix and toG problem , t )isthe Thus, after calculating ,possible itaxis is possible construct aG matrix and tothe reduce the z tobyconstruct problem of estimation ˆMMЕ ( zpi()rip thetomeasured echo signal usingthe preduce of estimation εˆ MME (zi ) along the axis z by the measured echo signal p using the Equations (8) and z isbycalculated problem(8)ofand estimation axis the measured signal Equations (10). If theεˆpropagation timethe of the pulse from theecho centers of thep of using the MMЕ ( zi ) along (10). If the propagation time of the pulse is calculated from the centers of the of piezo crystal tdel (zi ) as t ( z ) piezo crystal as a function of the depth of the point reflector, then the form can be of the of ( z ) centers Equations (8) (10). If the propagation time of the pulse is calculated from the del and i a function of the depth of the point reflector, then the form ε(z) can be estimated from the formula pof (tbe (considered zi ))depth estimated from formula . This of formula can be considered asthe a degenerate (z.i )This piezo crystal as formula aˆfunction thea degenerate point reflector, thenof form ε ( zmethod. ) can be del ( zi )  delthe εˆ del ( zi ) = p(tdelthe (tzdeli )) can as version the C–SAFT version of the6from C–SAFT method. Figure shows calculated six point-type reflectors as foraadegenerate distance εˆdel (echo zi ) = signal p(tdel ( zpi ())t). from estimated thethe formula This the formula can be considered Figure 6 front showsfaces the calculated echo signal the sixpair point-type reflectors for a distance 2.5 mm p ( tmm. ) from between the of the transducers is 120 The first of reflectors with coordinates version of the C–SAFT method. between the front faces of thecoefficients transducers1isand 120−mm. The first apair of reflectors with coordinates and 7.5Figure mm, the reflection 1 simulates crack a height of 5 mm, 6 shows the calculated echo signal from the six with point-type reflectors forsimilarly, a distance p -1 ( t )simulates 2.5 mm and 7.5 mm, the reflection coefficients 1 and a crack with a of 5 The mm, pulse of the second pair has coordinates 22.5 and 27.5 mm, and the third one 42.5 andheight 47.5 mm. between the frontpair faces of the transducers is27.5 120mm, mm. The first pair of reflectors withThe coordinates similarly, second coordinates and one 42.5 mm. the head the wave and onehas reflected from22.5 the and object bottom ofthe thethird sample wereand not47.5 calculated. Echoes 2.5 mm and 7.5 mm, the reflection coefficients 1 and -1 simulates a crack with a height pulse of the head wave and one reflected from the object bottom of the sample were not calculated. of 5 mm, from the second and third pairs are resolved and have a phase shift of 180 degrees. Echo signals from similarly, has coordinates 22.5 and 27.5 mm, and the third onedegrees. 42.5 andEcho 47.5 mm. The Echoes from the the second second pair and third pairs are resolved and have a phase shift of 180 the first pair of reflectors will not be resolved, so at low depths tdel (zi ) changes weakly. The amplitude pulsefrom of thethe head and one reflected from the objectsobottom the sample not calculated. t del ( ziwere ) changes signals firstwave pair of reflectors will not be resolved, at lowofdepths and shape of pulses from the reflectors are different due to the divergence of the ultrasonic beam and Echoes from the second and of third pairs aretheresolved have a phase of 180 degrees. Echo weakly. The amplitude and shape pulses from reflectorsand are different due toshift the divergence of the finite size of the piezoelectric plates. thesignals ultrasonic beam finite of the piezoelectric from theand firstthepair ofsize reflectors will not beplates. resolved, so at low depths t del ( zi ) changes Thus, after calculating

weakly. The amplitude and shape of pulses from the reflectors are different due to the divergence of the ultrasonic beam and the finite size of the 4 piezoelectric plates. 6 1, 2 4

1, 2

6

5

3

Figure 6. Echoes measurement in thein TOFD mode. mode. Figure 6. Echoes measurement the TOFD 3

Figure 7a shows the result of the evaluation of

ˆ

5

( z ) (black graph) and ˆ ( z ) (red

i del i Figure 7a shows the result of the evaluation of εˆММЕ MME ( zi ) (black graph) and εˆ del ( zi ) (red graph) Figure 6. Echoes measurement in the TOFD mode. p ( z , t ) graph) when the matrix is formed taking into account the echo shape calculated thedepth of G when the matrix G is formed taking into account the echo shape p(zi , ti) calculated forforthe depth of 45 mm. This the that fact that images thethird thirdpair pair (reflectors (reflectors 55and have thethe form 45 mm. This led to led thetofact the the images of of the and6)6) have form of a ˆpair zi )«side εˆdel zi )40%. 7a the shows theofresult ofreflectors the evaluation of εpair (black graph) and (red of a  Figure –function, and the image of the thesecond second has lobes» ofof thethe order of (of ММЕ ( δ–function, and image the reflectors ofofthe has «side lobes» order

Under ideal reconstruction of the values of the reflection coefficient εˆ(zshape must be equal p(example zi , t ) calculated graph) when the matrix G is formed taking into account the echo for the i ) in this

depth of 45 mm. This led to the fact that the images of the third pair (reflectors 5 and 6) have the form of a δ –function, and the image of the reflectors of the second pair has «side lobes» of the order of

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40%. Under ideal reconstruction of the values of the reflection coefficient

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εˆ(zi )

in this example10must of 22

εˆММЕ ( zi ) and εˆdel ( zi ) at the p( zi , t ) forofall ( z ) has no difference calculation of the 7b matrix of result echo shape depths. Theεˆ εˆdel Gthe εˆ MME (zi ) and to 1 or −1. Figure shows of the evaluation del ( zi ) iat the calculation of be equal to 1 or −1. Figure 7b shows the result of the evaluation of

the matrix G ofp(t) echo shape zi , tnonlinear ) for all depths. The εˆ del (zi ) has no difference from the shape tdel (p(t) zi ) from the shape except forp(the scale transformation according to the dependency except for the nonlinear scale transformation according to the dependency tdel (zi ), but the εˆ MME (zi ) , but the εˆММЕ ( zi ) has a resolution of more than five times higher than for εˆdel ( zi ) , which allows has a resolution of more than five times higher than for εˆ del (zi ), which allows one to resolve reflectors reflectors of 1 and 2. The amplitude the the reflector is 1 small the calculation of 1one andto2.resolve The amplitude the reflector is 1 small of since calculation of p(since z , t) did not take into i

p( zi , t ) the didfeature not take accountthe therefractive feature ofindex calculating thefirst refractive critical account of into calculating near the criticalindex anglenear [40].the It isfirst clear that such a convincing result (Figure 7b) obtained in the absence of measurement noise and operator angle [40]. It is clear that such a convincing result (Figure 7b) obtained in the absence of measurement noise. To obtain better images of reflectors 1 and 2, is necessary to2,bring together the emitting and noise and operator noise. To obtain better images ofitreflectors 1 and it is necessary to bring together receiving converters. the emitting and receiving converters.

Figure 7. 7. Image Image evaluation evaluation of of reflectors reflectors ee (z) (z) and and ee(z) (z)by byTOFD–echoes. TOFD–echoes. Figure

5. Improving Improving the the Image Image Quality Quality of of the the Reflectors ReflectorsReconstructed Reconstructedby bythe theMe MeMethod Methodon onaaSet Set of 5. Echoes of Echoes This section section provides providesexamples examplesof of the the use use of of the the ME ME method method to to restore restore the the reflectors reflectors image image from from This echo signals signals with withsuper-resolution super-resolutionand andincreased increasedsignal/noise signal/noise ratio (see (see section section Restoring Restoring the the image image echo of reflectors). reflectors). of 5.1. 5.1. Increasing Increasing the the Resolution Resolution of of an an Image Image 5.1.1. 5.1.1. Four Four Side Side Drilled Drilled Holes Holes Figure Figure 88 shows shows aa sketch sketchof ofthe theduralumin duraluminblock blockwith with0.5 0.5mm mmdiameter diameterside sidedrilled drilledholes. holes. Image Image of holes located at a depth of 36 and 38 mm with a distance between centers about 2 mm was restored. of holes located at a depth of 36 and 38 mm with a distance between centers about 2 mm was restored. For of the the echo echosignals signalsofofthe theused usedantenna antennaarray array MHz, elements, of For the the acquisition acquisition of (5 (5 MHz, 32 32 elements, thethe sizesize of the the piezo crystal is 0.9 × 10 mm, pitch 0.1 mm) on a plexiglass prism with a tilt angle of 20 degrees. piezo crystal is 0.9 × 10 mm, pitch 0.1 mm) on a plexiglass prism with a tilt angle of 20 degrees. The The antenna array prism are schematically shown in Figure antenna array andand prism are schematically shown in Figure 8. 8.

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11 of 22 1111ofof2222

Figure Figure8.8.Rescattering Rescatteringtest testconsists consistsfrom fromthree threegroups groupsof offour fourholes holesof of0.5 0.5mm mmdiameter. diameter. Figure 8. Rescattering test consists from three groups of four holes of 0.5 mm diameter.

Figure 9 shows the result of the reconstruction of reflectors by the C–SAFT method using 1024 Figure 9 shows the result of the reconstruction of reflectors by the C–SAFT method using 1024 Figure 9 shows the result ofresolution the reconstruction of reflectors by the C–SAFT 1024 (tt)).. Non-sufficient echoes Non-sufficient ofthe theimage image doesnot notallow allow determinemethod notonly onlyusing the type nm ( echoes ppnm resolution of does totodetermine not the type of p ( t ) echoes . Non-sufficient resolution of the image does not allow to determine not only the type nm of the reflector but also their quantity. On the image lines of black color are plotted holes’ contour. the reflector but also their quantity. On the image lines of black color are plotted holes’ contour.

of the reflector but also their quantity. On the image lines of black color are plotted holes’ contour.

Figure 9. The image of the group of the side drilled holes, closest to the surface, of the rescattering test, reconstructed by the ME method.

Figure 9. The image of of thethe group holes,closest closesttoto surface, of the rescattering Figure 9. The image groupofofthe theside side drilled drilled holes, thethe surface, of the rescattering test, test,Figure reconstructed by theME ME method. 9 shows the result of the reconstruction of the reflectors image by 400 echoes pnm (t) by reconstructed by the method.

the ME method ( α = 10.0 , μ =10 ) according to Equation (8). The longitudinal and frontal (t) the Figure 9 shows thethe result ofofthe reconstruction of the reflectors reflectors imagebyby400 400 echoes by Figure shows result the reconstruction echoes pnm p (tnm ) by resolution of9the image increased more than twofoldofinthe comparisonimage with the image of Figure 10. The − 5 −5 ME (α (=α10.0, µis=clearly according Equation (8). The longitudinal and frontalof resolution of the image MEmethod method , 10μ =)10 ) inaccording to Equation (8). The longitudinal and frontal = 10.0 of the three holes seen theto ME image. The amplitude of the indication the fourth the image increased more than twofold in comparison with the image of Figure 10. The image of the hole was enough to detect because of thein shading effect. resolution ofnot thelarge image increased moreit than twofold comparison with the image of Figure 10. The three holes is clearly seen in the ME image. The amplitude of the indication of the fourth hole was not image of the three holes is clearly seen in the ME image. The amplitude of the indication of the fourth large enough to detect it because of the shading effect. −5

hole was not large enough to detect it because of the shading effect.

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Figure 10. The image of the group of the side drilled holes, closest to the surface, of the rescattering test, reconstructed by the C–SAFT method.

5.1.2. 2.5d–Image of Models of Two Cracks in Welded Joint of du800 Type Model experiments were carried out with a welded joint of 800 mm diameter, in which seven artificial reflectors were made. This type of pipelines is typical for the nuclear power industry of the Russian Federation. Seven artificial reflectors were made in the weld. Reflectors with numbers 3 and 4 simulating the cracks have tilt 20 and −14 degrees to the axis z, a height of about 5 mm and the length along the weld about 34 mm. Reflector are milled grooves with a width of 1 mm. Figure 11 shows a picture of the weld, part of the scanning system and two antenna arrays installed on the prism and located approximately at the location of the reflectors 3 and 4. The reinforcement bead of the weld on the outer surface has been removed. The antenna array (5 MHz, 32 elements, piezo crystal size 0.9 × 10 mm, pitch 0.1 mm) on a rexolite prism with a 35-degree angle of inclination was used to acquire the echo signals. Entropy 2018, 20, x 13 of 22

11. The weld800 frommm 800 mm diameterpipeline pipeline and antenna arrays installed on the prisms Figure 11. Figure The weld from diameter andtwo two antenna arrays installed on the prisms are located approximately at the location of the grooves 3 and 4. The balloon shows the image of the are located approximately at the location of the grooves 3 and 4. The balloon shows the image of the end of the cutted weld. end of the cutted weld.

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Figure 12 on the left shows the C–SAFT image, and on the right the ME image reconstructed according to Equation (8) obtained by the P-side antenna array. The B-type images show the contours of the grooves 3 and 4. It can be seen that the coordinates of the indications of the reflector 4 differ from the contours of the grooves. This can be explained by the fact that, firstly, the weld surface is rough. And, secondly, the welded joint can introduce additional distortions due to its special acoustic properties. TheThe resolution of800 themm MEdiameter image in comparison the arrays C–SAFT image Figure 11. weld from pipeline and twowith antenna installed onhas the more prismsthan doubled. According to the D-type image, it can be assumed that the upper boundary of the groove are located approximately at the location of the grooves 3 and 4. The balloon shows the image of the 4 is rough, despite the fact that it is milled. end of the cutted weld.

Figure 12. 12. C–SAFT images (a) and ME images (b) for(b) the P-side the TdT scheme (a transverse Figure C–SAFT images (a) and ME images for thefor P-side foracoustic the TdT acoustic scheme wave is emitted, a transverse wave, reflected from thereflected defect, is received). The D- and C-type images (a transverse wave is emitted, a transverse wave, from the defect, is received). The are Dcombined to the maximum. and C-type images are combined to the maximum.

5.2. An An Increase Increase in in the the Signal-To-Noise Signal-To-NoiseRatio Ratioofofthe theImage Imagefor forthe theCase CaseofofLarge LargeAbsorption Absorption 5.2. In Section Section 3, 3,when whenusing usingaacomplex complexsignal, signal,ititwas waspossible possibleto toincrease increasethe theimage imagesignal-to-noise signal-to-noise In ratioby bymore morethan than20 20dB dBfor foraa11mm mmdiameter diameterplanar planarhole holeat at40 40mm mmdepth depthby bydeconvolution deconvolutionof ofeach each ratio echo signal. Reconstruction of the image by the method of ME from a set of echoes by Equation (8) echo signal. Reconstruction of the image by the method of ME from a set of echoes by Equation (8) also allows one to increase its quality. To acquire echoes by FMC, a 16-element antenna array with an operating frequency of 4 MHz and a step of 2.5 mm was used, each element of which was mounted on an individual prism with an angle of 45 degrees. Figure 13a shows the image obtained by the correlation method according to Equation (6). The image is characterized by a high level of noise, where is not particularly clearly visible indications of the flat-bottom holes. Signal to noise ratio in the ME image reconstructed according to Equation (8) (Figure 13b), compared with Figure 13a, increased by 16 dB, when measured within the area marked with a green square, and the resolution increased by at least by three times.

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15

15

250

20

250

20 200

200

25

3

25 150 z, мм

z, мм

150 30

30

100

4

35

100 35

50

40

-4

-2

0

2

4

6 x, мм

(a)

8

10

12

14

16

50

40

-4

-2

0

2

4

6 x, мм

8

10

12

14

16

(b)

Figure 13. SAFT- (a) and ME-images (b) for the TdT acoustic scheme. Figure 13. SAFT-(a) and ME-images (b) for the TdT acoustic scheme.

5.3. An Increase in the Signal-To-Noise Ratio in Materials with Structural Noise 5.3. An Increase in the Signal-To-Noise Ratio in Materials with Structural Noise There are materials in which so-called structural noise occurs at the boundaries of crystal grains There materials in whichofso-called structural occurs the boundaries crystal grains due to are multiple rescattering the probing pulses noise [41]. For suchatmaterials, it is notofpossible to increase due to multiple rescattering of the probing pulses [41]. For such materials, it is not possible to increase the signal-to-noise ratio due to the increase in the energy input into the material, since simultaneously the signal-to-noise ratioindue the increase in the energy inputfrom into the since simultaneously with the increase theto amplitude of the pulse reflected thematerial, defect, increases the amplitude of withthe thepulses increase in the amplitude of the pulse reflected from the defect, increases thematerials, amplitude reflected from the boundaries of the crystal grains. In coarse-grained theofmain the pulses reflected from the boundaries of the crystal grains. In coarse-grained materials, the main factor affecting the size of the minimum detectable reflector is not a small sensitivity, but a high level factor thenoise. size of minimum detectable reflector is not small sensitivity, butshort a high level [42]. ofaffecting structural Tothe reduce its level, it is recommended toameasure using very pulses of structural noise. To reduce its level, it is recommended to measure using very short pulses [42]. The reconstruction of the image of the reflectors by the ME method should increase the resolution, The reconstruction of the image of the reflectors by the ME method should increase the resolution, which is similar to the reduction of the echoes length and this should lead to a decrease in the level which similar tonoise. the reduction of the echoes length and this should lead to a decrease in the level of ofisstructural structuralLet noise. us demonstrate this by the example of austenitic weld joint testing. The sample on which the Let us demonstrate by the example of austenitic weld joint testing. sampleofon69which the an measurements werethis carried out was a fragment of the pipeline with aThe thickness mm, with measurements were carried out was a fragment of the pipeline with a thickness of 69 mm, with austenitic welded joint (Figure 14). From one edge on the border of the “weld seam-baseanmetal austenitic welded joint of (Figure 14).58From one edge ondiameter the border the “weld (perlite)” at depths 8, 33 and mm three 3.0 mm sideofdrilled holes seam-base were made,metal indicated (perlite)” at depths of 8, 33 and 58 mm three 3.0 mm diameter side drilled holes were made, indicated by numbers 4, 5 and 6. The weld area is marked with a translucent red polygon. The longitudinal by numbers 4, 5 and 6. The weld with a The translucent red polygon. The longitudinal wave velocity in the weld jointarea wasis setmarked to 5.6 mm/µs. measurements were carried out in the mode waveofvelocity in the weld joint was set to 5.6 mm/µs. The measurements were carried out in the mode triple scanning at five points with a step of 9.3 mm antenna array (5 MHz, 32 elements, piezocrystal of triple at five with aonstep of 9.3 mmprism antenna array (5 MHz, tilt 32 elements, sizescanning 0.9 × 10 mm, 0.1points mm pitch) a plexiglass with a 20-degree angle. Thepiezocrystal scanning range size 0.9 × 10 mm, 0.1 mm pitch) on a plexiglass prism with a 20-degree tilt angle. The scanning range is schematically shown in Figure 14 with a red arrow. One LdL acoustic scheme (the longitudinal is schematically shown inlongitudinal Figure 14 with a redreflected arrow. One LdL (thewas longitudinal wave wave is emitted, the wave, from theacoustic defect, isscheme received) used to reconstruct is emitted, the longitudinal wave, reflected from the defect, is received) was used to reconstruct the the images of reflectors. images of reflectors. Figure 15a shows the C–SAFT image of the boundary of the hole 6. The structural noise has a level of about 6 dB, against which one can see the indication of the hole, the contours of which are shown as black ellipse. Figure 15b presents the result of processing the same signals by the ME method according to Equation (8) after 15 iterations at the background coefficient µ = 10−6 . In comparison with the image obtained by the correlation method, the beam resolution of the image obtained by the ME method increased approximately three times, and the signal/noise ratio increased by more than 30 dB. However, the indication of the hole boundary on the ME image is split, due to the fact that when calculating the matrix G it is necessary to take into account different acoustic properties of the base metal and the weld joint.

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Figure 14. A view of a 69 mm pipeline sample with three side drilled holes.

Figure 15a shows the C–SAFT image of the boundary of the hole 6. The structural noise has a level of about 6 dB, against which one can see the indication of the hole, the contours of which are shown as black ellipse. Figure 15b presents the result of processing the same signals by the ME method according to Equation (8) after 15 iterations at the background coefficient μ = 1 0 − 6 . In comparison with the image obtained by the correlation method, the beam resolution of the image obtained by the ME method increased approximately three times, and the signal/noise ratio increased by more than 30 dB. However, the indication of the hole boundary on the ME image is split, due to the fact that Figure when calculating is necessary to take into account different acoustic G it sample 14. A ofthe aof 69amatrix mm pipeline with three sideside drilled holes. Figure 14.view A view 69 mm pipeline sample with three drilled holes. properties of the base metal and the weld joint. Figure 15a shows the C–SAFT image of the boundary of the hole 6. The structural noise has a level of about 6 dB, against which one can see the indication of the hole, the contours of which are shown as black ellipse. Figure 15b presents the result of processing the same signals by the ME method according to Equation (8) after 15 iterations at the background coefficient μ = 1 0 − 6 . In comparison with the image obtained by the correlation method, the beam resolution of the image obtained by the ME method increased approximately three times, and the signal/noise ratio increased by more than 30 dB. However, the indication of the hole boundary on the ME image is split, due to the fact that when calculating the matrix G it is necessary to take into account different acoustic properties of the base metal and the weld joint.

(a)

(b)

Figure 15. SAFT-(a) and ME-images (b) for the P-side for the LdL acoustic scheme.

6. Increasing of Echoes Acquisition Rate The disadvantage of acquiring echoes by double scanning is a large amount of measured data. With an increase in the number of elements Ne of the antenna array, the number of echoes increases as a square. Therefore, in spite of the fact that the echoes are measured in Ne cycles of the probing

(a)

(b)

6. Increasing of Echoes Acquisition Rate The disadvantage of acquiring echoes by double scanning is a large amount of measured data. With an increase in the number of elements

Ne of the antenna array, the number of echoes increases

asEntropy a square. Therefore, in spite of the fact that the echoes are measured in N e cycles of the probing 2018, 20, 621 16 of 22 signal, the transfer of a large volume of echoes from the measurement unit to the computer for image reconstruction may take a long time. signal, the transfer of a large volume of echoes from the measurement unit to the computer for image From the point of view of the theory of multichannel communication, the dual scan mode is reconstruction may take a long time. similar to the situation where users send a message in turn which is received by all users. The From the point of view of the theory of multichannel communication, the dual scan mode is consistent nature of the radiation of the probe pulse allows us to answer the question: “Who is the similar to the situation where users send a message in turn which is received by all users. The consistent source of the message?” This communication mode, when each pair of receiver-transmitter is nature of the radiation of the probe pulse allows us to answer the question: “Who is the source of allocated the whole spectrum or most of it for a selected period of time, is called time-division the message?” This communication mode, when each pair of receiver-transmitter is allocated the multiple access (TDMA) [43]. If all users could send their messages at the same time, and at the whole spectrum or most of it for a selected period of time, is called time-division multiple access reception, it would be possible to select all the messages and understand: “Who sent the (TDMA) [43]. If all users could send their messages at the same time, and at the reception, it would be message?”—it would essentially increase the rate of echo signal acquisition and decease their possible to select all the messages and understand: “Who sent the message?”—it would essentially amount. To solve this problem in the theory of multichannel communication, when the transmission increase the rate of echo signal acquisition and decease their amount. To solve this problem in the channels have a common frequency band but different code modulation, a code division multiple theory of multichannel communication, when the transmission channels have a common frequency access technology has been developed, which is called code division multiple access (CDMA) [43]. band but different code modulation, a code division multiple access technology has been developed, For its implementation, all elements of the antenna array are simultaneously excited, each with its which is called code division multiple access (CDMA) [43]. For its implementation, all elements of unique probing pulse sn (t ) , and echoes are simultaneously received by all elements of the antenna the antenna array are simultaneously excited, each with its unique probing pulse sn (t), and echoes pm(array. t) areSchematically, array. Schematically, received echoes measured in CDMA shown in Figure 16 on the rightin are simultaneously by all elements of themode antenna echoes measured CDMA in Figure 16 on the right and they can be represented by the expression: and they mode can beprepresented by the expression: m ( t ) are shown N N ee

pmp(mt()t )==  (t). . ∑ ppnn,m , m (t ) nn= =11

(17) (17)

Figure 16. The principle of increasing the speed of recording echo signals.

The first way to use this approach is to decode the echo signals pm (t) so that the m-th subscriber can make an evaluation of the echo signal pen,m (t) from the subscriber with the number n. For effective

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decoding of echo signals, the correlation function Rn,m (τ ) of a set of coding signals sn (t) intended for excitation of elements of the antenna array must have the following property: Rn,m (τ ) =

Z∞

sn (t)sm (t + τ )dt = δn,m δ(τ ),

n, m = 1, 2, . . . , Ne .

(18)

−∞

where δn,m is the Kronecker symbol. The set of signals having the property (18) is called orthogonal. Signals for practical use with this property are not known, but several sets of coding signals sn (t) have been developed, which to some extent approach the ideal set with the property (18). Sets of such signals are called pseudo-orthogonal. These include phase-manipulated signals encoded by a sequence of length Nc . The law of change of the chip phase can be determined by the Kasami codes [44], Gold [45], de Brain [46], in which the phase of each chip can take values 0 or 180 degrees. It is possible to use Zadov–Chu sequences with arbitrary phase for each chip [47]. A set of sequences with a random phase can also be considered as pseudo-orthogonal. Each chip can determine the phase of one period at the carrier frequency f c , but the option of random frequency measurement f c + δ f from chip to chip is possible. For the emitting of such signals to the arbitrary pulse generator of the flaw detector, higher requirements are imposed, in comparison with the usual shock excitation generator. N The first way to restore the image is as follows. After selecting a set of code signals {sn (t)}n=k 1 , their simultaneous pulsing and reception in one cycle, the measured echoes pm (t) must be decoded to evaluate the signals p(t), as if measurements were made in the FMC mode. By decoded echoes pen,m (t) using the C–SAFT method, one can restore the image (arrow and rectangle with a yellow fill in Figure 16). In Figure 16 the decoding procedure is likened to the passage of a beam of light through a prism. If we use the color analogy in Figure 16, the decoding with the correct selection of N signals {sn (t)}n=k 1 will allow the signals pn,m (t) from the total set pm (t) to be “colored” only in one color. Considering the circumstance that pulses of “different colors” can be very close to each other, the algorithm of decoding should provide a high resolution in time domain. A common method of decoding complex signals pm (t) is matched filtering with a code signal sn (t). Given that matched filtering in the time domain is equivalent to convolution [43], the decoding operation can be written as: pen,m (t) =

Z∞

pm (τ )sn (τ − t)dτ

(19)

−∞

This algorithm for compression of complex signals has a high speed and allows one to obtain images with a frequency of more than 10 Hz, but it does not allow to obtain a low level of interchannel noise and achieve the effect of super-resolution. For decoding of echoes by Equation (19), the level of interchannel noise, i.e., the average value of the cross-correlation function of signals in the set N

1

{sn (t)}n=k 1 , can be estimated as the value of ( Nc )− 2 . A more complex method of decoding simple or complex signals is based on the use of the maximum entropy method [37] and was considered in the section Processing of echo signals presented in a complex form. To obtain the estimation of echo signals by the ME method by measured pen,m (t) N for each sn (t) from a set of coding signals {sn (t)}n=k 1 , it is necessary to form a matrix G. The time of decoding of echoes pm (t) by the ME method is much longer than by the Formula (19), which will lead to decreasing the frequency of image formation. But for an automated system with post-processing, such a speed limit is not essential. The second way to obtain the image is to get the image of the reflectors immediately using the ME method without decoding pm (t), as described in section Restoring the image of reflectors. This method in Figure 16 corresponds to an arrow and a rectangle with a green fill. The efficiency of the algorithms for decoding echoes depends on the type of reflectors being reconstructed and their quantity, since this depends on how close to each other the echoes are located. That is, the quality of the decoding of echoes from a single point reflector can be very good [48],

The efficiency of the algorithms for decoding echoes depends on the type of reflectors being reconstructed and their quantity, since this depends on how close to each other the echoes are located. That is, the quality of the decoding of echoes from a single point reflector can be very good [48], but when processing echoes from many reflectors, the reconstructed image may end 18up with an Entropy 2018, 20, 621 of 22 unacceptably high level of inter-channel interference. Ideally, the use of CDMA technology will allow to measure for an antenna array of 128 elements only in one cycle instead of 128, and decrease the but when processing echoes from many reflectors, the reconstructed image may end up with an number of unacceptably echoes from 16348 128. high level to of inter-channel interference. Ideally, the use of CDMA technology will allow The model experiment was carried out on the only sample described in Section 4 with drilled to measure for an antenna array of 128 elements in one cycle instead of 128, and decrease the holes number of echoes 16,348 128. in the rescattering test, butfrom using thetoelectronic unit of the AUGUR-ART system, which allows pulsing The model experiment was carried out the sample described in Section 4 with drilled holesaccording in complex signals. As a set of complex signals, 32on( N ) phase-manipulated signals k  N e  32 the rescattering test, but using the electronic unit of the AUGUR-ART system, which allows pulsing

to the Kasami code were with length Each period had a signals frequency randomly 15N.e = complex signals. As a used set of complex signals, 32N(N 32) phase-manipulated according c  k = thethe Kasami used with length Nc = 15.echoes, Each period a frequency randomly distributed distributedtoin [5 ±code 2] were MHz band. To acquire an had antenna array was used (5 MHz, 32 in the [5 ± 2] MHz band. To acquire echoes, an antenna array was used (5 MHz, 32 elements, crystal elements, crystal size 0.9 × 10 mm, 0.1 mm pitch) on a rexolite prism with a slope angle of 35 degrees. size 0.9 × 10 mm, 0.1 mm pitch) on a rexolite prism with a slope angle of 35 degrees. Figure 17 Figure shows17an example of one complex probing (red graph) prepared forand study and shows an example of one complex probing signal signal (red graph) prepared for study emitted and received by the by equipment used (black Onecan can the limited bandwidth emitted and received the equipment used (blackgraph). graph). One seesee thatthat the limited bandwidth of the antenna array distorts the ideal shape of the probing signal: in the received echo signal, sharp of the antenna array distorts the ideal shape of the probing signal: in the received echo signal, sharp phase jumps are actually lost when the chip changes and the signal amplitude changes from chip phase jumps are actually lost when the chip changes and the signal amplitude changes from chip to to chip. chip.

17. An example prepared for emitting complex probing probing signal (the(the graph in red)inand theand the Figure 17. Figure An example prepared for emitting thethe complex signal graph red) measured echo signal (the graph in black). measured echo signal (the graph in black).

18a shows the image of the four side drilled holes reconstructed by the C–SAFT method in Figure 18aFigure shows the image of the four side drilled holes reconstructed by the C–SAFT method the set of echoes measured during the emission of a simple signal in the FMC mode. There are three in the set of echoes measured during the emission of a simple signal in the FMC mode. There are indications corresponding to the boundaries of the three holes, which are accurately resolved since the three indications corresponding to the wave. boundaries thethem, threetheholes, accurately resolved reconstruction made for transverse But apartoffrom pulseswhich scatteredare between the holes formed several false indications, which do not allow us to determine not just type scattered of reflectors, since the reconstruction made for transverse wave. But apart from them, the the pulses between but also their quantity. The signal-to-noise ratio of the image can be estimated as 28 dB. Figure 18b the holes formed several false indications, which do not allow us to determine not just the type of shows the image reconstructed by the correlation method according to Equation (6) during decoding reflectors, but also their quantity. The signal-to-noise ratio of the image can be estimated as 28 dB. by matched filtering. Due to the high level of inter-channel interference, the image quality due to Figure 18b ashows the image reconstructed correlation according to Equation low signal-to-noise ratio of 12 dB hasby anthe unacceptably lowmethod quality. Figure 18c shows the result(6) of during decoding by filtering. Due to thereflectors high level interference, the image thematched reconstruction of the image of the withof 32 inter-channel echoes pm (t) by the ME method according to quality − 5 Equation (8) (α = 25.0, µ = 10 of).12 ThedB longitudinal and frontal resolution the image increased due to a low signal-to-noise ratio has an unacceptably low of quality. Figure 18cmore shows the than twofold in comparison with the image of Figure 18a. On the ME image, the indications of themethod result of the reconstruction of the image of the reflectors with 32 echoes pm (t ) by the ME four holes are clearly visible, and one false indication with a commensurate amplitude. It should be

according to Equation (   25.0 andthan frontal resolutiontime of the , of  echoes  105 p).mThe recalled that the(8) acquisition time (t) is longitudinal in Ne times shorter the measurement in image FMC mode. data setin of comparison the measured echo signals as much increased more thanThe twofold with the is image ofsmaller. Figure 18a. On the ME image, the indications of the four holes are clearly visible, and one false indication with a commensurate amplitude. It should be recalled that the acquisition time of echoes pm (t ) is in N e times shorter than

the measurement time in FMC mode. The data set of the measured echo signals is as much smaller.

Entropy 2018, 20, 621 Entropy 2018, 20, x

19 of 22 19 of 22

Figure Figure18. 18.The Theimage imagerecovered recoveredby bythe themethod methodfrom fromthe theset setofofechoes echoesmeasured measuredby bythe theemission emissionofofaa simple simplesignal signalininthe theFMC FMCmode mode(a), (a),by bythe thecorrelation correlationmethod methodwith withdecoding decodingby bythe thematched matchedfiltering filtering (b) and the ME method without decoding (c). (b) and the ME method without decoding (c).

7.7.Conclusions Conclusions The Theuse useofofthe themaximum maximumentropy entropymethod methodallows allowsone onetotoimprove improvethe thequality qualityofofthe thereconstructed reconstructed image of the reflectors and thereby increase the reliability of ultrasonic non-destructive testing. image of the reflectors and thereby increase the reliability of ultrasonic non-destructive testing. TheThe use use of the ME method in the model experiments increased the resolution of the image by a factor of the ME method in the model experiments increased the resolution of the image by a factorof of three threein incomparison comparisonwith withthe thelinear linearreconstruction reconstructionmethods. methods.This Thismade madeititpossible possibletotomore moreaccurately accurately determine determinethe thetype typeand andnumber numberof ofreflectors. reflectors. Compression of complex signals bythe theME MEmethod method made it possible to increase the signal-toCompression of complex signals by made it possible to increase the signal-to-noise noise than the imaging in materials a high attenuation, whichit made it ratio ratio more more than by 20 by dB 20 fordB thefor imaging in materials with a with high attenuation, which made possible possible to detect reflectors at a great depth with superresolution. The ME method is equivalent to to detect reflectors at a great depth with superresolution. The ME method is equivalent to reducing reducing the duration of echolength, signalsand length, and this should thestructural level of structural noise. In the duration of echo signals this should reduce thereduce level of noise. In the model the model experiment, it was possible to reduce the level of structural noise of the reflector image by experiment, it was possible to reduce the level of structural noise of the reflector image by 12 dB. 12 dB.Since when processing echoes measured in TOFD mode, the shape of the echoes depends on Since when measured in method, TOFD mode, thethis shape the echoes on the the depth of theprocessing reflectors, aechoes variant of the ME taking intoofaccount, wasdepends proposed. In a depth of the reflectors, a variant of the ME method, taking this into account, was proposed. In a numerical experiment, the effectiveness of this approach is shown. numerical experiment, the effectiveness of this approach is shown. The possibility of using the ME method for obtaining a 2.5D-image of the reflectors in a The of using method for obtaining a 2.5D-image of the a reference referencepossibility block made fromthe 800ME mm diameter pipe. Application of the MEreflectors method in together with block made from 800 mm diameter pipe. Application of the ME method together with CDMA CDMA technology allowed to reduce the echoes acquisition time by FMC and obtain an image with technology allowed to model reduceexperiment the echoes acquisition by FMC andofobtain an image with superresolution. In the it was possible time to increase the rate echo signal registration superresolution. In the model experiment it was possible to increase the rate of echo signal registration for the 32-element elemental array by 32 times. With the increase in the number of

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for the 32-element elemental array by 32 times. With the increase in the number of elements of the antenna array, the proposed approach further increases the rate of registration of echoes. This can be very important for ultrasound medical diagnosis. Acknowledgments: The author expresses his gratitude to the director of the Research and Production Center for Nondestructive Testing ECHO + Vopilkin Alexei Kharitonovich for supporting this work. Conflicts of Interest: The author declares no conflict of interest.

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