ultrasound imaging algorithm: half-matrix focusing method

Research ArticleUltrasound Imaging Algorithm Half-Matrix Focusing MethodBased on Reciprocity

Xufei Guo 12 Yan Han1 and Pengfei Nie1

1Shanxi Key Laboratory of Signal Capturing amp Processing North University of China Taiyuan 030051 China2Lvliang University Lvliang 033000 China

Correspondence should be addressed to Xufei Guo gx_nuc163com

Received 30 September 2020 Revised 11 January 2021 Accepted 16 January 2021 Published 2 February 2021

Academic Editor (omas Schuster

Copyright copy 2021 Xufei Guo et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

(e ultrasonic phased array total focusing method (TFM) has the advantages of high imaging signal-to-noise ratio (SNR) and highdefect resolution but the problem of large amount data capturing and processing limits its practical industrial applications Toreduce the imaging calculation demand of the total focusing method a half-matrix focusing method (HFM) is proposed based onthe acoustic reciprocity (e method simplifies the calculation process of full-matrix data capturing (FMC) and total focusimaging (e experimental results show that the signal obtained by the linear array transceiver sensor is highly consistent and theimaging resolution and signal-to-noise ratio of the half-matrix focusing method are slightly lower than those of full-matrixfocusingmethod and higher than those of the B-scan imaging However compared with TFM data acquisition and computationalefficiency using the HFM have been improved significantly

1 Introduction

Ultrasonic phased array testing adopts multichannel delaytransceiver technology to emit each array element so as toachieve the angle deflection of the acoustic beam and focuson depth changes and electronic scanning which improvethe ability to detect defects [1ndash4] (e ultrasonic phasedarray imaging method based on postprocessing has beendeeply studied by scholars Holmes et al [5] first proposedthe concepts of full-matrix capture (FMC) and establishedthe full-focus imaging method (TFM) based on FMCCompared with the conventional ultrasonic phased arrayfocusing algorithm TFM can realize the synthetic focusingof any point in the detection region and the imaging qualityis obviously better [6ndash9] (is method has therefore beenextensively used in the fields of aviation nuclear powerhuman organs and so on [10ndash12]

However due to the large amount of full-matrix datainvolved TFM imaging calculations are time-consumingand have poor real-time processing in real time which limit

their industrial applications To improve imaging efficiencyof TFM Sutcliffe et al [13] proposed a method of hardwareacceleration of postprocessing using GPU Bouaziz et al[14 15] proposed a real-time method that uses parallelcomputing with multiple field-programmable gate arraysBut the cost will increase significantly by improving thehardware architecture Another research goal is to improvethe imaging algorithm while maintaining the image qualityHunter et al [16] introduced a generalization of the wave-number algorithm for Fourier-domain full-matrix imagingHu et al [17] proposed a sparse-TFM imaging method basedon sparse array optimization and new edge-directed inter-polation For the problem of limited acquisition speedMoreau et al [18] proposed an effective aperture methodbased on far-field approximation In order to improve theimaging efficiency Zhao et al [21] used the upper triangledata to complete the ultrasound focus imaging

In the article a half-matrix focusing method (HFM)based on transceiver sensor reciprocity is developed toimprove inspection efficiency (is method can effectively

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 8888469 11 pageshttpsdoiorg10115520218888469

reduce full-matrix data capturing and TFM imaging timewhile maintaining the image quality (e remainder of thispaper is organized as follows In Section 2 the FMC theoryand the imaging methods are introduced Section 3 presentsthe experimental methods and procedures for defect de-tection Moreover the experimental results of data pro-cessed with conventional TFM and HFM are analyzedSection 4 compares HFM imaging and B-scan imaging anddiscusses the subsequent research directions of the HFMimaging Finally the conclusions are given in Section 5

2 Methods

21 FMC and TFM Figure 1 shows the principle of full-matrix ultrasonic data capture [19 20] A linear arrayconsisting of N elements excites the ultrasonic wave insequence During each excitation the ultrasonic signal isreceived by every array element Finally an N times N A-scandata matrix is obtained HereAij(t) is the N times N A-scansignal emitted by element i and received by element j (eformat of full-matrix data is shown as follows

A

A11 A12 A1j A1N

A21 A22 A2j A2N

Aij Ai2 Aij AiN

AN1 AN2 ANj ANN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

Figure 2 shows the principle of total focus algorithm[19 20] Image reconstruction using full-matrix data isdivided into two steps First discretization of the inspectedregion into a grid is performed Secondly calculation of theamplitude intensity at a grid point P(xp zp) is calculated by

Ip xp zp1113872 1113873 1113944

N

i11113936N

j1Aij tij xp zp1113872 11138731113872 1113873 (2)

where tij(xp zp) is the time of flight starting from thetransmitter element i passing through the focusing pointP(xp zp) and arriving at the receiver element j (e time offlight is defined by

tij xp zp1113872 1113873

xi minus xp1113872 11138732

+ z2p

1113969

+

xj minus xp1113872 11138732

+ z2p

1113969

1113874 1113875

c

(3)

where c is the propagation velocity of ultrasound in themedium and (xi 0) and (xj 0) are the Cartesian coordinatesof the transmitter and the receiver respectively

22HFM Consider that the performance indicators of eachchannel of the linear ultrasonic array transducer have ex-cellent consistency (at is the signal that is transmittedfrom element i and received on element j is identical to the

signal transmitted on element j and received on element i It isnamed acoustic reciprocity According to the principle ofacoustic reciprocity the full-matrix data shown in equation (1)have good symmetry So the upper triangular or lower tri-angular matrix signal of the full-matrix data can be used incomputational imaging (is method is called the half-matrixfocus imaging (HFM) shown as follows (4) [21] or (5)

Ip xp zp1113872 1113873 1113944N

i11113936N

ji

Aij tij xp zp1113872 11138731113872 1113873 (4)

Ip xp zp1113872 1113873 1113944N

ij

1113944

N

j1Aij tij xp zp1113872 11138731113872 1113873 (5)

During data acquisition and imaging calculation usingthe half-matrix focusing method N times N sets of A-scan datawill be reduced to N times ((N + 1)2) Before imaging calcu-lation it is necessary to normalize the image data In thearticle the amplitude data are normalized to between minus128and 127 It can superimpose and offset the random noise ofpositive and negative values [21] (e amplitude is nor-malized by

Ip xp zp1113872 1113873 Ip xp zp1113872 1113873

Imax xp zp1113872 1113873times 127 Ip xp zp1113872 1113873ge 0

Ip xp zp1113872 1113873 minusIp xp zp1113872 1113873

Imin xp zp1113872 1113873times 128 Ip xp zp1113872 1113873lt 0

(6)

3 Simulation and Results

31 Experimental Conditions (e finite element method(FEM) was used to validate our proposed imaging algorithmand ABAQUS 2016 (DassaultVelizy-Villacoublay France)software was used to simulate the full-matrix detection ofdefects in a Q235 steel plate (e finite element model isshown in Figure 3 (e 32-element linear array transducerhad a central frequency f 5MHz with an interelementspacing of 05mm and an element width of 04mm Full-array probe parameters are listed in Table 1

(e defect positions in the steel plates are shown inFigure 4 (e TFM and HFM were tested on three finiteelement models of isotropic steel plates (100mmtimes 30mm)To suppress the interference of reflected waves each modelhas infinite border with a thickness of 5mm In model 4(a)there are three round through-hole defects (Nos 1 2 and 3)with diameters of 2mm Between each defect the lateralspacing was 5mm(e distance between defect No 1 and theupper edge of the steel plate model was 14mm In model4(b) there is an oval through-hole (No 4) with a long axis of4mm and a short axis of 2mm (e distance between thedefect and the upper edge of the steel plate model was19mm In model 4(c) there is a rectangular through-hole(No 5) defect with a length of 3mm and width of 1mm(e

2 Mathematical Problems in Engineering

distance between the defect and the upper edge of the steelplate model was 20mm

In this experiment multiple analysis steps were set Eachelement was excited by a 4-cycle sinusoidal signal modulatedby a Hanning window in sequence at each analysis step At

the same time every array element received ultrasonicsignal Finally data for 32 times 32 full matrix were captured(e signals were processed on MATLAB R2016a (Math-Works Natick MA USA)

32 Results and Analysis

321 Symmetry of Full-Matrix Data Figure 5 presents theamplitude data for 32 times 32 full matrix of round through-hole oval through-hole and rectangular through-hole re-spectively It can be seen that the data for full matrix werecaptured to be symmetrical (e result indirectly shows thatonly the upper or lower triangle data can be used for focusimaging that is half-matrix focus imaging (erefore the

(a) (b)

(c)

Figure 3 Finite element inspection model (the enlarged part is array chip) (a) round through-hole (b) oval through-hole (c) rectangularthrough-hole

Table 1 Experimental array parameters

Parameter ValueNumber of elements 32Element width 04mmElement pitch 01mmSampling time interval 2e-8sCenter frequency 5 MHzWave velocity 5800ms

Element 1 excited Element N excited

Full elements received Full elements received

1~N array elements are sequentially excited

Figure 1 Schematic diagram of full-matrix data capture

1 i j N0

x

z

P

(xi 0) (xj 0)

(xp zp)

Figure 2 Schematic diagram of the total focus algorithm

Mathematical Problems in Engineering 3

resolution of half-matrix imaging is very close to that of full-matrix focus imaging in theory

322 Consistency of the Captured Signal According to thereciprocity of the ultrasonic transceiver sensor the acousticsignal captured by each transceiver channel remains con-sistent(e data of transceiver array elements exchange wereextracted from full-matrix data At the same time the datawere analyzed and compared Here six sets of A-scan datawere randomly selected from full-matrix data Signals re-ceived by different channels are shown in Figures 6ndash8Figure 6 presents A-scan signal of defect Nos 1 2 and 3detection Figure 7 presents A-scan signal of defect No 4detection Figure 8 presents A-scan signal of defect No 5detection It can be seen that the transceiver element isexchanged and the collected signal basically coincides (isfully demonstrates the reciprocity of the ultrasonic trans-ceiver channel (is further illustrates that half-matrix datacan be used in efficient focus imaging

323 Comparison of Imaging Results In the section thedifference between the imaging results of TFM and HFMwas compared According to the principle of TFM andHFMwe performed imaging experiments on round through-holedefects oval through-hole defect and rectangular through-hole defect respectively (e imaging results are shown inFigures 9ndash11

Comparing the two-dimensional imaging performanceof the two algorithms the five defects can be clearly dis-played by TFM and HFM However it can be seen there isscattered noise around defects (mainly in the horizontaldirection) by using the half-matrix imaging method Andthe shape and size of the defect have changed slightly (is isdue to the small amount of data processed by the half-matriximaging algorithm and poor signal average effect

324 Image Synthesis A-Scan Signal and SNR In order toquantify the imaging performance of the two methodsimage synthesis A-scan signal is used as shown in Figure 12It can be seen that image synthesis signals of Nos 1ndash5 byTFM and HFM coincide (e amplitude and phase of thesignal are basically equal

For quantitatively analyzing the performance of the twoimaging methods the signal-to-noise ratio (SNR) is used toindicate the relationship between the defect signal andnoise (e definition of SNR for a defect can be expressed[21ndash23] by

SNR 20 times log10Amax

Aave

1113888 1113889 (7)

where Amax is the maximum value (peak-to-peak) of thedefect signal in the surrounding region and Aave is the

P1 P2 P3 P4 P32

5InfiniteborderUnitmm

12

35

5

5

5

15ϕ = 2

(a)

P1 P2 P3 P4 P32

5

Unitmm 4

19

24 Infiniteborder

(b)

20

31

55

Infiniteborder

P1 P2 P3 P4 P32

Unitmm

(c)

Figure 4 Schematic of defects in (a) round through-hole (b) oval through-hole (c) rectangular through-hole


00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

108 times 10ndash6

ndash108 times 10ndash6

540 times 10ndash7


000

2468101214161820222426283032

Array e

lemen

ts

Time (s)

Am

plitu

de (m

)

(a)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68

1012

1416

1820

2224

2628

3032

Array e

lemen

ts

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(b)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68101214

1618

2022

2426

2830

32Arra

y elem

ents

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(c)

Figure 5 Full-matrix data (a) round through-hole (b) oval through-hole (c) rectangular through-hole

10

08

06

04

02

00

00E + 00 50E ndash 06 10E ndash 05 15E ndash 05 20E ndash 05

Nor

mal

ized

ampl

itude

Time (s)

A24ndash19A19ndash24

(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A15ndash4A4ndash15

(b)

Figure 6 A-scan data of round through-hole (a) A24minus19 and A19minus24 (b) A15minus4 and A4minus15


average of noise signal Except for the defect signal the restof the data is regarded as a noise signal Here we analyzedthe SNR of five defects in the three steel plate models forTFM and HFM (e results are shown in Figure 13

(e SNR for different defects by using TFM is better thanthe SNR by using HFM But the range of difference in SNRof TFM and HFM is very small It can be seen that thedifference range is between 109 dB and 187 dB Since more

data are superimposed during total focus imaging itswaveform average effect is more obvious and the SNR isslightly higher

In terms of imaging calculation time the conditions are thatthe imaging pixel is set to 01mmtimes 01mm and the imagingpoint is set to 1024000 (e CPU model of computer is IntelCore i5-9300H (24GHz) and RAM is 8GB (e full-matrixfocus imaging takes 835 seconds while the half-matrix focus

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A18ndash2A2ndash18

(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A32ndash1A1ndash32

(b)

Figure 7 A-scan data of oval through-hole (a) A18minus2 and A2minus18 (b) A32minus1 and A1minus32

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00

00E ndash 00 50E ndash 06 10E ndash 05 15E ndash 05 20E ndash 05

Nor

mal

ized

ampl

itude

Time (s)


(b)

Figure 8 A-scan data of rectangular through-hole (a) A22minus21 and A21minus22 (b) A16minus13 and A13minus16


ndash108 ndash84 ndash6 ndash36 ndash12 12 36 6 84 108x (mm)

0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)

Figure 9 Focus imaging of round through-hole (a) TFM (b) HFM


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)

Figure 10 Focus imaging of oval through-hole (a) TFM (b) HFM


03

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)

Figure 11 Focus imaging of rectangular through-hole (a) TFM (b) HFM


imaging takes 41 seconds Compared with TFM the imagingtime by using HFM is approximately doubled and imagingefficiency greatly improved

4 Discussion

(e method used in the article is similar to that described inliterature [21] but the experimental results are different It canbe seen from the experimental results that compared with theTFM the disadvantage of the HFM is its low signal-to-noise ratio However compared with traditional ultrasoundB-scan imaging HFM imaging has a higher signal-to-noiseratio In the article under the same experimental conditions

B-scan imaging is performed on three kinds of defects andthe image obtained is shown in Figure 14 It can be seen thatthere is background noise in the image

In the literature [24 25] a microseismic waveformextraction method using unsupervised machine learningtechnique is proposed to improve the resolution of mi-croseismic imaging With reference to the method ofseismic imaging machine learning and waveform featureextraction methods can also be used in ultrasound half-matrix imaging to increase the signal-to-noise ratio andimprove imaging resolution For imaging efficiency theliterature [13] mentioned the use of GPU parallel accel-eration to reduce the time of ultrasound TFM imaging

40

20

0

ndash20

ndash40

00E + 00 10E ndash 06 20E ndash 06 30E ndash 06 40E ndash 06 50E ndash 06 60E ndash 06

Nor

mal

ized

ampl

itude

Full matrixHalf matrix

Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)

Figure 12 Image synthesis amplitude signal of defects (a) round through-hole (b) oval through-hole (c) rectangular through-hole


30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263

No1 No2 No3 No4 No5Defect


Figure 13 Comparison of signal-to-noise ratio of TFM and HFM


0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)

Figure 14 B-scan imaging of 3 different defects (a) round through-hole (b) oval through-hole (c) rectangular through-hole


(erefore the ultrasound HFM imaging can be combinedwith GPU parallel computing to further reduce the imagingtime (e two methods discussed above deserve furtherresearch

5 Conclusions

In this paper a half-matrix focusing method based ontransceiver sensor reciprocity is developed to focus imaging(e imaging quality and computational efficiency are ana-lyzed In terms of image performance the synthesizedsignals of the half-matrix focus imaging algorithm and thefull-matrix focus imaging algorithm have good consistencyBut the imaging resolution and SNR of TFM are slightlyhigher than those of HFM However the HFM can simplifythe data acquisition and imaging operation process of thefull-matrix focusing algorithm (e imaging time was ap-proximately doubled (e HFM provide an important ref-erence for the industrial application

Data Availability

(e data used to support the findings of this study areavailable from the corresponding author upon reasonablerequest

Disclosure

(e funders had role in the writing of the manuscript and inthe decision to publish the results

Conflicts of Interest

(e authors declare no conflicts of interest

Acknowledgments

(e authors thank Prof Han for his academic supervisionand personal support (is work was supported by theEmergency Management Project of Natural Science Foun-dation of China (Grant no 61842103) and Natural ScienceFoundation of Shanxi China (Grant nos 201801D121156and 201901D111165)

References

[1] C Li D Pain P D Wilcox and B W Drinkwater ldquoImagingcomposite material using ultrasonic arraysrdquo NDT amp E In-ternational vol 53 pp 8ndash17 2013

[2] C Yuan C Xie L Li F Zhang and S M Gubanski ldquoUl-trasonic phased array detection of internal defects in com-posite insulatorsrdquo IEEE Transactions on Dielectrics andElectrical Insulation vol 23 no 1 pp 525ndash531 2016

[3] X X Zhao T Gang and B X Zhang ldquoPrediction of radiationbeam fields from an array transducer with non-paraxial multi-Gaussian beam modelrdquo Artificial Intelligence in Federal Ad-ministrative Agencies vol 33 pp 475ndash480 2008

[4] H J Hu X Zhu C H Wang et al ldquoStretchable ultrasonictransducer array three dimensional imaging on complexsurfacesrdquo Science Advances vol 4 pp 1ndash11 2018

[5] C Holmes B W Drinkwater and P D Wilcox ldquoPost-processing of the full matrix of ultrasonic transmit-receivearray data for non-destructive evaluationrdquo NDT amp E Inter-national vol 38 no 8 pp 701ndash711 2005

[6] J Zhang BW Drinkwater and P DWilcox ldquoEffects of arraytransducer inconsistencies on total focusing method imagingperformancerdquo NDT amp E International vol 44 no 4pp 361ndash368 2011

[7] J Verkooijen and A Boulavinov ldquoSampling phased array anew technique for ultrasonic signal processing and imagingrdquoInsight - Non-Destructive Testing and Condition Monitoringvol 50 pp 153ndash157 2007

[8] A J Hunter B W Drinkwater and P D Wilcox ldquoAuto-focusing ultrasonic imagery for non-destructive testing andevaluation of specimens with complicated geometriesrdquo NDTamp E International vol 43 no 2 pp 78ndash85 2010

[9] A Velichko and P D Wilcox ldquoReversible back-propagationimaging algorithm for postprocessing of ultrasonic arraydatardquo IEEE Transactions on Ultrasonics Ferroelectrics andFrequency Control vol 56 no 11 pp 2492ndash2503 2009

[10] C J L Lane T K Dunhill B W Drinkwater andP D Wilcox ldquo3D ultrasonic inspection of anisotropicaerospace componentsrdquo Insight - Non-destructive Testing andCondition Monitoring vol 52 no 2 pp 72ndash77 2010

[11] J Russell R Long D Duxbury and P Cawley ldquoDevelopmentand implementation of a membrane-coupled conformablearray transducer for use in the nuclear industryrdquo Insight -Non-destructive Testing and Condition Monitoring vol 54no 7 pp 386ndash393 2012

[12] E Moghimirad C A Villagomez Hoyos A MahloojifarB Mohammadzadeh Asl and J A Jensen ldquoSynthetic apertureultrasound Fourier beamformation using virtual sourcesrdquoIEEE Transactions on Ultrasonics Ferroelectrics and Fre-quency Control vol 63 no 12 pp 2018ndash2030 2016

[13] M Sutcliffe M Weston B Dutton P Charlton andK Donne ldquoReal-time full matrix capture for ultrasonic non-destructive testing with acceleration of post-processingthrough graphic hardwarerdquo NDT amp E International vol 51pp 16ndash23 2012

[14] M Njiki A Elouardi S Bouaziz O Casula and O Roy ldquoAmulti-FPGA architecture-based real-time TFM ultrasoundimagingrdquo Journal of Real-Time Image Processing vol 16 no 2pp 505ndash521 2019

[15] S Bouaziz M Njiki A Elouardi O Casula and O Roy ldquoAreal-time implementation of the Total Focusing Method forrapid and precise diagnostic in nondestructive evaluationrdquo inProceedings of the IEEE 24th International Conference onApplication Specific Systems pp 245ndash248 Architectures andProcessors Washington DC USA June 2013

[16] A J Hunter B W Drinkwater and P D Wilcox ldquo(ewavenumber algorithm for full-matrix imaging using an ul-trasonic arrayrdquo IEEE Transactions on Ultrasonics Ferroelec-trics and Frequency Control vol 55 no 11 pp 2450ndash24622008

[17] H Hu J Du C Ye and X Li ldquoUltrasonic phased arraysparse-TFM imaging based on sparse array optimization andnew edge-directed interpolationrdquo Sensors vol 18 no 6p 1830 2018

[18] L Moreau B W Drinkwater and P D Wilcox ldquoUltrasonicimaging algorithms with limited transmission cycles for rapidnondestructive evaluationrdquo IEEE Transactions on UltrasonicsFerroelectrics and Frequency Control vol 56 no 9pp 1932ndash1944 2009


[19] J N Potter P DWilcox and A J Croxford ldquoDiffuse field fullmatrix capture for near surface ultrasonic imagingrdquo Ultra-sonics vol 82 pp 44ndash48 2018

[20] H Zhang Y Liu G Fan H Zhang W Zhu and Q ZhuldquoSparse-TFM imaging of lamb waves for the near-distancedefects in plate-like structuresrdquo Metals vol 9 no 5 p 5032019

[21] X Zhao T Z Qi Z Y Wang et al ldquoUltrasonic detectiontriangle matrix focus imaging algorithmrdquo Journal of Me-chanical Engineering vol 55 no 4 pp 19ndash24 2019

[22] H Zhang L Zeng G Fan H Zhang Q Zhu and W ZhuldquoInstantaneous phase coherence imaging for near-field defectsby ultrasonic phased array inspectionrdquo Sensors vol 20 no 3p 775 2020

[23] A Velichko and P D Wilcox ldquoAn analytical comparison ofultrasonic array imaging algorithmsrdquo 3e Journal of theAcoustical Society of America vol 127 no 4 pp 2377ndash23842010

[24] Y Chen ldquoFast waveform detection for microseismic imagingusing unsupervised machine learningrdquo Geophysical JournalInternational vol 215 no 2 pp 1185ndash1199 2018

[25] M S Omar and Y K Chen ldquoAutomatic waveform-basedsource-location imaging using deep learning extracted mi-croseismic signalsrdquo Geophysics vol 85 pp 171ndash183 2020


reduce full-matrix data capturing and TFM imaging timewhile maintaining the image quality (e remainder of thispaper is organized as follows In Section 2 the FMC theoryand the imaging methods are introduced Section 3 presentsthe experimental methods and procedures for defect de-tection Moreover the experimental results of data pro-cessed with conventional TFM and HFM are analyzedSection 4 compares HFM imaging and B-scan imaging anddiscusses the subsequent research directions of the HFMimaging Finally the conclusions are given in Section 5

2 Methods

21 FMC and TFM Figure 1 shows the principle of full-matrix ultrasonic data capture [19 20] A linear arrayconsisting of N elements excites the ultrasonic wave insequence During each excitation the ultrasonic signal isreceived by every array element Finally an N times N A-scandata matrix is obtained HereAij(t) is the N times N A-scansignal emitted by element i and received by element j (eformat of full-matrix data is shown as follows

A

A11 A12 A1j A1N

A21 A22 A2j A2N

Aij Ai2 Aij AiN

AN1 AN2 ANj ANN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

Figure 2 shows the principle of total focus algorithm[19 20] Image reconstruction using full-matrix data isdivided into two steps First discretization of the inspectedregion into a grid is performed Secondly calculation of theamplitude intensity at a grid point P(xp zp) is calculated by

Ip xp zp1113872 1113873 1113944

N

i11113936N

j1Aij tij xp zp1113872 11138731113872 1113873 (2)

where tij(xp zp) is the time of flight starting from thetransmitter element i passing through the focusing pointP(xp zp) and arriving at the receiver element j (e time offlight is defined by

tij xp zp1113872 1113873

xi minus xp1113872 11138732

+ z2p

1113969

+

xj minus xp1113872 11138732

+ z2p

1113969

1113874 1113875

c

(3)

where c is the propagation velocity of ultrasound in themedium and (xi 0) and (xj 0) are the Cartesian coordinatesof the transmitter and the receiver respectively

22HFM Consider that the performance indicators of eachchannel of the linear ultrasonic array transducer have ex-cellent consistency (at is the signal that is transmittedfrom element i and received on element j is identical to the

signal transmitted on element j and received on element i It isnamed acoustic reciprocity According to the principle ofacoustic reciprocity the full-matrix data shown in equation (1)have good symmetry So the upper triangular or lower tri-angular matrix signal of the full-matrix data can be used incomputational imaging (is method is called the half-matrixfocus imaging (HFM) shown as follows (4) [21] or (5)

Ip xp zp1113872 1113873 1113944N

i11113936N

ji

Aij tij xp zp1113872 11138731113872 1113873 (4)

Ip xp zp1113872 1113873 1113944N

ij

1113944

N

j1Aij tij xp zp1113872 11138731113872 1113873 (5)

During data acquisition and imaging calculation usingthe half-matrix focusing method N times N sets of A-scan datawill be reduced to N times ((N + 1)2) Before imaging calcu-lation it is necessary to normalize the image data In thearticle the amplitude data are normalized to between minus128and 127 It can superimpose and offset the random noise ofpositive and negative values [21] (e amplitude is nor-malized by

Ip xp zp1113872 1113873 Ip xp zp1113872 1113873

Imax xp zp1113872 1113873times 127 Ip xp zp1113872 1113873ge 0

Ip xp zp1113872 1113873 minusIp xp zp1113872 1113873

Imin xp zp1113872 1113873times 128 Ip xp zp1113872 1113873lt 0

(6)

3 Simulation and Results

31 Experimental Conditions (e finite element method(FEM) was used to validate our proposed imaging algorithmand ABAQUS 2016 (DassaultVelizy-Villacoublay France)software was used to simulate the full-matrix detection ofdefects in a Q235 steel plate (e finite element model isshown in Figure 3 (e 32-element linear array transducerhad a central frequency f 5MHz with an interelementspacing of 05mm and an element width of 04mm Full-array probe parameters are listed in Table 1

(e defect positions in the steel plates are shown inFigure 4 (e TFM and HFM were tested on three finiteelement models of isotropic steel plates (100mmtimes 30mm)To suppress the interference of reflected waves each modelhas infinite border with a thickness of 5mm In model 4(a)there are three round through-hole defects (Nos 1 2 and 3)with diameters of 2mm Between each defect the lateralspacing was 5mm(e distance between defect No 1 and theupper edge of the steel plate model was 14mm In model4(b) there is an oval through-hole (No 4) with a long axis of4mm and a short axis of 2mm (e distance between thedefect and the upper edge of the steel plate model was19mm In model 4(c) there is a rectangular through-hole(No 5) defect with a length of 3mm and width of 1mm(e







(a) (b)

(c)








1 i j N0

x

z

P

(xi 0) (xj 0)

(xp zp)










Aave

1113888 1113889 (7)


P1 P2 P3 P4 P32


12

35

5

5

5

15ϕ = 2

(a)

P1 P2 P3 P4 P32

5

Unitmm 4

19

24 Infiniteborder

(b)

20

31

55

Infiniteborder

P1 P2 P3 P4 P32

Unitmm

(c)



00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

108 times 10ndash6


540 times 10ndash7


000

2468101214161820222426283032

Array e

lemen

ts

Time (s)

Am

plitu

de (m

)

(a)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68

1012

1416

1820

2224

2628

3032

Array e

lemen

ts

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(b)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68101214

1618

2022

2426

2830

32Arra

y elem

ents

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(c)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A15ndash4A4ndash15

(b)







10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A18ndash2A2ndash18

(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A32ndash1A1ndash32

(b)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(b)




0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



03

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)




4 Discussion




40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)



30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References

































(a) (b)

(c)








1 i j N0

x

z

P

(xi 0) (xj 0)

(xp zp)










Aave

1113888 1113889 (7)


P1 P2 P3 P4 P32


12

35

5

5

5

15ϕ = 2

(a)

P1 P2 P3 P4 P32

5

Unitmm 4

19

24 Infiniteborder

(b)

20

31

55

Infiniteborder

P1 P2 P3 P4 P32

Unitmm

(c)



00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

108 times 10ndash6


540 times 10ndash7


000

2468101214161820222426283032

Array e

lemen

ts

Time (s)

Am

plitu

de (m

)

(a)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68

1012

1416

1820

2224

2628

3032

Array e

lemen

ts

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(b)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68101214

1618

2022

2426

2830

32Arra

y elem

ents

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(c)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A15ndash4A4ndash15

(b)







10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A18ndash2A2ndash18

(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A32ndash1A1ndash32

(b)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(b)




0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



03

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)




4 Discussion




40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)



30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References



































Aave

1113888 1113889 (7)


P1 P2 P3 P4 P32


12

35

5

5

5

15ϕ = 2

(a)

P1 P2 P3 P4 P32

5

Unitmm 4

19

24 Infiniteborder

(b)

20

31

55

Infiniteborder

P1 P2 P3 P4 P32

Unitmm

(c)



00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

108 times 10ndash6


540 times 10ndash7


000

2468101214161820222426283032

Array e

lemen

ts

Time (s)

Am

plitu

de (m

)

(a)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68

1012

1416

1820

2224

2628

3032

Array e

lemen

ts

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(b)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68101214

1618

2022

2426

2830

32Arra

y elem

ents

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(c)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A15ndash4A4ndash15

(b)







10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A18ndash2A2ndash18

(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A32ndash1A1ndash32

(b)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(b)




0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



03

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)




4 Discussion




40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)



30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References




























00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

108 times 10ndash6


540 times 10ndash7


000

2468101214161820222426283032

Array e

lemen

ts

Time (s)

Am

plitu

de (m

)

(a)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68

1012

1416

1820

2224

2628

3032

Array e

lemen

ts

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(b)

108 times 10ndash6


540 times 10ndash7


000

Am

plitu

de (m

)

24

68101214

1618

2022

2426

2830

32Arra

y elem

ents

00E

+ 0

0

10E

ndash 0

4

20E

ndash 0

4

30E

ndash 0

4

40E

ndash 0

4

50E

ndash 0

4

60E

ndash 0

4

Time (s)

(c)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A15ndash4A4ndash15

(b)







10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A18ndash2A2ndash18

(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A32ndash1A1ndash32

(b)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(b)




0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



03

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)




4 Discussion




40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)



30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References
































10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A18ndash2A2ndash18

(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)

A32ndash1A1ndash32

(b)


10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(a)

10

08

06

04

02

00


Nor

mal

ized

ampl

itude

Time (s)


(b)




0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



03

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)




4 Discussion




40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)



30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References





























0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)



03

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)




4 Discussion




40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)



30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References





























4 Discussion




40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(a)

0

40

20

ndash20

ndash40


Nor

mal

ized

ampl

itude


Time (s)

(b)


40

20

0

ndash20

ndash40


Nor

mal

ized

ampl

itude

Time (s)

(c)



30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References




























30

25

20

15

10

5

0

SNR

(dB)

25232381

2631

24262551

2302

2632 2612 2652 263





0

3

6

9

12

15

18

21

24

27

10

20

30

40

50

60

y (m

m)

(a)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(b)


0

3

6

9

12

15

18

21

24

27

y (m

m)

10

20

30

40

50

60

(c)




5 Conclusions


Data Availability


Disclosure




Acknowledgments


References





























5 Conclusions


Data Availability


Disclosure




Acknowledgments


References




























ultrasound imaging algorithm: half-matrix focusing method

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