PHASE SUPERPOSITION PROCESSING FOR ULTRASONIC IMAGING

Journal of Sound and Vibration (1996) 193(5), 1015–1021

PHASE SUPERPOSITION PROCESSING FOR ULTRASONIC IMAGING L. T, X. R. M, H. T  Z. X. G Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin 150001, Peoples Republic of China (Received 28 February 1995, and in final form 19 October 1995) In order to improve the resolution of defect reconstruction for non-destructive evaluation, a new phase superposition processing (PSP) method has been developed on the basis of a synthetic aperture focusing technique (SAFT). The proposed method synthesizes the magnitudes of phase-superposed delayed signal groups. A satisfactory image can be obtained by a simple algorithm processing time domain radio frequency signals directly. In this paper, the theory of PSP is introduced, and some simulation and experimental results illustrating the advantage of PSP are given. 7 1996 Academic Press Limited

1. INTRODUCTION

SAFT [1–3] is an important imaging technique for ultrasonic non-destructive evaluation based on the fundamental of synthetic aperture radar (SAR) [4–5]. High resolution images can be reconstructed by synthesizing signals acquired at different positions in an array. The key problem of SAFT is the enhancement of both the resolution and the SNR. As far as a pulse-echo line-array system is concerned, the theoretical lateral resolution of the traditional delay and sum (D and S) algorithm is lR/2L, where l is the wavelength, R the distance between the transducer array and the detecting point, L the whole length of effective synthetic aperture. Because the spread angle of a single probe is constrained by l/d (d denotes the lateral length of a single probe), the maximum attainable L is limited to lR/d, thus the best resolution of said system is theoretically d/2. On the other hand, the sensitivity of a single probe is dependent on the aperture. Therefore, reducing the aperture of a single probe for high resolution will inevitably influence the SNR of the result. The PSP technique described in this paper is an approach to increasing both the resolution and the SNR with little additional computing cost. The traditional D and S is developed with a pre-processing of phase superposing before summing. The processing theory for the case of a monochromatic signal, the lateral resolution, and the numerical simulation result of a point scatter, are described in section 2. In practical testing, a wide band linear array ultrasonic transducer is used, and the simple and fast algorithm enables a personal computer to perform the signal processing. An experimental cross-section imaging result provided in section 3 characterized the effectiveness of the technique. 1015 0022–460X/96/251015 + 07 $18.00/0

7 1996 Academic Press Limited

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1016

2. THEORY OF PSP

2.1.   Generally, a time harmonic scalar wave irradiated from a point source (0, 0) can be described as A(r, t) = A0 R(r) exp[i(vt − kr)],

(1)

where k = v/c = 2pf/c denotes the wave number of the wave with frequency f and propagation speed c in the testing medium, and R(r) is the coefficient representing the attenuation brought about by the transmission distance. Because only the second term of the index denotes the phase delay and influences the amplitude, exp(ivt) can be omitted in the following. On the other hand, R(r) can be regarded as a constant in the condition of far field approximation. Hence, the major term taken into account is only exp(−ikr). As to a transceiver position rj , the received signal scattered by a point defect rd is E(rj ) = exp[−ik(2=rd − rj =)].

(2)

The time delay processing of SAFT [1] is equivalent to weighting equation (2)by exp[−ik(2=r − rj )] in the frequency domain. In the case of two dimensions, (2N + 1) transducers are arranged from r−N to rN , and the synthetic reconstructing image of a point defect model is N

F(r) = s exp[i2k(=r − rj = − =rd − rj =)]

(3)

j = −N

where rj and rd denote the positions of the jth transceiver and the point defect respectively, as shown in Figure 1. Equation (3) indicates that the reconstructed image is derived by the sum of space waves with different space phases 2k(=r − rj = − =rd − rj =); thus the defect point rd is enhanced by these waves with zero phase at =r − rj = − =rd − rj =. For one receiver, the said locus is a circle. Instead of summing the delayed amplitudes directly, PSP pre-processes the kernel by superposing several different space phases: N − nm

F(r) =

s

j = −N + nm

$

n

%

exp i2k s (=r − rj + lm = − =rd − rj + lm =) . l = −n

(4)

Here m denotes the phase superposing interval and (2n + 1) the order of phase superposition. The key idea of the processing method is that the (2N + 1 − 2nm) magnitudes of phase-superposed groups of (2n + 1) delayed signals are synthesized instead of the (2N + 1) delayed signals in D and S. The two neighbor transducers in one group

Figure 1. Two-dimensional geometric diagram of a linear array transceiver and a defect point.

  

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are spaced m times the interval of two neighbor transducers in the array. Another form of equation (4) is N − nm

s

F(r) =

j = −N + nm

N − nm

s

=

j = −N + nm

6

7

n

t exp[i2k(=r − rj + lm = − =rd − rj + lm =)]

l = −n

6

7

n

t [E(rj + lm )P(r, rj + lm )] , l = −n

(5)

where E(rj ) is the received signal of the jth transceiver, and P(r, rj ) represents the phase delay processing. Equation (5) shows that PSP is convenient in practical testing, because the received time domain signals are employed without any transformation. 2.2.   Consider the amplitudes of the image, reconstructed by PSP, on the line y = R parallel to the transducers’ array y = 0 and passing an assumed scattering point (0, R). The (2N + 1) transducers are arranged as a line array from −Ndx to +Ndx; dx is the interval between two neighboring positions. Equation (4) is rewritten as N − nm

F(x, R)=

6

7

n

exp i2k s [zR 2 + [( j + lm)dx − x]2 + zR 2 + [( j + lm)dx]2 ] .

s

j = −N + nm

l = −n

(6)

The analytical solution of equation (6) is difficult to find, but one can roughly estimate the lateral resolution by substituting into the far field approximation zR 2 + a 2 = R + a 2/2R,

aR.

(7)

In this approximation equation (6) is N − nm

s

F(x, R) =

6

n

exp ik2 s

j = −N + nm

$

= exp i(2n + 1)k

l = −n

%

x2 R

$

%7

[( j + lm)dx − x]2 [( j + lm)dx]2 − 2R 2R

N − nm

s

j = −N + nm

$

exp i2(2n + 1)jk

dx · x R

%

(8)

The amplitude envelope of equation (8) is =F(x, R)= =

b

b

sin [(2N + 1 − 2nm)(2n + 1)(2p/lR)dx · x] . sin[(2n + 1)(2p/lR)dx · x]

(9)

The distance between the origin and the first zero point is nearly the half power width of the function described in equation (9), which can be regarded as the lateral resolution rPSP = l

lR 1 lR 1 lR = 1 , (2n + 1)2(2N + 1 − 2nm)dx nPS 2L' nPS 2L

(10)

where nPS denotes the order of phase superposing, L' = (2N + 1 − 2nm)dx, and L = (2N + 1)dx is the whole aperture of the transducers’ array. In equation (6), the second summing, i.e. the PSP, is equivalent to multiplying the signal’s frequency k by nPS , so the lateral resolution of PSP can be enhanced to 1/nPS of the D and S (rDS l = lR/2L). However,

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Figure 2. Simulating reconstructed images of a point defect by (a) D and S and (b) PSP.

the grouping of transducers reduces the number of magnitudes to be synthesized from (2N + 1) to (2N + 1 − 2nm), and thus the synthetic aperture from L = (2N + 1)dx to L' = (2N + 1 − 2nm)dx. According to this estimation of lateral resolution, it is shown that for 1/nPS enhancement of the lateral resolution, are needs Nnm, and it is easily derived that 2N + 1 q m(2n + 1) is needed at least for rPSP Q rDS l l . Although the theoretical estimation of the lateral resolution indicates that PSP is equivalent to frequency multiplication processing, the processing method is not the simple form of adding a multiple constant in the phase of equation (3) (this is a special case of equation (4) when m = 0), just like the phase multiplication in electromagnetic holography [6]. In practical ultrasonic imaging, the testing error and the random noise in the original signal usually cause great noise in the final image, and the high order correlation processing of different channel signals will enhance the SNR. The order and the interval of phase superposing employed in the following simulation and experiment are chosen according to the experimental result; the optimizing of parameters is expected in further research. 2.3.  Figure 2 shows the comparative simulation imaging results of a point defect model derived by PSP versus D and S, according to equation (4) and equation (3) respectively, which characterizes the greatly improved SNR and resolution of PSP. Some parameters are f = 3 MHz, c = 5900 m/s, N = 32, n = 1, m = 3, the interval of two neighboring transceivers of the discrete linear array is 1·5 mm, the vertical distance between the assumed defect point and the linear array is 100 mm, and the size of the imaging region is 14 mm × 14 mm. 3. EXPERIMENT

The time domain signal processing algorithm for a practical pulse-echo test can be expressed as N

F(r) = s j = −N

$

=r − rj =A

0

1%

2=r − rj = c

,

(11)

  

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Figure 3. Block diagram of experimental set-up. N − nm

F(r) =

s

j = −N + nm

6 $ n

t

l = −n

=r − rj + lm =A

0

1%7

2=r − rj + lm = c

,

(12)

these corresponding to D and S and PSP respectively [7], where A(t) is the received signal, =r − rj = denotes the compensation of the attenuation caused by the transit distance, and 2=r − rj =/c is the transit time. The transducers of the linear array are excited and irradiate ultrasonic pulses into the specimen (t is assumed to be 0) one by one, and the received signals are normalized by their maximal amplitudes and weighted by t, to compensate the attenuation of transit distance. Finally, the cross-section is reconstructed by summing the phase-superposed magnitudes. Figure 3 is the block diagram of the imaging system. In the experiment, a B-scan ultrasonic wide band piezoelectric transducer linear array with 3 MHz center frequency was used. As shown in Figure 4, the size of the probelet is 12 mm × 1·2 mm, the distance between the centers of two neighbor transducers is 1·5 mm, and the number of probelets is 66. The signal is sampled and processed by a personal computer with a 40 MHz A/D converter. Two 3 mm diameter and one 2 mm diameter (lower) cylindrical defects with anti-plane axes were machined in a steel specimen, in which the speed of an ultrasonic longitudinal wave is 5925 m/s. A 40 mm × 40 mm cross-section imaging region was assumed to include these defects. The center of the imaging region is 60 mm deep under the surface of the specimen. Figure 5 shows the comparison imaging results of said specimen achieved by D and S and PSP. It is obvious that PSP improves the resolution and SNR. 4. CONCLUSIONS

In the traditional D and S imaging method, the resolution of the image can be enhanced by reducing the dimension of transducer, as a small size single transducer provides a large

Figure 4. Structure of the B-scan linear array transducer: (a) diagram; (b) photograph.

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Figure 5. Comparative test results of defects (a) detected by PSP (b) D and S (c).

spread angle that can enlarge the valid aperture, but the SNR of the original signal will inevitably be influenced. Synthesizing the magnitudes of phase superimposed groups of delayed signals, instead of the delayed signals in D and S, the PSP, equivalent to frequency multiplying and high order correlating in the time domain, is an approach to improve the resolution. Furthermore, processing time domain signals directly enables the technique to be suitable for an irregular surface condition, since only the delay times and the corresponding amplitudes are employed in the processing. The method can overcome some problems to some extent. However, further research on the optimizing of the parameters is expected. ACKNOWLEDGMENTS

This research work is supported by the National Natural Science Foundation of China No. 19232031, Chinese Education Commission Science Foundation, and Chinese Education Commission Across Century Scientist Foundation. The B-scan linear-array probe was provided by Shantou Institute of Ultrasonic Instruments.

  

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