1. Introduction
Complex structures such as welds, joints, and pipe bends are critical components in massive buildings, bridges, and oil pipelines, where internal defects pose considerable threats [
1,
2,
3,
4]. Therefore, an effective and accurate nondestructive testing (NDT) technique to detect and characterize defects is essential to ensure the safety and reliability of these components. The ultrasonic phased array, as a well-established technique, is widely used in industrial NDT applications [
5,
6,
7,
8] due to its portability, reliability, and relatively high accuracy. Conventional ultrasonic phased array techniques are based on delay-and-sum (DAS) beamforming, which generates coherent ultrasonic beams by controlling the time delay of transmitters [
9,
10]. However, the limited information in the data in these solutions leads to low image contrast and distortion of the target object and severely restricts the applications for complex structures. The appearance of full-matrix capture (FMC) has solved this problem [
11]. FMC performs data acquisition by having all elements act as receivers while every successive element acts as a transmitter. Hence, FMC datasets contain more information about the objects, and the focusing process can be performed offline without applying delay laws during acquisition [
12].
On the basis of FMC, many algorithms have been developed independently. The total focusing method (TFM) [
13,
14] is a typical post-processing algorithm that uses all the FMC data to focus at every pixel in the region of interest (ROI). It can significantly improve detection sensitivity and spatial resolution and is considered to be the gold standard in ultrasonic imaging. However, the computational cost of TFM is substantial, as it is an extension of the DAS approach in the time domain [
15]. In order to resolve the computational issue, some researchers have sought for solutions in the frequency domain. Hunter et al. [
16] proposed a wavenumber algorithm for full-matrix imaging, which has superior computational performance. Nevertheless, this algorithm is only applicable for single-layered structures. To date, frequency-domain algorithms have been proposed to accommodate more complicated structures, such as the extended phase-shift migration (EPSM) method [
17] and multi-layered wavenumber algorithms [
18,
19]. These methods are adapted to multi-layered structures and have improved the computational efficiency.
The algorithms mentioned above are only valid for a multi-layered medium with a surface parallel to the phased array. In practice, however, complex structures such as welds and joints usually require oblique incidence detection to obtain excellent inspection coverage [
20,
21]. In this situation, the TFM needs to use the Fermat principle, calculating the refraction points at each interface to accurately estimate the wave propagation time in the multi-layered medium [
18]. However, this complicated step increases the computational cost dramatically. Several other approaches to full-matrix imaging have also been proposed for oblique incidence detection. Ray tracing methods [
22,
23] estimate the propagation time through iterative calculations, no matter how complicated the interface is, but the complexity of iterative calculations limits their efficiency. The root-mean-square (RMS) velocity [
24,
25] has been applied to reduce the calculation time. However, the approximation of RMS velocity introduces a huge error when the incidence angle of the array is overlarge. The virtual source TFM [
26,
27] can achieve total focusing through arbitrarily shaped interfaces. In addition, it reduces the processing complexity of time-of-flight calculations, leading to more efficient implementation of the TFM. Unfortunately, although many attempts have been made to improve the efficiency, these time-domain algorithms still cannot meet the high-resolution real-time imaging demand. To solve this, Lukomski proposed an FMC method based on phase-shift migration (PSM) [
28], implemented in a wavenumber domain. Unlike EPSM, this method allows for imaging of a multi-layered medium with lateral velocity variations, such as a medium that is not perpendicular to the surface of the arrays. The main advantage of the algorithm is a much higher efficiency than time-domain algorithms. Nevertheless, this algorithm cannot achieve a spatial resolution as high as the standard TFM, and the signal-to-noise ratio (SNR) is lower than for the TFM. Thus, a high-efficiency and high-accuracy algorithm for ultrasonic full-matrix imaging in oblique incidence detection on multi-layered structures is intensely desired.
In this paper, a modified wavenumber method for full-matrix imaging of multi-layered structures with oblique array incidence is proposed. The proposed method performs a coordinate rotation in the frequency domain to extrapolate the original wave field, which is parallel to the linear array, to the virtual measurement line that is parallel to the object’s surface. Then both transmission and reception wave-field extrapolation are performed in the rotated coordinate system, and an imaging condition is applied to obtain a total focused image of the multi-layered structure. The proposed method introduces an accurate coordinate transformation relation, so that it can deal with any incident angle without precision loss. It can be used in many applications, such as immersion detection of large objects with slightly curved surfaces and oblique incidence inspection of welds with an angled wedge. The simulations and experimental results show that the proposed method has high accuracy and efficiency.
The remainder of this paper is organized as follows. The method and the derivation of the proposed method are provided in
Section 2. The results and discussion illustrating the method performance are presented in
Section 3. Finally, the conclusions are given in
Section 4.
2. Method
2.1. Wave-Field Extrapolation for Multi-Layered Structures
In this subsection, the basic wave-field extrapolation is introduced, as well as the extension to multi-layered structures. Here, a linear array is assumed to detect a homogeneous medium with a sound velocity of
. The
-axis is directed along the interface, and the
-axis is perpendicular to the interface, as shown in
Figure 1. In FMC, the ultrasound waves excited by the
u-th element will propagate to the scatterers and the reflected waves will be received by all array elements. Let
be the ultrasound pressures reflected by the point scatterer. The propagation of the reflected pressures should satisfy the wave equation given in [
18]:
Meanwhile,
can be expanded in the wavenumber domain as in [
28]:
where
is the Fourier expansion of
,
is the wavenumber in the
x direction, and
is the angular frequency. By inserting Equation (2) into Equation (1), the Helmholtz equation can be obtained:
where
is the wavenumber component in the
z direction, defined as
The general solution of Equation (3) is given by [
17]:
where
and
represent the upward-traveling and downward-traveling waves, respectively. We assume that the array is placed at
and all reflectors are located in the half-space
. Thus, only upward-traveling waves can be recorded by the ultrasonic array. Considering the boundary condition
, which is the 2D Fourier transform of the received pressures
at depth
, the wave field at an arbitrary depth
can be extrapolated as in [
17]:
where
The sign function in Equation (7) guarantees that the
value represents the waves traveling in the
direction. Thus, the pressures
in time–space coordinates can be reconstructed by an inverse Fourier transform
In the case of multi-layered structures where the sound velocity varies along the
-axis (
Figure 1), Equation (6) cannot be applied directly. On the one hand, the phase factor
in Equation (6) is determined by the sound velocity of the medium, and it should be modified in order to extend it to a multi-layer case. On the other hand, the wave transmissions through each interface must be considered. Theoretically, the transmission coefficient between different layers is a complex function of the incident angle and the acoustic impedances of the materials [
29]. In practice, however, the transmission coefficient can be approximated as a constant based on the assumption of a narrow transducer beam [
30].
A typical multi-layered structure is schematically illustrated in
Figure 1. The sound velocity is different between layers, while it is homogeneous inside each layer. The layers are numbered
, where
and
denote the sound velocity and thickness of layer
, respectively. Using the narrow beam assumption, the extrapolated wave fields above and below the interface
are proportional:
where
and
represent the wave fields of the upper and lower side of the interface, respectively.
For a point scatterer within layer
, the wave field at the interface
is used as the boundary condition defining the solution within the layer. This gives us the solution
By proceeding in this way for the remaining layers, the wave field at depth
within layer
can be extrapolated as:
Since the focus is usually on the relative amplitudes within each layer, the amplitude scaling effect is insignificant in the image reconstruction and can reasonably be neglected [
19].
2.2. Oblique Incidence Compensation
The multi-layer algorithm presented above is only applicable for horizontal multi-layered structures as shown in
Figure 1. However, in practical phased array inspection, such as weld inspection, a wedge between the array and the object is used to create an oblique beam and obtain excellent coverage. Consequently, a tilt angle is generated between the normal direction of the transducer surface and the
-axis. The wave-field extrapolation of the multi-layered structures represented by Equation (11) is no longer applicable. Therefore, we propose an oblique incidence compensation method to extrapolate the measured wave field to a virtual measurement line that is parallel to the object’s surface.
Considering the case of oblique incidence, as shown in
Figure 2, a wedge with angle
is added between the linear array and the multi-layered structure. The sound velocity of the wedge is
. A classical Cartesian coordinate system is established using the center of the first array element as the origin. Another coordinate system
is obtained by rotating the coordinate system
by an angle
, so that the
-axis is parallel to the surface of the multi-layered structure. The tilt of the wedge is compensated for by extrapolating the measured wave field from
to the virtual measurement line
. The original element position
is extrapolated to the new position
, which can be expressed in the original system as in [
31]:
Thus, for the oblique incidence situation, the wave field along the virtual measurement line
can be obtained by inserting Equation (12) into Equation (8):
After combining the
and
terms, Equation (13) is rewritten as
where
Rearranging Equation (15), we obtain
With the integrals transformation, replacing
with
, Equation (14) is expressed as
where
and
After the rotation transformation, the wavenumber-domain wave field
given by Equation (17) can be used directly as the start point for wave-field extrapolation, giving
where
is the wavenumber along
-axis. Then, for the multi-layered structures, the wave field at depth
within layer
in an oblique incidence system can be extrapolated as:
2.3. Full-Matrix Imaging in Wavenumber Domain
In the case of FMC data, we consider the rotated coordinate system
, where the linear array is rotated to
. For simplicity, we assume that the
u-th element emits ultrasound waves at
, and the scatterer is located at
in the
-th layer of the oblique multi-layer structure. The ultrasound waves reach the scatterer at
, which is determined by the relative position of the scatterer and the element. At this moment, the reflected wave field from the scatterer is tightly focused and is not affected by the other scatterers. If there is an array element just above it to receive the reflected signals, an optimally focused image can be reconstructed [
17]. Then, the imaging condition becomes
instead of
. Thus, the focused image line at depth
can be reconstructed according to [
28]:
However, the computational burden of calculating
is very heavy for FMC data, especially in multi-layered structures. To solve this problem, we extend the received wave-field extrapolation to the transmitted wave field, to compensate for the propagation time. For the
u-th active element, the transmitted pressures can be written as
. Note that the transmitted pressures travel along the
direction. According to Equation (5),
should be set to zero, and the extrapolated wave field from the active array element
can be expressed as
where
is the spectrum of
. Combining the wave-field extrapolation and the tilt compensation operations in oblique multi-layered structures, we can obtain the extrapolated transmitted wave field at position
in the
-th layer, which is expressed as
where
and
It should be noted that Equations (26) and (27) differ from Equations (16) and (19) due to the physical differences between forward and backward propagations. Thus, the sign function in Equations (7) and (15) should be different when it is used in the transmitted wave-field extrapolation.
After extrapolating the transmitted wave field at the point of interest, the image can be reconstructed by taking a correlation between the forward-extrapolated wave field of
and the backward-extrapolated wave field of
, depth by depth. The correlation in the frequency domain can be written as
where
and
are the inverse Fourier transforms of
and
over
, respectively. The corresponding image condition is defined as
After reconstructing all B-scan images, the final image is obtained by superimposing the B-scan results:
where
represents the distribution of scatterers in the
coordinates, and
is the number of array elements.
2.4. Implementation
Figure 3 illustrates the implementation flowchart of the proposed algorithm. In summary, the algorithm consists of five main steps:
1. Performing a 2D Fourier transform of the received pressures and transmitted pressures .
2. Using Equations (17) and (25) for tilt compensation to extrapolate both received and transmitted wave fields from the coordinate system to .
3. Using Equations (21) and (24) for wave-field extrapolations to obtain both received and transmitted wave fields at depth in layer .
4. Image reconstruction at depth by taking a correlation between the two wave fields using Equation (29).
5. Obtaining the full-matrix image by superimposing all B-scan images using Equation (30).
It should be noted that the substitution of with is implemented by an interpolation in the domain. Note also that a frequency truncation is applied, as the frequency spectrum of ultrasonic signals is band-limited. After the 2D Fourier transform, the spectra and are truncated to obtain a subset corresponding to , where and are the lower and upper cutoff frequencies of the array, respectively. Therefore, the SNR of the image is improved due to the removal of high-frequency noise. The efficiency is also improved due to the dataset size reduction.
The proposed algorithm provides an accurate method for compensating for the oblique incidence of the linear array and achieving oblique incidence full-matrix imaging with a significant efficiency advantage compared with the time-domain TFM.
4. Discussion
The experimental results show that the proposed method can achieve oblique incidence full-matrix imaging of multi-layered structures. It has a great efficiency advantage compared with the TFM and can suppress the background noise. It also shows an improvement in lateral resolution.
However, the proposed method also has some limitations. Firstly, the imaging results are not satisfactory and readable when the defect is in an area that the acoustic beam cannot easily access, as shown in
Figure 12b. This is due to the existence of the refraction angle, resulting in the defect receiving very little acoustic energy. Secondly, although the proposed method can achieve full-matrix imaging at any tilt angle mathematically, the conversion of transverse and longitudinal waves in a solid medium must be considered in practical applications. When the tilt angle is too large, the longitudinal waves in a solid medium disappear. At this time, if the formula was still derived by using the velocity of longitudinal waves, incorrect results would appear. In this case, the effectiveness of imaging with other types of ultrasound waves, i.e., transverse waves, should be considered and tested in numerical simulations and laboratory experiments.
5. Conclusions
In this paper, a modified wavenumber method for full-matrix imaging of multi-layered structures with oblique array incidence was proposed. The proposed method utilizes a frequency-domain coordinate rotation to map the original wave field onto the virtual measurement line parallel to the object’s surface, performs both transmission and reception wave-field extrapolation in the rotated coordinate system, and finally obtains a total focused image of the multi-layered structure by applying a correlation imaging condition. Since the coordinate transformation is accurately defined in the mathematics, the proposed method can deal with any incident angle without precision loss. It can be used in many applications, such as immersion detection of large objects with slightly curved surfaces and oblique incidence inspection of welds with an angled wedge. The performance of the proposed method was evaluated by FMC imaging in an oblique incidence situation conducted both in simulation and experimentally. Compared with the TFM, the proposed method provides a more mathematically rigorous solution based on the wave equation, and it can suppress the artifacts presented in the TFM. In addition, the algorithm efficiency is significantly improved; for example, it only takes about 4.25 s to reconstruct an FMC dataset with a size of 4096 × 64 × 64 on an ordinary PC. It was demonstrated that the proposed method is superior to the TFM in both accuracy and efficiency.
In the future, the efficiency of the proposed method can be further improved to meet the demands of real-time imaging and can be extended to 3D cases. Furthermore, the TFM is shown to outperform the wavenumber algorithm at large angles relative to the array [
16], pointing the way for the improvement of the wavenumber algorithm.