###### Abstract  
Full-matrix capture (FMC)-based ultrasonic imaging provides good sensitivity to small defects in non-destructive testing and has gradually become a mainstream research topic. Many corresponding algorithms have been developed, e.g., the total focusing method (TFM). However, the efficiency of the TFM is limited, especially in multi-layered structures. Although the appearance of wavenumber algorithms, such as extended phase-shift migration (EPSM) methods, has improved imaging efficiency, these methods cannot be applied to cases with oblique incidence. Therefore, a modified wavenumber method for full-matrix imaging of multi-layered structures with oblique array incidence is proposed. This method performs a coordinate rotation in the frequency domain to adapt it to the oblique incidence. It then utilizes wave-field extrapolation to migrate the transmitting and receiving wave field to each imaging line, and a correlation imaging condition is used to reconstruct a total focused image. The proposed method can deal with any incident angle without precision loss. Moreover, it inherits the computational efficiency advantages of the wavenumber algorithms. The simulation and experimental results show that the proposed method performs better in terms of accuracy and efficiency than the TFM. Specifically, it is nearly 60 times faster than the TFM when processing an FMC dataset with a size of 4096 \(\times\) 64 \(\times\) 64.  
ultraconic array imaging; full-matrix imaging; wavenumber algorithm; multi-layered medium; oblique incidence +
Footnote †: journal: Applied Sciences  
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. [PERSON] 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, [PERSON] proposed an FMC method based on phase-shift migration (PSM) [28], implemented in a wavenumber domain.