# Research on Sparse Representation Method of Acoustic Microimaging Signals
## 1 Introduction  
AMI can evaluate the surface layer, sub-surface layer and internal structure of materials with non-destructive, precision and high sensitivity. It has been used in the fields of microelectronics [1], materials science [2], technical science [3], etc. AMI is particularly sensitive to area defects such as poor adhesion and closed cracks in multi-layer structures, making it especially suitable for failure analysis and reliability evaluation of electronic packages [4; 5; 6; 7]. The thickness of each layer of an electronic package is about micron to sub-millimeter level. Considering the penetration depth and resolution, the ultrasonic frequencies required for electronic packages detection are 20-200 MHz. Recently, electronic packages have been evolving towards ultra-miniaturization and ultra-high density, which creates challenges for their AMI. When the thickness of the layer is less than or equal to the length of the ultrasonic wave, the echo overlap and waveform distortion of the adjacent interface are caused by insufficient longitudinal resolution. As shown in Figure 1, when the top and bottom layers of the die are detected, the top echo \(x_{1}(t)\) and the bottom echo \(x_{2}(t)\) are overlapped with other echoes. While increasing the ultrasonic frequency can improve longitudinal resolution, it also means greater attenuation and lower penetration. In addition, the attenuation of high-frequency ultrasound results in a relatively low signal-to-noise ratio [8]. The above problems make the traditional time domain AMI methods no longer applicable, and the resolution limit of time domain AMI has been reached. Frequency domain AMI methods obtain single-frequency images by processing theFFT, which can reveal some features and defects below the resolution limit of time domain AMI, but the improvement of the resolution is limited because of the spectrum overlap and frequency shifting [9; 10]. Since ultrasonic echoes are non-stationary signals limited by time and frequency, the time-frequency analysis method is more suitable for the analysis of overlapping echoes [11]. Wavelet analysis-based deconvolution (WABAD) [12], continuous wavelet transform (CWT) [13], sparse representation (SR) [14] and other time-frequency domain AMI methods have been proposed to achieve super-resolution and high robustness. Among them, SR can adaptively select the atoms that match the signal from the atom dictionary and concentrate the information or energy contained in the signal on a few atoms, which can effectively reveal the time-frequency structure of the signal. Studies [13; 15] have shown that SR has certain advantages in solving the above problems and has been applied to resolution improvement [10; 16], noise suppression [17; 18], pulse detection [15; 19; 20], blind source separation [21], multi-mode guided wave separation [22], etc. The SR method uses an atom dictionary as a decomposition set [23], which is concise and adaptive. The key to successfully using the SR method is to select an appropriate atom dictionary [24]. Atom dictionaries can be divided into traditional dictionaries and learning dictionaries. The commonly used traditional dictionaries are the Chirplet dictionary [25], Daubechies dictionary [26] and Gabor dictionary [27; 28], which show a strong resemblance to ultrasonic echoes. Traditional dictionaries contain extensive time-frequency features, sufficient diversity and high redundancy, which are not suitable for processing overlapping echoes. [PERSON] et al. [24; 29] proposed an adaptive and interpolated Gabor dictionary for the analysis of complex ultrasonic echoes. Researchers have proposed learning dictionaries, such as the ICA learning dictionary [8], FOCUSS-CNDL learning dictionary [30] and K-SVD learning dictionary [31], to obtain a more sparse representation. Learning dictionaries obviously match the echo better, but its versatility is poor and engineering implementation is also difficult.