# Research on Sparse Representation Method of Acoustic Microimaging Signals
## 2 Materials and Methods
### General Formulation
#### 2.1.2 Sparse Representation  
Sparse representation takes the form of adaptively selecting atoms in the atom dictionary to approximate the signal using a sparse decomposition algorithm. Considering the sparsity of the solution, decomposition speed and accuracy, Matching pursuit (MP) [27] is one of the most commonly used sparse decomposition algorithms. The MP algorithm is a greedy algorithm to find sub-optimal sparse decomposition locally. Given that the atom dictionary \(F=\left\{\varphi_{\gamma}(t)\right\}_{\gamma\in\mathbb{G}}\), \(\varphi_{\gamma}\) is the atom, and \(G\) is the parameter set, through orthogonal projection in the dictionary \(F\), \(y(t)\) is decomposed into [23; 35]:  
\[y\ =\ \Big{\langle}R^{0}y,\ \ \varphi_{\gamma_{0}}\Big{\rangle}\varphi_{ \gamma_{0}}+R^{1}y, \tag{4}\]  
where \(R^{0}y=y\), \(R^{1}y\) is the residual signal after \(y\) is approximated in the \(\varphi_{\gamma_{0}}\) direction. In order to make \(R^{1}y\) as small as possible, the atom \(\varphi_{\gamma_{0}}\) should satisfy:  
\[|\langle y,\ \ \varphi_{\gamma_{0}}\rangle|=\max_{\gamma\in\mathbb{G}}| \langle y,\ \ \varphi_{\gamma}\rangle|, \tag{5}\]  
We perform decomposition \(m\) times, then,  
\[y=\sum_{i=0}^{m-1}\Big{\langle}R^{i}y,\ \ \varphi_{\gamma_{i}}\Big{\rangle} \varphi_{\gamma_{i}}+R^{m}y. \tag{6}\]  
The atoms in the atom dictionary have localized characteristics in time and frequency. It is shown that the Gabor dictionary can simulate AMI signals well and is the most commonly used dictionary for processing AMI signals [10; 15; 29]. The Gabor dictionary is defined as [27]:  
\[\varphi_{\gamma}=\frac{1}{\sqrt{s}}g\Big{(}\frac{t-u}{s}\Big{)}\cos(\omega t +\theta), \tag{7}\]where \(g(t)=e^{-\pi t^{2}}\) is the Gaussian window function, \(\gamma=(s,u,\omega,\theta)\) is the time-frequency parameter, \(s\) is the scale factor, \(u\) is the displacement factor, \(\omega\) is the frequency factor, and \(\theta\) is the phase factor. In practical applications, \(\gamma\) is discretized as [27; 36]:  
\[\gamma=\Big{(}a^{j},pa^{j}\Delta u,ka^{-j}\Delta\omega,q\Delta\theta\Big{)}. \tag{8}\]  
where \(a=2\), \(\Delta u=1/2\), \(\Delta\omega=\pi\), \(\Delta\theta=\pi/6\), \(0<j\leq\log_{2}N\), \(0\leq p\leq N2^{-j+1}\), \(0\leq k<2^{j+1}\), \(0\leq q\leq 12\). The number of atoms \(L_{\rm F}\) is determined by the signal length \(N\), and \(L_{\Phi}=52(N\log_{2}N+N-1)\).  
The energy of the echoes is mainly concentrated in small time-frequency subspaces, while the noise is distributed over the entire time-frequency plane. In the sparse decomposition process, the echoes are decomposed into several atoms, and the noise usually does not match any atoms. In order to achieve super-resolution and high robustness, the time-frequency window is determined according to the frequency of the transducer and the target interface or defect location in the sample. In addition, the atom with the center in the time-frequency window and the largest decomposition coefficient is selected. The selected atom and its decomposition coefficient are respectively used as the approximation of the incident pulse \(\varphi_{i}(t)\) and the reflection coefficient \(c_{i}\)[8; 10].