# Research on Sparse Representation Method of Acoustic Microimaging Signals
## 2 Materials and Methods
### Constraint Dictionary-Based Sparse Representation
#### 2.2.2 Algorithm Design  
The sparse representation based on the constraint dictionary is equivalent to an approximate solution of the following optimization problem:  
\[min||c||_{0}\qquad s.\ t.\qquad y=c\Phi. \tag{13}\]  
Its astringency has been be proved [27]. In this paper, the time-frequency parameters of the atom dictionary are constrained to form a constraint dictionary according to AMI signals. Then MP algorithm is utilized to adaptively choose atoms from the constraint dictionary to decompose the signal. Finally, atoms are selected according to the time-frequency window to reconstruct the echoes before overlap. The detailed algorithm design is as follows.  
1. Initialization: given the overlapping AMI signal \(y\), the constrained Gabor dictionary \(\Phi\) and the residual signal threshold \(\varepsilon\), set the initial solution support set \(\mathsf{S}^{0}=\varnothing\) and the initial residual signal \(R^{0}y=y\).
2. Discretization of constraint dictionary: the AMI signal is exemplified in Figure 3a, where \(x_{0}(t)\) is selected as the reference echo and the target echoes are \(x_{1}(t)\) and \(x_{2}(t)\). According to the constraints above, discretization is performed at a small discrete interval to achieve high resolution.  
Figure 2: Spatial sampling of the frequency factor before (**a**) and after (**b**) constraint.  
3. Matching pursuit (\(i=0,1,2,\ldots,m-1\)): 1. All the atoms \(\varphi_{\gamma}\) are sequentially inner producted with the residual signal \(R^{i}y\) to find the atom \(\varphi_{\gamma_{i}}\) whose decomposition coefficient satisfies \(c^{i}=\big{|}\big{\langle}R^{i}y,\ \varphi_{\gamma_{i}}\big{\rangle}\big{|}= \max_{\gamma\in\mathrm{G}}\big{|}\big{\langle}R^{i}y,\ \varphi_{\gamma}\big{\rangle}\big{|}\), and update the support set \(\mathrm{S}^{i+1}=\mathrm{S}^{i}\cup\{\varphi_{\gamma_{i}}\}\). (Since \(N\) inner product operations can be converted into a cross-correlation operation, FFT can be used to quickly implement cross-correlation operations to increase the calculation speed [23].) 2. Update residual signal: \(R^{i+1}y=R^{i}y-\big{\langle}R^{i}y,\ \varphi_{\gamma_{i}}\big{\rangle}\). 3. Stop condition: \(||R^{m}y||_{2}\leq\varepsilon\). 4. Selection and reconstruction: 1. As shown in Figure 3b, each atom \(\varphi_{\gamma_{i}}\) obtained by the matching iteration is represented by a Heisenberg box, and the decomposition coefficient is represented by the darkness of the box. The larger the coefficient, the darker the box. 2. According to the target echoes \(x_{1}(t)\) and \(x_{2}(t)\), atoms with the largest decomposition coefficients are selected as the time-frequency window, respectively. 3. The atom and its decomposition coefficient are considered as the approximation of the incident pulse \(\varphi_{i}(t)\) and the reflection coefficient \(c_{i}\)[8,10]. The atoms whose centers lie within the time-frequency window are chosen to reconstruct \(x_{1}(t)\) and \(x_{2}(t)\) before the overlap, i.e., \(x_{1}(t)=c^{0}\varphi_{\gamma_{0}}+c^{4}\varphi_{\gamma_{4}}\), \(x_{2}(t)=c^{1}\varphi_{\gamma_{1}}+c^{5}\varphi_{\gamma_{5}}\), resulting in Figure 3c. 4. Display the peak value of the target echo at each x-y position to generate a C-scan image.