## 3 Compressive Sampling Algorithm
Thus, if the requirements (1) \(\mathbf{\alpha}\) is \(K\)-sparse with respect to a specific basis, which means the signal is compressible, (2) the transfer matrix \(\mathbf{\Theta}=\mathbf{\Phi}\mathbf{\Psi}\) satisfies both the RIP and incoherence conditions, and (3) the smaller measurement number \(m\) is larger than the required number \(m>\upmu\cdot K\cdot\log(N/K)\) (generally, \(\upmu=4\)) are satisfied, then, \(\mathbf{\alpha}\), the sparse coefficients vector of the original signal \(\mathbf{x}\) can be exactly or approximately reconstructed from the random low-rate sampling measurement vector \(\mathbf{y}\) by solving a convex program. The \(l_{1}\)-norm optimization algorithm is a comprehensive recovered algorithm in CS process, which pursues for the least \(l_{1}\)-norm value (which is related to number of nonzero entries) of the sparse vector [27]. The \(l_{1}\)-norm optimization algorithm of CS is expressed as:  
\[\min||\mathbf{\alpha}||_{1}\ \text{subject to}\ \ ||\mathbf{\Theta}\mathbf{\alpha}- \mathbf{y}||_{2}<\varepsilon \tag{7}\]  
where \(||\mathbf{\alpha}||_{1}=\sum\limits_{i=0}^{\mathbf{N}}|\mathbf{\alpha}_{i}|\), \(||\cdot||_{1}\) is the \(\ell_{1}\)-norm, \(||\cdot||_{2}\) is the \(\ell_{2}\)-norm, and \(\varepsilon\) is the noise bound. As a practical example, considering the image of a smiley face with 716 valid pixels shown in Figure 1(a), each selected ray-path acquires part of the information of the desired image. To collect measurements for reconstruction, \(m\) ray-paths were employed to capture independently oriented values (Figure 1(b)). The measured information can be expressed as the inner product between a random Bernoulli distributed matrix and the original image. According to the CS theory presented previously, a \(\ell_{1}\)-minimization optimization approach was used to recover the desired image based on 200 random measurements. Figure 1(c) indicates the reconstructed result, which has great agreement with the original image. The flaws and holes are sparsely distributed in the structure (sparse in the spatial domain), which naturally satisfies the requirement of CS and makes the combination of CS and ultrasonic CT technique possible [32]. Thus, by applying the CS theory to the ultrasonic CT detection method, a number of measurement paths will be abandoned from the dense measurement net. Fewer measurement paths are selected from the traditional measurement net of the conventional ultrasonic CT method for reconstruction purposes. Furthermore, the sparse distribution of flaws and holes in the concrete structure means that the slowness of most microstructures is the standard slowness \(\mathbf{s_{0}}\). However, the slowness of microstructures that contain holes or flaws have non-zero values of the sparse property of \(\Delta\)s.  
In addition, in order to satisfy the RIP conditions, a random Bernoulli distributed matrix \(\boldsymbol{\Phi}\) is proposed to choose the measurement paths from the traditional ultrasonic CT measurement net in the measurement stage. The compressed information from the measurement of the structure is expressed as:  
\[ \begin{array}{l}\boldsymbol{T_{m}}=\boldsymbol{\Phi}\cdot\boldsymbol{T}= \boldsymbol{\Phi}\cdot\boldsymbol{A}\cdot(\boldsymbol{s_{0}}+\Delta\boldsymbol {s})\\ \boldsymbol{y}=\boldsymbol{\Phi}\cdot(\boldsymbol{T}-\boldsymbol{T_{0}})= \boldsymbol{\Phi}\cdot\boldsymbol{A}\cdot\Delta\boldsymbol{s}\end{array}  \tag{8}\]  
where, \(\boldsymbol{T_{m}}\) is the vector of measured travel time through the selected paths, \(\boldsymbol{\Phi}\) is a random Bernoulli distributed matrix representing the selected paths, \(\boldsymbol{A}\) is the matrix of travel length of all the microstructures, \(\boldsymbol{s_{0}}\) and \(\Delta\boldsymbol{s}\) are the standard slowness and the change in slowness of all the microstructures, \(\boldsymbol{T_{0}}\) is the standard travel time of the measurement paths, and \(\boldsymbol{y}\) is the change in the measured travel time compared to the standard travel time of the selected paths.