## 3 Compressive Sampling Algorithm  
The conventional reconstruction algorithm of ultrasonic CT has its own limitations, including redundant paths and a heavy workload caused by the dense net of measurement paths. To eliminate these drawbacks, in this paper, a novel reconstructed algorithm is proposed for reducing the number of measurements and speeding up the damage detection program based on CS. Compressive sampling is a new theory of information acquisition proposed by [PERSON] [26], [PERSON] [27], and [PERSON] et al. [28; 29] in 2006. It asserts that one certain signal can be recovered from far fewer samples or measurements than from the traditional Nyquist sampling methods. Assuming that a real-valued, finite-length, one-dimensional, discrete-time signal \(\mathbf{x}\in\mathbb{R}^{N}\) consists of \(x[n],n=1,2,\cdots,N\) (higher-dimensional data can be changed into a long one-dimensional vector by reshaping) and that the unknown signal \(\mathbf{x}\) is defined to be compressible when it can be expressed a \(K\)-sparse vector \(\mathbf{\alpha}\in\mathbb{R}^{N}\) (i.e., it has only \(K\ll N\) non-zero entries) with respect to a specific basis \(\mathbf{\
abla}\in\mathbb{R}^{N\times N}\), the unknown signal can be expressed as:  
\[\mathbf{x}=\mathbf{\
abla}\mathbf{\alpha} \tag{4}\]  
where, \(\mathbf{\alpha}\) is a coefficients vector with \(K\) nonzero entries (\(K\)-sparse) and \(\mathbf{\
abla}\) is the basis matrix representing a specific domain to make the unknown signal sparse. Then, the signal can be exactly recovered or approximated through a small number of linear measurements (\(K<m\ll N\)) to compress the information of the unknown signal \(\mathbf{x}\) to a measurement vector \(\mathbf{y}\), which can be expressed as:  
\[\mathbf{y}=\mathbf{\Phi}\mathbf{\
abla}\mathbf{\alpha}+\mathbf{n}=\mathbf{\Theta}\mathbf{\alpha}+\mathbf{n} \tag{5}\]where \(\mathbf{\Phi}\in\mathbb{R}^{m\times N}\) is the measurement matrix that has fewer rows than the columns, \(n\) denotes the noise during the measuring process, and \(\mathbf{\Theta}=\mathbf{\Phi}\mathbf{\Psi}\) is defined as the transfer matrix; thus, the rows of matrix \(\mathbf{\Theta}\) are much fewer than the columns of \(\mathbf{\Theta}\). Since the number of measurement is smaller than the number of samples in the original signal, it is an ill-posed problem solving for the unknown signal \(\mathbf{x}\) from the measured information \(\mathbf{y}\)[30; 31]. However, if the signal is compressible and sparse in the specific basis, as previously mentioned, then an exact or approximate recovery of \(\mathbf{\alpha}\) is possible by solving a \(l_{1}\)-norm optimization problem, which can be used to reconstruct the original unknown signal \(\mathbf{x}\). Furthermore, in order to reconstruct the unknown signal, the transfer matrix \(\mathbf{\Theta}\) is required to satisfy the following restricted isometry property (RIP) condition [27]:  
\[1-\delta\leq\frac{||\mathbf{\Theta}\mathbf{v}||_{2}}{||\mathbf{v}||_{2}}\leq 1+ \delta,\ \delta>0. \tag{6}\]  
where, \(\mathbf{v}\in\mathbb{R}^{N}\) is all \(K\)-sparse vector and \(\delta\). is an isometric constant, which is defined as the smallest value that satisfies Equation (6). Random Bernoulli matrices and random Gaussian matrices have been proved to satisfy RIP conditions and can be used to reconstruct the original signal with high probability [27] under an appropriate linear measurement condition. The number of required measurement is expressed as \(m>\upmu\cdot K\cdot\log(N/K)\), where \(\upmu\) is a constant which is dependent on the basis used for measurement. However, both the RIP and incoherence can be achieved with high probability simply by selecting \(\mathbf{\Phi}\) as a random matrix (Gaussian matrix or Bernoulli matrix).