# Overview of Compressed Sensing: Sensing Model, Reconstruction Algorithm, and Its Applications
## 2 Sensing Methods
### Semitensor-Product Compressed Sensing  
[PERSON] et al. presented the semitensor product (STP) of matrices that broke through the limitation of conventional matrix operations. They further proposed an evolution of the traditional matrix product [26; 27; 28; 29]. Traditional matrix multiplication should meet the limitations of matrix dimensions, that is, the column number of matrix \(A\) must equal the row number of matrix \(x\), as is shown in Figure 3. STP theory breaks through this limitation, and it can execute matrix multiplication when the dimensions of two matrices are unmatched. Moreover, STP maintains the main properties of ordinary matrix multiplication. Suppose that \(u\) is a row vector of dimension \(np\), and \(v\) is a column vector of dimension \(p\). By splitting \(u\) into \(p\) equal blocks, that is, \(u^{1},\cdots,u^{p}\), each part \(u^{i}\) is a row vector of dimension \(n\). The definition of STP, represented by \(\ltimes\), is  
\[u\ltimes v=\sum_{i=1}^{p}u^{i}v_{i}\in R^{1\times n}. \tag{20}\]  
Similarly,  
\[v^{T}\ltimes u^{T}=\sum_{i=1}^{p}v_{i}(u^{i})^{T}\in R^{n\times 1}\]  
Let \(A\in R^{m\times n}\) and \(B\in R^{p\times q}\). If either \(n\) is a factor of \(p\) or \(p\) is a factor of \(n\), then we define the semitensor product of \(A\) and \(B\) as follows:  
\[A\ltimes B=\left( \begin{array}{ccc}A_{1}\ltimes B^{1}&\cdots&A_{1}\ltimes B ^{q}\\ \vdots&\ddots&\vdots\\ A_{m}\ltimes B^{1}&\cdots&A_{m}\ltimes B^{q},\end{array} \right) \tag{21}\]  
where \(A_{i}\) is the \(i\)-th row of \(A\), and \(B^{i}\) is the \(j\)-th column of \(B\). Equivalently, we can also define the STP of \(A\) and \(B\) by using the Kronecker product:  
\[A\ltimes B=(A\otimes I_{t/n})(B\otimes I_{t/p}), \tag{22}\]  
where \(t\) is the least common multiple of \(n\) and \(p\), i.e., \(t=lcm(n,p)\). [PERSON] et al. proposed semitensor-product compressed sensing (STP-CS), which combined semitensor product and compressed sensing [30]. They analyzed STP-CS from a theoretical perspective to demonstrate that the sparse solution is unique with regard to spark and coherence. The RIP criterion is satisfied in the STP-CS model. There are many classical sensing matrices that can be used in STP-CS, such as the Gaussian, Bernoulli, and chaotic matrices. These classical matrices can be used in STP-CS because the RIP configuration of order \(k\) in STP-CS is equivalent to that in conventional CS. On the basis of the semitensor product, STP-CS uses a low-dimensional sensing matrix to compress high-dimensional signals. The storage space needed in STP-CS is greatly saved compared with that in block-compressed sensing (BCS) with small block size. The semitensor product can be used to improve the reconstruction algorithm to realize parallel reconstruction, which can simultaneously perform signal reconstruction in multiple CS decoders, resulting in a reduction in total reconstruction time. An application of STP-CS was presented to reduce calculation energy consumption, and it was applied to the communication of wireless sensor networks (WSNs) [31]. In terms of recovery quality, STP-CS is almost equal to conventional CS and CCS. [PERSON] et al. proposed a random-sampling method based on the STP-CS framework [32]. They used an improved iteratively reweighted least-squares (IRLS) algorithm to obtain the values of the sparse vector. Simulation results showed that their method could save at least one-quarter of the storage resources when ensuring reconstruction performance. The P-tensor product (PTP) was proposed on the basis of STP. It not only solved the dimensional-matching problem in matrix multiplication, but also provided a new method for angle calculation between different dimensional vectors [33]. For example, we can calculate the angle between a one- and a three-dimensional vector by PTP. PTP compensates for the limitations of STP when performing operations on vectors with different dimensions. In PTP, a smaller matrix is changed into a larger matrix, conforming to dimension matching by the tensor operation of matrix \(P\).