## 2 Research on 3D Temperature Field Reconstruction Algorithms
### Principles of Tucker Decomposition Algorithm  
Tucker decomposition, proposed by [PERSON] in 1963, is a mathematical technique that facilitates the transformation of an n-dimensional high-dimensional tensor into an n-dimensional low-dimensional core tensor along with n-factor matrices. Tucker decomposition is widely used in problems such as large-scale data processing [2]. The mathematical model for the decomposition process of an Nth-order tensor \(H\in\mathbb{C}^{m_{1},n_{2}\times n_{3}\times n_{4}}\) is expressed as follows [28]:  
\[H=C\times_{1}F_{(1)}\times_{2}F_{(2)}\times_{3}F_{(3)}\ldots\times_{N}F_{(N)} \tag{6}\]  
where \(C\) represents the core tensor obtained through the Tucker decomposition of tensor \(H\), \(\times_{N}\) denotes the N-mode product of the tensor, and \(F_{(N)}\) indicates the factor matrix of tensor \(H\) in the N-mode direction. During the process of reconstructing the 3D temperature distribution using acoustic temperature measurement, \(n_{1}\) sets of 3D temperature distributions are obtained, either through simulation calculations or instrument measurements. Subsequently, the TOF data for each temperature distribution plane are computed to form the prior dataset. To facilitate subsequent computation and analysis, the prior dataset is represented as a third-order tensor, denoted by \(TR\). Through Tucker decomposition, the prior dataset is decomposed into the form of a core tensor and the mode product of the decomposition factors, as shown below [28]:\[T_{R}=\left\{ \begin{array}{l}T_{1}\\ T_{2}\\ \vdots\\ T_{Z}\\ \vdots\\ T_{n_{1}}\end{array} \right\}=C_{t}\times_{1}F_{t(1)}\times_{2}F_{t(2)}\times_{3}F_ {t(3)}\times_{4}F_{t(4)} \tag{7}\]  
where \(\mathbf{C}_{t}\) represents the core tensor obtained from the Tucker decomposition of the prior dataset tensor \(\mathbf{TR}\), and \(\mathbf{C}_{t}\in\mathbb{C}^{m_{1\times}m_{2}\times m_{3}\times m_{4}}\). In addition, \(\mathbf{F}_{t(i)}\) denotes the decomposition factor matrix in the \(i\)-mode direction of the tensor \(\mathbf{TR}\), and \(\mathbf{F}_{t(i)}\in\mathbb{C}^{n_{i}\times m_{i}}\). The mode product of the tensors has the following property:  
\[C\times_{1}P\times_{2}Q=C\times_{2}Q\times_{1}P \tag{8}\]  
Based on Equation (8), Equation (7) can be transformed into the following form:  
\[\left\{ \begin{array}{l}T_{1}\\ T_{2}\\ \vdots\\ T_{Z}\\ \vdots\\ T_{n_{1}}\end{array} \right\}=C_{t}\times_{1}F_{t(1)}\times_{2}F_{t(2)}\times_ {3}F_{t(3)}\times_{4}F_{t(4)}\times_{1}\left\{ \begin{array}{l}f_{11}\\ f_{12}\\ \vdots\\ f_{1Z}\\ \vdots\\ f_{1n_{1}}\end{array} \right\} \tag{9}\]  
where \(\mathbf{f}_{1i}\) represents the \(i\)-th factor vector in the mode-1 factor matrix, and \(\mathbf{f}_{1i}\in\mathbb{C}^{1\times m_{1}}\). Considering Equation (9), the following equation can be deduced:  
\[T_{i}=A\times_{1}f_{1i} \tag{10}\]  
The coefficient tensor \(\mathbf{A}\) can be determined as follows: \(\mathbf{A}=\mathbf{C}_{t}\times_{2}F_{(2)}\times_{3}F_{(3)}\times_{4}F_{(4)}\), where \(\mathbf{A}\in\mathbb{C}^{m_{1\times}n_{2}\times n_{3}\times n_{4}}\). Here, the symbol \"\(\times_{i}\)\" represents the tensor product operation along the i-th dimension. The tensor product involves a multi-linear algebraic operation that generates a new tensor from two given tensors, where each element in the new tensor is the product of the corresponding elements from the original tensors [38; 39]. After performing the tensor product operation between \(\mathbf{C}_{t}\) and \(\mathbf{F}_{t(i)}\) along the \(i\)-th dimension, the resulting \(\mathbf{A}\) tensor has the following dimensions: \(\mathbf{A}\in\mathbb{C}^{m_{1\times}n_{2}\times n_{3}\times n_{4}}\). For the three-dimensional temperature distribution \(T_{x}\) to be reconstructed in the same combustion environment, it must also satisfy Formula (11) [38]:  
\[T_{x}=A\times_{1}f_{1x} \tag{11}\]  
where \(\mathbf{f}_{1x}\) indicates the factor vector of the first mode for the reconstruction of the 3D TOF distribution.