# Application of Tucker Decomposition in Temperature Distribution Reconstruction
## 2 Fundamental Measurement Principles
### Reconstruction Algorithm Based on Tucker Decomposition  
In the reconstruction of three-dimensional temperature distribution, the a priori dataset is composed of \(\mathbf{n_{1}}\) three-dimensional temperature field obtained by simulation calculation or measurement. In order to facilitate the subsequent calculation and analysis, the prior dataset is recorded as a fourth-order tensor \(\mathbf{TR}=\{\mathbf{T_{1}},\mathbf{T_{2}},\mathbf{T_{3}},\ldots,\mathbf{T_{i}},\ldots,\mathbf{T_{n }}\},\mathbf{T_{i}}\in\mathbb{C}^{n_{2}\times n_{3}\times n_{4}}\), \(\mathbf{TR}\in\mathbb{C}^{n_{1}\times n_{2}\times n_{3}\times n_{4}}\). The prior dataset is decomposed into the form of core tensor and decomposition factor mode product through Tucker decomposition, as follows:  
\[\mathbf{TR}=\left\{ \begin{array}{c}\mathbf{T_{1}}\\ \mathbf{T_{2}}\\ \vdots\\ \mathbf{T_{i}}\\ \vdots\\ \mathbf{T_{n_{1}}}\end{array} \right\}=\mathbf{C_{t}}\times_{1}\mathbf{F_{t(1)}}\times_{2} \mathbf{F_{t(2)}}\times_{3}\mathbf{F_{t(3)}}\times_{4}\mathbf{F_{t(4)}}, \tag{8}\]  
where \(\mathbf{C_{t}}\)--the core tensor of prior dataset tensor \(\mathbf{TR}\) Tucker decomposition, \(\mathbf{C_{t}}\in\mathbb{C}^{m_{1}\times m_{2}\times m_{3}\times m_{4}}\) and \(\mathbf{F_{t(i)}}\)--decomposition factor matrix in the \(i\)-module direction of tensor \(\mathbf{TR}\), \(\mathbf{F_{t(i)}}\in\mathbb{C}^{n_{i}\times m_{i}}\). \[\mathbf{C}\times_{1}\mathbf{P}\times_{2}\mathbf{Q}=\mathbf{C}\times_{2}\mathbf{Q}\times_{1}\mathbf{P} \tag{9}\]  
where \(\mathbf{P}\)--the decomposition factor matrix in the 1-mode direction and Q--the decomposition factor matrix in the 2-mode direction. According to Formula (10), Formula (9) is transformed into the following form:  
\[\mathbf{T}\mathbf{R}=\left\{ \begin{array}{c}\mathbf{T}_{\mathbf{1}}\\ \mathbf{T}_{\mathbf{2}}\\ \vdots\\ \vdots\\ \mathbf{T}_{\mathbf{n_{1}}}\end{array} \right\}=\mathbf{C}_{\mathbf{t}}\times_{1}\mathbf{F}_{\mathbf{t} (\mathbf{1})}\times_{2}\mathbf{F}_{\mathbf{t}(\mathbf{2})}\times_{3}\mathbf{F}_{\mathbf{t}(\mathbf{3})} \times_{4}\mathbf{F}_{\mathbf{t}(\mathbf{4})}\times_{1}\left\{ \begin{array}{c}\mathbf{f}_{ \mathbf{1}1}\\ \mathbf{f}_{\mathbf{1}2}\\ \vdots\\ \mathbf{f}_{\mathbf{1}i}\\ \vdots\\ \mathbf{f}_{\mathbf{1}m_{1}}\end{array} \right\}, \tag{10}\]  
where \(\mathbf{f}_{\mathbf{1}i}\)--the \(i\)th factor vector in the decomposition factor matrix in the direction of 1 modulus and \(\mathbf{f}_{\mathbf{1}i}\in\mathbb{C}^{1\times m_{1}}\). It can be seen from Formula (11):  
\[\mathbf{T}_{i}=\mathbf{A}\times_{1}\mathbf{f}_{\mathbf{1}i}, \tag{11}\]  
where \(\mathbf{A}\)--reconstruction coefficient tensor, \(\mathbf{A}=\mathbf{C}\times_{2}\mathbf{F}_{(2)}\times_{3}\mathbf{F}_{(3)}\times_{4}\mathbf{F}_{( \mathbf{4})}\) and \(\mathbf{A}\in\mathbb{C}^{m_{1}\times_{2}\times_{2}\times_{3}\times n_{4}}\). In the process of solving, for the three-dimensional temperature distribution to be reconstructed under the same combustion environment, the Equation (13) should also be satisfied:  
\[\mathbf{T}_{\mathbf{x}}=\mathbf{A}\times_{1}\mathbf{f}_{\mathbf{1}\mathbf{x}}, \tag{12}\]  
where \(\mathbf{f}_{\mathbf{1}\mathbf{x}}\)--the three-dimensional temperature distribution to be solved to reconstruct the 1-mode decomposition factor vector. According to the definition of model product, the Formula (13) is transformed into the form of vector and matrix multiplication:  
\[\mathbf{t}_{\mathbf{x}}=\big{(}\mathbf{f}_{\mathbf{1}\mathbf{x}}\times\mathbf{A}_{\mathbf{1}\textit{ unfold}}\big{)}^{T}, \tag{13}\]  
where \(\mathbf{t}_{\mathbf{x}}\)--first-order representation of three-dimensional temperature distribution; \(\mathbf{A}_{\mathbf{1}\textit{unfold}}\)--the expansion of the coefficient matrix A along the direction of 1 modulus and \(\mathbf{A}_{\mathbf{1}\textit{unfold}}\in\mathbb{C}^{(n_{2}\times n_{3}\times n_{4}) \times m_{1}}\).