# Reconstruction Optimization Algorithm of 3D Temperature Distribution Based on Tucker Decomposition
## 2 Fundamental Measurement Principles  
### Tucker Decomposition Reconstruction Algorithm  
The following is a brief introduction to the Tucker decomposition reconstruction algorithm. The detailed derivation process is shown in the previously published paper [9]. A prior dataset obtained by simulation calculations or by sensors is denoted as \(\mathbf{TR}=\{\mathbf{T}_{1},\mathbf{T}_{2},T_{3},\ldots,\mathbf{T}_{i},\ldots,\mathbf{T}_{n1}\}\), \(\mathbf{T}_{i}\in\mathbb{C}^{n_{2}\times n_{3}\times n_{4}},\mathbf{TR}\in\mathbb{C} ^{n_{1}\times n_{2}\times n_{3}\times n_{4}}\). Through Tucker decomposition and tensor properties, the prior dataset can be decomposed into the following form:  
\[\mathbf{TR}=\left\{ \begin{array}{c}\mathbf{T}_{1}\\ \mathbf{T}_{2}\\ \vdots\\ \mathbf{T}_{i}\\ \vdots\\ \mathbf{T}_{n_{1}}\end{array} \right\}=\mathbf{C}_{t}\times_{1}\mathbf{F}\times_{2}\mathbf{V} \times_{3}\mathbf{W}\times_{4}\mathbf{U}, \tag{1}\]where \(\mathbf{C}_{t}\in\mathbb{C}^{m_{1}\times m_{2}\times m_{3}\times m_{4}}\) is called the core tensor, \(\mathbf{U}\in\mathbb{C}^{n_{i}\times m_{i}}\), \(\mathbf{V}\in\mathbb{C}^{n_{i}\times m_{i}}\), \(\mathbf{W}\in\mathbb{C}^{n_{i}\times m_{i}}\) and \(\mathbf{F}\in\mathbb{C}^{n_{i}\times m_{i}}\) are four factor matrices. Because the tensor mode product conforms to this formula \(\mathbf{C}\times_{1}\mathbf{P}\times_{2}\mathbf{Q}=\mathbf{C}\times_{2}\mathbf{Q}\times_{1}\mathbf{P}\)[14], Formula (1) can be converted to the following form. \[\mathbf{T}\mathbf{R}=\left\{ \begin{array}{c}\mathbf{T}_{1}\\ \mathbf{T}_{2}\\ \vdots\\ \mathbf{T}_{i}\\ \vdots\\ \mathbf{T}_{n_{1}}\end{array} \right\}=\mathbf{C}_{t}\times_{2}\mathbf{V}\times_{3}\mathbf{W} \times_{4}\mathbf{U}\times_{1}\left\{ \begin{array}{c}\mathbf{f}_{11}\\ f_{12}\\ \vdots\\ \mathbf{f}_{1i}\\ \vdots\\ \mathbf{f}_{1n_{1}}\end{array} \right\}, \tag{2}\]  
where \(\mathbf{f}_{1i}\in\mathbb{C}^{1\times m_{1}}\) is the \(i\)th factor vector in the decomposition factor matrix in the 1 norm direction. From Formula (2), for the reconstruction under the same conditions, the temperature field should satisfy Formula (3):  
\[\mathbf{T}_{x}=\mathbf{A}\times_{1}\mathbf{f}_{1x}, \tag{3}\]  
According to the definition of the pattern product, Formula (3) can be converted to:  
\[\mathbf{t}_{x}=(\mathbf{f}_{1x}\times\mathbf{A}_{\text{1 unfold}})^{\text{T}}, \tag{4}\]  
where \(\mathbf{t}_{x}\) is the representation of the 3D temperature distribution by vector; \(\mathbf{A}_{\text{1 unfold}}\in\mathbb{C}^{(n_{2}\times n_{3}\times n_{4})\times m_ {1}}\) is the coefficient matrix of \(\mathbf{A}\) expanding in the direction of 1 mode. In the process of temperature measurement, when the sensor is arranged, the measurement matrix is determined:  
\[\mathbf{t}_{\text{M}}=\mathbf{M}\mathbf{t}_{x}, \tag{5}\]  
where \(\mathbf{t}_{\text{M}}\in\mathbf{R}^{k\times 1}\) is a column vector of temperature values measured by the sensor; \(\mathbf{M}\in\mathbf{R}^{k\times n}\) denotes sensor position. It can be derived from Equations (4) and (5):  
\[\mathbf{MA}_{\text{1 unfold}}\mathbf{f}_{1x}^{\text{T}}=\mathbf{t}_{\text{M}}, \tag{6}\]  
In Formula (6), \(\mathbf{t}_{\text{M}}\) is the measured value, \(\mathbf{M}\) is a known matrix, and \(\mathbf{A}_{\text{1 unfold}}\) is calculated by Tucker decomposition. Therefore, the solution becomes to establish a reconstruction error minimization function:  
\[J\left(\mathbf{f}_{1x}^{\text{T}}\right)=\text{arg}min\|\mathbf{t}_{\text{M}}-\mathbf{MA} _{\text{1 unfold}}^{\text{T}}\mathbf{f}_{1x}^{\text{T}}\|^{2}, \tag{7}\]  
where \(J\left(\mathbf{f}_{1x}^{\text{T}}\right)\) is the objective function.