# Reconstruction Optimization Algorithm of 3D Temperature Distribution Based on Tucker Decomposition
###### Abstract  
For the purpose of solving the large temperature field reconstruction error caused by different measuring point arrangements and the problem that the prior dataset cannot be built due to data loss or distortion in actual measurement, a three-dimensional temperature profile reconstruction optimization algorithm is proposed to repair the empirical dataset and optimize the arrangement of temperature measuring points based on Tucker decomposition, the minimum condition number method, the greedy algorithm, and the hill climbing algorithm. We used the Tucker decomposition algorithm to repair the missing data and obtain the complete prior dataset and the core tensor. By optimizing the dimension of the core tensor and the number and position of the measuring points calculated by the minimum condition number method, the greedy algorithm, and the mountain climbing algorithm, the real-time three-dimensional distribution of the temperature field is reconstructed. The results show that the Tucker decomposition optimization algorithm could accurately complete the prior dataset, and compared with the original algorithm, the proposed optimal placement algorithm improves the reconstruction accuracy by more than 20%. At the same time, the algorithm has strong robustness and anti-noise, and the relative error is less than 4.0% and 6.0% with different signal-to-noise ratios. It indicates that the proposed method can solve the problem of building an empirical dataset and 3D temperature distribution reconstruction more accurately and stably in industry. Tucker decomposition; prior dataset repair; three-dimensional temperature distribution optimization; optimal sensor placement; noise reduction +
Footnote †: journal: Medical Imaging  
1
Footnote 1: School of Energy, Power and Mechanical Engineering, North China Electric Power University,  
Beijing 102206, China 1, [PERSON]  
2 School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, China 2, [PERSON]  
1 School of Energy, Power and Mechanical Engineering, North China Electric Power University,  
Beijing 102206, China 1, and [PERSON]  
1 School of Energy, Power and Mechanical Engineering, North China Electric Power University,  
Be  
using the Tucker tensor decomposition to decompose the unknown tensor parameter, and then using the alternative optimization method to solve the optimization problems. [PERSON] et al. [8] used it to detect infrared small targets and estimate and eliminate the major principal components, enhancing the image contrast. In previous studies, we have reconstructed the three-dimensional temperature field by Tucker decomposition [9]. Although the temperature field distribution can be obtained by the Tucker decomposition reconstruction algorithm through a small amount of measurement data, different arrangements of measuring points have a great influence on the reconstruction accuracy due to the small amount of measurement data. At the same time, industrial data often have missing or distorted phenomena, resulting in incomplete empirical datasets, which cannot be directly reconstructed using the reconstruction algorithm proposed by the author before [9]. Most existing research, however, focused on how to use Tucker decomposition to analyze flow field distribution and image reconstruction, ignoring the establishment of empirical datasets and the selection of optimal measurement points. [PERSON] et al. [10] used it for wind field reconstruction predictions and to create a third-order prior database by CFD simulations. [PERSON] et al. [11] obtained the optimal parameter through the energy-cumulative method that computes the sum of eigenvalues one by one. This method cannot automatically select eigenvalues, and they must be manually added one by one until the cumulative energy achieves a specified ratio to all energy when using Tucker decomposition in hyperspectral change detection. Generally, Tucker decomposition can be regarded as high-order principal component analysis (PCA) [12].