# Application of Tucker Decomposition in Temperature Distribution Reconstruction
###### Abstract  
Constrained by cost, measuring conditions and excessive calculation, it is difficult to reconstruct a 3D real-time temperature field. For the purpose of solving these problems, a three-dimensional temperature distribution reconstruction algorithm based on Tucker decomposition algorithm is proposed. The Tucker decomposition algorithm is used to reduce the dimension of the measured data, and the processed core tensor is used for the temperature field reconstruction of sparse data. Theoretical analysis and simulations show that the proposed method is feasible; the overall optimization is realized by selecting the appropriate core tensor dimensions; and the reconstruction error is less than 3%. Results indicate that the proposed method can yield a reliable reconstruction solution and can be applied to real-time applications. Tucker decomposition; three-dimensional temperature distribution; core tensor; dimensionality reduction +
Footnote †: journal: Medical  
process [10; 11]. It is convenient and efficient, but it can only be used for single-point measurements. It is impossible to install enough equipment for detailed information. Non-contact measurement methods, such as acoustic measurement [12], laser diagnostic measurement [13] and radiation imaging methods [14], can be used to reconstruct the temperature field. These techniques have the advantages of a wide measurement range, fast dynamic response and little impact on the flame [15; 16]. However, these methods can generally measure the parameters on the line or surface, and many problems remain that may limit their application; for example, noise, vibration, and ash in the industrial environment may have a great impact on their accuracy. Furthermore, sensors involved in these techniques are difficult to install, and the measurement systems are complicated and expensive. All of these factors increase the difficulty of applying these devices. Numerical methods, such as computational fluid dynamics (CFD), can also be used to obtain accurate and detailed temperature information. However, numerical computations always involve a large amount of data; they are exceptionally time-consuming and not suitable for real-time applications. Therefore, it is significant to propose and develop an effective method to use these data. By extracting eigenvector by learning from experience dataset, principal component analysis (PCA)-- also known as Karhunen-Loeve (KL) expansion or proper orthogonal decomposition (POD)--can analyze and utilize the most essential characteristics of these data. As the PCA is convenient and efficient, it is used in many different domains, including signal analysis [17], data compression [18], image processing [19], chemical reaction analysis [20] and control systems [21]. In recent years, it has also been well used in 2D temperature distribution and wind field analysis and reconstruction. [PERSON] et al. [22] combines measurement and numerical algorithms based on PCA to obtain temperature distribution in real time by using feature vectors and measurement data. [PERSON] et al. [23] used PCA for real-time wind field reconstruction for large areas and improving the efficiency of sensor applications. In order to develop a 3D temperature field reconstruction method that can be used in real applications, inspired by the PCA, a three-dimensional temperature distribution reconstruction algorithm based on Tucker decomposition algorithm is proposed. Tucker decomposition was first proposed by [PERSON] in 1963 [24]. Since then, Tucker decomposition has been applied in many different fields, including data compression [25], image processing [26], signal analysis [27] and so on. [PERSON] et al. [28] used it for infrared small-target detection, and [PERSON] et al. [29] used it in wind velocity distribution reconstruction. The method has also been used in hyperspectral change detection [30].