## 1 Introduction  
In gas combustion systems, temperature information not only shows details about the burning process, but also reveals the equipment running status. It has significant meaning for system safety and efficiency. There are numerous methods for measuring temperature. For example, conventional temperature measuring equipment, such as thermal resistances and thermocouples, are widely used in the combustion process. These instruments are convenient and efficient, but they can only be used for single-point measurements. It is impossible to install enough equipment for detailed information [1]. New temperature measurement techniques [2; 3; 4], such as acoustic measurement and optical measurement, can be used to reconstruct the temperature field. Although these techniques have been used in some combustion systems, 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 techniques. Numerical methods can also be used to acquire temperature information. For example, computational fluid dynamics (CFD) methods are employed to get accurate and detailed temperature information. However, numerical computations always involve a large amount of data, so they are exceptionally time-consuming and not suitable for real-time applications. As mentioned above, in real applications, directly using a mass of empirical data or calculating data are difficult. How to make a use of these data has been a hot area of research in recent years. By using multi-resolution techniques and studying the hidden structure [5] of these data, the fractal-wavelet analysis [6; 7; 8] method can analyze and utilize the most essential characteristics of these data. As the fractal-wavelet analysis is convenient and efficient, it is used in many fields, such as image coding, texture discrimination, fractal analysis [8], distribution calculation [9], and antenna geometry analysis [10; 11]. In order to solve the temperature measurement problem, inspired by the fractal-wavelet analysis method, the aim of this study is to develop a two-dimensional temperature field reconstruction method that can be used in real applications. Some data reduction methods offer a new approach for solving these problems. These kinds of methods can derive a dimension reduction model with minimal information loss, which makes it possible to use CFD calculation data in real-time measurement. Among such methods, proper orthogonal decomposition (POD) is the most commonly used approach [12]. POD is an effective way to reduce data size--it is also known as Karhunen-Loeve (KL) expansion or principal components analysis (PCA). It was first discussed by Pearson in 1901 [13]. Since then, POD has been used in many different domains, including signal analysis [14], data compression [15], image processing [16; 17], chemical reaction analysis [18], and control systems [19; 20]. The POD method was first introduced in fluid mechanics by Lumley [21]. Since then, it has been widely used in flow field analysis [22; 23]. Allery used it to investigate particle dispersion [24], and Qin and Sun used it in flow field reconstruction [25; 26]. The method has also been used in oceanography [27] and aerodynamics studies [28; 29]. Furthermore, some researchers have developed optimization algorithms for sensor placement based on the POD method [27; 30; 31]. There are also some important studies on POD's theoretical basis; for example, Kirby gave detailed discussions on the POD method [12], and Lu et al. presented a review of the POD method in a variety of research areas [32]. In this study, a method based on POD is proposed for the rapid and reliable reconstruction of the temperature field. Simulation data and experimental data were used to test the proposed method.