## 1 Introduction  
In the field of railway transportation, the health state of wheels has a crucial impact on vehicle operation safety. Therefore, it is of great significance to monitor the generation and development of wheel defects in real time to enhance the safety and reliability of railway operation. Wheel flats are the most typical form of defect during long-term service of train wheels, and this defect induces the failure of both the vehicles and the infrastructures. Wheel flats are mainly the result of the following two aspects: (1) The anomalous wear of the wheel tread due to sudden braking of the moving vehicle [1] and (2) the local reduction of wheel-rail adhesion force, which is often caused by rail surface foreign matters, leading to complete sliding between the wheel and rail [2]. Wheels with flats create intermittent impact loads on trains and rails while moving, which strongly increases the dynamic responses of the vehicle-track coupling system [3]. In light of previous studies, the wheel-rail impact loads caused by wheel flats are far higher than the wheel-rail contact force under normal circumstances [4]. These high impact loads accelerate the aging of the vehicle and track components, inducing defects such as axle abnormal vibration, rail abrasion, and fastener fracture. In some extreme situations, the wheel-rail contact conditions change dramatically because of wheel flats, which directly threatens the operation safety. To solve this problem, railway operation departments generally take measures with respect to both precautions and renovation. In terms of precautions, at present, most passenger trains are equipped with advanced anti-sliding systems, which, to some extent, alleviate the frequency of wheel-rail sliding [5; 6]. Nevertheless, with increasing operating speed and axle load, wheel flats cannot be completely avoided. In addition, with regards to those freight trains without an anti-sliding system, the condition of the wheels is usually even worse, which has a significant impact on the long-term service of the trains and infrastructures. Concerning the repair of wheel flats, nowadays, wheel reprofiling is the most effective, and therefore the most popular approach for repairing wheel defects [7]. However, under the premise that the health state of wheels is difficult to monitor in real time, how to develop a reasonable wheel tread reprofiling strategy becomes an open issue. Excessive repairs not only cause a surge in management costs but also shorten the life of healthy wheels, since all the wheels are subject to grinding during a reprofiling procedure [8]. Therefore, it is of great importance to reprofile the wheelsets according to their real conditions. In recent years, researchers have proposed many different methods for in-service measuring of wheel defects, which have, generally, been divided into two types, on-board and wayside measurements [9]. Most on-board techniques are based on vibration, acoustic, image detection, and ultrasonic technologies. [PERSON], [PERSON] et al. [10] proposed a diagnosis method based on the time domain analysis of the train axial accelerations and introduced a new index to judge whether the wheel flats exist. [PERSON], [PERSON] et al. [11] put forward an acoustic emission (AE) signal-based detection approach. By installing an acoustic sensor near the wheel, anomalous wheel-rail noises excited by wheel flats are collected and, then, analyzed. [PERSON], [PERSON] et al. [12] used an infrared camera to capture alterations of temperature extension on the wheel tread to detect the surface cracks. A limitation for on-board methods is that this method is quite time-consuming since the detection is only implemented after the rail is heated. [PERSON], [PERSON] et al. [13] proposed an air-coupled ultrasonic method. They installed the sensors on the bogie approaching the wheels to measure the radial and circumferential defect of wheel. However, the high cost of equipment makes it difficult to promote this approach.