## 1 Introduction  
The high-speed train system is rapidly developing in China and, as a result of rolling contact and vibration impact, wheel polygonization, which is a type of railway-wheel out of roundness (OOR), is now a growing problem. Wheel-tread polygons have been detected with radial irregularity wavelengths from 0.12 to 3.4 m (which is approximately equal to the circumference of a wheel) and with 1 to 25 harmonic orders around the wheel circumference. In the early 1980s, [PERSON] described wheel polygons observed in the Dutch railway system [1]. Wheel polygons can be both dangerous and annoying: the serious Eschede accident of an Inter City Express train in Germany in the summer of 1998, in which 101 people were killed and 194 injured, was attributed to a fatigue crack in a wheel rim caused by a wheel polygon. In recent years, failures in vehicle system components have appeared in the Chinese high-speed railway system, such as gearbox cracks [2], bolt fracture at the axle-box cover and brake disk [3], and other fastener failures that are caused by high-frequency impact due to wheel polygons. In addition, a periodic wheel out-of-roundness leads to noise and vibration levels that annoy passengers, especially at certain train speeds. The maximum vertical wheel-rail contact force is increased by wheel OOR, leading to fatigue and reducing the life span of the track and vehicle components, such as wheelset bearings, gearbox, bolts, fastenings, and sleepers [4]. Trained workers are required to analyze wheel polygons by means of a roughness-measuring system such as m|Wheel, which is time-consuming and not particularly precise. Thus, it is essential to develop procedures for real-time, on-line detection of OOR and to find suitable countermeasures. Different approaches for detecting OOR have been reported, including vibration-impact measurement, displacement determination, image and ultrasonic telemetering, laser sensing, and noise detection. For example, [PERSON] and [PERSON] [5] discussed the Muller-BBM wheel monitoring system allowing for the detection of wheel tread damages. The system, with sensors installed at several track sections to record rail vibration, can identify wheel flats or polygons of all approaching vehicles in an early stage. Much research has been devoted to detect wheel defects based on wheel-rail interaction and track response [6; 7; 8]. [PERSON] and [PERSON] [9] compared three methods of predicting impact force on a railway track-like structure using both finite element and experimental techniques. The results indicated that measuring strain was an effective method of reconstructing dynamic impact force, but that efficacy would be reduced as the impact force increased. [PERSON] et al. [10] described a real-time out-of-roundness wheel-defects monitoring system based on track strain response by using Fiber Bragg Grating sensors. Wayside monitoring systems based on acoustic [11] and ultrasonic [12] sensors have also been employed to detect wheel conditions. However, these sensors can be influenced by electromagnetic interference under a railroad environment. Besides, polygonal-wheel or local spill-wheel defects cannot be detected. Sensors based on Laser-PSD (Position Sensing Detector) technology to measuring out-of-roundness wheel set on-line was reported in [13]. The system can determine the size of wheel out-of-roundness by detecting the displacement of a light spot in PSD based on track dynamics. However, the Laser-PSD sensor requires a precision setup, which is difficult in practice. For most of these measurement systems, the sensors are positioned on rails or sleepers, mainly for OOR detection at speeds below 200 km-h\({}^{-1}\). Vehicle vibrations caused by wheel and track defects affect axle-box accelerations, so some research has been done on using axle-box accelerations to identify short track defects, such as squats, rail spalling, rail corrugation, and track irregularities in several wavelength ranges, and also for more general track surveying.