## 2 Related Works  
[PERSON] and [PERSON] [15] performed robot state estimation with particle filter in a 3D state space. However, the conventional particle filter suffers the problem of a high computational cost when a particle filter is applied in high-dimensional state space. [PERSON] et al. [6] combined a particle filter and a visual significance model to solve the sudden movement tracking problem. By embedding a visual significance model into particle filter, their method was efficient in handling abrupt motion tracking problems. However, their particle filter method was also applied in a 3D state space so that it suffers the same problem as the conventional particle filter. In addition, their method is not suitable for estimating high-dimensional humanoid robot joint angles. [PERSON] [7] proposed a novel correction particle filter (CPF) for robust visual tracking. Similarly to the method in [6], the CPF method was efficient at handling partial occlusion and scale variation for object tracking; however, it is not suitable for constructing a real-time humanoid robot imitation system.  
[PERSON] et al. [16] proposed an approach to detect 2D pose of people in an image. [PERSON] et al. [17] presented a real-time method to capture the full global 3D skeletal pose of a human using a single RGB camera. Those studies used a convolutional neural network (CNN) method to estimate human pose. Unlike those studies, we aimed to estimate a humanoid robot's pose instead of a human's pose. There is no necessity to build complex neural networks to capture human poses for the purpose of estimating humanoid robot joint angles. A RGB-D camera, such as Kinect, is more convenient for us to capture human poses.  
[PERSON] et al. [18] used a RBG-D camera to extract human motion features and developed a hidden semi-Markov model (HSMM) based approach to evaluate the performance of people affected by motor disabilities. They proposed a HSMM based method to model human motions for the purpose of evaluating the correctness of rehabilitation exercises. Unlike their research, for the purpose of constructing a real-time imitation system with low computational cost, we combine GPDM and particle filter to estimate high-dimensional humanoid robot joint angles. We first model humanoid robot motions in a compact state space using GPDM. Then, the estimation is performed in a GPDM learned compact state space, and finally, we restore the joint angles from the compact state space to the high-dimensional humanoid robot joint angle space.  
[PERSON] and [PERSON] [8] proved that a high-dimensional state space can be replaced by a low-dimensional latent space learned by dimension reduction. They utilized an autoencoder, a type of artificial neural network, to acquire a 3D state space from the original high-dimensional space of the humanoid robot joint angles. A particle filter was then applied to the resulting 3D state space, achieving state estimation of the humanoid robot with a relatively low computational cost. Unfortunately, the autoencoder constructed the 3D state space to replace the high-dimensional space without the required motion models for particle state tracking. Therefore, they assumed that motion in the 3D space was random according to a Gaussian distribution. An appropriate motion model realizes efficient state tracking from current state \(s_{t}\) to next state \(s_{t+1}\).  
Humanoid robot dynamical motion models were learned in the research by [PERSON] and [PERSON] [14] by using kernel PCA-GPDM. In their research, they only constructed 3D humanoid robot motion models, but the learned motion models were not verified. In this paper, we combine the particle filter and GPDM to realize real-time humanoid robot imitation with low computational cost.