# Ordered Subspace Clustering for Complex Non-Rigid Motion by 3D Reconstruction
## 1 Introduction  
Modeling and analysis of non-rigid motions from image sequence are challenging problems in computer vision due to complex deformable pattern and shape structure, for example, dynamic scenes, human body activities, expressive or talking faces, etc. NRSfM tries to restore 3D non-rigid features and camera movements out of 2D point tracks collected by a monocular camera, which has received increasing attentions in the related community. Generally, two classes of current popular NRSfM methods roughly are: shape basis factorization and correspondence. The shape basis methods [1; 2; 3] assume principal components of non-rigid motion and then utilizes different shape bases to give a linear representation. The correspondence approaches [4; 5] aim to reconstruct 3D motions from dense points or each pixel in the image sequence, which generally need spatial constraints as regularizers. Although current NRSfM methods work well in reconstructing simple non-rigid deformations, these NRSfM methods still have problems when it comes to practical scenarios with complex non-rigid shape variations and different kinds of motions, such as human activities of sitting, walking, bending, dancing etc. Recently, [PERSON] et al. [6; 7] adopted a simple subspace to model non-rigid 3D shapes and proposed a \"prior-free\" method for NRSfM problem, where there is no prior assumption about the non-rigid or camera motions. In fact, the method can be regarded as an extension of the Robust Principal Components Analysis (RPCA) method [8]. RPCA aims to recover low-rank subspaces from noisy data. [PERSON]'s method makes the same assumption as RPCA, however, [PERSON]'s method is oriented to 3D data reconstructed from the known 2D information. Unfortunately, the method suffers from low accuracy in reconstructing complex and various non-rigid motion. Although this method has been improved by iterative shape clustering [9], which reconstructs 3D shapes and clusters the ones recurrently, however, the improved method still faces problems for dealing with the complex non-rigid motion. Considering complex non-rigid motion in NRSfM, [PERSON] et al. [10] followed [PERSON] et al. [6; 7] and proposed a method named complex non-rigid motion 3D reconstruction by union of subspaces (CNRMS), which reconstructs 3D complex non-rigid motion from 2D image sequence with relative camera motions. In this method, the clustering for 3D non-rigid motion is simultaneously implemented by a union of subspaces. It is considered that the union specifics of the individual subspaces are more suitable to model the complex and various non-rigid motion. The experimental results show that the method has higher clustering accuracy and better reconstruction results compared with the method in [6]. Though introducing a subspace clustering method into the NRSfM model brings considerable improvement for both 3D reconstruction and motion segmentation, current NRSfM methods usually apply fundamental subspace clustering theory. However, the clustering research has had many successful applications in computer vision, pattern recognition, and image processing [11; 12; 13], especially spectral clustering of subspace clustering methods by affinity matrix have better performance. For example, the segmentation accuracy in [14] is three percentage points higher than RPCA, since the constraint of affinity matrix is helpful for data representation [13; 14; 15]. According to the representative clustering methods [16; 17; 18; 19; 20], it is interpreted that to obtain good clustering results, the intrinsic property of the data should be explored, and the feasible structure of the affinity matrix should also been considered. However, the current methods of NRSfM take no account of these factors. On one hand, the sequence property of non-rigid motion has not been considered in current methods, which is ubiquitous in motion segmentation and other applications involved in sequential data.