# Ordered Subspace Clustering for Complex Non-Rigid Motion by 3D Reconstruction
## 2 Related Works  
We briefly summarize the representative methods of NRSfM, particularly the NRSfM method in [6; 10].  
In recent works, [PERSON] et al. [6] proposed a method of NRSfM which adopted a low-dimensional subspace to model the non-rigid 3D shapes, which is similar to the principle of RPCA, proposed by [PERSON] et al. [8]. The RPCA approach usually represents the noisy 2D data derived from a low-dimension subspace, where the clean data is recovered by the following objective function,  
\[\min_{D,E}\lVert D\rVert_{*}+\lambda\lVert E\rVert_{l},\text{ s.t. }X=D+E, \tag{1}\]  
where \(D\) is the clean data of low-dimension subspace with low-rank constraint modeled by nuclear normal \(\lVert\cdot\rVert_{*}\). \(E\) denotes residual noise, and \(X\) denotes corrupted data with noise. \(\lVert E\rVert_{l}\) denotes error norm, when \(l=1\) it is the sparse error and \(l=1,2\) it denotes group sparse error. The error penalty parameter is noted by \(\lambda>0\).  
The NRSfM model proposed by [PERSON] et al. [6] can be regarded as an extension of RPCA, and the overall model is described below,  
\[\min_{X,E}\lVert X\rVert_{*}+\lambda\lVert E\rVert_{2},\text{ s.t. }W=RX^{\#}+E, \tag{2}\]  
where \(W\in R^{2F\times N}\) is the known 2D information of N points, which also can be seen in the sequences projected from 3D points coordinates noted by \(X^{\#}\in R^{3F\times N}\), here, the number of 3D points is \(N\) and there are totally \(F\) frames. \(X\in R^{F\times 3N}\) is a reshape of \(X^{\#}\). The projection matrix is denoted by \(R\in R^{2F\times 3F}\), which can be pre-computed out of the 2D sequences by the methods in [6; 23]. In this method, the \(l_{2}\) norm is selected for the error \(E\) under the assumption of Gaussian noise. It can be replaced by other norms as in RPCA. It is called a \"prior-free\" method as there is no prior assumption on non-rigid framework or camera motions.  
[PERSON] et al. [10] argued that [PERSON] et al.'s method [6] assumed the data sampled from a subspace and suffered the same low accuracy as RPCA, when it is applied to complex and various motion reconstruction. [PERSON] et al. proposed a NRSfM method by modeling complex and various non-rigid motion as subspace set inspired from subspace clustering method, the low rank representation (LRR) [14; 15], and the model is described below,  
\[\min_{X,Z,E}\lVert Z\rVert_{*}+\lambda\lVert E\rVert_{l}+\gamma\lVert X\rVert _{*},\text{ s.t. }W=RX^{\#}+E,X=XZ. \tag{3}\]  
where \(X=XZ\) stands for subspace clustering constraint which automatically enforces structure of subspace set \(X\), with \(Z\), stands for a matrix of low rank coefficients. \(W=RX^{\#}+E\) constrains the 3D reconstruction out of 2D projections, that is, \(W\) to \(X\). The penalty parameter for \(\lVert X\rVert_{*}\) is \(\gamma\) and the penalty parameter of \(\lVert E\rVert_{l}\) is \(\lambda\), \(\lambda>0\). It is shown that the method simultaneously reconstructs and clusters the 3D complex non-rigid motions \(X\), which involved a subspace set by low-rank affinity matrix \(Z\).  
Recently, ordered subspace clustering (OSC) [16] and the ordered subspace clustering with block-diagonal priors (QOSC) [17] is proposed. We try to model the sequential property of sequence data in the view of subspace clustering.