A Review of Subspace Segmentation: Problem, Nonlinear Approximations, and Applications to Motion Segmentation

Algorithm 3

Subspace segmentation.

Require: The 𝐷×𝑁 data matrix 𝑊 whose columns are drawn from subspaces of dimension 𝑑

Ensure: Clustering of the feature points.

1: Compute the SVD of  𝑊 as in (8).

2: Estimate the rank of 𝑊 (denoted by 𝑟) if it is not known. For example, using (9) or any other appropriate choice.

3: Compute 𝑉_𝑟^𝑡 consisting of the first 𝑟 rows of 𝑉^𝑡.

4: Normalize the columns of 𝑉_𝑟^𝑡.

5: Replace the data matrix 𝑊 with 𝑉_𝑟^𝑡.

6: Find the angle between the column vectors of 𝑊 and represent it as a matrix.

 {i.e., arccos (𝑊^𝑡𝑊).}

7: Sort the angles and find the closest neighbors of column vector.

8: for all Column vector 𝑥_𝑖 of 𝑊  do

9:  Find the local subspace for the set consisting of 𝑥_𝑖 and 𝑘 neighbors (see (10)).

  {Theoretically, 𝑘 is at least 𝑑−1. We can use the least square approximation for the subspace

   (see the section Local Subspace Estimation). Let 𝐴_𝑖 denote the matrix whose columns form

  an orthonormal bases for the local subspace associated with 𝑥_𝑖.}

10: end for

11: for 𝑖=1 to 𝑁 do

12:   for 𝑗=1 to 𝑁 do

13:   define 𝐻=(𝑑_𝑖𝑗)=𝑥_𝑗−𝐴^𝑡_𝑖𝑥_𝑗𝑝+𝑥_𝑖−𝐴^𝑡_𝑗𝑥_𝑖𝑝/2

14:  end for

15: end for {Build the distance matrix}

16: Sort the entries of the 𝑁×𝑁 matrix 𝐻 from smallest to highest values into the vector ℎ and set the threshold 𝜂 to

  the value of the 𝑇th entry of the sorted and normalized vector ℎ, where 𝑇 is such that

  𝜒_[𝑇,𝑁2]−ℎ₂is minimized, and where 𝜒_[𝑇,𝑁2] is the characteristic function of the discrete set 𝑇,𝑁².

17: Construct a similarity matrix Ξ by setting all entries of 𝐻 less than threshold 𝜂 to 1 and by setting all

 other entries to 0. {Build the binary similarity matrix}

18: Normalize the rows of Ξ using 𝑙₁-norm.

19: Perform SVD Ξ^𝑡 = 𝑈_𝑛Σ_𝑛𝑉_𝑛^𝑡.

20: Cluster the columns of Σ_𝑛𝑉_𝑛^𝑡 using k-means. Σ_𝑛𝑉_𝑛^𝑡 is the projection on to the span of  𝑈_𝑛.