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

Algorithm 1

Subspace segmentation—row echelon form approach—no noise.

Require: 𝐷×𝑁 data matrix W.

1: Find rref (W) of W.

2: Find Brref (W) of W by setting all non-zero entries of rref (W) to 1.

3: for all 𝑗 from 1 to 𝑁 do

4:  Pick the 𝑗th column 𝑏_𝑗 of Brref (W).

5:  if 𝑏_𝑗 is pivot then

6:   continue

7:  end if

8:  for all 𝑖 from 1 to 𝑗−1 do

9:  if 𝑏_𝑖 is non-pivot and 𝑏_𝑖,𝑏_𝑗 > 0 then

10:    Place {𝑏_𝑖,𝑏_𝑗} in the same cluster 𝐶_𝑖.

11:     break

12:  end if

13:   end for

14: end for

15: for all 𝐶_𝑖 do

16:  Pick any 𝑏∈𝐶_𝑖.

17:  Separate 𝑏 into unit vectors 𝑢¹_𝑖,…,𝑢^𝑑𝑖_𝑖. {These vectors form a basis for a subspace

    𝑆_𝑖 with dimension 𝑑_𝑖.}

18:  for all 𝑘 from 1 to 𝑁 do

19:   if 𝑏_𝑘∈{𝑢¹_𝑖,…,𝑢^𝑑𝑖_𝑖} then

20:    Place 𝑏_𝑘 in the same cluster 𝐶_𝑖. {This is for handling pivot columns.}

21:   end if

22:  end for

23:  Place the corresponding columns in W into the same cluster 𝐖_𝑖.

24: end for

25: Renumber indices 𝑖’s of 𝑆_𝑖 starting from 1.