Apply GPCA to Motion Segmentation - Intelligent Video Event Analysis and Understanding - page 38

To N data points, we have,

c

⎛ ⎞

n 0

⎛ ⎞

1 x ... x ⎜ ⎟

⎜

1 1

⎟

⎜ ⎟

L c ≡ ⎜ ⎟ = 0 ,

(A6)

⎜ ⎟

n

c

⎜ ⎜ ⎟ ⎟

n
N

n − 1

⎜ ⎜ ⎟ ⎟

1 x ... x

⎝

N

⎠

1

⎝ ⎠

n LR ×+

Nn ( )

cR +

n 1

where ∈ and ∈ . Usually, the group number is estimated as,

{ }

n = min i : λλε

+

< ,

(A7)

i 1 i

where λ is the i th singular value of L , which is the collection of the first i +1

columns of L , and ¶ is a given threshold that depends on the noise level.
After solving the coefficient vector c of Eq.(A6), we can compute the n roots of
()

i

{} n
ii

n px , which correspond to the n cluster centers

μ . Finally, the segmenta-

= 1

tion of the date is obtained by,

2

i = arg min ( x − μ ) .

(A8)

j

j = 1,..., n

The scheme of Eq.(A5-A8) is called as the Polysegment algorithm in [13].