GSVD-based LDA algorithm for hand gesture recognition
1

From (7) ∼ (9) set Hb and Hw as follows.

H b = N s [ m ¯ 1 ( 1 ) m ¯ t ( l ) , , m ¯ D ( l ) m ¯ t ( l ) ] , H w = [ A 1 ( 1 ) m ¯ 1 ( 1 ) 1 ¯ N s T , , A D ( 1 ) m ¯ D ( 1 ) 1 ¯ N s T ] .

Here, 1 ¯ N s is the 1-vector of size Ns. Note that Sb(l) =HbHbH and Sw(l)=HwHwH are satisfied.

2

Compute the SVD(singular value decomposition) of matrix Z which is composed of (Hb,Hw).

Z = [ H b H H b H ] C D ( N s + 1 ) × M ( N M ) , Z = P [ Λ 0 0 0 ] U H .

Here, Λ is a diagonal matrix with effective rank s, P C D ( N s + 1 ) × D ( N s + 1 ) and U C M ( N M ) × M ( N M ) are orthogonal matrices.

3

Partition the matrix P as

P = [ P 11 P 12 P 21 P 22 ] ,

where P11C(D×s), P 12 C D × ( D ( N s + 1 ) s ) , P 21 C D N s × s , and P 22 C D N s × ( D ( N s + 1 ) s ) are submatrices.

Compute the orthogonal matrix V from the SVD of P11.

P 11 = W V H .
4

Compute X as

X = U [ Λ 1   V 0 0   V ] .

Then set transformation matrix G ¯ ( l ) as

G ¯ ( l ) = [ [ X ] 1 , [ X ] 2 , , [ X ] M D ] C M ( N M ) × M D ,

where [X]i is ith column of matrix X.