The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
287 
w T Sjw = a T K t a (14) 
where 
< 15 > 
< l7 > 
where K h is the kernel interclasses scatter matrix, K u is the 
kernel intraclasses scatter matrix, and K t is the total scatter 
matrix. All of three matrixes are nonnegative matrixes, and their 
sizes are NxN. 
From Equation (12) and (13), Fisher discriminant function (4) 
can be expressed as 
J» = 
a T K h a 
a T K w a 
(18) 
where a is a nonzero vector. The orthogonal constraint 
condition can be expressed as 
w] Wj = a] <f> T fydj = a]Kdj = 0, Vz * j; i, j = 1,L , d 
So, the Model I can be expressed by kernel matrixes as 
max(J 1 , (a)) 
Model II 
a Ka = 0,j = 1,L ,r 
(19) 
aeR 7 
That is to say that, if we know the first T discriminant 
vectors a, ,L ,a r , the r + 1 discriminant vector a r+{ can be got 
through resolving the above optimization problem, a, is the 
eigenvector corresponding to the maximal eigenvalue of 
eigenfunction K h a = AK w a . If {a,,a 2 ,L ,a d } is from the 
Model Hand {rv^w^L ,w d } is from Model /, the relationship 
between them is 
h>,. ='£ j a *0(x k ) = 0a i ,i = 1,L ,d (20) 
k=1 
where </> = (^(.x, ),L ,<f>(x N j) . 
In Baudat’s literature (Baudat et al., 2000), instead of./,(H>), 
they used J 2 (w) 
r , w> T S* w 
J 2^)= 
w S“w 
Correspondingly, the Model I of GDA can be rewritten as 
max(J 2 (H’)) 
Model/: \ w^.n> = 0,j = 1,L ,r 
w&H 
and the Model //of GDA can be rewritten as 
max^i«)) 
Model II: \ 
a r j Ka = 0,y = 1,L ,r 
a g R n 
(21) 
(22) 
For Model I with J 2 («) , if we have known the first 
r (r > 1) discriminant vectors, the « r+1 can be gotten by 
resolving the following eigenfunction. 
rK b a r+1 = AK t a r+l (23) 
where f = /- KA 7 (AKK~' KA T ) AKK~ ] , / is an identity 
matrix. A = (a, ,« 2 ,L ,a r ) T .Because w is an identity vector in 
Model /, W 7 W = a' Ka = 1 .If a has been known, a should 
be standardized by dividing yja 7 Ka j . 
In the feature space H , if a group of discriminant vectors 
[h> p m> 2 ,L ,»v d } have been known, for the sample tj>(x) , its 
discriminant feature is 
w,<t>(x) = Y, a i <!>(x k )(f>(x) = Y j a-k{x k ,x) = aj$ x (24) 
k=1 k=l 
where is kernel vector of the input sample x . 
The transformation function of GDA is 
y = W T <fcx) = [w Xi w v L , Wjfftx) 
= [a,,a 2 >L , a d ] T £ x 
where y is the feature extracted by GDA which has 
d dimensions. 
2.2 Kernel Function 
Basing on the theory of kernel function, once a kernel function 
k(x,y) accords with Mercer theorem, then it corresponds to 
a inner product kernel function, mapping function and feature 
space in a certain space. In fact, to change kernel parameter is 
to implicitly change mapping function in order to change the 
complexity of distribution in sample sub-space. There are three 
kinds of kernel that are usually used. 
(1) Dimensional polynomial kernel of degree d 
k(x,y) = [(x-y) + p] d 
where p and d are custom parameters. If /7 = 0 and d = 1, it 
will be called linear kernel function. 
(2) Radial basis function (RBF) kernel 
f 
k(x,y) = ex p 
V 
<7 
/ 
where <7 2 > 0. 
(3) Neural Network kernel function 
k(x, j) = tanh(//(x: • j) + v) 
where p and v are parameters. Different from polynomial 
kernel and RBF kernel, the neural network kernel accords with 
the Mercer theorem only when (p, v) are certain values. 
2.3 Flow of Feature Extraction based on GDA 
According to Baudat’s literature (Baudat et al., 2000), we select 
J 2 {w) as the Fisher discriminant function, through the analysis 
above, the steps of feature extraction based on generalized 
discriminant are described as follows. 
(1) Select the kernel function &(•,•) and its parameters, and 
the amount d of the feature will be extracted.