Full text: Mapping without the sun

2.1 PCA Algorithm (James R. Carr, 1998) 
2.3 Image Fusion 
Given M intercorrelated image bands, one form of principal 
components analysis can proceed by assembling a matrix, S , 
M x M in size and symmetrical, such that S(i,l) is the 
variance for band i, and S(i, j) = S(j, /) is the 
covariance between bands i and j . Variance can be 
computed as 
VAR(x) = 
f N \ 
N N 
N T. x i ~'L x ‘'L x i <2 - 1) 
/=i /=i ¿=1 
in which N is the total number of pixels, X, collectively 
representing a particular image band. This equation compares 
to that for covariance between two image bands, X and y : 
" N i N N 
COV(x,y) = 
/=1 ¿=1 
N N 
• Y Z v . v r Z'X 1 '. < 2 - 2 > 
i=1 /=1 /=1 
where N is also the total number of pixels, X, for an image 
band, and y , for the other image band. 
Suppose a scene multispectral image is to be transformed using 
principal components analysis of seven (M) bands. In this 
case, a matrix, S , of size 7x7 is computed. Diagonal 
entries of this matrix consist of the seven variances for each 
band. Additionally, there are 21 off-diagonal entries, 
symmetrical above and below the diagonal, representing the 
covariances for all possible two-band combination from the 
group of 7 bands. 
2.2 Image Transformation 
Once the eigenvectors of the matrix, S , are computed for PCA 
transformation, they can be used to image transformations. This 
procedure is straightforward. Let X represent a matrix 
whose columns are the M eigenvectors. The size of X , in 
this case, is M x M . Further, let A denote the total 
collection of multispectral or hyperspectral imagery. In this 
case, A is of size, N X M , where N is the total number 
of pixels per image plane (for example, N is 262,144 if the 
image plane is 512x512), and M is the number of spectral 
bands. Then, a simple matrix multiplication is obtained: 
A’ = AX (2-3) 
where the first column of A' is the first principal 
components image, the second column of A' is the second 
principal components image, and so on (James R. Carr, 1998). 
Suppose the eigenvectors, 17 1 , U 2 ,A ,U M , are sorted by 
descend order according the eigenvalues of the matrix S , 
and X = [Wj, U 2 ,A , U M ] . Then, A = \ Y^, Y 2 , A , 
Y u ], A = [A„A 2 ,A,A m ]. 
Let the histogram of the panchromatic image match with that of 
the first principal component image. And the first principal 
component was replaced by the matched image. Accordingly, 
the fused image can be obtained by reconstructing the images 
through inverse PCA transformation. 
3.1 2DPCA 
Let X denote an n-dimensional unitary column vector, and 
image A , an 171ХП random matrix, project onto X by 
the following linear transformation 
Y = AX (3-D 
Thus, we obtain an m-dimensional projected vector Y , which 
is called the projected feature vector of image A . The total 
scatter of the projected samples can be introduced to measure 
the discriminatory power of the projection vector X , and can 
be characterized by the trace of the covariance matrix of the 
projected feature vectors. 
J(X) = tr(S x ) (3-2) 
S x = E(Y — EY)(Y - EY) T 
= E[AX - E(AX)][AX - E(AX)] r 
= E[(A - EA)X)][{A - EA)X] T 
tr(S x ) = X T [E(A-EA) T (A-EA)]X (3-3) 
C, =E[(A-EA) T (A-EA)] (3-4) 
The matrix C t , which is an 17 X n nonnegative definite 
matrix, is called the image covariance (scatter) matrix. Suppose 
that there are M training image samples in total, the j th 
training image is denoted by an 171 xn matrix Aj 
( j = 1, 2, Л , M), and the average image of all training 
samples is denoted by A . Then, C t can be evaluated by 
C -=T^(Aj-AfiAj-A) (3-5) 
Alternatively, the criterion in (3-2) can be expressed by 
J(X) = X T C,X (3-6) 
where X is a unitary column vector. This criterion is called 
the generalized total scatter criterion. The unitary vector X 
that maximizes the criterion is called the optimal projection 
axis. Intuitively, this means that the total scatter of the 
projected samples is maximized after the projection of an 
image matrix onto X . 
The optimal projection axis X opt is the unitary vector that 
maximizes the total scatter, i.e., the eigenvector of C t 
corresponding to the largest eigenvalue. In general, it is not 
enough to have only one optimal projection axis, but usually 
need to select a set of projection axes. Therefore, we select the

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