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e a suitable 
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e Karhunen- 
ng the 
(Multi 
li picture 
ether 
Xp.•*X„). 
k, 
the PCT 
nt to N 
Y = Q 
k ^ 
0 ) 
■1 
The transformation Q is a rotation to the axes of the principal components 
of the sample set }, a sub-set of all picture-point vectors { X.} 
In the statistically ideal case, { S}would be identical to{ X }, since one 
would then have the best estimate, G, of the covariance matrix of { X }: 
M 
C: 
C. . = 
ij 
E 
k=1 
ik 
m. )(X, - m . )/m 
i' Jk j" 
( 2 ) 
where m. is the mean of X. 
l l 
The new principal component axis can as well be determined by a few well 
chosen samples of classes. C is in that case not a true covariance matrix 
and eigen vectors will be more important than precise values of variances. 
Equation (2) implies that the second order moments of ()T- "m) should be 
computed, relative to each pair of old axes i and j, each in the range from 
1 to N. The eigen values and eigen vectors are then computed from this 
second order moment matrix C. If Q is the column eigen vector matrix of C, 
then transformation to the PC axis is given by equation (1). Now, if 
are the eigen values in decreasing order, a property of Q is 
X > 
that 
• • • • v'V 
N 
G'= Q 1 CQ =A = 
A, 0 
o 1 A, 
Lo o 
...(3) 
The covariance matrix C' of the transformed set {S }is a diagonal matrix A.. 
Each eigen value A corresponds to the variance of a new PC picture, {Y.} 
the variance being related to the amount of contrast. Since the eigen 1 
values are ordered such that A^ is the smallest, minimal total variance 
would be lost by leaving out the new pictures N, N-1, etc. As for the eigen 
values An» A-^_,| ’ etc., since the signal to noise ratio increases from a minimum 
value, data compression is performed with little loss of information. Fig. 1, 
taken from the M.Sc. thesis of SHARIFI, [ 1 ] , shows how rapidly the eigen 
values decrease with increasing i in a practical case. Depending on the 
acceptable noise level, pictures 4 do 12 or 6 to 12 can be. left out without 
perceptible loss of a useful signal 
It is also often desirable to re 
scale the PC pictures. .This implies 
that not only is visible variation 
increased, but the noise as well 
For further illustrations of PCT’s, 
see LANGREBE [ 2] and EARL and TAYLOR[ 3] 
Figure 1.. 
The variance in the principal component 
pictures as a function of the rank 
order.