June 2014
Intermediate to advanced
576 pages
16h 43m
English
book
Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition
by Morton J. Canty
Content preview from Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition
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A
Mathematical Too l s
A.1 Cholesky decomposition
Cholesky decomposition is used in some of the IDL and Python routines in this
book to solve generalized eigenvalue problems associated with the maximum
autocorrelation factor (MAF) and maximum noise fra ction (MNF) transfor-
mations as well as with canonical correlation analysis. We sketch its justifi-
cation in the following.
THEOREM A.1
If the p × p matrix A is symmetric positive definite and if the p × q matrix
B, where q ≤ p, has rank q, then B
⊤
AB is positive definite and symmetric.
Proof. Choose any q-dimensional vector y 6= 0 and let x = By. We can
write this as
x = y
1
b
1
+ . . . + y
q
b
q
,
where b
i
is the
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