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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The discrete wavelet transform 95
In the case of the Haar wavelets, their de termination turns out to be quite
easy because there is a s imple correspondence between the basis functions
(φ, ψ) and the space of 2
n
-component vector s (Strang, 1989). Consider for
instance n = 2. Then the correspondence is
φ
0,0
=
1
1
1
1
, φ
1,0
=
1
1
0
0
, φ
1,1
=
0
0
1
1
, φ
2,0
=
1
0
0
0
, . . .
and
ψ
0,0
=
1
1
−1
−1
, ψ
1,0
=
1
−1
0
0
, ψ
1,1
=
0
0
1
−1
.
Thus the orthogonal basis B
2
may be represented equivalently by the mutually
orthogo nal vectors
B
2
=
1
1
1
1
,
1
1
−1
−1
,
1
−1
0
0
,
0
0
1
−1
.
This gives us a more convenient representation of the expansion in Equation
(3.21), namely
¯g = B
n
ˆg, (3.22)
where ¯g =
g(0) . . . g(c − 1)
⊤
is a column vector of the
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