Image Processing Reference
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than all alternative bases. This should be true on equal footing, i.e., when no more
than
N
basis vectors are used in each of the expansions
f
k
. Accordingly, we are
interested in minimizing the error, by varying
B
N
for the same observation set
O
:
K
K
f
k
f
k
,
f
k
−
f
k
2
=
f
k
−
f
k
−
k
k
K
f
k
,
f
k
−
f
k
,
f
k
=
f
k
,
f
k
+
2
k
K
f
k
2
−
f
k
,
f
k
f
k
2
+
=
2
(15.10)
k
By substituting Eqs. (15.4) and (15.5) in the last term of this equation, we then obtain:
K
M
N
f
k
2
=
k
f
k
2
−
f
k
2
+
f
k
−
2
f
k
(
m
)
ψ
m
,
f
k
(
m
)
ψ
m
m
=1
m
=1
k
N
M
N
=
T
+
k
f
k
2
−
2
f
k
(
m
)
ψ
m
+
f
k
(
m
)
ψ
m
,
f
k
(
m
)
ψ
m
m
=1
m
=
N
+1
m
=1
M
N
=
T
+
k
f
k
2
−
f
k
,
f
k
−
2
2
f
k
(
m
)
ψ
m
,
f
k
(
m
)
ψ
m
(15.11)
m
=1
m
=
N
+1
where
T
=
k
2
f
k
(15.12)
is constant w.r.t. the changes of
B
M
because the length of
f
k
is the same (a zero-
order tensor) in every basis. In Eq. (15.11), the first scalar product term
f
k
,
f
k
,
can
f
k
2
, whereas the second scalar product term vanishes because of
orthogonality of the involved
be identified as
ψ
m
, to yield
K
K
1
K
T
K
−
1
K
f
k
2
=
f
k
2
f
k
−
(15.13)
k
k
Since
T
is a constant, minimizing this expression is equivalent to maximizing
K
K
K
N
f
k
2
=
f
k
,
f
k
=
|
f
k
(
m
)
|
2
k
k
=1
k
=1
m
=1
K
N
K
N
2
=
=
|
ψ
m
,
f
k
|
ψ
m
,
f
k
f
k
,
ψ
m
(15.14)
k
=1
m
=0
k
=1
m
=0
The scalar product of the vector space, the Euclidean
E
M
, to which both the vectors
of
T
m
f
k
, so that the highest bound of
O
and
B
M
belong, is given by
ψ
m
,
f
m
=
ψ
