Image Processing Reference
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to be recovered is reasonably well approximated by members of this family,
the estimation procedure so constructed will work well on the actual data we
have measured. The PDFT can therefore be viewed as a (somewhat general-
ized) statistical procedure, in the sense that the method is optimized for a
family (an
ensemble
) of possibilities. Similar methods are used in numerical
quadrature, in which one attempts to estimate the integral of a function from
sampled data; estimates are calculated to be exact on a family, say, of polyno-
mials, and then applied to the measured data.
In the discussion above describing the PDFT, we assumed that the family
H
is an
N
-dimensional linear subspace; assuming that the matrix
D
is invert-
ible, we solve the matrix equation
b
(
x
) =
Da
(
x
) for
a
(
x
). When
H
is a more com-
plicated family, perhaps having no finite parameterization, we seek a least
squares solution of
b
(
x
) =
Da
(
x
). It can happen that the vector
b
(
x
) has one
entry for each member of the family
H
making it an infinite dimensional vec-
tor. We then consider
D
T
b
(
x
) =
D
T
Da
(
x
); the vector
D
T
b
(
x
) involves a sum over
all the members of
H
, but the result
D
T
b
(
x
) is an
N
-dimensional vector. In this
case, we would model
D
T
b
(
x
) and
D
T
D
directly, rather than performing the
infinite summation.
A standard problem in numerical analysis is the estimation of a definite
integral, say
I
b
=∫ ()d
, from finitely many values of the function
g
, say
g tt
a
gt
1
…
. The usual approach is to take as our estimate of
I
a linear com-
bination of the data values; that is, let our estimate be
I
given by
(),,()
gt
N
N
∑
1
ˆ
I
=
a gt
()
nn
n
=
1
,
…
should be. One way to determine these coefficients is to select them so that
the estimate
I
is exactly equal to
I
for some set of special functions,
g
, such as
polynomials of degree
N
− 1 or less. Now let us apply this philosophy to the
Fourier transform estimation problem.
Our problem now is to estimate
The next step is to determine what the values of the coefficients
a
N
∞
∫
1
2π
fx
()
=
F
()exp(
ω
i
ω d
xx
)
−∞
1
…
. For each fixed value of
x
this is an integral
estimation problem. So for each fixed value of
x
, we determine a set of coef-
ficients that depend on
x
, say
ax
from the data
F
( ,,(
ω
F
ω
)
N
, and take as our estimate of
f
(
x
)
1
(),, ()
…
ax
N
the quantity
N
∑
1
fx
()
=
axF
()()
ω
n
n
n
=
The next step is to select a special class of functions for which this estima-
tion procedure must work perfectly and determine the coefficients needed to
have that.
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