Image Processing Reference
In-Depth Information
we show the sampled interpolation function to be used in the up-sampling. The
up-sampled signal (shown as the magenta samples in top, right) is obtained by
convolving the two discrete signals shown at the bottom. Evidently, multipli-
cations with zeros do not need to be executed if efficiency is cared for in the
implementation.
9.2 The Gaussian as Interpolator
When resampling a discrete signal, its continuous version is resampled. The continu-
ous signal is in turn obtained by placing the interpolation functions at the grid points
and summing them up with weights which are the corresponding discrete signal val-
ues. Although this happens in the experimenter's imagination, the implications of
it in practice are sizeable because this reasoning influences the discrete filters to be
used. Here it is important to note that we generally do not know the original signal
but only know its samples. Because of this, any reconstruction is a guess translates
to making an assumption on, and a motivation for the interpolation function.
The interpolation function that is commanded by the band-limited signal theory
in conjunction with Fourier transform theory is the sinc(
t
) function, Eq. (6.27). As
pointed out in Sect. 8.1, this choice brings us to the frontier where practical interest
conflicts with purely theoretical reasoning. A reconciliation can be reached by not
insisting on the strict “limitedness” in the frequency domain in exchange for smaller
and direction-isotropic filters (in 2D and higher dimensions). We keep the idea of
displaced tentlike functions as a way to reconstruct the original function, but these
functions are subjected to a different condition than strict band-limited requirement.
Instead, the latter will be replaced by smoothness and compactness in both domains,
i.e., we will require from the interpolation function
μ
0
that not only
μ
0
but also its
FT is “compact”, in addition to being “smooth”.
We will suggest a
Gaussian
to be used as an interpolator, primarily for reasons
called for by 2D and higher dimensional image analysis applications. However, to
elucidate why the Gaussian family is interesting, we will use 1D in the discussions,
for simplicity.
Definition 9.1.
With
0
<σ
2
<
∞
, the functions :
x
2
2
σ
2
1
√
2
πσ
2
g
(
x
)=
exp(
−
)
(9.11)
and their amplified versions
A · g
(
x
)
, where
A
is a constant, constitute a family of
functions, called
Gaussians
.
The Gaussian in Eq. (9.11) is positive and has an area of 1, guaranteed by the constant
factor
1
√
2
πσ
2
. These qualities make it a
probability distribution function
, the Normal
distribution.
A Gaussian with a fixed
3
σ
is a decreasing function that converges to zero with
increased
x
.
A property of the Gaussian family is that it is invariant under the Fourier
3
With increasing
σ
a Gaussian converges functionally to the constant
A
.
