Image Processing Reference
In-Depth Information
In 1920s, when investigating the nature of matter in quantum mechanics, it was
discovered via this inequality that joint measurement accuracies of certain physically
observable pairs, e.g., position and momentum, time and energy, are limited [103].
Later, Gabor used similar ideas in information science and showed that the joint
time-frequency plane is also granular and cannot be sampled with smaller areas
than the above inequality affords [77]. The principle obtained by interpreting the
inequality came to be known as
Heisenberg uncertainty
,the
uncertainty principle
or
even as the
indeterminacy principle
. The principle is a cornerstone of both quantum
mechanics and information science.
In the case of a Gaussian, we have
Δ
(
g
)=
σ
because Eq. (9.11) is already
a known probability distribution function having the variance
σ
2
. Likewise,
G
is a
normal distribution with variance 1
/σ
2
, yielding
Δ
(
g
)=1
/σ
. Consequently, the
uncertainty inequality is fulfilled with equality by the Gaussian family. For signal
analysis, this outcome means that the Gaussian pair is the most compact Fourier
transform pair. Because of this, extracting a local signal around a point by multiply-
ing with a “window” signal will always introduce high-frequency components to the
extracted signal. The Gaussian windows will produce the most compact local signal
with the least amount of contributions from the high-frequency components. This
is also useful when integrating (averaging) a local signal. Using another function
than Gaussian with the same width will yield an average that is more significantly
influenced by the high frequencies.
9.3 Optimizing the Gaussian Interpolator
Although we have not been precise about the interpolator we used, in Sect. 9.1 we
showed the principles of how the continuous interpolators perform up- and down-
sampling of the discrete signals. Furthermore, we suggested using the Gaussians as
interpolators in the previous section, but we did not precisely state how the variance
σ
2
should be chosen as a function of
κ
, which we will discuss next.
In down-sampling, the Gaussian will not only be used as an interpolator, but at
the same time also it will be used as a lowpass filter to reduce the high frequencies
while retaining the low frequencies as intactly as possible. Therefore, a Gaussian
must mimic or approximate the
characteristic function
:
χ
(
ω
)=
1
,
B
2
<ω<
B
2
−
0
,
otherwise.
To fix the ideas, we choose
B/
2, determining the symmetric band of interest around
the DC component, as
B
2
π
2
. The quantity
B
is the bandwidth of interest that we
wish to retain, which includes also the negative frequencies. The particular choice of
B
above allows a size reduction of the signal with the factor
κ
=
2
B
=2.InFig.9.3
the correponding characteristic (box) function is shown. The Gaussian
G
(
ω
)=exp
=
ω
2
2(
σ
)
2
−
(9.18)
