Image Processing Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
−0.2
−64
−56
−48
−40
−32
−24
−16
−8
0
8
16
24
32
40
48
56
Fig. 8.4. The cross sections (along the horizontal axes) of the interpolators shown in Fig. 8.5
and 8.2. Note that both of the latter 2D interpolators have bandwidth of
8
(horizontally). The
circular region interpolator, the
Besinc function
, is shown in
blue
, whereas the quadratic region
interpolator, the sinc function, is shown in red
It is straightforward to show that both in 2D as well as higher dimensions, character-
istic functions defined on such rectangular volumes always lead to products of sinc
functions of the cartesian variables in the “other” domain.
Admittedly, the sinc functions are suggested by the theory as the ideal interpola-
tion functions for 1D band-limited signals, and they can be easily extended to higher
dimensions. However, from this, one should not hasten to conclude that the sinc func-
tions are the ideal (and the best) interpolation functions in practice. This is far from
the truth because of the various shortcomings of the sinc functions. To start with,
the sinc functions are not ideal interpolators for 2D and higher dimensions even in
theory, since it is most probable that natural
N
D signals to be processed will have
characteristic functions (in the
-domain) that do not have direction preference, e.g.,
the average camera images in 2D are best described by characteristic functions with
boundaries that are circles. In 2D, such a circular characteristic fu
nction re
sults in an
interpolation function that is obtained by substituting
π
ω
=
ω
x
+
ω
y
into a 1D
ω
function, as below:
)=
2
J
1
(
π
ω
)
π
Besinc(
ω
(8.11)
ω
where
J
1
is a
Bessel function
(of the first-order of the first kind). Since it is the
analogue of the sinc function, we call Eq. (8.11)
Besinc function
. The interpolator
function is illustrated by Fig. 8.3, whereas its profile through the origin, the Besinc
function, is given in Fig. 8.4.
