Image Processing Reference
In-Depth Information
8.3 Sampling Band-Enlarging Operators
Here we discuss how to discretely carry out certain operations that will finitely en-
large the volume in which a band-limited function is non-zero, i.e., the result is also
a band-limited function, albeit with a different band limitation. There is no general
formula for nonlinear operators, although with some care many can be discretized
as long as they do not enlarge the frequency support to infinity, i.e., the volume of
frequencies in which
F
is zero diminishes to zero after the operation.
Multiplication of two images. Assume that we know the discrete values of two
images
f
and
g
on the same grid
r
l
and we need the discrete values of
fg
. This is
usually referred to as a (pointwise)
multiplication of two images
. An intuitive ap-
proximation is
(
fg
)(
r
l
)=
f
(
r
l
)
g
(
r
l
)
.
(8.21)
However, the use of the original grid
r
l
poses a problem because the resulting dis-
crete function on the right-hand side of Eq. (8.21) may not represent the continuous
(
fg
)(
r
) faithfully enough, although
f
(
r
l
) and
g
(
r
l
) may do so with their respec-
tive functions
f
(
r
) and
g
(
r
). The problem is traced to the fact that the continuous
fg
has a Fourier spectrum that is wider than each of its constituent functions. The
multiplication
fg
in the spatial domain is equivalent to the convolution
F
G
in the
frequency domain to the effect that the result will be “wider” than each of
F
and
G
.
Particularly undesirable effects may result if the sum of the bandwidths of
F
and
G
is larger than
π
in one or more of their dimensions, (
x
1
,x
2
,
∗
x
N
)
T
. Discretizing
such a product of images at the sampling rate of the original grid will yield aliasing
errors because of the violation of the fundamental sampling conditions given by the
Nyquist theorem. Thus, reconstructing
fg
from
f
(
r
l
)
g
(
r
l
) will contain unacceptable
errors.
To avoid a violation of the Nyquist theorem, discrete images containing very high
frequencies should not be multiplied with each other directly first, but they should be
up-sampled with a factor 2 to force the highest frequency components to be less than
π/
2. Multiplying such images is then risk-free if the product is sampled at the rate
of the new, finer grid. Although the continuous product has a larger bandwidth than
its constituents, the bandwidth is not as large as it can violate the Nyquist theorem
upon sampling on the new grid. A lossy, but faster alternative to this procedure is,
prior to multiplication, to apply a lowpass filtering to the image to make sure that
no significant frequency components exist outside of the central square having the
width and height
π
in the spectrum. Squaring, being a special case of multiplication
of two digital images, is subject to the same reasoning, i.e., no significant power
of high-frequency components above the central square with height and width of
π
should exist upon squaring. The reasoning is extended to
N
D in a straightforward
fashion.
Because
···
∂f
∂x
i
has the same bandwidth as
f
, products and squares of the partial
derivative images must not contain components outside of the central square with
height and width of
π
.
Rotation of an image. The
rotation of an image
with the arbitrary angle of
θ
rep-
resents a linear operator. It can be achieved by replacing
r
in Eq. (8.6) or equivalently
