Image Processing Reference
In-Depth Information
taken to be the width of an approximating rectangular pulse with height h (0) and
the same area. It is easy to show that
1
1
0
1
2
k
W
=
hxdx
()
=
,
(2.8)
h
h
()
Nk
1
2
k
which is exactly half the width of the main lobe of h ( x ).
The right-hand side of Equation 2.8 is known as the Fourier pixel size, in
contrast to the usual image pixel size
x can be made arbitrarily
small using any signal interpolation schemes, but image resolution is fundamen-
tally limited to 1/( N
x . Note that
k ). Another implication of Equation 2.8 is that W h and N
cannot be reduced simultaneously; in other words, improving image resolution
and reducing the number of measured data points cannot be achieved simulta-
neously.
In addition to a loss of resolution in , the convolution operation in
Equation 2.6 also results in the well-known Gibbs ringing artifact in . This
artifact manifests itself as spurious ringing around sharp edges, as illustrated in
Figure 2.1. The maximum undershoot or overshoot of the spurious ringing is
about 9% of the intensity discontinuity and is independent of the number of data
points used in the reconstruction. The frequency of oscillation, however, increases
as more data points are used. For this reason, when a large number of data points
is used in practice, the spurious ringing does not cover an appreciable distance
in the reconstructed image and thus becomes invisible.
ˆ ()
ρ
x
ˆ ()
ρ
x
N = 32
N = 64
FIGURE 2.1 Gibbs ringing artifacts.
Search WWH ::




Custom Search