Image analysis – Image enhancement or restoration – Image filter
Reexamination Certificate
2000-03-28
2004-07-13
Mehta, Bhavesh M. (Department: 2625)
Image analysis
Image enhancement or restoration
Image filter
C382S264000, C382S300000
Reexamination Certificate
active
06763143
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention relates to a method of and a system for processing a signal, and more particularly to a method of and a system for processing a signal in which deterioration in high frequency components of a signal, which gives rise to a problem especially in a digital signal processing involving linear interpolation, is corrected.
2. Description of the Related Art
When carrying out frame rate conversion of an animation image, rotation of an image, or enlargement/contraction of an image is carried out on a.sampled image signal or when pitch conversion is carried out on a sampled sound signal, signal components for points other than sampling points (interpolating points) are required. As a method of obtaining signal components for points other than sampling points, there have been known methods in which linear interpolation is employed as disclosed, for instance, in Japanese Unexamined Patent Publication No. 2(1990)-294784 and Journal of Japanese Academy of Printing (vol.32, No.5, 1995, “Introduction to Prepress”).
However, it has been known that linear interpolation generates low-pass characteristics which depend upon the interpolating positions. Analysis of low-pass characteristics generated by linear interpolation will be described on interpolation of a one-dimensional signal by way of example.
A one-dimensional signal shown in
FIGS. 5A and 5B
whose frequency band is limited to −&pgr;/T<&ohgr;<&pgr;/T will be discussed, hereinbelow. In
FIG. 5A
, f(t) is the model one-dimensional signal and F(&ohgr;) shown in
FIG. 5B
represents frequency components of the one-dimensional signal f(t). Rendering the one-dimensional signal f(t) discrete at sampling cycles T is equivalent to multiplying the one-dimensional signal f(t) by an impulse train such as represented by the following formula (1).
s
(
t
)
=
∑
∞
k
=
-
∞
δ
(
t
-
kT
)
(
1
)
When thus rendered discrete, the one-dimensional signal f(t) is represented by the following formula (2).
fs
(
t
)
=
f
(
t
)
·
s
(
t
)
=
∑
∞
k
=
-
∞
f
(
t
)
·
δ
(
t
-
kT
)
(
2
)
Frequency components of the signal obtained by rendering discrete the original one-dimensional signal f(t) are obtained by Fourier transformation of fs(t) by the formula (2) and are represented by the following formula (3)
Fs
(
ω
)
=
υ
∼
{
f
(
t
)
·
s
(
t
)
}
=
1
T
∑
∞
k
=
-
∞
F
(
ω
-
k
2
π
T
)
(
3
)
As can be understood from formula (3), the spectrum of the signal obtained by rendering discrete the continuous signal f(t) is infinite repetition at cycles of 2&pgr;/T of the spectrum of the original signal f(t) as shown in
FIGS. 6A and 6B
. Accordingly, it will be understood that, by extracting the frequency components in the range of −&pgr;/T<&ohgr;<&pgr;/T by an ideal low-pass filter, a signal Fsb(&ohgr;) represented by the following formula (4) is obtained, and the original continuous signal f(t) can be completely restored from the signal which has been rendered discrete by inverse Fourier transformation of the signal Fsb(&ohgr;). The ideal low-pass filter is defined to be a filter which outputs signal components in a frequency band of |f| &pgr;/T and cuts signal components in a frequency band of |f|>&pgr;/T. The characteristics of the ideal low-pass filter are as shown in FIG.
7
.
Fsb
(
ω
)
=
1
T
F
(
ω
)
(
4
)
Influence of linear interpolation on the frequency characteristics of a signal will be discussed, hereinbelow. In the following, a discrete signal obtained when the sampling timings of the original signal are retarded by &tgr;·T are approximated by linear interpolation of fs(t). Then by carrying out frequency analysis on the interpolation signal obtained by the linear interpolation, influence of linear interpolation on the frequency characteristics of a signal will be studied. The interpolation signal is as shown in FIG.
8
and represented by fs′(t) in the following formula (5).
fs
′
(
t
)
=
[
(
1
-
τ
)
f
(
t
-
τ
T
)
+
τ
f
{
t
+
(
1
-
τ
)
T
}
]
·
∑
∞
k
=
-
∞
δ
{
t
-
(
k
+
τ
)
T
}
(
5
)
Fourier transformation of the formula (5) gives the following formula (6).
Fs
′
(
ω
)
=
1
T
{
(
1
-
τ
)
F
(
ω
)
·
ⅇ
-
j
τ
T
ω
+
τ
F
(
ω
)
ⅇ
j
(
1
-
τ
)
T
ω
}
*
∑
∞
k
=
-
∞
δ
(
ω
-
k
2
π
T
)
·
ⅇ
-
j
τ
T
ω
=
1
T
[
{
(
1
-
τ
)
+
τⅇ
j
T
ω
}
·
F
(
ω
)
·
ⅇ
-
j
τ
T
ω
]
*
[
ⅇ
-
j
τ
T
ω
·
∑
∞
k
=
-
∞
δ
(
ω
-
k
2
π
T
)
]
=
1
T
∑
∞
k
=
-
∞
{
1
-
τ
+
τⅇ
j
T
(
ω
-
k
2
π
T
)
}
·
F
(
ω
-
k
2
π
T
)
·
ⅇ
j
τ
T
(
ω
-
k
2
π
T
)
·
ⅇ
-
j
2
π
k
τ
=
1
T
∑
∞
k
=
-
∞
{
1
-
τ
+
τⅇ
j
T
(
ω
-
k
2
π
T
)
}
·
F
(
ω
-
k
2
π
T
)
·
ⅇ
-
j
τ
T
ω
(
6
)
The signal represented by formula (6) is still in the form of infinite repetition at cycles of 2&pgr;/T of the signal whose frequency band is limited to −&pgr;/T<&ohgr;<&pgr;/T. When extracting the frequency components in the range of −&pgr;/T<&ohgr;<&pgr;/T by an ideal low-pass filter, a signal Fsb′(&ohgr;) represented by the following formula (7) is obtained, and a signal which is close to the original continuous signal in spectrum can be restored. However, unlike the signal represented by formula (4), the signal represented by formula (7) is a signal obtained by applying band-pass characteristics represented by filtering characteristics of the following formula (8) to the spectrum of the original signal.
Fsb
′
(
ω
)
=
1
T
(
1
-
τ
+
τⅇ
j
T
ω
)
·
ⅇ
-
j
τ
T
ω
·
F
(
ω
)
(
7
)
filter
(
ω
,
τ
)
=
1
T
(
1
-
τ
+
τ
ⅇ
j
T
ω
)
·
ⅇ
-
j
τ
T
ω
(
8
)
The gain characteristics of the filter represented by formula (8) are as shown by the following formula (9).
&LeftBracketingBar;
filter
(
ω
,
τ
)
&RightBracketingBar;
=
1
T
&LeftBracketingBar;
1
-
τ
+
τ
ⅇ
j
T
ω
&RightBracketingBar;
=
1
T
{
(
1
-
τ
)
2
+
τ
2
+
2
τ
(
1
-
τ
)
cos
T
ω
}
1/2
(
9
)
The formula in { } on the right side of formula (9) can be changed to the following formula (9a). Since 2(1−cos T&ohgr;)≧0, formula (9) is minimized when &tgr;=½ and maximized when &tgr;=0 or 1. That is, attenuation is maximized when &tgr;=½, and is nu
Mehta Bhavesh M.
Nixon & Peabody LLP
Riso Kagaku Corporation
Strege John
Studebaker Donald R.
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