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Image resolution limits resulting from mechanical
vibrations. Part Ill: numerical calculation of
modulation transfer function
0. Hadar
M. Fisher
N. S. Kopeika, MEMBER SPIE
Ben-Gurion University of the Negev
Department of Electrical and Computer
Engineering
Beer-Sheva, Israel
Abstract. Low-frequency mechanical vibrations are a significant problem
in robotics, machine vision, and practical reconnaissance where primary
image vibrations involve random process blur radii. They cannot be described by an analytical MTF. A method of numerical calculation of MTF,
relevant in principle to any type of image motion, is presented. It is demonstrated here for linear, high, and low vibration frequencies. The method
yields the expected closed form solutions for linear and high-frequency
motion. The low-vibration-frequency situation involves random process
blur radii and MTFs that can only be handled statistically since no closed
form solution is possible. This is illustrated here. Comparisons are made
to a closed form approximate MTF solution suggested previously for lowfrequency motion. Agreement between that analytical approximation and
exact MTF calculated numerically is generally good, especially for relatively large and linear motion blur radius situations. For nonlinear short
exposure motion, MTF levels off at relatively high nonzero values and
never approaches zero. Such situations yield a two-fold benefit: (1) larger
spatial frequency bandwidth and (2) higher MTF values at all spatial frequencies since MTF does not approach zero.
Subject terms: reconnaissance; robotics; machine vision; modulation transfer
function; vibrations; image motion.
Optical Engineering 31(3), 581 -589 (March 1992).
1 Introduction
much less than that achievable with the expensive high-
In many high-resolution vehicular or airborne imaging sys-
resolution sensor.
Vibration has a number of effects on the sensors, one of
sensors, resolution is limited by image motion and, as a
result, the high-resolution capability of the sensor may be
which is the excitation of the support structures for the
optical elements. This effect can be reduced by mounting
the sensor on vibration isolators that filter out the higher
tems and in robotic systems, despite the use of high-quality
wasted. One of the important factors that affect the performance of reconnaissance systems is sensor angular velocities during image recording. The primary contributors to
these unwanted angular velocities are
velocity of the aircraft relative to the earth
2. low-frequency aircraft angular motions
3 . vibration-induced angular velocities.
1.
In normal reconnaissance and robotics, the sensor moves
during the exposure. Some of the resulting image motion
can be removed, but not all of it. The residual motion blurs
the image, and usually this blur becomes the limiting factor
for many high-quality imaging systems.
It is quite useful and important to be able to model the
expected image degradation as part of system analysis. As
a result of such analysis, one can make system design much
more cost-effective; it makes no sense, for example, to
utilize an expensive, high-resolution sensor in a situation
where vibrational blur limits image quality to resolution
Paper 06011 received Jan. 3, 1991; revised manuscript received Aug. 15, 1991;
accepted for publication Aug. 20, 1991.
1992 Society of Photo-Optical Instrumentation Engineers. 0091-3286192/$2.00.
frequency vibrations where resonant frequencies for the sup-
port structures are located. Typically, the isolators cut off
between 10 and 20 Hz with peaking of response at a slightly
lower frequency. The major effects of mechanical vibrations
in limiting image resolution often derive from the lowvibration-frequency components because of their large amplitude.
The low-vibration-frequency situation is complex because, as demonstrated below, the blur radius is a random
process. In imaging system design, modulation transfer
function (MTF) is a convenient engineering tool. The overall
system MTF is generally limited by the MTF of the weakest
link. In systems involving image vibration or motion, this
weakest link is often the blur caused by the image vibration
or motion, rather than that resulting from optical or electronic components. The formulation of such image blur into
an MTF-type format is thus very convenient for system
design and system analysis purposes, and is the subject of
this paper. Image motion can take many forms. Here, numerical calculation of MTFs to describe image quality will
be considered for uniform linear motion, sinusoidal vibrations at high vibration frequencies, and sinusoidal vibrations
at low vibration frequencies. The analysis presented here is
most pertinent to photographic or those types of CCD systems where all picture elements are exposed simultaneously.
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HADAR, FISHER, and KOPEIKA
The decrease of MTF with increasing spatial frequency
signifies contrast degradation at higher spatial frequencies.
At some relatively high spatial frequency, system MTF has
decreased to such a low value of contrast that it is below
the threshold contrast function of the observer or machine
at the output. This means that such high-spatial-frequency
content of the image cannot be resolved by the observer
Thus the new pattern has the same shape as the original but
l2
with a phase lead determined by
By definition, the modulation contrast in the image plane
(with motion) is
MC1 =
BOlTfVte
because of the poor contrast. The spatial frequency at which
system MTF is just equal to the threshold contrast of the
observer or machine defines the maximum useful spatial
frequency content of the system, called here f,-max.
The existence of MTF for frequencies beyond the cutoff
frequency is sometimes referred to as spurious or false resolution.1 This is an interesting phenomenon because it sug-
gests, falsely, that blur radius is smaller than actual blur
radius.
2 MTFs of Image Motion
Image motion and the resulting blur arise because of relative
movement between the object or scene and the viewing
system. This system may result from translational velocity
or vibrations or both.
MTF of Linear Motion
Degradation of image quality as a result of motion in the
image plane can take several forms. For example, if motion
is linear at a constant velocity V in the image plane, then
for an exposure time te resulting noncircular blur radius d
in that direction is of spatial extent Vte. In order to find the
modulation transfer function for this image motion, we need
to know the modulation of the intensity pattern of the image
and of the object. As a simple mathematical model, an image
with a sinusoidal luminance pattern,
Bm siniifVte=Bm
— slnc(irfVte)
B0
,
(6)
and the modulation contrast function (MCF), which here is
also sine wave response or MTF, is, by virtue of Eqs. (2)
and (6),
MTF = MCF =
—i = Isinc(irfVte)
MC0
,
(7)
wherefis the spatial frequency.2 Note that this goes to zero
whenfVte 1 . This is the point at which the image blur Vte
equals the reciprocal of the spatial frequency frrnax. Spatial
frequencies higher than (Vte) are analogous to blur radii
smaller than Vte in the spatial domain. Since such blur radii
would be smaller than the actual minimum blur radius , they
and spatial frequencies higher thanf,.ax cannot exist. These
high spatial frequencies are an example of ' 'false resolution.' '1
2.1
i(x) = Bo + Bm cos2'rrfx(t)
(1)
will be considered, where f is spatial frequency, x(t) is the
motion function for spatial coordinate x, and B0 and Bm are
constant.
The modulation contrast (MC) of the image without motion is thus
2.2 MTF of Sinusoidal Motion
The sinusoidal image motion is important in aircraft and
vehicles because of turbines and motors that give rise to
mechanical vibrations. In robotics and machine vision, linear motion is almost always accompanied by vibrations that
are often close to being sinusoidal. The sinusoidal motion
can be prevented in principle by proper design; in practice,
however, it is often the most serious source ofimage motion.
The problem is much more prevalent and serious in aircraft than in spacecraft because of large rotating turbines,
motors , and generators . The structures also vibrate because
of buffeting by airstreams. These motions can be minimized
by using vibration isolators or gyro-stabilized camera plat-
rm1 The vibration amplitudes in damped or stabilized
systems are of very low amplitude, although not low enough
so as not to impair resolution.
Degradation of image quality as a result of sinusoidal
motion depends on the ratio of exposure time te to the period
MC0 =
(2)
B0
If image motion is linear, then
of the sinusoidal motion T0. In this case, it is necessary to
distinguish two categories:
1.
x(t)=xo+vt
(3)
and the new luminance distribution is
i(x, t) = Bo + Bm cos2'rrf(xo + Vt)
high-frequency vibration, where the exposure period
is long compared to the period of the simple harmonic
motion (te>T0)
2. low-frequency vibration, where the exposure period
is short compared to this period (te<T0).
(4)
The exposure of any point is proportional to the average of
the intensity over the interval of the exposure time te. Thus,
e
i(x, t) = — J [Bo + Bm cos2f(xo + Vt) dt
/
= Bo + Bm 51fl(lTfVte) cos2'nf( x0 + Vte
fVte
\ —2
(5)
Quantification of the low-frequency vibrational image blur
radius d is much more complicated, however, because it
depends on the initial phase of the oscillatory motion as
well as on the instant and duration of the time exposure,
both of which are often random processes.
2.2.1 High-frequency vibrations
The case of relatively high-frequency oscillatory motion is
defined as concerning a vibration in which one or more
complete vibration cycles (To) fall within the exposure pe-
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATION
F
dmin D 1
dmax
2w
w
0
f2\fte\1
- cos j;;) ) j '
2D sin [
() () ] .
(10)
(11)
Average and maximum achievable resolutions have been
analyzed,3 and statistics that can be used to define resolution
Cl)
a
limits derived from mechanical vibrations have been
computed3 and verified experimentally.4 In general, the low-
frequency-vibration case causes more severe degradation
than the high-frequency case because vibration amplitude
(a)
generally decreases with increasing temporal frequency . The
low-vibration-frequency MTF approximation in Ref. 3 assumes uniform motion because there are many linear portions of the sine wave motion for short exposure times . The
MTF is obtained from Eq. (7) by substituting d for uniform
motion blur radius Vte . The blur radius d is a random variable
that depends on the time instant t, as seen in Fig. 1(a).
The MTF approximation3 is sinc('rrfd).
I
3 Numerical Analysis of Image Motion MTF
TIME(s)
(b)
Fig. 1 Image motion and blur radius for te/T00.1 (a)single frequency; (b) double frequency.
nod. The method of analysis is similar to that used for
uniform motion. The motion function is
2irt
x(t)=xo+D cos— ,
and the MTF is given2 by
M(f)=Jo(2irfD) ,
(9)
where D is maximum vibration amplitude and the subscript
S 15
for sinusoidal motion.
2.2.2 Low-frequency vibrations
This type of image motion is characterized by a relatively
long vibrational period T0, which is longer than the time
exposure. This means image blur takes place only during a
portion of the vibration period rather than during the whole
vibration period, as in the previous case. Image blurring at
low vibration frequencies (te<T) 5 a random process. In
this case, the amount of blur that occurs for a given te
depends on
relative exposure time te/T and the blur radius d. Each
MTF is compared to the analytical sinc(iifd) function
approximation3 with the corresponding blur radius. The MTF
for each t is different. The following examination is for
(8)
T0
As shown above, degradation of image quality as a result
of image motion can be described by an MTF. The MTF
for sinusoidal vibration at low vibration frequencies has not
been examined previous to Ref. 3 . Characterizing this lowfrequency random process analytically is complicated. For
each t there is a different blur radius and MTF curve, even
for constant te . In this paper, low-vibration-frequency MTFs
are obtained via the same conceptual method as that for
Eqs. (1) and (2) but with a numerical solution because of
the complexity. The MTF is obtained here as a function of
when (tx) during the cycle the picture was taken.
The time is random. As seen in Fig. 1(a), minimum blur
occurs when exposure takes place at a vibration extremum,
whereas maximum blur occurs when the exposure is centered at x(t) = 0. In all cases, the shorter the time exposure,
the smaller the blur radius.
Minimum and maximum blur radii are
single and second harmonic low-frequency vibrations . In
the latter case, image motion and blur radius are given in
Fig. 1(b). All the calculations described below were obtamed numerically using a VAX 8300 computer.
Method
The MTF is obtained for each t by moving the time exposure te on the time axis from zero to T0 and computing
for each t the appropriate MTF. In each interval te the
modulation contrast function (MCF) was obtained by dividing the modulation contrast of the vibrating image by
that of the static image. The modulation contrast is calculated via the computer for sinusoidal luminance patterns of
varying spatial frequency. Image motion is given by
3.1
x(t)=xo+f(t)
,
(12)
wheref(t) is a general image motion function and the image
intensity varies with time as
i(x, t)
Bo + Bm cos[2'rrfx(t)J = Bo + Bm{cos(2lTfxo)
x cos[2lTff(t)] — sin(2iifxo) sin[2lTff(t)I} .
(13)
For the vibrating image, the mean intensity over the exOPTICAL ENGINEERING / March 1 992 / Vol. 31 No. 3 / 583
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HADAR, FISHER, and KOPEIKA
posure period te can be computed from the integral
tx +
i(x, t) =
i(x, t) dt = B0 + [Bm cos(2fxo)]
IBm sin(2irfxo)l
x A(txf)[
te
F-.
(14)
]B(txf) ,
where
t +te
A(t, f) = Xj. cos[2ff(t)I dt
tx
B(t,f)=
SPATIAL FREQUENCY
(15)
txe
+t
sin[2ff(t)] dt
The integral limits vary with t , which itself varies between
zero and T0. The average of light intensity i depends on the
beginning of the exposure time t, spatial frequency f, and
Fig. 2 MTF calculated numerically for linear motion of blur radius
d'O.5, 1 , and 1.5.
are the expected theoretical Bessel function results in Ref.
2 . Figure 3 shows the theoretical and numerical results for
te 8T0 and te 8 ST0. All graphs are identical. These re-
the location XO.
suits for linear motion and high vibration frequency support
The time functions A(t, f) and B(t, f) were computed
numerically using a VAX 8300 computer with MATLAB
software. Each t during the oscillation period and each
the approach presented here and suggest it is valid for all
types of one-dimensional motion, including low vibration
frequencies . In looking back to previous experimental results,4 it is clear that despite the randomness of instants of
exposure, the experimental results do support the numerical
calculation results described below , particularly for small
blur radii that pertain to nonlinear motion corresponding to
dmin in Fig. 1(a).
spatial frequency f as a parameter yield the MCF for given
relative exposure time (teIT). The MCF function was calculated according to vibrating image modulation contrast
('max Imin)/(Imax + 'mm) , and 'max and 'mm were calculated
by moving XO from 0 to 1/f and finding the maximum and
minimum light intensities in this range. Here, f is varying.
Substituting B0 = Bm = 1 in Eq. (13) yields unity modulation contrast for the static image. As a result, MCF, equal
to modulation contrast of the vibrating image divided by
that of the static image, is the modulation contrast of the
vibrating image. For sinusoidal luminance patterns,
MCF=MTF.
Limiting resolution occurs when the overall system MTF
is equal to the threshold contrast required at the output
(frrnax).
The above method was used to find the MTF for various
kinds of image motion—linear motion and high- and lowfrequency sinusoidal motions.
3.1 .1 Linear motion
The MTF function for linear motion [Eq. (7)J was calculated
by the same method shown above. The results are exactly
the same as the theoretical result. The MTF depends on the
blur radius d = Vte . Figure 2 shows MTFs for d = 0.5 , 1,
and 1 .5 . The theoretical and numerical results are identical.
3.1 .2 High-frequency vibration
The MTF for this kind of vibration was calculated for two
cases. In the first case, the interval of integration is te = nTo,
and in the second, te = (n + 112)To, where n is an integer
number. The purpose is to show that neglect of half a motion
period (To) does not influence MTF results because in both
cases the blur radius is the peak-to-peak displacement 2D.
This results from the fact that te>T0. The results obtained
3.1 .3 Low-frequency vibration
For low-frequency vibrations, where te<T, the resolution
is limited by the blur radius d. Image blurring is a random
process so thatfrmax 5 a variable depending on tand limited
by lid. The criterion used to findfrmax is shown here. The
normalized MTF decreases from zero spatial frequency
monotonically with spatial frequency until a break point
occurs, denoted here asfri . The frequency at this point was
chosen to be frmax• This choice is based on the condition
that this frequency is smaller than lid; ffri 5 higher than
lid, then frmax lid. This is consistent with the idea that
fiax cannot be smaller than actual blur radius. MTF at
such high frequencies is defined as false resolution.
An error parameter is defined here as
en= (d1
dr') x 100% ,
(16)
where dl is the frequency at which sinc('rrfd) is zero for
a given t. If this error is negative , then frmax 11.
3.2 Results and Discussion
The method for numerically calculating MTFs for all types
of image motion presented above shows excellent agreement
with closed form MTFs that can be determined analytically
for linear and high-mechanical-frequency sinusoidal motion. We now use this method to calculate MTFs for random
584 / OPTICAL ENGINEERING / March 1992 / Vol. 31 No. 3
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATION
'U
'IF-
SPATIAL FREQUENCY
(a)
SPATIAL FREQUENCY
(b)
SPATIAL FREQUENCY
(a)
SPATIAL FREQUENCY
(b)
Fig. 3 MTF calculated numerically for high sinusoidal vibration frequencies, where (a) te/To 8 and (b) te/T0 8.5.
Fig. 4 Average MTF for te/To 0.05: (a) single frequency; (b) double
frequency.
blur radii derived from low-frequency sinusoidal image
motion.
measured with two parameters, the mean-square-error (MSE)
and the error parameter defined in Eq . (16) . These param-
The following results refer to single and double lowfrequency vibrations {x =xO + D i[coswt + cos(2wt)]}. For each
vibration frequency, the results shown in Figs. 4 through 8
are for several relative exposure times (te/T). For example,
for one frequency vibration at 2.5 Hz, teIT0 0.05 , 0. 1,
and 0. 15. For two vibration frequencies at 2.5 Hz and 5
Hz, te/T00.05, 0.1, and 0.15.
For each relative exposure time te/T only a few graphs
are presented from a full series . These include one at mmimum blur radius , another at maximum blur radius , and the
average MTF for each te/T0. Each MTF is compared to the
sinc('rrfd) function (MTF approximation),3 which is a function of the blur radius d that is shown on the graph above
each MTF curve.
Note that throughout, spatial frequency is in units recip-
Agreement between sinc and actual MTF functions was
eters are calculated for spatial frequencies below false resolution, i.e. , f<frmax. The blur radius d is inversely proportional tOfrmax. Al50,frmax depends on the relative exposure
time te/T0.
For the case of minimum blur radius d in Figs. 7(a) and
8(a), frmax 5 at its highest value [Figs. 7(b) and 8(b)I . Blur
shape for small blur radii is not linear with motion, and the
value of MTF atfrmax is not zero as expectedfrom the sinc
approximation but is relatively high (MTF'0.32). This corresponds to relatively high contrast at the spatial frequency
equal to the reciprocal of the actual blur radius, even though
that spatial frequency is the highest physically possible.
These results are supported experimentally by Figs. 5(j) and
5(k) of Ref. 4, but their significance was not noticed then.
In Fig. 5 of Ref. 4, as d decreases, the experimental MTF
rocal to those of d. The average MTF sinc(iifd) depends
curves resemble more and more Fig. 7(b) here for dmin. The
on the average blur radius d for the given relative exposure
time. In each of Figs. 4 through 8, te/T0 is constant but t
varies from 0 to To. Despite the averaging, each of these
figures illustrates the randomness of MTF corresponding to
implication is that spatial detail corresponding to spatial
the randomness of t.
frequency frmax can be seen with good contrast, but spatial
detail corresponding to frequencies just above frmax cannot
be resolved at all because they relate to blur radii smaller
than those that physically exist.
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HADAR, FISHER, and KOPEIKA
L)
C,,
SPATIAL FREQUENCY
SPATIAL FREQUENCY
(a)
(a)
SPA11AL FREQUENCY
SPATIAL FREQUENCY
(b)
(b)
Fig. 5 Average MTF for te/To=O.1 : (a) single frequency; (b) double
frequency.
Fig. 6 Average MTF for te/To — 0.15: (a) single frequency; (b) double
frequency.
For maximum blur radius dmax defined in Eq. (1 1), frmax
corresponding to curve d in Figs. 7(c) and 8(c) is minimum
[Fig. 7(d)]. The blur shape is much more linear with motion
and the MTF seems to be very close to zero at frmax• For
d1
example, for single frequency vibration te/Tø 0. 1 ,
is equal to or greater than that due to linear motion blur
the
maximum blur radius is O.618,frmax 5 1 .501, and the MTF
is equal to 0.0654% at this frequency. This blur seems to
be very linear and gives good agreement between actual
MTF and the sinc function approximation; MSE is very
low. On the other hand, the minimum blur radius is 0.0489,
frrnax 5 18.8, and the MTF is equal to 32.37% at this frequency. It appears that if the blur radius is small, there are
two benefits: (1) frrnax 5 increased and (2) MTF is higher
at all frequencies and, as a result of the nonlinear motion,
does not approach zero atfrmax. This means that the motion
causing the blur is very important, rather than only the blur
radius. This surprising result becomes apparent when the
transfer functions due to linear motion blur shape are compared with those due to nonlinear motion blur shape for the
same values of blur radius. For example, the same blur
radius is given for two cases in Fig. 9, but d2 is more linear
so its MTF is much closer to the sinc('rrfd) approximation.
On the other hand, the motion giving rise to the blur radius
in Fig. 9 is very nonlinear, and its MTF is much higher
than the sinc(irfd) approximation. This result has been suggested previously on the basis of theoretical considerations .
In all cases, the MTF due to nonlinear motion blur shape
shape.
The graphs of the average MTF were computed for a
specific te/T0 and compared to the sincQrrfd) approximation
for average blur radius d. The average blur radius d increases
with te/T, andfrm therefore decreases. Also, it seems that
agreement with the sinc('rrfd) approximation improves as
te/Tø increases (Figs. 4 through 6).
For two frequency vibrations, the results seem to be very
similar to single frequency vibrations. The comparison between them is given in Table 1 . Since d is constant in Table
1 , increasing relative exposure te/T implies greater nonlinearity of motion, i.e. , exposure takes place near an extremum of the sine wave motion. Consequently, MTF atfrmax
increases.
A very important practical conclusion that can be drawn
from Figs. 4 through 8 is that the use of sinc(lTfd) function
as an inverse filter for reconstructing the image is much
more accurate for large blur radii. [The sinc(irfd) function
586 / OPTICAL ENGINEERING / March 1992 / Vol. 31 No. 3
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATION
I
0.8
one freq.
0.6
te/To = 0.1
0.9
d= 4.8943e-02
0.8
0.4
0.7
0.2
0.6
0
0.5
-0.2
0.4
-0.4
0.3
-0.6
0.2
-0.8
0.1
_10
0.05
0.1
0.2
0.15
0.25
0.3
0.35
0.4
e=7.492%
\'\
,,,..
,
jose = 4.9673e-03
MTF(frmax) = 0.3237
frmax= 18.8
SC(pi*d) - MTF
,,,,,
I- false resolution ->
,,,,,,,,,
5
10
15
20
25
SPATIALFREQUENCY
TIME (s)
(b)
(a)
0.8
one freq.
0.6
d=0.618
teffl = 0.1
\
0.4
0.2
0
I
....
-0.2
-0.4
-0.6
-0.8
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
04
lIME (s)
(c)
SPATIAL FREQUENCY
(d)
Fig. 7 Minimum blur radius and MTF for (a) and (b) minimum blur radius and (C) and (d) maximum
blur radius for single frequency: te/To=O.1.
has also been shown experimentally4 to be much more accurate for small blur radii than the Bessel function expression, Eq. (9), which is often used by optical engineers.I
3.3 Effect of Motion Amplitude on Vibration MTF
Up to this point, only cases in which motion amplitude was
of a constant value D = 1 have been considered. Now, situations in which t , te , and T0 are constant but D varies are
presented. Intuitively, one would expect that as D increases,
blur radius d would also increase andfrmax would decrease.
Resolution is poorer. Indeed, this is verified by the MTF
calculations shown in Figs. 10 and 1 1 . In the former, exposures are centered at t = To/4, and blur radii are maxima
(d= dmax) 3fld essentially linear, thereby giving rise to MTFs
that strongly resemble sinc functions. It is clear from Fig.
10 that as D increases, frmax decreases. On the other hand,
in Fig. 1 1 , MTFs for exposures centered essentially at T0/2
are presented. Here, blur radii are minima and much more
nonlinear. Here too, as D increases, frmax essentially decreases, but because of the nonlinearity and the non-sinclike form, quantitative dependences of D on frmax are not
quite so clear. In summary, motion amplitude is certainly
a critical factor in final image resolution. This is important
as regards low-frequency mechanical vibrations, which generally are much less controlled by stabilization systems.
4 Conclusions
A general method for numerically calculating MTFs for
various types of image motion has been presented and dem-
onstrated for uniform and sinusoidal image motion. The
latter applies to mechanical vibrations . For exposures that
are relatively long compared to vibration period, MTF is in
closed form. For short exposures, or low-frequency vibra-
tions, the MTF is a random process that depends on the
portion of the sine wave vibration in which the exposure
takes place. Actual MTFs of single and double lowfrequency vibrations have been analyzed and compared to
the sinc(irfd) approximation.3 In most cases, there was
good agreement between the two functions, especially for
large and linear blur radius values . The effects of motion
amplitude have also been considered, and resultant MTFs
have been numerically calculated. They agree with intuitive
expectations that as D increases image quality decreases.
For minimum blur radii, where image motion is noticeably
nonlinear, the MTF not only goes to higher spatial frequencies but also levels off and actually reaches frmax at a
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HADAR, FISHER, and KOPEIKA
two fret
0.8
d 7.1020e-02
te/'Fo —0.1
0.6
0.4
0.2
-0.2
-0.4
0.05
0.1
0.15
0.2
0.25
0.3
0.35
C
4
TIME(s)
SPATIAL FREQUENCY
(a)
(b)
two freq.
d=0.8155
0.8
te/To = 0.1
0.6
0.4
0.2
-0.2
-0.4
0
0.05
0.1
0150.2
0.25
0.3
0.35
0.4
TIME(s)
SPATIAL FREQUENCY
(c)
(d)
Fig. 8 Minimum blur radius and MTF for (a) and (b) minimum blur radius and (c) and (d) maximum
blur radius for double frequency: te/To=O.1.
one freq.
tefl'o = 0.05
two freq.
dl=0.4
d2=0.3826
teiTo =0.05
0.5
-0.5
d2
—1
0.05
0.1
0.2
0.15
0.25
0.3
0.35
0.4
TIME (s)
SPATIAL FREQUENCY
(a)
(b)
Fig. 9 MTF for constant blur radius dO.39.
588 / OPTICAL ENGINEERING / March 1 992 / Vol. 31 No. 3
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATION
Table 1 Comparison between MTFs for single and double frequency
vibrations; dO.31.
MTF (f')
rm
tjf0
one freq.
0.0039
0.1579
0.3495
3.2
2.9
2.8
+
+
+
0.1114
0.3488
0.3934
3.1
1.4
1.4
0.05
0.1
0.15
0.05
0.1
0.15
two freq.
image motion. This research can be very useful in image
processing and reconstruction as well as in imaging system
design and analysis.
Acknowledgment
This work was partially supported by the Paul Ivanier Center
for Robotics and Production Management.
+
+
+
References
1.
N. Jensen, Optical and Photographic Reconnaissance Systems, John
Wiley & Sons, New York (1968).
2. T. Trott, ''The effects of motion on resolution,'Photogramm.
'
Eng.
26, 819—827
3.
4.
(1960).
D. Wulich and N. S. Kopeika, "Image resolution limits resulting from
mechanical vibrations," Opt. Eng. 26, 529—533 (1987).
S. Rudoler, 0. Hadar, M. Fisher, and N. S. Kopeika, "Image reso-
lution limits resulting from mechanical vibrations. Part II: experiment,"
Opt. Eng. 30(5),
' 577—589 (1991).
5 . S. C . Som, 'Analysis of the effect of linear smear on photographic
images," J. Opt. Soc. Am. 61, 859—864 (July 1971).
Ofer Hadar received in 1990 the BSc degree in electrical and computer engineering
from Ben-Gurion University of the Negev.
He is now an MSc student and research
assistant in the electro-optics program. His
current research interest is the influence of
time domain impulse response motion on
image quality. He also has worked on developing a method to calculate the MTF
SPATIAL FREQUENCY
Fig. 1 0 MTF affected by motion amplitude—linear blur, teITo
function of image vibration in real time. Hadar
is a member of IEEE.
0.1.
Moshe Fisher received in 1990 the BSc
degree in electrical engineering from BenGurion University of the Negev. He is now
an MSc student and research assistant in
the Department of Electrical and Computer
Engineering at Ben-Gurion University of the
Negev. His current interest is digital image
restoration.
C)
z
N. S. Kopeika received the BS, MS, and
PhD degrees in electrical engineering from
I,
:y':j .
t
SPATIAL FREQUENCY
Fig. 1 1 MTF affected by motion amplitude—nonlinear blur, te/To 0.1.
high level of contrast, thereby improving resolution and
image quality considerably. These surprising numerical calculation results for small blur radii actually agree with experimental results obtained previously in Fig. 5 of Ref. 4,
where as d decreases , the experimental MTF curves resemble more and more the numerical calculations for
shown
here in Fig. 7(b). The methods and approach presented here
can be quite useful for calculating MTFs of all types of
.J
I
,
the University of Pennsylvania, Philadelphia, in 1966, 1968, and 1972, respectively. His PhD dissertation, supported by
a NASA Fellowship, dealt with detection of
millimeter waves by glow discharge plasmas and the utilization of such devices for
detection and recording of millimeter wave
holograms. In 1973 he joined the Depart-
ment of Electrical and Computer Engi-
neering, Ben-Gurion University of the Negev, Beer-Sheva, Israel,
where he is a professor and department chairman. In 1978/1979 he
was a visiting associate professor in the department of Electrical
Engineering, University of Delaware, Newark. He has published over
70 journal papers and has been particularly active in research of
time response and impedance of properties of plasmas. He also
authored a general unified theory to explain EM wave-plasma interactions all across the electromagnetic spectrum. Recently, he has
contributed towards characterizing the open atmosphere in terms of
an MTF with which to describe eftects of weather on image propagation. Kopeika is a Senior Member of IEEE and a member of SPIE,
OSA, and the Laser and Electrooptics Society of Israel.
OPTICAL ENGINEERING / March 1 992 / Vol. 31 No. 3 / 589
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