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Image resolution limits resulting from mechanical vibrations. II - Experiment
Article in Optical Engineering · May 1991
DOI: 10.1117/12.55843
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Image resolution limits resulting from mechanical vibrations.
Part 2: Experiment
S. Rudoler
0. Hadar
M. Fisher
Abstract. A theoretical model, developed by Wulich and Kopeika that gives
the MTF for flow vibration frequency sinusoidal image motion applicable
N. S. Kopeika, MEMBER SPIE
Ben-Gurion University of the Negev
Department of Electrical and Computer
Engineering
Beer-Sheva, Israel lL-84105
to reconnaissance, robotics, and computer vision, is evaluated experimentally to determine (1) accuracy of the MTF model and the validity of
assumptions upon which it is based, (2) accuracy of "lucky shot" theoretical analysis to determine the number of independent images required
to obtain at least one good quality image, and (3) accuracy of prediction
for average blur radius. In most cases agreement between theory and
experiment is quite good. Discrepancies are not too great and are attributed to problems with underlying theoretical assumptions where uniform
linear motion cannot be assumed. The theory and experiment here are
confined to low-frequency sinusoidal vibration where blur radius and spatial frequency content are random processes.
Subject terms: image motion; modulation transfer function; mechanical vibrations.
Optical Engineering 30(5), 577-589 (May 1991).
CONTENTS
1. INTRODUCTION
1. Introduction
2. Experimental setup
2.1. Resolution chart
2.2. Camera
2.3. ITEX Series 2000 Image Processor
2.4. VAXstation II/GPX
3. Experimental MTF measurement
4. Lucky shot analysis
5. Average blur radius
6. Conclusions
7. Acknowledgment
8. References
Paper 2893 received March 28, 1990; revised manuscript received Sept. 26,
1990; accepted for publication Nov. 25, 1990. This paper is a revision of a
paper presented at the SPIE conference Airborne Reconnaissance XIV, July
1990, San Diego, Calif. The paper presented there appears (unrefereed) in SPIE
Proceedings Vol. 1342.
1990 Society of Photo-Optical Instrumentation Engineers.
Image motion as a result of vibrations is often the limiting factor
in image resolution for moving systems, such as in reconnaissance, robotics, computer vision, etc. It is quite useful to model
the expected image degradation as part of system analysis for
two reasons:
1.
To 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 much less than
that achievable with the sensor.
2. To permit image processing so as to improve the final
image; knowledge of a vibrational motion MTF can
strongly facilitate the image processing and perhaps
even reduce stabilization requirements and, hence, cost.
OPTICAL ENGINEERING / May 1991 / Vol. 30 No. 5 / 577
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RUDOLER, HADAR, FISHER, KOPEIKA
tx + te
zw
I—
w
0
-J
0
C/)
0
Fig. 1 . Blur radius das a function of exposure time te for exposures shorter than vibration period T0.
The effects of sinusoidal type image motion, which often results
upon a probability distribution function calculated for sinusoidal
from mechanical vibrations, can be divided into high and low
vibrational frequencies. In the first case, the time exposure is
frequency of image motion and time exposure such that te <
longer than the vibrational period and the image blur is therefore
the entire peak-to-peak translation ofthe image. Assuming image
motion
T0. As a result of this probability distribution function, average
image blur was found to be almost directly proportional to the
relative time exposure te/To; i.e. ,2
; 3.57D
x(t) = D cos2irtIT0 ,
(2)
,
(3)
Minimum and maximum blur radii are
the MTF for the high vibration frequency case is1
M5(f) = Jo(2ifD)
.
T0
(1)
where Jo is the zero-order Bessel function,fis spatial frequency,
and D is the amplitude of the sinusoidal displacement. The total
blur radius for the high vibration frequency case is the peak-topeak displacement, which is 2D.
The low vibration frequency situation, as shown in Fig. I ,
involves time exposures te shorter than the vibration period T0.
In this case, the blur radius d is a random process that depends
upon the time the exposure takes place. This type of blur is
often more damaging than the high vibration frequency blur
because in real-life situations D for low vibration frequencies is
in many cases much greater than D for high vibration frequencies.2 Equation (2) is inappropriate for the low-frequency blur,
although it has been used often in the past for want of a better
MTF model. A theoretical model was developed recently that
describes statistically random process , low-frequency blur radii
and spatial frequency content.2 This includes best case, worst
case, and average image degradation. In addition, a lucky shot
model was developed that predicts the number of independent
time exposures required to obtain at least one picture with a
given probability Q so that its blur radius will be less than a
given value of d. This is also equivalent to obtaining at least
one picture with a spatial frequency content greater than a given
spatial frequency bandwidth.2 The theoretical model was based
dmin D[ 1 — cos(—
/2i)
\ T0 ()
]
dmax
,
2D sn[() ()]
(5)
respectively. As seen in Fig. 1 , minimum blur occurs when
exposure takes place at a vibration extremum, while 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.
The translation of the mechanical data on vibration to the spatial
frequency domain was accomplished using the assumption that
for short time exposures the image velocity is essentially uniform, where the MTF for linear motion is1'2
M,(f) =
sinc'rrfd .
(6)
The purpose of this paper is to report experimental results of
the theoretical analysis for low mechanical frequency sinusoidal
image motion.2 The experiments centered on the legitimacy of
utilizing Eq. (6) for short exposure sinusoidal image motion,
the accuracy of the lucky shot analysis prediction of the number
of independent images required to obtain at least one good image
when d < d0 'yd with a probability Q, and the accuracy of
578 / OPTICAL ENGINEERING / May 1 991 / Vol. 30 No. 5
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATIONS. PART 2: EXPERIMENT
Resolution
chart
Motion
Vibration
Platform
Oscilloscope
Function Generator
Fig. 2. Experimental setup.
.62.5
.189
.256
.370
.556
.455
.286
Lii
i.00
.833 1.43
platform. Fortunately, the back and forth motion of the machine
is sinusoidal. By attaching the resolution chart to the shaker, it
was possible to vibrate it sinusoidally in the horizontal direction.
The shaker has a very small frequency range that effectively
0.08.
limited trials to te/T0
2.1. Resolution chart
Figure 3 is the resolution chart used in these experiments. The
numbers across the top of the chart represent the frequency in
lines per millimeter (1pm) of each of the line sets. This chart
was made using Cricket Draw software and an Apple laser printer.
The resolution chart was vibrated sinusoidally by the shaker. A
piezoelectric motion sensor was attached to the vibration plat-
form so the motion of the platform could be displayed and
measured. This signal was sent to a vibration meter that amplified
it and sent it on to an oscilloscope. A Bruel and Kjaer AccelFig. 3. Resolution chart. Spatial frequencies in lines per millimeter.
Eq. (3) for predicting average blur radius. The essential exper-
imental setup is described in Sec. 2. The experimental MTF
measurements are presented in Sec . 3 , the accuracy of the lucky
shot analysis in Sec. 4, and the determination of average blur
radius in Sec. 5, followed by conclusions in Sec. 6.
2. EXPERIMENTAL SETUP
Figure 2 describes the main test setup used for these experiments.
Unfortunately, an easily controllable vibration table was not
available. Instead a chemistry shaker was used as a vibration
erometer Type 4370 and Vibration Meter Type 25 1 1 were used.
The vibration meter sent the oscilloscope a voltage proportional
to the position of the moving resolution chart (Hameg Digital
Storage Scope HM 205-2). The y-axis of the scope represented
the position of the chart, while the x-axis displayed time. It was
therefore possible to measure the position of the chart, as well
as the amplitude and frequency of vibration. If a picture of the
scope display was taken, the received image was only that which
the scope beam traced during the exposure time. Since the
y-component of that trace was the motion of the vibration platform, it was possible to measure d, the movement of the resolution chart, during the exposure time. The vibration platform,
sensor, and amplifier essentially generate a sinusoidal wave on
the scope. This signal can be simulated by using a function
generator instead of the vibration equipment. The advantage to
OPTICAL ENGINEERING / May 1991 / VI. 30 No. 5 / 579
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RUDOLER, HADAR, FISHER, KOPEIKA
this simulated motion is that measurement can be taken from
the scope for a wide range of vibration frequencies not available
from the shaker. The drawback, of course, is that there is no
moving resolution chart to photograph using this method. As
will be seen both the actual vibration equipment and the function
generator were useful in these experiments . An IEC Model F3 1
function generator was used.
hardware are written in C on the VAX and call up the ITEX C
library of routines. The microVAX is also used for data analysis.
In addition to the C compiler and the ITEX library, the VAXLAB
signal processing software was used extensively to perform such
functions as fast Fourier transforms and graphing.
3. EXPERIMENTAL MTF MEASUREMENT
2.2. Camera
A Cohu CCD camera was used in this experiment along with a
Fuji optical lens with adjustable focus and aperture. The camera
has a fixed exposure time of 20 ms over which the incoming
light is integrated. The CCD is a 754 x 488 pixel semiconductor
matrix. Each pixel integrates the incoming light for 20 ms. During the next 20 ms, the data is shifted into a buffer and written
out serially onto the communication channel . (Each pixel is
transferred as an analog level, no A/D conversion takes place
in the camera, and the signal is essentially time multiplexed and
sent to ITEX hardware.) The output of the camera is in video
format. During this 20 ms interval while the data is being transferred, the CCD array is integrating the incoming light for the
next picture. Thus, during each 20 ms interval, one picture is
being taken while the last picture is being sent to the ITEX
hardware. One other important aspect of the CCD array is that
the whole array is exposed simultaneously. One can assume that
each pixel was exposed to the same motion. This is consistent
with the theoretical analysis in Ref. 2. A much more complicated
situation occurs in some systems where each pixel is exposed
at a different time.
2.3. ITEX Series 2000 Image Processor
The ITEX hardware receives the video signal from the camera
and can either send it directly to the image display or encode it
for processing and storage. The ITEX 2000 has special image
processing hardware and firmware as well as a full library of C
software. It is accessed and controlled through a VAX computer
and all its functions are called from C . It has some local memory,
but stores data on the VAX drives. Note that the ITEX uses an
interlaced format. Every image that is sent to the display is
actually the combination of two 20 ms pictures. It displays the
first picture on even rows, and the second picture on odd rows.
This gives a 40 ms image. It is simple enough for the computer
to separate the two pictures, however, and use one or the other.
In the work discussed here, the even rows were used. The resolution of the ITEX encoder is 512 x 256. It fills a 512 x 512
buffer using the interlaced method described earlier. Because
the vibration table moves in the horizontal direction in these
tests, advantage was made of higher horizontal resolution (in
both the camera and the encoder). The accuracy of the encoder
is 8 bits, which is 256 gray levels.
2.4. VAXstation II/GPX
A microVAX workstation operates under VMS and is the user's
interface to all the imaging hardware. Programs to use the ITEX
580
Initially an attempt was made to take a picture of the moving
resolution chart, and to record the amount of motion d simultaneously. While the motion is being measured continuously,
recording it during the exact interval that an image was being
taken proved problematic . While the ITEX hardware uses the
sync pulse of the camera to decide when to snap a shot, it does
not send a signal out defining which sync pulse it began its
picture. An attempt was made, using two cameras, to first photograph the chart and then the scope (where the motion is displayed). If there were a constant delay between the two pictures,
it would have been possible to calculate d from the second
picture. Unfortunately, the delay between the two images varies
making this method impractical.
Because of the problem in measuring d, the only way to
obtain an estimate of d was by looking at the blurred image of
the resolution chart. By assuming approximately linear motion
and using a set of line stripes where the image is approximately
uniformly gray, it is possible to estimate d, assuming that d is
approximately equal to one resolution chart image stripe line
width at this point. This method can be seen in Fig. 4 for a
motion of 2 mm. This only gives an approximation ofd because
only 1 1 sets of lines exist and at times the motion is nonlinear.
In other words , the contrast becomes zero when d = 1/f, because
at that point one black and white line pair appears uniformly
gray. Using this method of calculating d, it is possible to correlate blur radius, MTF (sine wave response) and MCF (square
wave response) for each image. Accordingly, software was written to calculate the MCF and MTF of each image. A standard
image of the still resolution chart was used in this calculation.
It was therefore not necessary to consider the MTF of the rest
of the imaging system, because the only difference between the
static image and the moving image was the motion itself. Using
the results of this process it was possible to plot MTF, MCF,
and sinc(irfd) from Eq. (6) for comparison purposes.
The results of this experiment for blur radii d equal to the 1 1
different available resolution chart spatial line frequencies are
shown in Figs. 5(a) through 5(k): The MTF is the noisy line
that extends for the entire x-axis. This plot was made by dividing
the FFT of the image of the vibrating resolution chart by the
FF1.' of the image of the still resolution chart. It thus is a "noisy"
version of the sine wave response. The MCF or square wave
response is the dashed line. This is the modulation contrast of
the vibrating image divided by the modulation contrast of the
still image. The MCF is only taken at the frequencies conesponding to the 11 line pairs on the resolution chart. For large
blurs, where only the widest line pair could be resolved, only
one data point exists, because the MCF at all the other line pairs
is zero (this is the case in Fig. 5). An ideal sinc function is also
shown on the chart. The blur radius d chosen to plot the sinc(irfd)
function on each graph is the width of the smallest line pair that
/ OPTICAL ENGINEERING / May 1991 / Vol. 30 No. 5
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATIONS. PART 2: EXPERIMENT
could be resolved. The estimate of d for which the sinc function
was plotted was limited to a line width on the resolution chart.
Although the MTF is a very noisy signal, the results shown
in Fig. 5 clearly show that it is reasonably close to the predicted
transfer function. The fact that the MTF is so jagged is an effect
of using the FFT. This is a well known result due to the noise
generated by the discreteness of the FFT.3 For the smallest and
the largest blur radii [Fig. 5(k) and Fig. 5(a), respectively] the
2 mm
4mm
>-
MTF is not that close to the sinc function, especially for the
smallest. This is because the use of the sinc function as an MTF
here is based on the assumption of linear motion during a very
short exposure of sinusoidal motion. However, when d is very
small and approaches dmin, the implication is that the picture
was taken at an extremum of the sine wave motion where the
motion is highly nonlinear. Also, for too large d the assumption
of linear motion and uniform velocity is inaccurate because of
the relatively long exposure te/T that can include both an extremum and a linear portion of the sine wave. Consequently,
we should expect that measured MTF in such cases is not quite
identical to a sinc function. What is surprising is that despite
the randomness it is not too much different from a sinc function.
The sinc function is certainly a better approximation than the
widely used Bessel function as shown in Fig. 6.
The MCF (measured square wave response) is generally slightly
offset from the MTF and the sinc function in Fig. 5 , but has
roughly the same slope. This is probably due to the method used
to calculate the MCF that chooses the minimum and maximum
intensity points across each black-white line pair, thereby ignoring blurring that might occur at the edges of each white line,
and therefore slightly overestimating MCF. The method used to
calculate the FFT should be mentioned. A single array of data
taken horizontally across the chart was analyzed. This means
that the analysis is one-dimensional. The data beyond which
resolution was zero (including false resolution) was blocked out.
In other words, if only the first three line pairs up to 0.286 1pm
could be resolved, all the data from the fourth line pair on was
deleted (moving to the left). To keep the array 512 pixels long,
so it could be used by the FFT, all the deleted pixels were filled
with a white value. A comparison of measured MCF (square
wave response) with MTF expressions in Eqs. (2) and (6) is
shown in Fig. 6 where Eq. (2) is inappropriate because te < T0
and Eq. (6) is an approximation for low vibration frequency
MTF. In Fig. 6 vibration frequency is a parameter and te 15 again
equal to 20 ms. Despite the randomness of blur radius, square
wave response is generally higher than sine wave response, as
it should be. Of importance here is that square wave response
is noticeably closer to the sinc function approximation than to
the inappropriate Besselfunction, which has often been used in
the past. In Fig. 5, te/T0 0.05. In Fig. 6, 0.02 < teIT0 <
0.08.
Again, for the longest and shortest relative time exposures
(te/T0), the sinc function MTF approximation in Fig. 6 is less
accurate than for medium relative time exposures, as discussed
with regard to Fig. 5. As te/T increases, blur radii increase and
the spatial frequency content is more limited. The Bessel function in Eq. (2) is not a function of the random variable d but
rather of D, which is constant. The sinc function MTF approximation is a function of d and therefore more closely resembles
the measured MCF (Fig. 6) and MTF (Fig. 5) without noise. In
general, the larger the blur radius, the closer the resemblance
[H
(a)
r
'
i
.
I kA
t'
______
(b)
Fig. 4. (a) unbiurred image: static images of 0.5 (left) and 1 (right)
1pm (or 0.25 and 0.5 1pm spatial frequencies), respectively. (b) effect
of 2 mm blur radius (i.e., 2 mm movement during exposure) on 0.5
(left) and 1 (right) 1pm spatial frequencies. In the latter case, 0.5 1pm
are reduced to one uniform shade of gray with zero contrast.
dID decreases
Eqs. (2) and (6). However, as d*
because of relatively short exposures or time exposures centered
between
at extrema of the vibration [when d* approaches d1 in Eq.
(4)], the sinc function becomes a better MTF approximation than
the constant Bessel function. In any event, the blur radius is a
random variable that can vary greatly even for a given exposure
time, depending upon the portion of the vibrational sine wave
function at which the exposure takes place. This randomness of
the low-frequency sinusoidal motion MTF can be seen in Fig. 7.
Here, T0 is the reciprocal of the vibration frequency and te 5
again 20 ms. Therefore, Figs. 7(a) and 7(b) represent MTF curves
where te/T and d* are constant in each graph. Yet each curve
in a given graph represents a different MTF corresponding to a
different instant of time or portion of the sine wave, over which
the exposure takes place. Thus, for low-frequency vibrations
there is no unique MTF. Conclusions of these MTF measurements for short-exposure low frequency sinusoidal image motion
are:
1.
of the randomness of the instant of exposure,
there is no unique MTF (Fig. 7).
Because
2. In many short-exposure cases the sinc function MTF
approximation for linear motion is a good, reasonable
approximation for sinusoidal motion (Figs. 5 and 6).
3.
When the sinc function MTF curve is less identical to
actual MTF, it is still a lot better approximation at low
mechanical frequencies than the widely used Bessel
function (Figs. 6 and 7), which really is appropriate for
high vibration frequencies only.
OPTICAL ENGINEERING / May 1 991 / Vol. 30 No. 5 / 581
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RUDOLER, HADAR, FISHER, KOPEIKA
0
2
C
B 1.5
2
2
Spatial frequency
(b)
(a)
02
0
2
U-
5
I—
5
Spatial trequancy
Spatial frequency
(dl
(c)
U-
02
0
S
I-
2
U-
F-
S
Ot
0.3
0.2
Spatial trequency
(e)
(f)
Fig. 5. Comparison of MTF (noisy solid curve), predicted MTF (sinc function prediction—solid
curve), and measured MCF (square wave response—dashed curve) forvarious blur radii d. Spatial
frequency is in lines per millimeter.
582 / OPTICAL ENGINEERING / May 1991 / Vol. 30 No. 5
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATIONS. PART 2: EXPERIMENT
IL
C
0
U.
0.2
0.3
0.4
0.5
Spatiat frequency
sp.t frequency
(h)
(9)
.--- -
j
d0.9mm
25L
2
0
0
1.5
1'
F
h)i !l
051
0L____T...
04
06
vti.;Itt!It)
Spatial frequency
Spatial frequency
(j)
(I)
0
Spatial frequency
(k)
Fig. 5 continued.
OPTICAL ENGINEERING / May 1 991 / Vol. 30 No. 5 / 583
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RUDOLER, HADAR, FISHER, KOPEIKA
—'—
1 .2
d=2.7mm
1 -_.=
d = 1.8mm
''SiNC
'\\\
0.8-
J'\ \
0
0.8
2.5
\'\ \\ \
0.6k
0
0.6
\\
'"
0.4
0.4
0.2
0.2
C
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Spatial frequency
Spatial frequency
(a)
(b)
1.2
.--lNCT_.
\\
\'\\.
d=l2mm
SINC_______________—
1
"
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0.8 L
0.8
0
'N
0.6
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0.4k
\\\
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0
\:::.\ \
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---,,
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...----L..
0.1
0.2
.
0.3
.
.
.
.
0.4
0.5
0.6
0.7
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Spatial frequency
(c)
0—..--0
\\\\ \\,
'
\
\
0.2
\\
\\
\"\\
\
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-.--
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0.2 -
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d=0.7mm
, --\\ \
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06
1
0.4
0.6
0.8
.
.—'---.-.-.-..-- 1
1.2
1.4
1.6
Spatial frequency
(d)
Fig. 6. Comparison of measured MCF (square wave response) with MTF expressions in Eq. (2) and
Eq. (6) for different blur radii with vibration frequency as a parameter. Spatial frequency is in lines
per millimeter.
4. LUCKY SHOT ANALYSIS
A completely different method of measuring d was also used.
In this process it was decided not to try and capture the image
of the resolution chart, but just to photograph the scope display.
The received image is just the path that the scope beam traced
during the 20 ms exposure time. This approach is demonstrated
for a 20 ms exposure time in Figs. 8(a) through 8(c). It is easy
to see that although both traces in Figs. 8(b) and 8(c) are the
same time length (x-axis), they have very different blur radii
d(y-axis). By measuring the y-component of length of the trace,
an accurate measure of d can be obtained. Instead of measuring
d in millimeters or in volts, however, d can be measured by the
584
number of pixels occupied by the trace in the image. If a full
cycle is then stored on the scope, D (the amplitude of the sine
wave) can also be pictured and measured in pixels. Since it is
only the relative blur d* that is of concern, it can be determined
as (d pixels)/(D pixels). This leads to a fairly accurate method
of measuring relative blur (d*). The one problem with this method
is that because of interlace the accuracy of the measurement is
2 pixels at each end of the trace for a total accuracy of
pixels. If the trace is small, less than 40 pixels, this error can
become significant. One last problem with this method is that
for small blur radii, when the beam does not travel very far, its
image tends to "bloom", that is it gets slightly longer and
thicker, because the beam is moving slowly and the CCD in-
/ OPTICAL ENGINEERING / May 1991 / Vol. 30 No. 5
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATIONS. PART 2: EXPERIMENT
MCF for 1 hz vibrations
UC-,
0
(a)
U-
Spatial frequency
-I_
(b)
(a)
(c)
U-
0
Fig. 8. Various blur radii for a 20 ms exposure: (a) full cycle, (b) lucky
shot, and (c) bad shot.
0
U-
tegrates its luminance. This effect degrades further the accuracy
Spatial frequency
(b)
of measurements for small values of d.
Software was written to apply the scope trace method of
measuring d. It allowed about 500 shots to be taken. This large
number of shots is important so that a statistically valid number
of samples can be taken. After 500 blur radii are measured for
a specific te/T0, d* 5 calculated and then the probability of
obtaining a lucky shot for a given value of d is calculated,
where a lucky shot is an image with relative blur radius less
than 4 Finally, the number of shots required to guarantee (with
Q percent of confidence) such a lucky shot is derived. From the
samples taken it is calculated
P(d)=,
U-
0
0
(7)
where
U-
P(d) =
d*
d=
Spatial frequency
(c)
b=
B=
Probability that
d* < d
relative blur
desired maximum relative blur
number
of shots with d* < d
total number of shots taken
then (1 — Q) = [1
—
P(d)1', where N is the number of shots
required to have at least 1 shot where d* < d (a lucky shot)
Fig. 7. Comparison of measured MCFs with MTF expressions in Eq. (2)
and Eq. (6) for blur radii and relative time response (t/t0) constant in
each graph where t = 20 ms: (a) T0 = 1 000 ms and (b) T 333 ms.
Spatial frequency is in lines per millimeter.
and Q is the desired confidence, finally giving
ln(1 — Q)
N= ln[1
—
P(d)J
(9)
OPTICAL ENGINEERING / May 1991 / Vol. 30 No. 5 / 585
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RUDOLER, HADAR, FISHER, KOPEIKA
Vibration Platform vs. Theory
Vibration Platform vs. Function Generator
teIro
•G Mc
0.05
2.5hz
Q.=O.8
.* Fcn. Gen.
(e!ro = 0.05
- Machine
2.5hz Q=o.8
-4- Theory
15
13
•1
C,)
0
0
(I)
U)
0
0
.0
C)
.0
z
z
0
E
E
11
9
7
5
3
0 05
0.25
0.15
0.35
0.05
0.25
C.15
0.35
Relative Blur
Relative B(ur
(b)
(a)
Fig. 9. Experimental comparison between vibration platform (machine) and function generator
for lucky shot experimental analysis.
Table 1 . Vibration frequencies and methods for experimental lucky
shot analysis.
FREOUENCY
:tctQ
VIBRATION SOURCE
D(PIXELS)
2.5 Hz
0.05
Vibration Platform
114
2.5 Hz
0.05
Function Generator
106
5.0 Hz
0.10
FunctionGenerator
189
7.5Hz
0.15
FunctionGenerator
189
10.0 Hz
0.20
Function Generator
189
These results are compared with theoretical predictions in Ref. 2
to see how closely they agree. As mentioned above, the vibration
platform was only capable of motion at one of the frequencies
of interest (teIT0 0.05). To test other vibrations frequencies,
it was necessary to use simulated vibrations supplied by the
function generator. To validate this approach, the statistical results of actual vibrations were compared with the signal-generated
pseudovibrations. They agreed very well, thus validating this
approach. Table I shows at which frequencies and with what
method data were taken. In each of these cases d was measured
for 500 statistically independent shots. The amplitude of the sine
wave D is also shown, measured in pixels.
The results of the two 2.5 Hz experiments agreed closely
both with theory and the function generator approach, as shown
in Fig. 9, thus justifying the use of only the function generator
for vibration frequencies . A software program was also written
to simulate the picture taking process. This program used a
586
randomly generated starting exposure time t and measured change
in sine wave amplitude during the chosen exposure time. A large
number of samples can be generated quickly this way and compared with the theoretical vibration platform and function generator results. If all the data agrees, that verifies the simulation
and the data can be used to study more complex functions , such
as vibrations involving the sum of two sine waves.
Graphs comparing the probability for a lucky shot using the
different methods and theoretical predictions from Ref. 2 are
shown in Figs. 10(a) through 10(h). All figures are for Q = 80%
confidence. To be 80% certain to take at least one picture with
a relative blur less than d0ID for a given relative exposure time
teIT, one reads along the x-axis to that relative blur and finds
the corresponding point on the y-axis to know how many shots
must be taken. For example, using Fig. 9(a), with a 2.5 Hz sine
wave and a camera that takes a 20 ms image, to be 80% certain
that the relative blur is less than 0. 15 in at least one shot, both
theory and experimental results predict that seven shots must be
taken.
As seen in Fig. 9 the theoretical lucky shot analysis is wellcorroborated experimentally.
5. AVERAGE BLUR RADIUS
Reference 2 also predicts that for short-time exposure the average
blur d will be 3.57Dte/T0. This proved incorrect as shown in
Table 2. It is clear that the experimental values of d are about
12 to 15% higher than expected, and that this difference between
theory and experiment varies with relative time exposure te/Tø,
as shown in Fig. 1 1 . Accordingly, the theoretical model was
/ OPTICAL ENGINEERING / May 1 991 / Vol. 30 No. 5
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATIONS. PART 2: EXPERIMENT
reexamined and, by using MATLAB software to minimize least
mean square error, the relative blur radius was determined to
be
Fig. 10 where d* is larger than Eq. (10) for short exposures.
For longer exposures this trend becomes reversed. Figure 1 1
shows that Eq. (10) is quite close to the actual measurements
of the average blur radius for relatively short exposures . Eq. (10)
-
I* _ . _ 3 7433
•
teIT0 .
(10)
D
-
A comparison between measured d* and Eq. (10) is shown in
does not explain all of the discrepancies in Table 2, but a good
deal of them. The rest is attributed to experimental errors and
to difficulty in measuring d* , including the uncertainty of 4
pixels described earlier. Equation (10) is also limited to short
Function Generator vs. Theory
(eITO
2.5
"1
0
(1)
0
0
.0
E
z
•G
0.05
h 0 0.8
Computer SimuLation vs. Theory
Fcn. Gen.
teTIO = 0.05
Theory
15
15
13
13
11
Cr,
11
C')
9
0
9
C)
.0
7
E
z
ii HL1
7
5
5
3
3
0.05
0.15
0.25
Relative Blur
(a)
teIro
5.0 hz
Co
0
0
.0
E
z:3
0.10
Q 0.8
0.35
(b)
Function Generator vs. Theory
0
•— Simulation
C
-C
Relative Blur
C,
.o. Theory
2.5 hz Q = 0.8
Computer Simulation vs. Theory
-a-- Theory
te/TO= 0.10
5.0 liz 0 = 0.8
Fcn. Con.
14
14
12
12
10
C/,
0
-0- Theory
Simulation
10
C')
8
0
8
C)
.0
6
E
z:3
6
4
4
2
2
0
0
0.7
RelatIve Blur
(c)
0.1
0.2
0.3
0.4
0.5
Relative Blur
0.6
0.7
(d)
Fig. 1 0. Lucky shot comparison between theory and experiment (continued on next page).
OPTICAL ENGINEERING / May 1 991 / Vol. 30 No. 5 / 587
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RUDOLER, HADAR, FISHER, KOPEIKA
Function Generator vs. Theory
Camputersimutation vs. Theory
-°- Theory
teTro 0.15
7.5 hz
tefTO= 0.15
Fcn. Gen.
0 0.8
-a- fleocy
Q = 0.8
7.5 hz
Sirnutation
15
13
U)
0
0
-C
-C
0
00
E
0
'11
C))
(1
9
C)
.0
z:
E
z
:3
7
5
3
0.8
0.6
0.4
0.2
0.2
1.0
0.6
0.4
Relative BLur
(f)
(0)
Function Generator vs. Theory
Computer Simulation vs. Theory
020
te/TO
Theocy
10hz Qa0.8
1.0
0.8
Relative Blur
te/TO = 0.20
-':3- Theory
-4- Fcn. Gen.
10hz 0=0.8
.4-S Sknutation
13
11
Ct)
0
0
-C
-C
C')
C))
0
0
C)
.0
.0
z:3
z
C)
9
7
E
E
:3
5
3
I
0.3
0.5
0.7
0.3
0.9
Relative Blur
0.5
0.7
Relative Blur
0.9
(h)
(g)
Fig. 1 0 continued.
exposures. It is clear that as te/Tø approaches unity, d must
approach 2D, as seen in Fig. 1 . Of course, for such a long time
exposure, the assumption of uniform motion is certainly not valid
for a sine wave.
vibration on image quality and support the MTF model developed there within the limitations illustrated in Sec. 3. The predictions of the number of shots required to get a lucky shot were
found to be extremely accurate. Although the data diverge for
small relative blur (or a large number of shots), this is not
6. CONCLUSIONS
The graphs shown in Figs. 5 , 6, and 8 clearly validate the
predictions in Ref. 2 concerning the influence of slow sinusoidal
surprising for two reasons. The first is the problems involved in
measuring small values of d accurately, as mentioned earlier.
The second reason is that since there are fewer events with small
588 / OPTICAL ENGINEERING / May 1 991 / Vol. 30 No. 5
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IMAGE RESOLUTION LIMITS RESULTING FROM MECHANICAL VIBRATIONS. PART 2: EXPERIMENT
7. ACKNOWLEDGMENT
Table 2. Comparison of theoretical and average blur radii.
FREOUENCY
VIBRATION SOURCE
MEASURED
3..3LOie/Io 3.24311e/T_o
This research was partially supported by the Paul Ivanier Center
for Robotics and Production Management.
VibrationPiatfomi
23.5
20.3
21.34
2.5 Hz
Function Generator
23.6
18.9
19.84
5.0 Hz
Function Generator
79.3
67.5
70.75
7.5 Hz
Function Generator
114
99.2
106.12
8. REFERENCES
10.0 Hz
FunctionGenerator
151
135
141.5
1
2.5 Hz
. T. Trott, ''The effects of motion in resolution,' ' Photogramm. Eng. 26, 819—
827 (1960).
2. D. Wulich and N. S. Kopeika, "Image resolution limits resulting from mechanical vibrations," Opt. Eng. 26, 529—533 (1987).
3. F. Wahl, Digital Image Signal Processing, Artech House, Boston, 1987.
N. S. Kopeika received the BS, MS. and Ph.D.
degrees in electrical engineering from the
University of Pennsylvania, Philadelphia, in
,
,,
'a
' j
t:'t..
tion of such devices for detection and recording of millimeter wave holograms. In 1973 he
joined the Department of Electrical and Com,t -,
.
.
.1
puter Engineering, Ben-Gurion University of
the Negev, Beer-Sheva, Israel, where he is a professor and department chairman. During 1978-1979 he was a visiting associate professor in the department of Electrical Engineering, University of Delaware, Newark. His research interests include atmospheric optics,
effects of surface phenomena on optoelectric device properties, optical communication, electronic properties of plasmas, laser breakdown of gases, the optogalvanic effect, electromagnetic wave-plasma
interaction in various portions of the EM spectrum, and utilization of
such phenomena in EM wave detectors and photopreionization lasers. He has published over 70 journal papers and has been particuIarly active in research of time response and impedance of properties
of plasmas. In addition, he is the author of 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 a modulation transfer function with
which to describe effects 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.
.
te/To
Fig. 11. Comparison between average relative blur radius d* in Eq.
(10) (lower curve) and experimental measurement (upper curve).
values of d, the statistics are less valid at this point. If a larger
number of samples were taken, the results likely would converge
slightly for smaller relative blur. The theoretical prediction for
average blur radius was fairly close to the experimental measurement, with some undershooting of it. A revised and more
accurate value is described by Eq. (10).
,
1966, 1968, and 1972, respectively. His Ph.D.
dissertation, supported by a NASA Fellowship, dealt with detection of millimeter waves
by glow discharge plasmas and the utiliza-
t
OPTICAL ENGINEERING / May 1 991 / Vol. 30 No. 5 / 589
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