Research on random targets for modulation transfer function (MTF) evaluation
began in 1986 at the Center for Education and Research in Optics and Lasers
(CREOL) located at the University of Central Florida. (1)
Under the direction of Dr. Glenn D. Boreman, research was conducted on the use
of integrating spheres to produce highly uniform laser speckle. (2)
Dr. Boreman demonstrated that the exit port of the integrating sphere could be
used with a specially designed mask to produce laser speckle with a known power
spectral density.
Al Ducharme began his Ph.D. research on MTF evaluation using random targets in
1990. His initial work built on Dr. Boreman's premise that an integrating
sphere and mask could be designed that would generate a random target with
known spatial frequency content. (3) The
first projected random targets had a narrow-band power distribution with a
center frequency inversely proportional to the distance between the image plane
and integrating sphere. Since, the square root of the measured power
divided by the input power, these narrow-band targets can be used to measure
the MTF of a detector array. The MTF is measured over a range of
frequencies by moving the detector array through a range of distances.
Other masks were developed but the narrow-band target received the widest
acceptance.
The one drawback of this method of target generation is that it is
expensive. They system consisted of a visible laser diode with a constant
current source, translation stage, integrating sphere, mask, and miscellaneous
optical mounting hardware. Dr. Ducharme's Ph.D. dissertation concentrated
on minimizing the complexity of the technique. (4)
The first improvement was the reduction of the entire system to a single
hologram of narrow-band laser speckle. (5) The
resulting hologram can be used in a much simpler playback configuration
consisting of only a few components. This hologram is now known as as the Projected
Random Test Pattern (P-RTP) The second improvement came from the
realization that a digital image could be created with a uniform or white-noise
power spectrum. (6) The image can be imaged
onto a detector array to measure the MTF over a wide range of frequencies using
a single measurement (collection of measurements over a range of
distances is not needed). The digital image is now known as the Imaged
Random Test Pattern (I-RTP). These patterns are part of the
standard MTF target product line available from P.I.O.
Section 2: Modulation Transfer Function
The MTF of an optical system quantifies the
ability of the system to resolve or transfer spatial frequencies. To
understand this, imagine an imaging system such as the human eye observing
waves of the ocean while the observer is riding in an aircraft, Assume
that the altitude of the aircraft is increasing linearly and that the waves are
of constant length (from peak-to-peak) and height, As the aircraft climbs
the observer will begin to resolve the individual waves until finally the ocean
appears flat. The altitude of the aircraft can be used to determine the
spatial frequency in cycles/mm of waves on the retina of the eye.
Therefore, the ability to resolve the waves plotted with respect to this
frequency would in essence be the MTF of the human eye.
Although this is a simplified example it
illustrates the following expression:
(1)
where M is the modulation
depth expressed as
(2)
and E is irradiance.
Once the value of M image
is measured experimentally, the MTF value of the imaging system
for the spatial frequency of the object can be calculated, The MTF is
measured over a range or frequencies using a series of targets.
It is also acceptable to work in the frequency
domain rather than the spatial domain. This is done using a fast Fourier
transform (FFT) of the digitally recorded image. The absolute value of
the result is then squared to yield the power spectral density (PSD) of the
image, S image. The MTF can then be
evaluated using
(3)
Working in the frequency domain is an extremely
useful technique since more complex targets can be used to measure MTF.
Specifically, a random target with a known power spectrum can measure MTF for
several frequencies simultaneously. This offers the following advantage
over the other MTF measurement techniques:
A major problem encountered in measurement o MTF
in sampled-image systems is that the image modulation depth depends on the
position of the deterministic target with respect to the image sampling
locations. A random target of known spatial-frequency content allows
measurement of a shift invariant MTF because the information of the test target
has random position with the respect to the sampling sites. The results
is an extremely repeatable MTF measurement technique. (6)
Another item to remember when making MTF
measurements is that all the systems components contribute to the final
performance of your system. This cascading effect is expressed:
MTF system =
MTF framegrabber x MTF cabling x
MTF detector x MTF lemses
(4)
Therefore, all of the planned system
components should be used during the MTF testing.
1951
Section 3: Using the Imaged
Random Test Pattern
The I-RTP is a square grayscale image with an
uncorrelated two-dimensional random pattern. This pattern has a uniform
bandlimited white-noise PSD. The bandwidth is equal to the number of
pixels in a single dimension divided by two. Since the spectrum of the
I-RTP is white (i.e., equal power at all frequencies) the square root of the
image PSD is equivalent to the MTF of the imaging system. Yes, it
is that simple. The MTF is measured for hundreds of spatial frequencies
in two-dimensions using a single measurement.
Figure 1. Sample Imaged
Random Test Pattern.
Section 3.1: Measurement
Configuration
To use the I-RTP you will need the target,
imaging system and detector array, uniform lighting, and a method to digitally
record and process a captured image. These items should be configured as
shown in Fig. 2.
Figure 2. I-RTP measurement
configuration.
The uniform light source is not shown in Fig. 2
but should be positioned so that image of the I-RTP has a uniform mean value.
This is not critical but can add amomalous low frequency components to the
measured PSD. Capturing an image of white paper and evaluating the digitally
recorded pixel values can be useful when trying to obtain uniform illumination.
There may be some applications where the system includes an illumination
source. For this case, this source should be used when measuring the MTF of the
system since the illumination will contribute to the overall system
performance.
Section 3.2: Measurement Procedure>
The image of the target should be adjusted so
that the field -of -view (FOV) of the imaging system is aligned with the
dash-dot lines shown in Fig. 2. Ideally the target should be imaged 1:1
onto the detector array. This will enable the widest range of test
spatial frequencies. An image (color or black-and-white) should be
captured using a framegrabbing device. Preferably the framegrabber should
be the one that is typically used in the application so that the MTF is
representative of the entire system (i.e., the MTF described by Eq. 4).
If the detector array is rectangular, which is usually the case, the edge of
the target should be imaged onto the dimension of interest. The digital image
of the target should be stored in a format that does not utilize compression
(i.e., BMP rather than JPEG). In addition, if a color image with several
color planes (e.g., Red 8-bit, Blue 8-bit, and Green 8-bit) is captured then
only a single plane should be used in processing.
3.3 Data Processing
In order to process the image you will need a
standard FFT algorithm available in most programming languages. The first
step in the program must be to convert the image to a double precision 2D
array. Then the mean value of the array is subtracted from the
image. If you are familiar with Fourier theory, this last step will
eliminate the dominant frequency component at 0 cycles/mm. A program
should be written that performs the following diagram on the N x
x N y image of the target.
Figure 3. Flow diagram of
algorithm to determine PSP estimate.
In words, the
processing algorithm forms a PSD estimate of the image in one dimensions.
The estimate is the result of the average of Ny spectums.
The resulting PSD array will have Nx -1. Since the
PSD is inherently positive frequencies , 0 to Nx /2-1,
and negative frequencies, Nx /2 to Nx-1.
Since the PSD is inherently symmetric we can discard the negative
frequencies. The resulting array will have Nx/2
vaues. The final processing step is to scale teh PSD array so that it
represents the MTF of the system under test. This is done by evaluationg
Eq. 3 now expressed
(5)
since the input power is
1 for all frequencies.
The final three steps
are used to scale the data into MTF units. First, the MTF array must be
normalized to the lowest frequency component in the array with the exclusion of
the value in array index value 0. This is acceptable since the lowest
frequencies will be passed through the system with little or no
attenuation. The MTF curve should be smooth near 0 cycles/mm and the
normalization should make the low frequencies equal to 1.0. Second, you
must take the square root of the PSD yielding the MTF y-axis array.
Finally, the frequency- or x-axis array must be created with units of
cycles/mm. This is done using the following expression:
(6)
where dx
is the width of a single detector in the x dimension, ξ(I)
is the x-axis array, and I is the array index value from 0 to Nx/2-1.
The resulting y-axis
MTF array is plotted against the x-axis frequency array. This plot
represents the MTF of your system in the x dimension. The same
procedure can be repeated for the y dimension to evaluate the MTF of
your system in the orthgonal axis.
Section 3.4:
Specification of 1-RTP
Parameter
Value
Light wavelenght range
Lowest Test Frequency
Highest Frequency
Physical Form
Size
400nm-to-800nm
1/Width of detector array
(cycles/mm)
600/Width of detector arrat
(cycles/mm)
Matte paper on
foam core
6" x 6"
Section 4: Using the
Projected Random Test Pattern
The P-PTP is a hologram of a random target
produced by exposing holographic film with narrowband laser speckle.
Illuminating the hologram with a coherent light source such as a visible laser
pointer replays the P-PTP. When .illuminated at the recording angle, a
real image of the narrow-band laser speckle is projected onto a detector array with
no intervening optics. This has the advantage of eliminating the
MTF of the lens system, which may be desirable in some
applications. The difference between the I-RTP and the P-RTP is that the
P-RTP tests one spatial frequency for each captured image in only the
horizontal dimension. Translating the detector array with respect to the
hologram tests additional frequencies.
Figure 4. Sample Projected
Random Test Pattern. The domination spatial frequency can be seen in the
horizontal dimension of the image.
Section 4.1: Measurement Configuration
To use the P-RTP you will need the target, a
laser source, a method to translate the detector system, a negative lens, and a
method to digitally record and process a captured image. These items
should be configured as shown in Fig.5.
Figure 5. P-RTP measurement
configuration.
The negative lens is used to expand the
collimated beam from the laser source. The angle between the illumination
and target is approximately 40 degrees and must be adjusted during
set-up. The angle between the target and optical axis of the projected
target is equal to the illumination angle. The orientation and incident
sides are marked on the hologram.
Section 4.2: Measurement Procedure
The image of the target should be adjusted so that the field-of-view (FOV) of
the imaging system is aligned with the dotted lines shown in Fig.5. An
image (color or black-and-white) should be captured using a frame grabbing
device. Preferably the frame grabber should be the one that is typically
used in the application so that the MTF is representative of the entire system
(i.e., the MTF described by Eq. 4). If the detector array is rectangular,
which is usually the case, the edge of the target should be imaged onto the
dimension of interest. The digital image of the target should be stored
in a format that does not utilize compression (i.e., BMP rather than
JPEG). In addition, if the color image with several color planes (e.g.,
Red 8-bit, Blue 8-bit, and Green 8-bit) then only a single plane should be used
in the processing.
The distance, Z, between the hologram and detector determines the
spatial frequency being tested. The relationship is expressed by
(7)
Therefore, high frequencies are tested close to
the hologram and low frequencies are tested far from the hologram. This
is analogous to the effect observed when moving an overhead projector away from
a screen. The image gets larger as the projector is moved farther away
from the screen. The exact distance does not need to be measured since
the tested frequency will be evident in the processed PSD..
Section 4.3: Data Processing
The PSD of the P-RTP target is narrow-band meaning that it has a known spectral
power that is concentrated over a small range of frequencies. The
theoretical P-RTP PSD is shown in Fig. 6.
Figure 6. Theoretical PSD of
P-RTP.
This figure illustrates the movement of the narrowband peak as Z is
varied. The beauty of this technique is that the power of the triangular
peak is always 1/2 of the central peak. As a result. Eq. 3 can be
expressed as
To begin making an MTF measurement you must position the detector array until
an image of a single speckle covers half of the image. After an image has
been digitally captured and stored, the detector array is moved towards the
hologram until approximately 2 speckles can be seen in the image. This
same distance increment is used until the target no longer illuminates the
entire detector array. Each stored image is then processed using
the method described in section 3.3 up to the point before you take the square
root of the PSD estimate. Instead, the square root of the height of the
narrowband peak divided by 1/2 is evaluated. The result is the modulation
transfer at the index value of the PSD array. The corresponding
frequency will be evaluated when all MTF values are measured. The MTF
value is stored in an array of sizes Nx/2 at the same
index value where the peak value was positioned. Once all of the images
have been processed the resulting MTF array should be plotted against the
x-axis or frequency array described in section 3.3. The MTF of the
detector array in the orthogonal axis can be measured by repeating this process
with the detector array rotated 90 degrees.
Section 4.4: Specification
of P-RTP
Parameter
Value
Light wavelenght range
Lowest Test Frequency
Highest Frequency
Physical Form
Size
600nm-to-700nm
1/Width of detector array
(cycles/mm)
>600/Width of detector arrat
(cycles/mm) *
35 mm projection slide
2" x 2"
* Depends on how close the
detector array can be positioned from the hologram.
Section 5: References
1. G. D. Boreman and E.
Dereniak, "Method for measuring transfer function of changed-coupled devices
using laser speckle." Opt. Eng., 24, pp. 148-150 (1986)
2. G. D. Boreman, Y. Sun, and
A. James, "Generation of laser speckle with an integrating sphere." Opt. Eng.,
29. pp. 339-342 (1990)
3. M. Sensiper, G. D. Boreman,
A. D. Ducharme, and D. R. Snyder, "MTF testing of detector arrays using
narrowband laser speckle," Opt. Eng., 32(2), pp. 395-400 (1993).
4. A. D. Ducharme,
"Signal-processing and flux-transfer issues for laser speckle." Ph.D
Dessertation, Univ. Central Florida (1994).
5. A. D. Ducharme and G. D.
Boreman, "Holographic elements for the modulation transfer function testing of
detector arrays," Opt. Eng. 34(8), pp. 2455-2458, (1995).
6. A. Daniels, G. D. Boreman,
A. D. Ducharme, "Random transparency targets for modulation transfer function
measurement in the visible and infrared regions," Opt. Eng., 34(3),
pp. 860-868 (1995).
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