Improving MTF Measurements of Under-sampled Optical Systems

Improving MTF measurements of under-sampled optical systems Joseph D. LaVeigne

a, Stephen D. Burks

b

aSanta Barbara Infrared, Inc., 30 S Calle Cesar Chavez, Santa Barbara, CA, USA 93103;

bNVESD, 10221 Burbeck Road, Fort Belvoir, VA 22060-5806

ABSTRACT

The modulation transfer function (MTF) of optical systems is often derived by taking the Fourier transform (FT) of a

measured line spread function. Recently, methods of performing Fourier transforms that are common in infrared

spectroscopy have been applied to MTF calculations. Proper apodization and phase correction have been shown to

improve MTF calculations in optical systems. In this paper these methods as well as another filtering algorithm based

on phase are applied to under-sampled optical systems. Results, both with and without the additional processing are

presented and the differences are discussed.

Keywords: MTF, FFT, modulation transfer function, Fourier transform, phase correction

1. INTRODUCTION

The pre-sample Modulation Transfer Function (MTF), consisting of the optics, detector array, and digitization

electronics, is one of the most important measurements in objectively determining an electro-optical system’s

performance. In order to precisely model an electro-optical system’s performance, it is ideal to determine which model

elements are resolution limited and which model elements are noise or sensitivity limited.

In most standard procedures for collecting an MTF, multiple consecutive frames of a tilted edge are collected and

averaged in order to produce a super-resolved edge response function. Unfortunately, in the presence of fixed pattern

noise, it becomes a difficult task to correctly measure an MTF that does not contain artifacts such as ringing (non-

monotonic decreasing as frequency increases) or a noise floor that prevents the MTF from trending towards zero at the

cutoff.

Also, in modeling a sensor’s performance with electro-optical modeling software like NVThermIP, the system contrast

threshold function uses individual inputs from resolution and noise measurements. If the resolution values are altered

due to the existence of fixed pattern noise, the system performance is in effect being doubly penalized for this noise.

It also makes sense to want to predict with the highest fidelity possible the pre-sample MTF. As many current electro-

optical systems incorporate staring arrays, this means that there is a potential for them to be under-sampled. As under-

sampled imaging systems typically have an MTF response past the Nyquist rate, this means that an accurate prediction

of any spurious response calculations will also need to rely on an accurate measurement of the pre-sample MTF.

1.1 MTF Measurements

Deriving a meaningful MTF from imperfect experimental data can be challenging. Noise and detector imperfections can

produce artifacts that make the extraction of a good MTF curve difficult to say the least, especially as the cutoff

frequency is approached. One of the most common problems is that by taking the magnitude of the optical transfer

function (OTF) any measured noise is rectified, leading to a frequency spectrum that never gets to zero at high

frequencies. In many IR systems, it is difficult to maximize the signal to noise ratio without clipping the system. If the

signal to noise ratio is too low, the system MTF will have additional artifacts due to noise. If the signal to noise ratio is

too high, clipping will lead to a nonsensical MTF.

As mentioned above, it is important to have an accurate pre-sample MTF in order to accurately predict a system’s

performance. If the measured pre-sample MTF never reaches a valid cutoff point (where it instead trends to a fixed

modulation value greater than zero), then it is difficult for a system tester to determine how the actual MTF behaves. For

instance, the system could cut off at the first point where the MTF approaches this MTF floor, or this MTF floor could

be included in the results until the half sample rate. In predictive models, such as NVTherm IP, these choices will

greatly affect the overall calculated sensor performance.

Previous work has been performed studying random and fixed pattern noise in MTF measurements, including

comparisons of using the line spread function (LSF) and edge spread function (ESF) to calculate MTF as well as the use

of super-resolution to overcome aliasing present in systems where the detector under-samples the optical system

response[1]. Earlier studies

[2] by the authors have also investigated how applying techniques such as apodization and

phase correction[3-6]

to the Fourier transform can improve the accuracy of MTF measurements that contain a significant

amount of noise. In those studies it was commented that phase correction may not be effective in removing artifacts due

to fixed patterns in a tilted edge or slit MTF measurement, especially if those patterns exhibits symmetry. In the

following section a similar method that is effective in removing artifacts due to fixed patterns, regardless of symmetry,

will be discussed.

2. DERIVING MTF FROM THE LSF

One of the more common methods of measuring MTF is through the use of a tilted edge target. The edge spread

function is sampled along the edge as well as across the edge in order to decrease the sample spacing and reduce

aliasing. The LSF is generated by taking the numerical derivative of the super-resolved edge spread function. Noise or

fixed pattern errors in the LSF can lead to artifacts in the derived MTF. While some fixed patterns can be easily

removed through background subtraction, performing that subtraction is not always convenient and that method is

generally not effective on errors in gain. This discussion will focus on methods for deriving MTF from the LSF that can

reduce artifacts, particularly those generated by fixed patterns, and hence improve MTF measurements.

2.1 The Super-Resolved Line Spread Function

One common way of generating a super-resolved LSF (SRLSF) is by first generating a super-resolved ESF (SRESF) and

then taking its derivative. Another method to consider would be to first take the derivative of the edge image and then

generate a SRLSF from the resulting tilted LSF. While the anti-aliasing effects are retained by sampling more often,

taking the derivative first does have the effect of incurring an additional MTF contribution due to the derivative

sampling. This sampling contribution can be easily modeled as a sinc function[7] )/)sin()(sinc( xxx = and removed.

Thus, taking the derivative first can be done without incurring a penalty, and allow some other processing to be done as

discussed in Section 2.3. After the processing is complete, there is the option of integrating to regain the ESF and avoid

the additional sampling contribution. This procedure may introduce additional artifacts, and it will not be covered in this

publication.

2.2 LSF, Pattern Noise and Phase

One of the benefits to be had by taking the derivative first is that it allows pre-processing of the LSF image. One of the

first benefits to be seen is an immunity to horizontal patterns. Since the derivative is taken on each line separately,

horizontal patterns are easily removed. Vertical patterns can also be managed, though the process is somewhat more

complicated than simply taking a derivative. The basic concept is to use knowledge of the phase relationship between

the individual LSFs to filter out fixed patterns. For simplicity, consider the following example: A staring array has a

fixed pattern of columns over which it is measuring an ideal tilted slit. If each LSF along the slit in the image were

aligned, the a reciprocal tilt would be imposed on the fixed pattern (see Figure 1). Ignoring the pattern, the MTF of each

line should be the same. The magnitude of the contribution due to the fixed pattern in each line should also be the same,

with the difference in each line solely being the phase of the fixed pattern contribution. Because the phase of any fixed

pattern noise will shift by a known amount that depends on the angle of tilt of the edge, it is possible to create a digital

filter to remove the contribution from that noise. Using this technique along with the techniques described earlier can

lead to improvements on measured MTF.

2.3 Theory

The same naming convention will be used as in the previous publication, notably a MTF derived by multiplying the OTF

by its complex conjugate will be referred to as a power spectrum and a MTF derived by performing the phase correction

algorithm presented previously will be referred to as a phase corrected MTF. In addition, a super-resolved LSF created

by taking the derivate of a super-resolved ESF will be called a post-derivative LSF, while one calculated by first

generating a tilted LSF by taking the derivative of each line and then extracting the super-resolved LSF will be referred

to as a pre-derived LSF. In all of the examples below, the regions of interest used to extract the LSFs and ESFs were

judiciously chosen such that additional apodization was not necessary.

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Figure 1: The concept of phase filtering. Consider a tilted slit as shown above. The plots

follow six of the individual lines in the image. The first shows the slit without the vertical pattern.

The second shows the slit with the pattern, the dotted lines follow some of the features in the

vertical pattern. The third plot shows the lines after they have been phase shifted so the slit peaks

line up. Note that the dotted lines which track the pattern are tilted at an angle equal and opposite

to the original angle of the slit.

As mentioned previously, removing horizontal patterns can be accomplished by taking the derivative of each line in an

ESF. Dealing with vertical patterns requires more work and is presented below. The Fourier transform convention used

is the following:

Forward transform:

dxexhfH ifx

x ∫∞

∞−

−= π2)()( , (1)

Inverse transform:

. x

ifx

x dfefHxh ∫∞

∞−

= π2)()( , (2)

Define ),( yxh as the tilted LSF response of an optical system. Define ),( yxg as a fixed pattern in the detector array

used to measure the LSF. The as-measured frame from the sensor would then be:

),(),( yxgyxh + . (3)

Consider an array with only a vertical fixed pattern on the output (such as an array with variability in its column

amplifiers). In such a case, ),( yxg would be constant in the y direction. Assuming the optical system under test is

well behaved in the vicinity of where the LSF is being measured, the LSF should be the same pattern just shifted along

the edge or slit being used to measure it. So, define a new variable )tan(' θyxx += that follows the angle θ of the

edge or slit. In such a case, the Fourier transform of the LSF in 'x coordinates without the added detector pattern should

be constant in the y direction, or

)(),( '' xx fHyfH = . (4)

The FT of the fixed pattern in x is also a constant. Shifting to 'x coordinates can be accomplished by adding a phase to

the FT:

))(,(),( ' yfGyfG xxx φ= , (5)

where

θπ

φ tan2

)( ''

yfy x

x = . (6)

In the ideal case that the only remaining errors are due to the fixed pattern, any vertical variation in the frequency data

in the vertical direction must be due to that fixed pattern and have the above mentioned relationship between phase and

position. Performing a second FT in the vertical direction yields:

),()( '' yxx fff GGGGHHHH + . (7)

Because ),( ' yfG x can be expressed as a phase variation that is linear in y as a function of fx (as mentioned above),

performing a subsequent FT in the y direction places all the variation in ),( ' yx ffGGGG on a line

θtanG 'xy ff = , (8)

where G is the magnitude of the contribution due to the pattern. All that remains to do is to remove the components

introduced by the fixed pattern and then perform two inverse transforms to revert back to a filtered version of the

original LSF. In practice, real data is not quite so nice and the LSFs along the slit or edge are not exactly the same due

to sampling, other errors and optical variations. Because of these variations, the LSF curves along the slit or edge are

not exactly the same and have some contribution along the line to be filtered. Typically these contributions are well

behaved and relatively small in systems where the fixed pattern is the dominant error. However, removal of all of the

spectral contribution along the line to be filtered can introduce unwanted artifacts. To avoid this, the components along

the line are replaced by interpolation using the data on either side of the line. The one point where this fails is at zero

frequency, which by definition does not change. Again, in practice, things differ from the ideal. In this case, as the

frequency approaches zero, the filtering algorithm becomes less effective. This result is not unexpected as the filter

needs on the order of a full cycle to be effective and since the variation is frequency dependent, more samples must be

collected along the slit or edge in order to filter effectively at low frequencies.

2.4 Phase Filtering algorithm

The algorithm steps are as follows:

1) Collect an ESF image.

2) Take derivative across the edge.

3) Fit the peak of each line with a parabolic curve to get better estimate of where the edge is centered on each line.

4) Fit the centers to a line to get the best estimate of line position and calculate the edge angle.

5) FFT each line to get individual OTFs

6) Phase shift each line such that all phases are zero at the edge position for that line.

7) FFT the resulting columns to show variations as the edge is progressed.

8) Convert to amplitude and phase and filter the out of phase components.

9) Convert back to real and imaginary coordinates and inverse FFT back to individual OTFs

10) Phase shift each line back

11) Inverse FFT back to LSFs

12) Derive the SRLSF

13) Calculate MTF (using phase correction algorithms discussed in the previous publication)

14) Correct resulting MTF for additional MTF incurred through the ESF derivative.

3. RESULTS AND DISCUSSION

The algorithms described above have been implemented into IRWindows4, the latest version of SBIR’s automated

Electro-Optical test software. They were then used to process a series of images in order to demonstrate how the

algorithm works and its effectiveness on artifacts due to horizontal, vertical and random patterns. In order to have a

known reference, one of the real systems has been degraded with synthetic patterns in order to have a known result as a

comparison.

3.1 Vertical Pattern Example

In order to better demonstrate the phase filtering portion of the algorithm the following example is given: Consider a

tilted synthetic LSF of Gaussian profile. Column artifacts are added with a mean amplitude of approximately 5% of the

peak amplitude of the LSF. Figure 2 follows the example LSF through the algorithm. Image (a) is the tilted LSF used.

Images (b) and (c) are the real and imaginary parts after the phase shift to align each LST and then the horizontal FFT of

each line to yield the OTF. Image (d) is the result of the vertical FFT of the OTFs. The narrow lines at the same angle

as the “slit” are the out of phase contributions to be filtered. The amplitude and phase are replaced with interpolated

values based on uncontaminated nearby pixels. Image (e) shows the result of the filtering. Figure 3 shows profiles of a

single line near the center of the image before and after is has been processed through the phase filtering algorithm.

a b c d e

Figure 2 Tracking the example through the algorithm. Image (a) is at step 2. Images (b) and

(c) are after step 6. Image (d) is after step 7 and Image (e) is after step 8.

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Figure 3 Result of phase filtering in the Vertical Pattern Example. The plots are profiles of a

horizontal line through the center of the slit in the vertical pattern example. The filtered line

shows how the algorithm is effective in removing high frequency artifacts, but is less effective at

removing those of lower frequency.

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Figure 4 Vertical Pattern Example MTF. These plots show the results of processing the

vertical pattern example image with various algorithms. The combination of phase filtering and

phase correction effectively removes all the artifacts at higher frequencies, while none are

particularly effective at removing the low frequency artifacts.

Figure 4 shows the resulting MTFs as calculated with variations on the processing to show the effects of each. The

partial edge line is based on using the minimum of 7 lines necessary to move one sample position over in order to

generate the super-resolved LSF. The unfiltered curve takes all possible SRLSFs from the image and averages them.

Note the improvement at low frequencies, but none at high frequencies where the curve is dominated by the pattern that

is present in each line. The pure line is the reference curve. Note that all other curves have significant and comparable

deviations from the pure cure at low frequencies. Coherent variations through the columns of the LSF are very difficult

ot remove without the addition of other artifacts. The filtered power spectrum (PS) and phase-corrected (PC) lines show

the results after phase filtering followed by either a power spectrum or phase correction, the latter of which makes

further improvements on the resulting MTF.

3.2 System 1

System 1 is a 2nd generation cooled InSb mid-wave staring array. It has a ratio of sigma vh to sigma tvh of 0.2, and it

had a signal to noise ratio of over 1200 to 1 in the collection of the edge response function. The edge response was

collected from a 14-bit digital port from the sensor. The image used along with a horizontal profile though the target is

shown in Figure 5. A comparison of the results from the standard SRMTF and one generated with the phase filtering

algorithm is shown in Figure 6. Because the signal-to-noise ratio (SNR) in the measurement was so high, the phase

filtering algorithm will not show a significant improvement over the standard super-resolved edge method. In fact, most

of the differences in Figure 6 are due to phase correction. This system can be used to demonstrate the effectiveness of

the algorithms described in this paper. In order to show how phase filtering affects various types of “noise”, the original

high quality image was degraded by the addition of horizontal, vertical and random noise. Each is addressed separately

below.

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Figure 5 Target used for System 1. The image of a half moon target above was used to derive

the MTF in System 1. The plot to the right is a horizontal profile through the center of the target.

Figure 7 shows the results of processing through a system with horizontal “noise” added. Note the similarities in all of

the curves. Horizontal noise tends to cycle with a period equal to moving one sample across as you traverse up the edge.

This creates a peak at the sample spacing, as seen in many of the MTF curves from real systems.

Figure 8 shows the improvement gained by using phase filtering on vertical noise. Note that despite the poor MTF

derived from the SRESF first method, the original MTF is recovered with the phase filtering and correction algorithms.

Phase correction is quite useful at removing random noise, though it requires more care and has the potential for

introducing more artifacts if its parameters are chosen poorly. The phase filtering algorithm is not very effective against

random noise, but it is effective against pattern noise and requires little or no adjustment in normal use.

Figure 9 shows the results from having all three types of noise added to a system in equal amplitudes. The filtering and

correction algorithms are very effective in improving the MTF and allowing the retrieval high quality results despite the

additional noise.

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Figure 6 UnDegraded MTF from System 1. The plots in this figure show the similarity

between the pre and post derivative methods as used on System 1.

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Figure 7 System1 Horizotal. These plots show the results of processing the System 1 image

through the phase filtering algorithm after the addition of a horizontal pattern.

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Figure 8 System 1, Vertical. These plots show the result of processing the System 1 image after

the addition of a vertical pattern. Note how well the filtering algorithm recovers the initial MTF

while the standard ESF method has significant artifacts.

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Figure 9 Horizontal, Vertical and Random. The plots above show the results of processing the

System 1 image after the addition of equally weighted horizontal, vertical and random patterns.

Again, phase filtering and phase correction are effective in recovering the original MTF.

3.3 System 2

System 2 is a 2nd generation scanning sampled photovoltaic array. Because it is a scanning array, its noise is dominated

by uncorrelated random-spatial-temporal (tvh) noise. It had an SNR of 101.3 to 1 in its collection of the edge response

function. The edge response was collected from an 8-bit RS-170 port from the sensor. The results of the collecting the

ESF first and processing through phase correction and filtering are shown in Figure 11. Although the noise is mostly

random, there is some horizontal patterning evident in the image, but little or no evidence of vertical patterns. Because

the noise is largely due to random and horizontal contributions, there is no improvement from the phase filtering,

although taking the LSF first does offer a slight improvement at higher frequencies. However, applying phase correction

does improve the high frequency MTF and remove the noise rectification.

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Figure 10 System 2 Image and Profile. The image of a half moon target above was used to

derive the MTF of System 2. The plot on the right is a horizontal profile through the center of the

target image.

3.4 Discussion

The above examples show the potential usefulness of the phase filtering algorithm in combination with phase correction.

The two methods act in complimentary ways to help reduce artifacts introduced into MTF measurements due to non-

ideal experimental data. The phase filtering algorithm may be considered indirect in concept and is certainly tedious in

its application. However, with careful attention to detail, a useful algorithm can be developed and added to a repertoire

of tools to be used to improve experimental data.

This tool gives EO system measurers a means for calculating an MTF measurement in the presence of correlated noise.

It is possible to get an approximation of the pre-sample MTF performance by either assuming a behavior of the detector

and optics or by turning the system gain to as low of a value as possible while imaging an edge target. These

assumptions are not perfect, and in the latter case, many systems will not allow for the imagery to be gained at all. The

ability to directly calculate an MTF without noise interference is a huge help for determining separable (resolution

components and noise components) system performance.

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Figure 11 System 2 MTF. The plots above show the results of processing the data from System

2 using phase filtering and phase correction. The phase filtering has a modest effect, primarily due

to the removal of a modest contribution due to a weak horizontal pattern. The further application

of phase correction finally removes all of the rectification.

4. SUMMARY

Phase information should not be discarded as it often is in many applications of the Fourier transform. Instead, careful

application of phase information and known geometry can result in significant improvements in MTF measurements.

The phase filtering method presented above in conjunction with phase correction were shown to be very effective in

removing fixed pattern and random noise in tilted edge or tilted slit MTF measurements. Thermal modeling software

like NVTherm IP is quite sensitive to the performance of the pre-sample MTF, so it is useful to have a calculation of the

measurement that has as high of fidelity as possible.

5. ACKNOWLEDGEMENTS

The authors would like to thank David Tanner, Kevin Schehrer and Alan Irwin for insightful conversations.

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