SIMG-503
Senior Research
Quality of Signal Detection Using an
Approximate Matched Filter
Final Report
Derek J. Walvoord
Chester F. Carlson Center for Imaging Science
Rochester Institute of Technology
May 2002
Table of Contents
Abstract
3
Copyright
4
Acknowledgements
5
Background and Theory
6
Methods
8
Results
Comparison of Matched Filter Attibutes
10
Matched Filtering on the Archimedes
13
Discussion
15
Conclusion
16
References
16
Appendix
17
2
Abstract
Spatial matched filtering is an imaging task that seeks to detect the occurrences
and determine the locations of copies of a known signal on an unknown background.
The aim of this research is to determine the quality of results obtained using an
approximate form of the “ideal” matched filter. Synthetic data was used to test the quality
of signal detection with the new filter as a function of filter order for both noise content
and degree of rotation. These experiments were replicated using two classical matched
filter designs to provide a standard for comparison. Results in this procedure have
demonstrated that the approximate matched filter is less sensitive to noise than the
“ideal” matched filter, while preserving the narrow correlation peak. Lastly, the three
filters were tested on images of the Archimedes Palimpsest to determine the efficiency
of the approximate matched filter on real data.
3
Copyright ©2002 Chester F. Carlson Center for Imaging Science
Rochester Institute of Technology
Rochester, NY 14623-5604
This work is copyrighted and may not be reproduced in whole or part without permission
of the Chester F. Carlson Center for Imaging Science at the Rochester Institute of
Technology.
This report is accepted in partial fulfillment of the requirements of the course SIMG-503
Senior Research.
Title: Quality
of Signal Detection Using an Approximate Matched Filter
Author: Derek J. Walvoord
Project Advisor: Roger L. Easton, Ph.D.
SIMG-503 Instructor: Anthony Vodacek, Ph.D.
4
- Acknowledgements I would like to thank Dr. Roger L. Easton for being an outstanding advisor to this
research project, and the Archimedes Palimpsest Imaging Team for use of many images
from the Archimedes Palimpsest. Also, thank you to Keith Knox for his shared
perspective on this work.
5
Background and Theory
Spatial matched filtering is a method for detecting the occurrences and locations
of a known signal on an unknown background. The scheme actuates an “alarm” of some
sort whenever the known signal f(x) is detected in the unknown background. The
measured signal contains the known signal f(x) at an unknown location x0 and (usually)
uncorrelated noise n(x):
g ( x ) = f ( x − x 0 ) + n( x )
(1)
In the space domain, the ideal output of a matched filter m(x) is a Dirac delta
function at the location of the target:
g (x ) * m(x ) = [ f ( x − x0 ) + n( x )]* m(x )
(2)
The frequency-domain representation of the ideal matched filter M(ξ) would generate a
linear phase factor:
G[ξ ] ⋅ M [ξ ] = F [ξ ] ⋅ e −2πiξx 0 ⋅ M [ξ ] + N [ξ ] ⋅ M [ξ ]
= e −2πiξx0 ⋅ 1[ξ ] + 0[ξ ]
(3)
Two conditions must be satisfied at all frequencies to generate this ideal output:
F [ξ ] ⋅ M [ξ ] = 1[ξ ]
(4)
N [ξ ] ⋅ M [ξ ] = 0[ξ ]
(5)
The first criterion requires that F(ξ) ≠ 0 at all ξ. Therefore, the only possible noise
spectrum is N(ξ) = 0. If Equations 4 and 5 are true, the transfer function of the ideal
matched filter is the reciprocal of the spectrum of the known signal:
M [ξ ] =
1
F [ξ ]
(6)
The denominator in the “ideal” matched filter transfer function M(ξ) would amplify any
wideband noise in the measured signal g(x) at spatial frequencies where 0<|F(ξ)|<1. This
amplified noise would be “spread” throughout the space domain via the inverse Fourier
transform. The output generated by the ideal matched filter is a Dirac delta function
centered at the desired coordinate x = x0 with additional non-zero amplitudes at other
coordinates. Because this “ideal” matched filter is very sensitive to noise, it is more often
used as a condition for construction useful filters rather than as a practical tool.
If noise is not present, the “realistic” matched filter produces the convolution of
g(x) with f*(-x) in the space domain (Equation 7). This is commonly known as cross
6
correlation. The product of G(ξ) with the complex conjugate of the Fourier transform of
the known signal F*(ξ) (Equation 8).
m( x) = f ∗ (− x )
(7)
M [ξ ] = F ∗ [ξ ]
(8)
The “realistic” matched usually produces a “correlation peak” with a better signal-tonoise ratio power ratio than the ideal matched filter if no noise is present, but the peak is
also broader. However, the correlation function is constrained by its sensitivity to
changes in g(x) and f(x). Also, the area of correlation peak is the maximum amplitude of
the output and twice the width of the finite support of f(x) is the support of the correlation
peak.
An approximate matched filter (see Appendix for derivation) can be derived using
a Taylor-series expansion to approximate the denominator of the ideal matched filter in
Equation 6. The rationale for the approximation of the denominator is to decrease the
amplification of noise in the filtering process while providing a narrower correlation peak
than the output of the “realistic” matched filter. Furthermore, the approximation using a
Taylor-series expansion yields a transfer function for the filter that may be truncated at
any order.
7
Methods
As stated previously, the primary goal of this research is to assess the benefit of
the new process compared to classical matched filtering. Synthetic 128x128 pixel
images were generated to use as targets and backgrounds. The unaltered target image
contains a white λ character centered on a 50% gray level background. Figure 1 shows
the target image and examples of the backgrounds used in the filtering process.
Target (Reference Image)
Added Gaussian Noise (σ = 50)
Rotated Character (4º)
Figure 1 – Examples of the Synthetic Images Generated for Matched Filtering
These images were used to compare the performance of the approximate
matched filter at different orders to the “ideal” and “realistic” matched filters. A metric for
the “quality” of signal detection is needed to compare the output correlation peaks. In
this experiment, “quality” is defined as the ratio of the maximum amplitude of the
correlation peak to the magnitude of the full width at half maximum calculated in the
vertical and horizontal direction.
Performance vs. Additive Gaussian Noise
The initial step in characterizing the approximate matched filter was
determination of “quality” for different levels of Gaussian noise. Programs were created
in IDLTM to perform the necessary filtering processes and quality calculations. This
procedure begins by adding a selected amount of Gaussian noise to the background
image. Fourier-domain matched filtering is then performed with each of the matched
filters and for selected orders of the approximate matched fitler. The output correlation
images become the input to another program that in turn, determines peak maxima and
calculates “quality” statistics.
Performance vs. Degree of Rotation
A similar process was then performed to begin characterization of the
approximate matched filter for the degree of rotation. The procedure differs in that a set
8
of background images was generated in PhotoshopTM by rotating the λ character at 0.2º
intervals.
Performance on Real Data
After assessing the performance on synthetic images, the filters were tested on
real images of characters from the Archimedes Palimpsest, an overwritten manuscript
that contains multiple inks and mold. The text has degraded significantly due to age, and
the resulting images are noisy. Characters from the overwriting (the Euchologion text)
were successfully matched using each filter to observe the difference in output
correlation peaks from real data.
The underwriting (the Archimedes text) is more difficult to match and therefore,
image processing techniques were used a priori to partially remove the parchment and
overwriting. Results from this technique, known as push-button processing are shown in
Figure 2.
Figure 2 - Push-button Processing and Brightness/Contrast Adjustments to Enhance Underwriting
9
The red channel of a tungsten illuminated image and the green channel of an ultraviolet
illuminated image were balanced and subtracted to create the processed image that was
enhanced further by adjusting its brightness and contrast. Median filtering of both
target and background images aided the matching of character edges as opposed to the
noise pattern of the target.
Results
Comparison of Matched Filter Attributes
Shown below in Figure 3 are surface plots of a few output correlation peaks, with
and without Gaussian noise added to the background image before the filtering process.
Note the difference in noise sensitivity between the “ideal” and “realistic” matched filters.
st
th
“Ideal”
“Realistic”
Approximate (1 order)
Approximate (1000 order)
Matched Filter
Matched Filter
Matched Filter
Matched Filter
Background signal with no added Gaussian noise (σ = 0)
Background signal with added Gaussian noise (σ = 50)
Figure 3 – Examples of Output Correlation Peaks With Each Matched Filter
Figures 4 through Figure 7 show the falloff rates for the two classical matched
filters and the approximate matched at different orders for added Gaussian noise and
degree of rotation. These results were determined using the images in Figure 1 (and
other backgrounds at different levels of noise and rotation) and the quality metric defined
for this experiment.
10
Matched Filter Falloff Rates for Added Gaussian Noise
Quality Metric
200
180
Approximate (order=1)
160
"Ideal"
140
"Realistic"
120
100
80
60
40
20
90
95
10
0
80
85
70
75
60
65
50
55
40
45
30
35
20
25
5
10
15
0
0
Gaussian Noise (σ)
Figure 4 – Quality Metric vs. Gaussian Noise for each Match Filter
Order of Approximation Falloff Rates for Noise
90
Approximate (order=1)
80
Approximate (order=3)
Quality Metric
70
Approximate (order=100)
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100 110 120 130 140 150
Gaussian Noise (σ)
Figure 5 – Quality Metric vs. Gaussian Noise for Different Orders of the Approximate Matched Filter
11
Matched Filter Falloff Rates for Degree of Rotation
Quality Metric
200
180
Approximate (order=1)
160
"Ideal"
140
"Realistic"
120
100
80
60
40
20
4.8
4.4
4
3.6
3.2
2.8
2.4
2
1.6
1.2
0.8
0.4
0
0
Degree of Rotation
Figure 6 – Quality Metric vs. Degree of Rotation for each Matched Filter
Approximation Falloff Rates for Rotation
Approximate (order=1)
90
80
Approximate (order=3)
Quality Metric
70
Approximate
(order=100)
60
50
40
30
20
10
4.
8
4.
4
4
3.
6
3.
2
2.
8
2.
4
2
1.
6
1.
2
0.
8
0.
4
0
0
Degree of Rotation
Figure 7 – Quality Metric vs. Degree of Rotation for Different Orders of the Approximate Matched Filter
12
Matched Filtering on the Archimedes Palimpsest
The results of correlation obtained from each filter when matching a “real”
character are shown in Figure 8. An extracted τ character was used as the “reference”
for a page of text taken from the red channel of the image under tungsten illumination.
Figure 8 - Output Correlation Peaks Using Real Data for Each Matched Filter
Effect of Median Filtering
Median filtering is a technique for reducing the effect of blurring on an image. The
median of the gray values in a defined neighborhood replaces the gray value of the
center pixel. This “sharpens” the edges and improves the quality of the correlation peak.
The use of median filtering on the push-button target and background reduced
the number of false alarms in filtering with the approximate matched filter. Figure 9
shows this reduction in false alarms (shown in pink) while maintaining the same number
of correct detections (shown in green) in matching a τ character to Archimedes text.
13
Before Median Filtering
After Median Filtering
Push-button processed background of underwriting (Archimedes text)
False alarms (pink) and correct detections (green) after matched filtering
Figure 9 – False Alarms Before and After Median Filtering (1st Order Approximate Matched Filter)
14
Discussion
This project successfully demonstrated the benefits of the approximate matched
filter over classical matched filtering algorithms in some respect. Conversely, these
results are characteristic of not only the matched filter attributes, but of the synthetic
images and the quality metric used in this research. For example, the falloff rates for the
different orders of the approximate matched filter in Figure 5 fluctuate through the overall
decline. This is most likely due to the rising noise level and the particular definition of the
quality metric. The synthetic images used in this research were dominated by lowfrequency components. This resulted in slower falloff rates than would occur for images
where high-frequency components dominate. With the low-frequency images and the
quality metric, results show that the correlation peak obtained using the first-order
approximate matched filter exhibits a better quality than the “realistic” matched filter for
Gaussian noise with σ<150 (Figure 4). Similarly, the “quality” of output peak of the firstorder approximate matched filter was as good or better than that of the “realistic”
matched filter for rotation angles ≤5°. In both tests, the falloff rate was slower than the
“ideal” matched filter and higher orders of approximation showed little effect (Figures 5 &
7). Finally, a possible improvement of the approximation of the “ideal” matched filter
would account for correlated noise in the background image (such as “pink” or “blue”
noise).
When applied to “real” data, the initial matching of overwritten characters when
testing the filters. The “ideal” matched filter heavily amplifies the noise in the image and
thus hides the correlation peak (Figure 8). The first-order approximate matched filter
produced the only usable results when matching characters to the underwriting of the
Archimedes Palimpsest. The “ideal” and “realistic” matched filters produced results
similar to those on overwritten characters (Figure 8), except that the wide correlation
peaks of the “realistic” matched filter nearly overlapped. Median filtering the target and
background image before implementing the first-order approximate matched filter proved
useful, as it significantly reduced the number of false alarms (Figure 9). The noise level
was reduced and therefore, was amplified less during the filtering process.
15
Conclusion
The performance of the approximate matched filter when compared against two
classical matched filter designs yielded results that suggest it will be useful in future
applications. Further characterization should include targets dominated by different
frequency ranges, as well as establishment of an improved measure of “quality” that
accounts for the rising noise floor. Finally, the approximate matched filter could be
improved to account for correlated noise.
References
Castleman, Kenneth R. Digital Image Processing. New Jersey: Prentice-Hall, Inc., 1979.
Easton, Roger L. (Course Notes)
Gaskill, Jack D. Linear Systems, Fourier Transforms, and Optics. New York: John Wiley
& Sons, Inc., 1978.
Gonzalez, R.C., Woods, R.E. Digital Image Processing. New York: AddisonWesley Publishing Company, 1993.
16
Appendix
Derivation of the Approximate Matched Filter
The "ideal" matched filter can be expressed as:
M [ξ ,η ] =
1
1
=
⋅ e − iΦ{F [ξ ,η ]}
F [ξ ,η ] F [ξ ,η ]
The magnitude of the reference is rewritten as a sum of terms:
F [ξ ,η ] = F [ξ ,η ] − Fmax + Fmax
F [ξ ,η ] = F [ξ ,η ] ⋅ e iΦ{F [ξ ,η ]}
F [ξ ,η ] = [ F [ξ ,η ] − Fmax + Fmax ]⋅ e iΦ{F [ξ ,η ]}
Now in a more useful representation:
 F
− F [ξ ,η ]   iΦ{F [ξ ,η ]}
 ⋅ e
F [ξ ,η ] = Fmax ⋅ 1 −  max

F
max
 
 
This allows us to write the "ideal" matched filter as:
 F
− F [ξ ,η ]  
1
1

M [ξ ,η ] =
=
⋅ e − iΦ{F [ξ ,η ]} ⋅ 1 −  max

F [ξ ,η ] Fmax
F
max
 
 
−1
A Taylor series expansion can be used to approximate the last term of the above
equation.
The Taylor series expansion for f (t ) =
f (t ) =
∞
1
when | t | < 1 is as follows:
1− t
1
=
tn
1 − t n=0
∑
This result yields an expression for the transfer function of an approximate matched
filter:
∞ F
e − iΦ{F [ξ ,η ]}
max − F [ξ ,η ] 
⋅
M [ξ ,η ] =


Fmax
Fmax

n = 0
∑
n
17
The expansion may be truncated at any order. For example, truncating the expanded
term at the first order:
M [ξ ,η ]1 =
=
F
− F (ξ ,η ) 
e − iΦ{F [ξ ,η ]} 
⋅ 1 + max

Fmax
Fmax


e − iΦ{F [ξ ,η ]}  Fmax + Fmax − F (ξ ,η ) 
⋅

Fmax
Fmax



2⋅ F
max − F (ξ ,η ) 
= e − iΦ{F [ξ ,η ]} ⋅ 
2


Fmax


After normalization, the first-order filter in the frequency domain is:
M [ξ ,η ]1 = e − iΦ{F [ξ ,η ]} ⋅ (2 − F [ξ ,η ] )
18