SIMG-503

Senior Research

QQuuaalliittyy ooff SSiiggnnaall DDeetteeccttiioonn UUssiinngg aann

AApppprrooxxiimmaattee MMaattcchheedd FFiilltteerr

Final Report

Derek J. Walvoord

Chester F. Carlson Center for Imaging Science

Rochester Institute of Technology

May 2002

2

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

3

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.

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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.

5

- 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.

6

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):

( ) ( ) ( )xnxxfxg +−= 0 (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:

( ) ( ) ( ) ( )[ ] ( )xmxnxxfxmxg ** 0 +−= (2)

The frequency-domain representation of the ideal matched filter M(ξ) would generate a

linear phase factor:

[ ] [ ] [ ] [ ] [ ] [ ]ξξξξξξ ξπ MNMeFMG xi ⋅+⋅⋅=⋅ − 02

[ ] [ ]ξξξπ 0102 +⋅= − xie

(3)

Two conditions must be satisfied at all frequencies to generate this ideal output:

[ ] [ ] [ ]ξξξ 1=⋅MF (4)

[ ] [ ] [ ]ξξξ 0=⋅MN (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:

[ ] [ ]ξξ

FM 1=

(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

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correlation. The product of G(ξ) with the complex conjugate of the Fourier transform of

the known signal F*(ξ) (Equation 8).

( )xfxm −= ∗)( (7)

[ ] [ ]ξξ ∗= FM (8)

The �realistic� matched usually produces a �correlation peak� with a better signal-to-

noise 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.

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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

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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

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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.

�Ideal�

Matched Filter

�Realistic�

Matched Filter

Approximate (1st order)

Matched Filter

Approximate (1000th order)

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.

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Matched Filter Falloff Rates for Added Gaussian Noise

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Gaussian Noise (σ)

Qua

lity

Met

ricApproximate (order=1)

"Ideal"

"Realistic"

Figure 4 � Quality Metric vs. Gaussian Noise for each Match Filter

Order of Approximation Falloff Rates for Noise

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Gaussian Noise (σ)

Qua

lity

Met

ric

Approximate (order=1)

Approximate (order=3)

Approximate (order=100)

Figure 5 � Quality Metric vs. Gaussian Noise for Different Orders of the Approximate Matched Filter

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Matched Filter Falloff Rates for Degree of Rotation

0

20

40

60

80

100

120

140

160

180

200

0

0.4

0.8

1.2

1.6 2

2.4

2.8

3.2

3.6 4

4.4

4.8

Degree of Rotation

Qua

lity

Met

ricApproximate (order=1)

"Ideal"

"Realistic"

Figure 6 � Quality Metric vs. Degree of Rotation for each Matched Filter

Approximation Falloff Rates for Rotation

0

10

20

30

40

50

60

70

80

90

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8

Degree of Rotation

Qua

lity

Met

ric

Approximate (order=1)

Approximate (order=3)

Approximate(order=100)

Figure 7 � Quality Metric vs. Degree of Rotation for Different Orders of the Approximate Matched Filter

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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.

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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)

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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 low-

frequency 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 first-

order 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.

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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: Addison-

Wesley Publishing Company, 1993.

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Appendix

Derivation of the Approximate Matched Filter The "ideal" matched filter can be expressed as:

[ ] [ ] [ ][ ]{ }ηξ

ηξηξηξ ,

,1

,1, Fie

FFM Φ−⋅==

The magnitude of the reference is rewritten as a sum of terms:

[ ] [ ] maxmax,, FFFF +−= ηξηξ

[ ] [ ] [ ]{ }ηξηξηξ ,,, FieFF Φ⋅=

[ ] [ ][ ] [ ]{ }ηξηξηξ ,

maxmax,, FieFFFF Φ⋅+−= Now in a more useful representation:

[ ] [ ] [ ]{ }ηξηξηξ ,

max

maxmax

,1, Fie

FFF

FF Φ⋅

−−⋅=

This allows us to write the "ideal" matched filter as:

[ ] [ ][ ]{ } [ ] 1

max

max,

max

,11

,1,

−−

−−⋅⋅==

FFF

eFF

M Fi ηξηξ

ηξ ηξΦ

A Taylor series expansion can be used to approximate the last term of the above equation.

The Taylor series expansion for ( )t

tf−

=1

1 when | t | < 1 is as follows:

( ) ∑∞

==

−=

011

n

ntt

tf

This result yields an expression for the transfer function of an approximate matched filter:

[ ][ ]{ } [ ] n

n

Fi

FFF

FeM ∑

∞

=

−

−⋅=

0 max

max

max

, ,,

ηξηξ

ηξΦ

18

The expansion may be truncated at any order. For example, truncating the expanded term at the first order:

[ ][ ]{ } ( )

−+⋅=

−

max

max

max

,

1,

1,FFF

FeM

Fi ηξηξ

ηξΦ

[ ]{ } ( )

−+⋅=

−

max

maxmax

max

, ,F

FFFF

e Fi ηξηξΦ

[ ]{ } ( )

−⋅⋅= −

2max

max, ,2

F

FFe Fi ηξηξΦ

After normalization, the first-order filter in the frequency domain is:

[ ] [ ]{ } [ ]( )ηξηξ ηξ ,2, ,1 FeM Fi −⋅= − Φ