Modified filter synthetic discriminant functions for improved optical correlator performance

Modified filter synthetic discriminant functionsfor improved optical correlator performance

R. K. Wang, C. R. Chatwin, and M. Y. Huang

By the filter modulation operator ., the modified filter synthetic discriminant function permits

advantageous preprocessing of individual training-set images that are used in a linear combination to

construct the filter synthetic discriminant function, which applies a modulation operator A/ to the

synthetic discriminant function. A relaxation algorithm is used to satisfy the equal correlation peaks

rule in the correlator output plane. As the filter modulation operators A/ and A' can be given any

functional form, the modified filter synthetic discriminant function design proposed is sufficiently general

to be described as a unified filter modulation synthetic discriminant function design.

Key words: Optical correlation, synthetic discriminant function, binary phase-only filter, unified filter.

Introduction

Optical systems, using holographic matched spatialfilters,' are able to recognize multiple occurrences ofan object in the presence of noise. However, imagedistortions due to scale mismatch, in-plane rotation,out-of-plane rotation, etc., restrict effective applica-tions. The phase-only filter2 (POF) and the binaryphase-only filter3 (BPOF) are particularly sensitive inthis regard.4 In general, for recognition of three-dimensional objects from two-dimensional projec-tions, it is necessary for a correlator to be invariant toall distortions while still being able to maintain gooddiscrimination between similar objects. These areconflicting requirements that we can address byexploiting synthetic discriminant functions5 (SDF's).

Conventional SDF's6 produce very broad outputcorrelation peaks and as a result do not achieve a100% success rate in discriminating one class ofimages from another. To solve this problem, Hornerand Gianino7 introduced the phase-only SDF filters;these filters produce very sharp correlation peaks butdo not satisfy the equal correlation peaks (ECP) rulein the output plane.8 Jared910 recently proposed aniterative relaxation algorithm, which enables thedesign of phase-only SDF's and binary phase-onlySDF's to satisfy the ECP rule. Their idea was toensure that the filter-encoding constraints are taken

The authors are with the Laser and Optical Systems EngineeringGroup, Department of Mechanical Engineering, University of

Glasgow, Glasgow G12 8QQ, UK.Received 2 March 1994; revised manuscript received 2 May 1994.

0003-6935/94/327646-09$06.00/0.3 1994 Optical Society of America.

into account at the design stage; this was called thefilter SDF (fSDF). In contrast to fully complexclassical matched filters and conventional SDF's,fSDF filters can be implemented on commerciallyavailable spatial light modulators (SLM's) for use inan optical correlator and are therefore of particularinterest for real-time optical pattern-recognition sys-tems employing rapidly updateable SLM's.

Young and Chatwin 1 pointed out that, for commonobjects, the concentration of most of the energy in thecentral zone of the spectrum, i.e., low-frequencycomponents, is responsible for the poor correlationperformance. However, the fSDF implements thefilter-modulation constraints (i.e., the BPOF) on theconventional SDF, which is a linear combination ofthe training-set images. There is no doubt that theconventional SDF is dominated by the lower-fre-quency components of individual training-set images.Thus this degrades the correlation performance evenwhen the filter-modulation constraints are imple-mented on the composite image.

To improve correlation performance, we give fur-ther consideration to the filter-encoding constraintapplied to the SDF construction. The idea is tosynthesize the SDF from the linear combination of aset of training images that are already filter modu-lated, i.e., preprocessed, so that the constructed SDFis dominated by the higher, not the lower, frequencycomponents of the individual training-set image.The filter-encoding constraint is then applied to theSDF. For convenience this is called the modifiedfSDF (MfSDF).

An initial investigation to compare the perfor-mance of the MfSDF filter with that of the fSDF filter

7646 APPLIED OPTICS / Vol. 33, No. 32 / 10 November 1994

with modulators X and XK set to be a binary phase-only filter (BPOF) and a phase-only filter (POF),respectively, shows that the BPOF/MfSDF canachieve distortion invariance to in-plane object rota-tions of up to 650, whereas the BPOF/fSDF filterattains only 45°. Furthermore, the MfSDF givesbetter discrimination between target and nontargetobjects and delivers a better signal-to-noise ratio(SNR). With the filter modulator X/ set to be aBPOF, the MfSDF can be implemented for real-timeoptical correlation with commercially available bi-nary phase-only spatial light modulators; the modula-tor XK was selected to be a POF to permit an initialinvestigation of the MfSDF design, but it is not theoptimal choice. The operatorff may also be altered;however, for optical implementations this is con-strained by the development of commercially avail-able spatial light modulator technology.

Inclusion of Filter Modulation for Synthetic DiscriminantFunction Design

Background

The fSDF technique9 1012 begins with a set of centeredtraining images, t(x, y), n = 0, 1, . . . , k, spanning thedesired distortion-invariant feature range. This im-age set is used to construct the fSDF, s'(x, y), for agiven filter modulation, XK. The desired peak-correla-tion response of s'(x, y) is a constant, Cn, for eachtraining image t(x, y):

f| tn(x, y)s'*(x, y)dxdy = t(x,y) s'(x, y) = n, (1)

where the integral is taken over the area of the inputfield. The function s'(x, y) includes the filter modula-tion, A, through the equation

s' (x, y) = -- 4[s(x, y)], (2)

where F is the Fourier-transform operator. Thepurpose of the fSDF procedure is to determine thefunction s(x, y) that solves Eq. (1) when given aparticular modulation function, AK. The functions(x, y) is chosen to be a linear combination of thetraining images:

k

s(x, y) = antn(x, y). (3)n=O

A general fSDF synthesis equation results from sub-stituting Eqs. (3) and (2) into Eq. (1):

-k

tn (X, y) [k W antn(X, Y)] = Cn- (4)n=O

For the POF's and BPOF's, Eq. (4) is a system ofnonlinear equations that may be solved with aniterative procedure13 based on the Newton-Raphsonalgorithm. The filter coefficients, a, are constrainedto be real and are initialized to give the desired

response vector, c; the accuracy of the filter coeffi-cients is improved by the use of the iteration formula

i+1 i + - nan = a + cI[n -COkmiJ (5)

where i is the iteration number, a is a dampingconstant, and m' is the modulus of the peak-correla-tion response of image t(x, y) with the filter con-structed with ai.

Further ConsiderationsWhen a high-pass filter modulation of XK is applied tothe composite function s(x, y), the higher-frequencycomponents of s(x, y) are enhanced. Unfortunately,these higher-frequency components are not optimallyrelated to the high frequencies of individual training-set images. As a result, this limits correlation perfor-mance, such as discrimination ability and the correla-tion-peak-to-secondary-peak ratio (PSR), defined byEq. (13). To overcome this problem, we must applythe filter modulation to individual training-set im-ages; thus the function s(x, y) is modified to be

k

s(x, y) = Y an7 l.,f[tn(x, y)j,n=O

(6)

where X denotes filter modulation of individualtraining-set images. The requirements of Eqs. (1)and (2) are still met.

The general modified SDF synthesis equation isnow rewritten as

tn(x, y) 0 I 1 { an,7-'YVY[tn(X, y) = Cn.n=O

(7)

Using the correlation theorem, we can write a simpleform of this equation in the frequency domain as

ff T(u, v) _ anl/[Tn(u, v)]| dudv = n, (8)

where T(u, v) is the Fourier transform of t(x, y).The filter modulation, AK, A, can be specified to take

on any desired form. When both ,' and At are theclassical matched-filter modulation, the MfSDF re-duces to the conventional SDF design. When .V isthe classical matched-filter modulation and , is freeto be defined, the MfSDF reduces to the fSDF design.Thus the MfSDF is a more generalized filter modula-tion SDF, in which the computational search space issufficiently constrained to permit an optimal solutionto be found in a reasonable time. The MfSDF is asubset of the generalized SDF's presented by Bahriand Vijaya Kumar.14

Whether or not the MfSDF produces ECP for allthe training-set images depends on the choice of themodulation operators AK and If the modulationoperator AK is nonlinear, the ECP rule will be broken.For example, if POF or BPOF modulation is imple-mented as the operator, the ECP rule will be

10 November 1994 / Vol. 33, No. 32 / APPLIED OPTICS 7647

corrupted, and the resulting spread in peak valueswill significantly lower the lowest-correlation-peak-to-secondary-peak ratio [worst-case discrimination PSRWgiven by Eq. (13)] for the training-set images. Thisproblem is resolved by use of iterative techniques tofind a coefficient vector a that yields equal correlationpeaks for all the training-set images. The relaxationalgorithm given by Eq. (5) was used with somesuccess herein and by others9 10' 1 215; while there is notheoretical guarantee that this algorithm will con-verge to a solution, it was successful in nearly allcases studied. An alternative successive algorithmis given by Bahri and Vijaya Kumar.1 6

Considerations for Real-time Optical Implementation

Some filter-modulation methods, such as the POF,tuneable edge-enhancement filters,17 and nonlinearfilters,'8 presuppose the use of continuous amplitudeand phase information of the objects in the opticalcorrelator. However, since programmable continu-ous amplitude and phase spatial light modulators arenot currently available and also appear difficult tofabricate, the utility of such correlators is severelylimited. However, programmable SLM's limited tobinary quantization of amplitude and/or phase arecommercially available. Thus the MfSDF can beimplemented for real-time optical correlation withthe filter modulator AK set as BPOF. This still leavesthe modulation operator At free to be defined, and itcan be given any type of advantageous modulation;hence this is a big advantage over the fSDF approach.Therefore it is possible to find a MfSDF with anoptimal choice of the modulation operator A tomaximize the overall performance capability of thefilter, given the limitations of current SLM's. Thisapproach is aimed at hybrid correlation system ar-rangements, as suggested by Young et al.' 9 Theirrecently proposed high-speed hybrid optical/digitalcorrelator system would employ the MfSDF with theAK operator selected to be BPOF and the X operatorfree to be defined.

Simulation and Results

In general one can optimize the individual imagefilter performance for the MfSDF by tuning thebandpass implemented by the modulation operator Aton each image. As this is an extremely large simula-tion task, we decided to choose the POF as themodulation operator A, as the POF has many of thedesirable features required of A. This choice ofoperator gives a good indication of the relative meritsof the MfSDF over the fSDF. Thus the SDF wasconstructed with POF modulation of the individualtraining-set images; hence this gives a higher weight-ing to image high-frequency components. The modu-lation operator AK was specified to be a BPOF, as thismay be implemented in real time on commerciallyavailable SLM's. Thus we constructed the MfSDFfrom the POF-SDF by using the relaxation algorithmto satisfy the ECP rule at the output plane.

In this study, the performance of the filter distor-tion-invariant range is considered. As the extent of

the distortion range increases, the number of trainingimages necessary to cover, effectively, the distortionrange, increases. Generally, the PSR of a filterdecreases as the filter is made increasingly moredistortion invariant. This behavior has a direct im-pact on the design and the applicability of opticalpattern-recognition systems. It is advantageous todesign a distortion-invariant filter with training im-ages spaced as widely as possible, while still maintain-ing an effective correlation-peak-to-secondary-peakratio at the output plane. This ensures successfulsystem performance while minimizing the amount ofredundant information encoded into the SDF's. Fur-thermore, the computational effort expended duringthe design process is directly reduced with fewertraining images.

The training-set images used in the simulationsconsists of in-plane rotated images of the Bradleyarmored personnel carrier (APC) vehicle. Each im-age is centered and normalized to unit energy. TheAPC vehicle was rotated from 00 to 90° in incrementsof 10; the images were encoded with a resolution of128 x 128 pixels. Views of the vehicle at 0°, 300, 600,and 900 are given in Fig. 1. At 00 the Bradley APCvehicle is 90 pixels in length and 40 pixels high.

Filter Construction

Figure 2 shows the filter design procedure of theBPOF/MfSDF constructed from the phase-onlymodulated individual training-set images, in whichten in-plane rotated vehicle training-set images areencoded, i.e., one image every 50, from 0° to 450. Acoefficient vector a, which gives equal correlationpeaks for all training set images, was determined byuse of the iteration algorithm given by Eq. (5).Figure 2(a) shows the composite image S (u, v) in thefrequency domain; it was constructed from the linearcombination of the phase-only modulated training-set images:

k

S(u, v) = E afl-tPOF[T(U, v)]. (9)n=O

From Fig. 2(a), it is obvious that the higher-frequencyinformation is enhanced and the lower-frequencydata is attenuated. Thus the higher frequencies ofindividual training-set images give a greater contribu-tion to the composite image.

In order to produce the binary version of theMfSDF, we apply the BPOF modulation constraint tothe above composite image. In this study, the BPOFwas chosen to be20

(10)Sb(U, V) = + Re[S(u, v) > 0Sb(, ) _ 1 otherwise

The generalized BPOF derived by Flannery et al. 21

includes the angle of a thresholding line in thecomplex plane and the offset from a center referencepoint of the pattern transformed in constructingthe filter, which may be used to choose optimally the


(a)

(b)

(C)

(d)Fig. 1. Examples of the in-plane rotated images of the BradleyAPC vehicle: (a), (b), (c), (d) the vehicle views at 0, 300, 600, and900, respectively.

thresholding line angle of the BPOF in the MfSDFconstruction. Equation (10) was selected for conve-nience; it is only one point of a continuum of thresh-old line angles, i.e., line angle 0 = 0.

According to Eq. (10), the real part of the compositeimage S(u, v) is of interest to produce the BPOF/

(b)

(C)Fig. 2. Filter design procedure of the MfSDF-BPOF made fromthe phase-only training-set images in the frequency domain: (a),(b), (c) the composite image S(u, v), the real part of the compositeimage S(u, v), and the final binary version of the MfSDF, respec-tively.

MfSDF. Figure 2(b) shows the real part of thecomposite image in the frequency domain, which isclearly dominated by the higher frequencies of indi-vidual training-set images. Furthermore, after theiteration to achieve ECP's for all the training-setimages, it can be seen from Figs. 2(a) and 2(b) that thecomposite image S(u, v) is a bandpass-type image.Finally, the BPOF/MfSDF is shown in Fig. 2(c), thiscan be encoded onto a binary phase-only SLM toimplement updateable correlation at video framerates.

For comparison, when the procedure is used toconstruct a BPOF/fSDF, for the same ten training-set images, the results are shown in Fig. 3. Figures3(a) and 3(b) are the composite image S(u, v), which isa linear combination of the ten training-set images,and the real part of the composite image S(u, v),


(a)

training images of the Bradley APC vehicle separatedby a rotation increment of 5. For example, theBPOF/MfSDF and the BPOF/fSDF designed forinvariance to in-plane rotation over a distortion rangeof 450 are constructed from ten training-set images.After construction, filters are correlated with images,spanning their entire design range, at every 10 interval.The peak-correlation intensity is measured for eachinput image, along with the peak-clutter, or secondary-peak intensity. The peak response is defined to bethe intensity of the maximum correlation peak occur-ring at the output plane:

P = Max I c(x, y) 12} = I c(xo, yo) 1, (11)

where (xo, yo) is the position of the maximum correla-tion peak at the correlator output plane. The second-ary peak was defined to be the highest intensity pointin the correlation plane at points at least three pixelsaway from the correlation peak P, with the additionalcondition that c(x, y) must be a local maximum at thesecondary peak' 3:

C = Maxtl c(x, y) I'l,

such that Ix - x0 I > 3, IY -yo I > 3,ac acx =-= ax ay

(C)

Fig. 3. Filter design procedure of the fSDF-BPOF made directlyfrom training-set images in the frequency domain: (a), (b), (c) thecomposite image S(u, v), the real part of the composite imageS(u, v), and final binary version of the fSDF, respectively.

respectively. The comparison illustrates that thecomposite image S(u, v) of the fSDF is dominated bythe very-low-frequency content of the individual train-ing-set images and has little energy at high frequen-cies. The binary version of the fSDF is shown in Fig.3(c). From Figs. 2(c) and 3(c), it can be seen that themidband frequencies (between low and high fre-quency) in the BPOF/MfSDF are richer and morecomplicated than that of the BPOF/fSDF, whichresults from the bandpass-type composite imageS(u, v) shown in Fig. 2(b); other than the midbandfrequency differences, the two filters are similar.This characteristic of the BPOF/MfSDF improvescorrelation performance.

Distortion Range of Filters

Filters designed to be invariant to in-plane rotation,for distortion ranges up to 90°, are constructed with

(12)

This guarantees that a smooth primary peak base willnot be counted as clutter. The peak-correlation andpeak-clutter responses (secondary peaks) were mea-sured over the specified distortion range for a givenSDF. Figure 4(a) shows the peak-correlation andpeak-clutter responses for a BPOF/MfSDF con-structed from in-plane rotated training images with adistortion range from 00 to 45°. The training imagesused are 50 apart. BPOF filters are extremely sensi-tive to distortion of the image; experience has shownthat a training-set image separation of 50 is probablythe maximum that can be used for the cases studied.The 50 interval is adequate to ensure that for theMfSDF the correlation-peak intensity is adequatelyseparated from the maximum-clutter peak. A simi-lar graph for the BPOF/fSDF is shown in Fig. 4(b).From Fig. 4 it is clear that the BPOF/MfSDF filter isinvariant to distortion over the range from 00 to 450,whereas the BPOF/fSDF does not give completeinvariance as one of the peak-clutter responses ex-ceeds the minimum correlation-peak response.

The ratio P /C defines the correlation-peak-to-secondary-peak ratio22 (PSR). The worst-case ratioof PSR across the distortion range is:

PSR =Pminmax

(13)

This is used to qualify the rotation invariance of anSDF, where Pmin is the lowest correlation-peak valueand Cma, denotes the highest secondary-peak valueacross the distortion range. If PSRW > 1 for an


(a)

(b)

6

5

0Z

q

M

0

6

5

5-

c 4

3

2

5 10 15 20 25Distortion Range (Deg.)

(a)

30 35 40

5.0

4.5

R 4.0-

.a 35 -

X 3.05-

5 2.5-0

- 2.0-

1 1.5-

0 1.0

0.5-

45

0 5 10 15 20 25 30 35 40 45Distortion Range (Deg.)

(b)

Fig. 4. Peak-correlation and peak-clutter responses with a distor-tion range from 00 to 450: (a) binary phase-only MfSDF, (b) binaryphase-only fSDF.

SDF, there will be no ambiguity in extracting thecorrelation peaks from clutter anywhere over thedistortion range; hence a simple thresholder may beused to evaluate the correlation response to an inputimage.

The PSRW data for the BPOF/MfSDF and theBPOF/fSDF with distortion ranges of up to 900 aredisplayed by Figs. 5 and 6, respectively. We obtainedthese results by making a filter from the 0 and 50training-set images and testing it on images from 00to 5 at 1 intervals and finding the PSRW value, thenmaking a filter from the 0, 50, and 100 training-setimages and testing it on images from 0 to 100 at 10intervals and finding the PSRW value, then continuingthis process up to 900. From Fig. 5 it can be seenthat the value of PSRW drops below 1 for a distortionrange of approximately 65°-70° for the BPOF/MfSDF,whereas Fig. 6 shows that this occurs at 450 for theBPOF/fSDF. Therefore it can be concluded that aBPOF/MfSDF with X set to phase-only modulationgives distortion invariance for in-plane rotations upto at least 650; whereas a BPOF/fSDF achieves only450 of rotation invariance. Thus the MfSDF methodcan be designed to be a distortion-invariant filter withtraining images spaced at greater intervals than the

0 10 20 30 40 50 60 70 80 90Distortion Ranges (Deg.)

Fig. 5. Worst-correlation-peak-to-secondary-peak ratio (PSRW)over the distortion ranges from o0 to 90° for the MfSDF method.

fSDF. For distortion invariance over a 1800 range,with the information content minimized, theBPOF/MfSDF needs three filters; whereas four fil-ters are necessary for the fSDF. Thus the binaryphase-only filter constructed utilizing the MfSDFmethod is more suitable than the fSDF method forhigh-speed hybrid optical correlation, as proposed byYoung et al.' 9

To clarify the behavior of the two filters, we plot theaverage modulus of the correlation peak and averageSNR (i.e., average over the distortion range) in Figs. 7and 8, respectively, as functions of the distortionranges. A range value of 0 corresponds to a filtermatched to a single image. The SNR (signal-to-noise ratio) is defined as the ratio of the correlation-peak response to the rms response outside the 50%correlation-peak intensity.2 3 It can be seen fromFig. 7 that the BPOF/fSDF filters produce the higherpeaks for every distortion range. However, this lossin correlation-peak height is compensated by thesharpness of the correlation peaks, which can be seenfrom Fig. 8 to give a better SNR for every distortionrange.

5.0

4.5

R 4.0

2 3.5

X 3.0

Z 2.5

': 2.0

-Oo 1.5

1.0

0.5

0.00 1 0 20 30 40 50 60

Distortion Ranges (Deg.)70 80 90

Fig. 6. Worst-correlation-peak-to-secondary-peak ratio (PSR.)over the distortion ranges from 00 to 900 for the fSDF method.


-a1- Corelation PeakI- - Clutter Peak

. . X. . ..

I

..............

.]

I1

1

C 0.40

o0.3

75

0.1

,

0 1 0 20 30 40 50 60 70 80 90

Distortion Ranges (Deg.)

Fig. 7. Average modulus of the correlation peaks over the distor-

tion ranges up to 900.

Target Discrimination

The fine structure of the target object image trans-lates into high-frequency components in the fre-quency domain that contribute greatly to the discrimi-nation ability of the filter. The MfSDF design placesa greater weighting on the higher frequencies ofindividual training-set images incorporated into thefilter than does the fSDF. Thus it is important toverify that the MfSDF filters give better discrimina-tion between the target and a second nontargetobject, of similar size and shape, than do the fSDFfilters. This test is performed by comparison of thediscrimination values of the BPOF/MfSDF andBPOF/fSDF to the input images of both the BradleyAPC vehicle shown in Fig. 1 and a similarly scaledand oriented Abrams MI tank, shown in Fig. 9. Thediscrimination capability of the filter is defined as

PtDC =-X

Fig. 9. Abrams MI tank at the orientation angle of 00.

taken as the target input image and the Abrams MItank as the nontarget input image. The correlation-peak responses of the target and nontarget images areevaluated from 0° to 90° at 50 increments of rotationfor both the BPOF's/MfSDF's and the BPOF's/fSDF's; the average values of the filter discriminationability are illustrated by Fig. 10(a). For instance, weobtain the discrimination value at 100 by making afilter from the 00, 50, and 10° training-set images andtesting it on 00, 5°, and 10° input images (target and

8

700

0.->' 6

C50

.g 4

0 3!

(14)2

where Pt is the intensity of the correlation peak forthe target input and Pn, is for the nontarget imageinput. In the simulations the Bradley APC vehicle is

20

18

16

14

12

10

e 8

¢ 6

0

-'

0

a

8

2

0

Distortion ranges (Deg.)(a)

0 1 0 20 30 40 50 60 70 80 90Distortion ranges (Deg.)0 --I I, ' ' _ . , . , 1 . , . , . (b )

0 10 20 30 40 50 60 70 80 90 Fig. 10. Discrimination capability of filters over the distortion

Distortion Ranges (Deg.) ranges up to 900; the target and nontarget inputs are at the same

Fig. 8. Average values of the signal-to-noise ratio (SNR) over the rotation angle: (a) average value of inputs at 50 increments, (b)

distortion ranges up to 900. average value of inputs at 10 intervals.


3

(a)

(b)

I

.4

(c)

(d)Fig. 11. Three-dimensional representations of the correlationfunctions obtained from correlating the target image and thenontarget image at the rotation angle of 400 with both MfSDF andfSDF designed for 400 in-plane vehicle rotation invariance: (a), (b)the results from the MfSDF filter for the target and the nontargetinput images, respectively; (c), (d) the results from the fSDF filter.

nontarget) and then finding the average value ofthese three discrimination values. It can be seenfrom Fig. 10(a) that, with the exception of onemarginal point at 300, the MfSDF delivers superiordiscrimination performance. However, the inputsat 5 intervals are representative of the training-setimages only, not the full distortion range. Thus thefilter discrimination ability was also tested at 10intervals; the results from averaging of discrimina-tion values are plotted in Fig. 10(b). Hence weobtained the discrimination value at 100 by making a

filter from the 00, 50, and 100 training-set images andtesting it on input images (target and nontarget) from00 to 100 at 1 intervals and then finding the averagevalue of these discrimination values. Figure 10(b)shows that the MfSDF still delivers slightly betterdiscrimination performance over the whole distortionrange than does the fSDF. If the training set in-cluded images a 1 intervals, better discriminationwould result.

The correlation functions of the target image andthe nontarget image at a rotation angle of 400 withboth the BPOF/MfSDF and the BPOF/fSDF, de-signed for 400 in-plane vehicle rotation invariance,are shown in Fig. 11. These correlation functionsgive an excellent illustration of filter discriminationperformance. To show the discrimination ability ofthe filters clearly, we normalized all nontarget corre-lation functions to the target autocorrelation peakheight.

Conclusions

This study introduces a flexible tuning algorithm tothe filter-modulation constraints of SDF design.Different from the fSDF design, the MfSDF designconsiders the filter-modulation constraint of X on theindividual training-set images. The linear combina-tion image function of the filter-modulated training-set images may therefore incorporate the higher-frequency components of the individual training-setimages. We then obtain the final composite image,i.e., the MfSDF filter, by applying the filter-modula-tion constraint X to the linear combination imagefunction. As the filter-modulation operators Yl andA4X can be given any functional form, the MfSDFdesign proposed herein is sufficiently general to bedescribed as a unified filter-modulation SDF design.

With the filter modulator set to BPOF, theMfSDF can be implemented for real-time opticalcorrelation with currently available programmableSLM's. A significant advantage of the BPOF/MfSDFis that the filter modulation X is free to be definedand can be used to apply any desirable modulationmethod. Therefore it is possible to construct aMfSDF with an optimal choice of the modulationoperator X to optimalize the overall performancecapabilities of the filter given the limitations of thecurrently available SLM's.

An initial investigation to compare the MfSDFfilters' performance, with modulators 9 and X beingchosen to be a BPOF and a POF, respectively, withthat of the fSDF filters has been completed. Com-puter simulations show that the BPOF/MfSDF fil-ters can achieve distortion invariance to in-planerotations of up to at least 650, whereas the BPOF/fSDF filters attain only 450 in the case studied.Therefore the BPOF/MfSDF method can be designedto be a distortion-invariant filter with training imagesspaced at larger distortion increments than theBPOF/fSDF method. Hence the BPOF/MfSDFmethod needs less filters to cover a distortion-invariant range of 1800 than does the BPOF/fSDFmethod. For the case studied the computer simula-


tions also show that the BPOF/MfSDF filters havethe better signal-to-noise ratio and target discrimina-tion ability when compared with the BPOF/fSDFfilters. The slightly lower correlation peaks achievedwith the MfSDF filter correlations does not cause anydetection difficulty because of the good peak sharpness.The MfSDF demonstrates better discrimination abil-ity between target and nontarget objects. Thus over-all, its performance is better than that of the fSDF.The effects of input additive noise and partiallystructured clutter on filter performance are currentlyunder investigation and are thus not reported herein.

The choice of the modulator X to be a POF wasmade to permit an initial investigation of the MfSDFdesign; it is not the optimal choice. Further re-search is required to obtain optimal performancefrom the MfSDF filters; this will be the focus of futureresearch.

References1. A. VanderLugt, "Signal detection by complex spatial filter-

ing," IEEE Trans. Inf. Theory IT-10, 139-145 (1964).2. J. L. Horner and P. D. Gianino, "Phase-only matched filter-

ing," Appl. Opt. 23, 812-816 (1984).

3. J. L. Horner and J. R. Leger, "Pattern recognition with binaryphase-only filter," Appl. Opt. 24, 609-611 (1985).

4. B. Javidi, S. F. Odeh, and Y. F. Chen, "Rotation and scale

sensitivies of the binary phase-only filter," Opt. Commun. 65,233-238 (1988).

5. C. F. Hester and D. Casasent, "Multivariant technique for

multiclass pattern recognition," Appl. Opt. 19, 1758-1761(1980).

6. D. Casasent, "Unified synthetic discriminant function compu-tational formulation," Appl. Opt. 25, 2343-2350 (1986).

7. J. L. Horner and P. D. Gianino, "Applying the phase-only filter

concept to the synthetic discriminant function correlationfilter," Appl. Opt. 24, 851-855 (1985).

8. D. Casasent and W. Rozzi, "Computer-generated and phase-

only synthetic discriminant function filters," Appl. Opt. 25,3767-3772 (1986).

9. D. Jared and D. Ennis, "Inclusion of filter modulation in the

synthetic discriminant function construction," Appl. Opt. 28,232-239 (1989).

10. D. Jared, "Distortion range of filter synthetic discriminantfunction binary phase-only filters," Appl. Opt. 28, 4335-4339(1989).

11. R. C. D. Young and C. R. Chatwin, "Design and simulation of a

synthetic discriminant function filter in an updateable photore-fractive correlator," in Optical Pattern Recognition III, D.Casasent and T. H. Chao, eds., Proc. Soc. Photo-Opt. Instrum.

Eng. 1701,239-263 (1992).12. M. B. Reid, P. W. Ma, J. D. Downie, and E. 0. Choa,

"Experimental verification of modified synthetic discriminantfunction filters for rotation invariance," Appl. Opt. 29, 1209-1214(1990).

13. G. F. Schils and D. W. Sweeney, "Iterative technique for the

synthesis of optical-correlator filters," J. Opt. Soc. Am. A 3,

1433-1442 (1986).14. Z. Bahri and B. V. K. Vijaya Kumar, "Generalized synthetic

discriminant functions," J. Opt. Soc. Am. A 5, 562-571 (1988).

15. J. D. Downie, "Case study of binary and ternary syntheticdiscriminant function filters with similar in-class and out-of-class images," Opt. Eng. 32, 561-570 (1993).

16. Z. Bahri and B. V. K. Vijaya Kumar, "Binary phase-only

synthetic discriminant functions designed using the successiveforcing algorithm," in Hybrid Image and Signal Processing II,D. P. Casasent and A. G. Tescher, eds., Proc. Soc. Photo-Opt.Instrum. Eng. 1297, 188-193 (1990).

17. C. R. Chatwin, R. K. Wang, and R. C. D. Young, "Tuneable

edge enhancement filtering," submitted to Opt. Lasers Eng.18. 0. K. Ersoy, Y. Yoon, N. Keshava, and D. Zimmerman, "Non-

linear matched filtering," Opt. Eng. 29, 1002-1012 (1990).19. R. C. D. Young, C. R. Chatwin, and B. F. Scott, "High-speed

hybrid optical/digital correlator system," Opt. Eng. 32, 2609-2615 (1993).

20. D. Psaltis, E. Paek, and S. Venkatesh, "Optical image correla-tion with a binary spatial light modulator," Opt. Eng. 23,698-704 (1984).

21. D. L. Flannery, J. S. Loomis, and M. E. Milkovich, "Design

elements of binary phase-only correlation filter," Appl. Opt.27, 4231-4235 (1988).

22. M. A. Flavin and J. L. Horner, "Amplitude encoded phase-onlyfilters," Appl. Opt. 28, 1692-1696 (1989).

23. J. L. Horner and H. 0. Bartelt, "Two-bit correlation," Appl.Opt. 24, 2889-2893 (1985).


Modified filter synthetic discriminant functions for improved optical correlator performance

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