Correlation filters for target detection in a Markovmodel background clutter
B. V. K. Vijaya Kumar, David P. Casasent, and Abhijit Mahalanobis
The performance of distortion-invariant correlation filters in the presence of background clutter is addressed.Background images are modeled as Markov noise processes, and a synthesis procedure for the optimal filter isdescribed. It is shown that spatially filtering the training set images eliminates the need for the inversion oflarge noise covariance matrices, thus leading to a computationally efficient filter realization. The effect oferrors (in the estimation of clutter correlation coefficient) on filter performance is theoretically analyzed, anda bound on the relative degradation of the SNR due to such errors is presented.
The use of spatial filters for pattern recognition hasreceived much attention in the recent past.'-8 It iswell known that the matched spatial filter (MSF) isoptimal for the detection of a specific image in thepresence of white noise.9 The MSF is easily imple-mented in an optical correlator and hence is attractivefor optical pattern recognition. However, the perfor-mance of the MSF degrades severely in the presence ofgeometric distortions. Generalized filtersl 3 have,therefore, been proposed to combat the effect of imagedistortions. We refer to all such filters as correlationfilters, since they are designed for use in an opticalcorrelator.10 The most well known of such correlationfilters is the synthetic discriminant function (SDF)and its variations.l' In this paper, we analyze theperformance of the minimum variance synthetic dis-criminant function (MVSDF)12 in the presence ofMarkov modeled background clutter. Gu and Lee5
have proposed a technique to improve the noise perfor-mance of correlation filters. However, in this method,sample images of background noise are treated astraining images, and important statistics of the noiseprocess are ignored.
Until now, the performance of correlation filters inthe presence of structured background clutter has notbeen addressed explicitly. We model the backgroundas a Markov noise process and analyze the effect of
The authors are with Carnegie Mellon University, Department ofElectrical & Computer Engineering, Center for Excellence in Opti-cal Data Processing, Pittsburgh, Pennsylvania 15213.
clutter on filter performance. Section II is a review ofthe MVSDF synthesis process. The model adoptedfor describing clutter in realistic images is discussed inSec. III. A modified synthesis technique for MVSDFsbased on 2-D Markov noise models is also described.The results of computer simulation are presented inSec. IV. The effect of an error in the correlationcoefficient estimate is analyzed in Sec. V, and a theo-retical bound on performance loss is obtained. Thisanalysis is carried out using a 1-D model for the sake ofsimplicity. Finally, experimental data on perfor-mance loss using the 2-D model are compared to thepredicted bounds. Section VI is a summary of theresults presented in this paper.
II. Background
We represent M X M images by the vectors x.These vectors are obtained by lexicographically order-ing the rows of the images. Furthermore, the matrix Xis defined so that its columns are the training set imagevectors xi, i.e.,
X = [xx 2 ,XN], (1)
where N is the total number of training images used.The vector xi is, therefore, of length d, and the matrixX is d X N, where d M2 . We wish to synthesize afilter vector h so that
XTh = u, (2)
where u is the specified constraint vector.Once h is determined, we apply an appropriate
threshold to the inner product of the input image andthe filter h [which satisfies Eq. (2)] and decide on theclass of the data. A particular solution for the filtervector h is
This is the conventional SDF and has been provedoptimal for target detection in the presence of additivewhite noise.12 In the presence of additive colorednoise with covariance matrix C, the minimum outputvariance is achieved by'2 choosing the filter to be
h = C-lX(XTC-lX)-lu. (4)
We model background clutter as an additive noiseprocess. Since the MVSDF maximizes output SNRby minimizing the output variance, it is the optimalfilter for target detection in background clutter. Inthis paper we seek an appropriate model for the back-ground and use this model to approximate C that isneeded in MVSDF synthesis.
In Sec. III we describe a Markov model for the back-ground noise process that results in a special form forthe matrix C. One problem associated with theMVSDF is that the matrix C in Eq. (4) is of very largedimensions (d X d dimensional) and in general is diffi-cult to invert. The choice of a Markov noise back-ground model allows us to compute the inverse of thecovariance matrix with relative ease.
We note that C-' is a tridiagonal matrix and is readilydetermined if the parameter p is known. The MVSDFcan then be synthesized with the matrix in Eq. (8).Recall that this is the 1-D model and does not take intoaccount the cross-correlation between the rows of animage. The generalization of this to the 2-D case hasbeen proposed by Arcese et al.14 We now summarizethe 2-D Markov model and demonstrate the process offilter synthesis by means of an example. The effect oferror in the estimation of p on filter performance istheoretically analyzed in Sec. V using a 1-D noise mod-el for the sake of simplicity. Experimental data (usingthe 2-D model) is then compared to the theoreticalresults to evaluate the effectiveness of the bounds.
Images are often modeled as Markov processes tocapture interpixel correlation. Cross and Jain 5 havemodeled textured images as random Markov process-es, and Pratt' 6 has suggested the use of 2-D Markovmodels for image registration via correlation. It hasbeen shown,'6 assuming row/column separability andand a Gauss-Markov process, that the cross-covari-ance between the kth and mth row is given by thematrix
Ill. Filter Synthesis Procedure
The results in this section provide a summary of theprevious work done to model background structure inreal images and discuss the new use of these models forthe synthesis of optimal SDF filters for distortion in-variance in the presence of cluttered noise. Ben-Yosefet al.'3 has experimentally shown the validity of a 1-Dexponential model for spatial correlation in IR images.The correlation function is modeled in the 1-D case(for stationary data) as
Zkm = P k-mIXkk. 1 < k,m < M. (9)
The Covariance matrix for 2-D Markov data can nowbe written as the M2 X M2 matrix
2llP22
C = P2233
M-12
P211
22
PX33
P2111
P2 2
233
P 2MM PM-3M
pM l111
pM-222
... pM-3233 , (10)
2MM _
RX(T) = E[x(t)x(t + -r)] = k exp(-ITra), (5)
where k is a constant, 1/a is the correlation length, andX- is the shift variable. Setting k = 1 (with no loss ofgenerality) and exp(-a) = p (the correlation coeffi-cient) we obtain
RX(T) = pITl1 (6)
The covariance matrix C(ij) = Rx(li - il) is, therefore,given by
P p d-l
P 1
c = p2 p ........... *(7)
pd-1
This is the 1-D Markov covariance matrix and isToeplitz. Moreover, the inverse of this matrix (re-quired for our optimal filter) can be analytically com-puted in terms of the parameter p and is given by'4
where M is the number of rows and columns in theimage. In this case, the covariance matrix is blockToeplitz, and its inverse can be computed quite easilyanalytically.
To compute the inverse matrix, we first note that theM X M matrix Xkk is the same for all k and identical tothe matrix in Eq. (7) except for its dimensionality.Denoting all 2kk by 2, we can write the elements of thematrix as
2kk(ii) = 2(ij) = Pli-J 1 < i,j < M. (11)
The inverse matrix 2-l is, therefore, of the same formas in Eq. (8). It can be shown that the inverse of thecovariance matrix for the 2-D case is
(1 + p')'-1-p2 1C-1 = 1 _p2 L
1 P
0- p2-1
(1 + 2)2-1
0
... o
.i
(12)
(8) where -1 is of the form in Eq. (8). I and 0 in Eq. (12)represent the M X M identity and zero matrices, re-spectively. Therefore, C-' is a d X d(d = M2) block
Toeplitz matrix, and its special structure can be ex-ploited to reduce computations during filter synthesis.
To see how this may by achieved, recall that the M2
dimensional vector x represents the discrete 2-D imagex(ij) and that h in Eq. (4) can be interpreted as com-posed of two filters; a prewhitener followed by theconventional SDF17 designed for the prewhitener out-put. We will show that multiplying a vector (say x) bythe matrix C-1 in Eq. (12) is analogous to simply con-volving the corresponding image x(ij) with a 3 X 3window. First, we note that if
for 2 < i • (M - 1), where x(i) and z(i) denote the ithrow of the images x(ij) and z(ij) and hence are treatedas row vectors in Eq. (14). Equations (13) and (14)describe the filtered or preprocessed images. In termsof the actual image pixels of z(ij) and x(ij) aftersubstituting the expression for 3-l in Eq. (8) into Eq.(14) we obtain
is a 3 X 3 mask operator. For p = 0, w(ij) = (ij) andz(ij) = x(ij). Thus, for uncorrelated (white) noise,the filtering or preprocessing of x by C-' does not alterthe training images x(ij), and the MVSDF is identicalto the conventional SDF (which is optimal'2 for whitenoise). When x(ij) = K is a constant image, from Eq.(15) we find z(ij) = K (1 - p)
2/(l + p)
2 is also aconstant image of scaled amplitude. However, as papproaches 1, the filtered constant image amplitudedecreases because, for these p values, the highly corre-lated noise looks very much like the constant imageitself.
Thus the matrix vector multiplications in Eq. (13)can be reduced to an image filtering process using theresult in Eq. (16) and thus can be achieved in a compu-tationally efficient process. We note that w(ij) is a 3X 3 operator that depends only on p. To synthesizethe optimal filter for background clutter, we thus needto estimate p and convolve the training images with the
operator w(ij) given in Eq. (17). The resulting imagesz(ij) can be considered to be the columns of the matrixZ. Although Z = C'1X [from Eq. (13)], the columns ofZ can be obtained directly by convolving the trainingimages x(ij) with w(ij) in Eq. (16), and the computa-tion of C-l is not required to obtain Z from X. Replac-ing the matrix product C'1X in Eq. (4) by Z we obtain h= Z(XTZ) 1lu. The matrix XTZ is N X N (where N isthe number of training images) and is easily invertible.Since Z can be computed without the inversion of alarge noise covariance matrix, the optimal filter h inEq. (4) can be readily synthesized. The inversion of ad X d covariance matrix (which in general is verydifficult) is thus avoided. In the next section, wesynthesize the MVSDF for a Markov noise model andexperimentally evaluate its performance.
It is important to note that for stationary noise theminimization of the variance at the filter output mini-mizes the variance at all other points in the correlationplane. This follows as a consequence of the propertiesof stationary stochastic processes. Insight may begained from the fact that for stationary noise process-es, the variance is identical at all points. Since Mar-kov noise is a stationary process, the MVSDF de-scribed in this paper minimizes the variance at allpoints in the output correlation plane and not just atthe point controlled by the constraints in Eq. (2).
IV. Example
The images of a tank were available for computersimulation of the optimal filter using a Markov noisemodel. The tank images were digitized every 100 at anaspect angle of 150 to obtain thirty-six grey level im-ages each with 256 X 256 pixels. The images were thendecimated to 32 X 32 pixels to reduce storage require-ments. Nine of these images (taken every 400) arepresented in Fig. 1. A 128 X 128 section of a full 512- X512-pixel IR real background image is shown in Fig. 2.This image was used as the background for the tankimages in Fig. 1. We thus seek to synthesize the opti-mal filter for detecting the tank images in Fig. 1 in thepresence of the background scenery in Fig. 2. Thisrepresents the first attempt at distortion-invariantpattern recognition in which realistic clutter parame-ters were included in the filter synthesis.
The spatial 2-D autocorrelation of the image in Fig. 2was computed. Assuming stationary statistics, thespatial autocorrelation of a 2-D sequence of randomnumbers is an estimate of the statistical autocorrela-tion of that sequence. For example, the value at theorigin of the deterministic autocorrelation is an esti-mate of Elx(m) - x(m)} = RX(0), where E-} denotes theexpectation and R.,(r) is the statistical autocorrelationfunction of the discrete random process x(m). Thusthe value of the deterministic autocorrelation r pointsaway from the origin is an estimate of R,(r). We shalldenote the 2-D spatial autocorrelation by G(ij) andthe statistical autocorrelation by R,(ij).
Recall that R,(ij) is assumed to have an exponential(Markov) form and is a function of p only. Our filtersynthesis requires that we estimate p. Toward this
Fig. 1. Training set images of nine tanks at 150 aspect angle.
end, we formed the squared error between RX(ij) andG.,(ij) as
E = 1/d2 . [R.(ij) - GX(ij)]2 (18)
and numerically found a p value for which this errorwas minimum. The value thus obtained is a mini-mum mean squared error (MSE) estimate of p. Nu-merical optimization routines must be employed forthe minimization of the mean squared error E. Weused the IMSL18 routine ZXMWD in our simulations.We then synthesized the MVSDF using and thesimplified procedure derived in Sec. III. A value of =0.238 was obtained for the image in Fig. 2 using theMMSE procedure.
In practice, we would either assume p values fortypical background or estimate p from the full (targetplus background) image as noted above. The same pvalue could be used for a number of successive imageframes of a given region. The training set of tankimages chosen for filter synthesis to achieve distrotioninvariance consisted of the nine images in Fig. 1. Thefilter output for each of the training images was speci-fied to be 1.0. This specifies the elements of u in Eq.(4). Fifteen background images of size 64 X 64 weregenerated from the 128 X 128 image in Fig. 2. Thesewere obtained by shifting the image in Fig. 2 by 4 pixelsand selecting the central 64 X 64 section. The fifteenbackground images thus generated are essentially un-correlated since p4
= (0.238)4 = 0.003. The broadside
view of the tank (a training set image) was then super-imposed on each of the fifteen 64 X 64-pixel back-ground images to produce fifteen test images. Eachimage was then correlated with the fitler (synthesizedusing ), and the variation in the output correlationpeak value was noted. The average squared error(ASE) for the fifteen test images was found to be 1.129X 10-2. This is small compared to the desired value of1.0 and corresponds to a correlation peak SNR of -20dB.
We next synthesized the MVSDF for ten values of pfrom 0.138 to 0.338 in steps of 0.02. These values wereclose to the b = 0.238 estimate. Each of the ten result-ing filters was correlated with fifteen test images withthe broadside tank in fifteen different clutter back-grounds, and the corresponding ASE was noted. TheASE values obtained for the various values of p arelisted in Table I. All ASE values are quite low with thesmallest output ASE obtained for p = 0.218. Thisindicates that the MSE estimate of p (i.e., b = 0.238) isin error. However, the relative degradation of ASEdue to this error (in correlation coefficient p) is 1.129/1.127 = 1.002, which is very small (0.2%).
We have shown that the MVSDF can be easily syn-thesized for practical Markov noise models. The esti-mation of the correlation coefficient is required prior
Table 1. Average Squared Error in Filter Output for Different Values of p
to filter synthesis. An error in this parameter estimatewill cause a loss in filter performance (higher outputvariance). The error should be kept as small as possi-ble to achieve the best possible results. We used nu-merical MMSE techniques for estimating the correla-tion coefficient p. The spatial autocorrelation of asample background image was used in the estimationprocess. Other parameter estimation technqiues canalso be employed. However, most estimation tech-niques are likely to yield results with some error. It is,therefore, important to address the effect of an error inthe estimate of p on the output variance of theMVSDF.
An error in the correlation coefficient p may beviewed as an error in the correlation length 1/a. Thequantization of the input image due to a limited num-ber of grey levels on the spatial light modulator (SLM)will also alter the statistics of the background. Theeffect of this can also be analyzed by considering theerror in the estimate of p and its effects. In Sec. V, wecarry out a theoretical analysis using a simplified 1-DMarkov model (ignoring the correlation between im-age rows) and compare the results to data obtainedusing the more complete 2-D Markov noise model.
V. Correlation Length Error Analysis
We have shown that if background clutter is mod-eled as a Markov noise process, the optimal filter canbe synthesized without the inversion of a large covari-ance matrix. However, the correlation coefficient pmust be estimated to determine a model that best fitsthe background noise process. The performance ofthe filter depends on how well a Markov model candescribe the noise process and the accuracy of the pparameter estimate. In general, most parameter esti-mation techniques yield results with some error (due tononzero variance of the estimates), and it is, therefore,important to determine the effect of this error on filterperformance.
As stated earlier in Sec. IV, a more practical concernis the SLM image quantization due to a limited num-ber of grey levels. The parameter p is estimated off-line prior to filter synthesis from representative back-ground images. The filter is then implemented in anoptical correlator. The limited number of grey levelsof the input SLM affects the autocorrelation functionof the background clutter and hence alters its statis-tics. In other words, quantization of the backgroundclutter changes the correlation coefficient (and hencethe correlation length) of the noise process. We nowtheoretically analyze the effect of this error on filterperformance in terms of SNR degradation.
It is appropriate at this point to consider the impli-cation of loss in SNR. Since background clutter is anoise process, it affects the value of the correlationpeak. The desired correlation peak value is specifiedby the user during filter synthesis and is used fordistortion-invariant pattern recognition. The degra-dation of the correlation peak in the presence of back-ground clutter causes classification errors and thusmust be kept to a minimum. SNR at the correlation
peak is a measure of this degradation, the extent towhich the correlation peak varies from the user speci-fied level. A loss of SNR implies reduced systemperformance and is, therefore, undesirable.
Assume that an estimate p of p can be obtained withan error 6 so that p = p + 6. We wish to analyze theperformance of the filter h synthesized with p (insteadof p). Since we seek to maximize output SNR (mini-mize error in the filter output) in the presence of back-ground clutter, we focus attention on the SNR degra-dation due to 6 (the error in the estimate for p).
To analyze the error effect, we must analyze the termXTC-1X in Eq. (4). The inverse of the covariancematrix C for the 1-D case is given in Eq. (8). We beginour analysis by approximating XTC-1X in terms of thedata XT X and p separately. This will then allow us tovary p to quantify o-2effects. To obtain an approxima-tion of XTC-1X, we assume that for sufficiently large d
gmm,(O) -t gm(1) -t g.,.(2) (19)
where gmm(k) is the cross-correlation function of themth and nth tank training image vectors, and k is thelag or shift variable in the cross-correlation computa-tion. Equation (19) states that the expection of thecross-correlation function values up to 2 pixels awayfrom the origin are approximately the same. Thisapproximation is based on the assumption that realis-tic data are essentially low pass in nature. The valid-ity of this assumption will be evaluated later by meansof an example.
Let a = (1 + p 2 )/(l - p 2) and b = -p/(l - p2 )
represent the elements of C-' in Eq. (8). We can nowrewrite the tridiagonal Teoplitz matrix C-' and obtain
where XI is obtained by shifting the columns of Xdown by one element and Xt is obtained by shifting thecolumns up by one element. The elements of XTX;and X7 Xt are the inner products of the original imagevectors, and those vectors shifted up or down by oneelement. The approximation in (19) states that
XTX , xTxt XTX1. (21)
This simplifies Eq. (20) to
(22)X 7 C-1 X (a + 2b)XTX = (1 X)2 X = p- p2 + P
For convenience, we denote XTX by V (the vector innerproduct matrix) and write Eq. (22) as
(23)XTC- 1X I P V.1 + This approximation will be used frequently in our
analysis. To obtain an idea of the accuracy of thisapproximation, we consider the expression for the out-put variance of the ideal filter. It can be shown'2 thatthe output variance of the optimal filter (synthesizedwith the exact parameter p) is given by
Using the result in Eq. (23) in Eq. (24), we may approx-imate o.2 in terms of V and p as
2 ( + P uTV-1u.(1 - p)
(25)
The exact value of a2 in Eq. (24) and the approximatevalue in Eq. (25) are plotted in Figs. 3 and 4 as afunction of p. The data base of nine tank images (Fig.1) was used to determine the V matrix. As shown inFigs. 3 and 4, the approximation error is acceptablysmall over a large range of p values. The actual (solidline) and approximate (dotted line) variance values areshown in Fig. 3 for 0.0 < p < 1.0. The two curvesmatch well for small values of p but diverge for p valuesclose to 1.0. This occurs because the covariance ma-trix C is ill-conditioned for large values of p, and use ofthe analytical inverse for the computation of a2 in Eq.(24) is not accurate. Since the covariance matrix can-not be inverted for large p, filter synthesis in such casesis impractical. We, therefore, limit our attention tosmaller values of p. This is practical since the p valuesfor real backgrounds are small. The approximate andactual variance values for 0.0 < p < 0.6 are shown inFig. 4. The error in the variance due to our approxi-mation is seen to be <8% for p < 0.5 and <17% for p •0.6. The approximation error is data dependent andsmaller for images with dominant low frequency com-ponents. This is true since the approximation in Eq.(19) is more accurate for such images.
Since variance is inversely related to SNR, we nextfind a suitable expression for the output variance 2obtained with the filter synthesized with , (i.e., whenan error is present in the parameter estimate) andcompare it with the approximate expression in Eq.(25). To do this, we express 62 in terms of u' 1V-'u, p,and its error and then obtain data term independentresults and conclusions.
We first derive an expression for the inverse covari-ance matrix C-' (obtained with p). This can be ex-pressed in terms of the exact C-1 and an error matrix Aas
-= 1- (p + 6)2 I - (p + )
where
O -1
-1 +2p= O' -1 ...
LO 0
(26)
0 0
+ i+ 2 -1-1 0 .
and y = (1 - p2)/(1 - [p + ]2) and : = /(1 - [p + ]2)are scalars.
We now derive an expression for 2 (when p is inerror by ) using 1i and C-'. The filter hI synthesizedwith the covariance matrix C is
h = C X(XT C lX)f1 u. (28)
For the matrix C, we substitute p + 6 in Eq. (23) toobtain
VariaxceX lo1
2
1.8 -
1.6 -
1.4 -1.2 -
1 -
0.8 -0.6-
0.4 -
0.2 -
0 0.1 0.2 .3I I I 00 .01 0. 2 0.3 0.4 0.5 0.6
Fig. 3. Comparison of actual and approximated variance values for0.0 < p < 1.0.
36 -32 -
28 -
Solid - ActualDotted - Approximated
24 -
Variance 20-x lo, 16 -
12 -
8 -4 -0-
F rI0 0.2 0.4 0.6 0.8 1
Fig. 4. Comparison of actal and approximated variance values for0.0 < p < 0.6.
XTC-IX = (1 - P - ) V.1 + + 1 (29)
Using this approximation, Eq. (28) is simplified to
i(1 - -+ ) ClXV u,and the corresponding variance is given by
a2 = TCh.
Substituting Eq. (30) into Eq. (31) yields
2 = (1 + P + ) UV XTCtCClXVtU.
(30)
(31)
(32)
We now substitute Eq. (26) for C-' into Eq. (32) anddescribe the variance Z-2 as
Expanding the brackets in Eq. (33) yields three terms,each of which can be further simplified. We includeall factors except the first
1 P + ad
factor in these three terms. We consider each termseparately. To simplify the first term containing C-',we use the approximation in Eq. (23) and write thisterm as
A bound on 2 may now be obtained by adding theexpressions in Eqs. (34), (37), and (40) [and then in-serting the sum in Eq. (33)]. This yields
a2 I +P+ ) [A(6) + B(b) + C(6)]uTV-tu, (42)
0.4 0.6Fractional Error in p
Fig. 5. Plot of upper bound on relative variance degradation as afunction of fractional error in p.
y2 UTrV-lXTC-lXV-Vu = 2 1 - P uTVlu = A(b)uTV-tu, (34)+ P
where
A(a) = Y2(1 - p)/(1 + p). (35)
Now consider the second term containing XTAX. Thematrix A is also tridiagonal as shown in Eq. (27).Therefore, the assumptions made to simplify XTC-1Xin Eqs. (22) and (23) also hold for XTAX, and we write
Finally, consider the third term in Eq. (34) contain-ing ACA. Since approximations such as (22) are diffi-cult to justify for this term, we determine the eigenva-lues of the covariance matrix C and the error matrix Ato provide an upper bound on the worst-case SNRdegradation due to the error 6. Assuming that both 6and p are positive, the maximum eigenvalues of A andC can be shown' 9 (for large values of d) to be
XAax = 6 + 2(p + 1), Xc (+ P) (39)
We use the fact that xTCx < XmaxxTx to place an upperbound on the full third term as
# 2 uTV lXTACAXVlu s fl2 (X\xC)[UTV 'X TAI [AXV 'uJ
= 02 (4man)(UTVlXT)A 2 (XV1U)
< ,2 (\max)(XAa) 2UTVlXTXVlU
< p2(AMaX)2(cmax)UTV"U
(6 + 2[1 + p])2 (1 + p) UTV-lu(1 - p)
- C()uTV'1u, (40)
where
(1 -p)
where A(6), B(6), and C(6) are as defined in Eqs. (35),(38), and (41). The ratio of the variance &2 in Eq. (41)with an error 6 in p to the ideal variance or2 is
(6) S (1 + p + 6i)2(1 -) [A(6) + 0(6) + C(6)]. (43)
Substituting the expressions for A(6), B(6), C(6), 1, andy, we obtain the following expression for the bound interms of p and 6:
The bound in (44) is plotted in Fig. 5 in decibels fordifferent values of p and the fractional error 6/p. Notethat for 6 = 0.0 the ratio bound is 1.0 (or 0 dB), indicat-ing that the optimal filter has been realized.
The average squared error of the filter output wascomputed in Sec. IV for various values of p, and a MSEestimate of p = 0.238 was obtained. Tests showed thatthe variation in filter output was actually smaller for p= 0.218, and, therefore, the MSE parameter estimate p= 0.238 is in error. This is expected since the back-ground noise process may not be exactly Markov andstationary. As stated in Sec. IV, the relative degrada-tion is SNR (see Table I) using p = 0.238 rather than b= 0.218 is 1.129/1.127 = 1.002 = 0.015 dB. The theo-retical upper bound for p = 0.218 and the 6 = 0.02 ofthese data (a 9% fractional error in p) is 0.038 dB ascalcualted from (44). Hence the observed degradationin SNR of 0.015 dB is well below the theoretical upperbound predicted by (44).
The bound in (44) is plotted in Fig. 5 for p = 0.01,0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, and 0.50.The only assumption in deriving this bound is that thetraining images have significant low frequency compo-nents. This is necessary for the approximation in Eq.(22) to hold. If this approximation is valid, the curvesin Fig. 5 represent an upper bound on the relative SNRloss (in decibels) due to an error (6) in the correlationcoefficient p. The relative SNR degradation is de-fined as the ratio of the minimum variance attainable(a2) with an ideal p estimate, and the output varianceactually realized (2) when the estimate b of p has anerror 6. This ratio is plotted on a decibel scale vs thefractional error in p (i.e., the ratio 6/p) in Fig. 5 fordifferent p values (the different curves shown in thefigure).
In Fig. 5 we see that the relative loss in SNR isnegligible for small values of p (uncorrelated noise)even when the error 6 is a significant fraction of thecorrelation coefficient (i.e., 6/p approaches 1.0). A 3-dB loss in SNR is realized for progressively smallerfractional errors as p becomes larger. For p 0.50, afractional error of 10% (6/p 0.1) results in a 3-dB
loss in SNR. Thus, even for moderately large values ofp, a significant amount of error in the parameter esti-mate can be tolerated.
VI. Conclusions
We addressed the performance of distortion-invari-ant correlation filters in the presence of backgroundclutter. We modeled background images as samplerealizations of a Markov noise process. This simpli-fied the synthesis of the optimal filter (MVSDF) byeliminating the need to invert large noise covariancematrices. The MVSDF can be synthesized using a 3 X3 mask operator to spatially filter the training images.This operator depends on the correlation coefficient pof the Markov model used to describe the backgroundclutter.
We also addressed the effect of quantization andparameter estimation errors on filter performance. Atheoretical bound on SNR loss was obtained using a 1-D Markov noise model. Actual data obtained for the2-D Markov model were then compared to the theoret-ical bound values. It was shown that a significantamount of error in the estimation of the clutter back-ground correlation coefficient can be tolerated formoderately large correlation coefficient values.
References
1. H. J. Caulfield and M. H. Weinberg, "Computer Recognition of2-D Patterns Using Generalized Matched Filters," Appl. Opt.21, 1699-1704 (1982).
2. Y. N. Hsu and H. H. Arsenault, "Optical Pattern RecognitionUsing Circular Harmonic Expansion," Appl. Opt. 21,4016-4019(1982).
3. D. Casasent, "Unified Synthetic Discriminant Function Com-putational Formulation," Appl. Opt. 23, 1620-1627 (1984).
4. R. R. Kallman, "Construction of Low Noise Optical CorrelationFilters," Appl. Opt. 25, 1032-1033 (1986).
5. Z. H. Gu and S. H. Lee, "Classification of Multi-Class StochasticImages Buried in Additive Noise," Proc. Soc. Photo-Opt. In-strum. Eng. 700, 44-51 (1986).
6. J. L. Horner and P. D. Gianino, "Phase Only Matched Filter-ing," Appl. Opt. 23, 812-816 (1984).
7. G. F. Schils and D. W. Sweeny, "Iterative Technique for theSynthesis of Optical Correlation Filters," J. Opt. Soc. Am. A 3,1433-1442 (1986).
8. D. Casasent and D. Psaltis, "Position, Rotation and Scale Invari-ant Optical Correlation," Appl. Opt. 15, 1795-1799 (1976).
9. H. L. Van Trees, Detection, Estimation, and Modulation The-ory: Part I Wiley, New York, (1968).
10. A. B. Vander Lugt, "Signal Detection by Complex MatchedSpatial Filtering," IEEE Trans. Inf. Theory IT-10, 139-145(1964).
11. D. Casasent and W. T. Chang, "Correlation Synthetic Discrimi-nant Functions," Appl. Opt. 25, 2343-2350 (1986).
12. B. V. K. Vijaya Kumar, "Minimum Variance Synthetic Dis-criminant Functions," J. Opt. Soc. Am. A 3, 1579-1584 (1986).
13. N. Ben-Yosef, K. Wilner, S. Simhony, and G. Feigin, "Measure-ment and Analysis of 2-D Infrared Natural Background," Appl.Opt. 24, 2109-2113 (1985).
14. A. Arcese, P. H. Mengert, and E. W. Trombini, "Image Detec-tion through Bipolar Correlation," IEEE Trans. Inf. Theory IT-16, 534-541 (1970).
15. G. R. Cross and A. K. Jain, "Markov Random Fields TextureModels," IEEE Trans. Pattern Anal. Machine Intell. PAMI 5,25-39 (1983).
16. W. K. Pratt, "Correlation Techniques of Image Registrations,"IEEE Trans. Aerosp. Electron. Syst. AES-10, 353-358 (1974).
17. B. V. K. Vijaya Kumar and A. Mahalanobis, "Alternate Inter-pretation of Minimum Variance Synthetic Discriminant Func-tions," Appl. Opt. 25, 2484-2485 (1986).
18. International Mathematics and Statistics Library, IMSL Inc.,Houston, TX.
19. A. K. Jain, "A Fast-Karhunen-Loeve Transform for a Class ofRandom Processes," IEEE Trans. Commun. COM-24, 1023-1029 (1976).
The authors gratefully acknowledge the support ofGeneral Dynamics Valley Systems Division for thisresearch through their Independent Research and De-velopment Funds.
