1. INTRODUCTIONPattern recognition, especially correlation-based algo-rithms for pattern recognition, has been an area of exten-sive research over the years. The traditional way to de-sign correlation filters has been to make filters thatoptimize different criteria.1–6 More recently, othersignal-processing techniques have found their way intothe optical pattern recognition community. Maximumlikelihood (ML), maximum a posteriori, and Bayes esti-mation have been used successfully to design powerful al-gorithms (processors) that are optimal from a statisticalperspective.7–10 The optimal processors are derived indifferent, but similar, ways. What they all have in com-mon is that they take the statistical distribution of thegray levels of the pixels in the image into account. Theprocessors have so far been derived from image modelsbased on particular well-behaved and well-known distri-butions such as the distributions of the exponential fam-ily.
As most image-processing systems are dealing with dis-crete data (the image intensity, for example, is oftencoded on 8 bits), image models based on continuous dis-tributions of gray levels are only approximations of thediscrete gray-level distribution in the actual image. Inpractice, this is no limitation if we are dealing with im-ages with hundreds of discrete levels; however, if we haveimages with a small number of levels we might be worriedabout the mismatch between the processor model and theactual discrete data. In some cases we would perhapsalso like to be able to model real gray-level distributions,that is, distributions that are not necessarily limited tothe group of continuous distributions that we have ana-lytical expressions for. It is thus clear that optimal pro-cessors that can be derived independent of the gray-leveldistribution and of the number of discrete levels in the in-put image would be useful. How to derive this set of pro-cessors is the topic of this paper.
In Section 2 we start by having a look at the imagemodel that will be used throughout the paper. In Sub-
sections 3.A and 3.B we go on to derive the optimal pro-cessor for two and three levels. In Subsection 3.C welook at the processor for images with N levels before inSubsection 3.D we illustrate the performance of the pro-cessor for five levels on images with an arbitrary distribu-tion as well as on images with a discrete Gaussian distri-bution. In Subsection 3.E we give an explanation of theresults that are presented. Section 4 contains a discus-sion of the results, and conclusions are presented in Sec-tion 5.
2. IMAGE MODELWe consider an image model made up of two distinct re-gions, the target region and the background region (this isthe statistically-independent-region model proposed byGoudail and Refregier10). It can be written mathemati-cally as
si 5 nitwi2j 1 ni
b~1 2 wi2j!, (1)
where nt is the noise on the target and nb is the noise onthe background and w is the so-called window functionsuch that w is equal to 1 within the target and equal to 0outside. The window function w was originally intro-duced by Javidi et al.8
The noise on the target and the background can be ofany arbitrary distribution. We have so far not imposedany limitation on the form that it can assume. If we as-sume a well-known continuous distribution, we can go onas described in Refs. 10–12 to obtain processors that areoptimal for Gaussian, x2, or other types of noise.13 How-ever, if we are facing images with an arbitrary distribu-tion or images with a low number of levels, we have toadopt a slightly different approach. A binary image willhave a distribution with spikes at the two points 0 and 1.This distribution of course cannot be modeled with anytraditional continuous model, but we can model it by us-ing the two discrete levels14 of Bernoulli noise, nt and nb ,which in the binary case will be given by
2000 Optical Society of America
H. Sjoberg and B. Noharet Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 1983
nit 5 H 1 with probability a
0 with probability 1 2 a, (2)
nib 5 H 1 with probability m
0 with probability 1 2 m. (3)
One can, of course, model any discrete distribution inthe same way. For an image with N levels, nt and nb willbe given by
nit 5 5
1 with probability a1
2 with probability a2
3 with probability a3
] ]
N with probability aN
0 with probability 1 2 (i51
N
a i
, (4)
nib 5 5
1 with probability m1
2 with probability m2
3 with probability m3
] ]
N with probability mN
0 with probability 1 2 (i51
N
m i
. (5)
It is easy to see that if N → ` and the spacing betweenthe discrete levels approaches zero we obtain a continu-ous distribution.
3. OPTIMAL PROCESSORSWe will now derive processors that are optimal for inputimages with discrete gray levels in a ML sense—ML pro-cessors. We will start by having a look at the ML proces-sor for binary images that was introduced in Ref. 14.
A. Maximum-Likelihood Processor for Two-Level InputImagesThe ML processor for two-level images is derived by maxi-mizing
L~ j ! 5 )iPwj
@sia1 1 ~1 2 si!~1 2 a1!#
3 )i¹wj
@sim1 1 ~1 2 si!~1 2 m1!#, (6)
where j is the position (in one dimension) of the targetthat we would like to estimate and where we have as-sumed independent pixel values. The derivations caneasily be extended to two dimensions, but we have forsimplicity used one-dimensional notation in this paper.This gives, after some straightforward calculations,
l~ j ! 5 ln L~ j !
5 Nr@ln~1 2 a1! 2 ln~1 2 m1!#
1 N ln~1 2 m1! 1 Ns@ln m1 1 ln~1 2 m1!#
1 w * s@ln a1 2 ln~1 2 a1!
2 ln m1 1 ln~1 2 m1!#, (7)
where N is the number of pixels in the image, Nr is thenumber of pixels in the reference, and Ns is the sum ofthe pixels in the input image. w * s means w correlatedwith s. If we now use ML estimation to estimate the nui-sance parameters a1 and m1 , we obtain
a1j 5
~w * s !j
Nr, (8)
m1j 5
Ns 2 ~w * s !j
N 2 Nr. (9)
What is interesting to note here is that the processorl( j) is a function of one correlation product, (w * s) j .Note also that the nuisance parameters are functions of jthrough the correlation factors.
The performance of this processor was reported in Ref.14 and will not be illustrated here. Instead we will gostraight on to have a look at the ML processor for imageswith three discrete levels.
B. Maximum-Likelihood Processor for Three-LevelInput ImagesThe ML processor for three levels is obtained by maximiz-ing
L~ j ! 5 )iPwj
F ~1 2 si!~2 2 si!
2~1 2 a1 2 a2!
1 ~2 2 si!sia1 1~1 2 si!si
22a2G
• )i¹wj
F ~1 2 si!~2 2 si!
2~1 2 m1 2 m2!
1 ~2 2 si!sim1 1~1 2 si!si
22m2G . (10)
Simple calculations give
l~ j ! 5 ln L~ j ! 5 p1~w * s !j 1 p2~w * s2!j
1 p3Nr 1 p4Ns 1 p5Ns2, (11)
where
p1 5 2 ln a1 232 ln~1 2 a1 2 a2! 2
12 ln a2 2 2 ln m1
132 ln~1 2 m1 2 m2! 1
12 ln m2 ,
p2 512 ln~1 2 a1 2 a2! 2 ln a1 1
12 ln a2 2
12 ln~1
2 m1 2 m2! 1 ln m1 212 ln m2 ,
p3 5 ln~1 2 a1 2 a2! 2 ln~1 2 m1 2 m2!,
p4 5 2 ln m1 232 ln~1 2 m1 2 m2! 2
12 ln m2 ,
p5 512 ln ~1 2 m1 2 m2! 2 ln m1 1
12 ln m2 . (12)
1984 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 H. Sjoberg and B. Noharet
s2 is the image obtained by squaring the pixels in s. Ns2
is the sum of the pixels in s2.Once again we use ML estimation to estimate the nui-
Fig. 1. Probability of correct location versus varying values of m.a 5 0.3.
Fig. 2. Probability of correct location versus varying values of m.a 5 0.3.
sance parameters and obtain
a ij 5
4~w * s !j 2 2~w * s2!j
2Nr, (13)
a2j 5
2~w * s !j 1 ~w * s2!j
2Nr, (14)
m1j 5
4~w * s !j 2 2~w * s2!j 2 4Ns 1 2Ns2
2Nr 2 2N, (15)
m2j 5
2~w * s !j 1 ~w * s2!j 1 Ns 2 Ns2
2Nr 2 2N. (16)
The performance of the processor for three levels is il-lustrated in Fig. 1. In these simulations we set a15 a2 5 a and m1 5 m2 5 m. In the figure we show theprobability of correct location plotted as a function of m.The value of a has been set to 0.3. We can see the drop inperformance as the values of m and a are equal. Theprobability of correct location was calculated over 500generated images for each value of m. The target is con-sidered to be correctly located if the estimated position,that is, the value of j, is exactly the same (same pixel) asthe true position. In Fig. 2 we show the results when a1 ,a2 , m1 , and m2 have different values. The parameters ofthe target are never exactly the same as the parametersof the background, and the performance does not drop tozero (a1 5 0.3, a2 5 0.4, m2 5 m1 2 0.1). In Fig. 3 weshow examples of images with different values of m, i.e.,a1 5 a2 5 a and m1 5 m2 5 m.
Fig. 3. (a) Scene with airplane. a 5 0.3 and m 5 0.06. (b) a 5 0.3 and m 5 0.16. (c) a 5 0.3 and m 5 0.26. (d) a 5 0.3 andm 5 0.36. (e) a 5 0.3 and m 5 0.46. (f) Binary reference target w.
H. Sjoberg and B. Noharet Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 1985
Once again what is important to note is that the opti-mal processor for three-level input images is a function oftwo correlation products, w * s and w * s2.
C. Maximum-Likelihood Processor for N-Level InputImagesThe trend that we saw from the processors for two andthree levels can in fact be generalized to an arbitrarynumber of levels, as we show in Table 1.
As any probability distribution can be written in theform of Eqs. (4) and (5) (either discretely or in the limit),what we have given here is a general form of the optimalML processor for the statistically-independent-regionmodel and white noise with arbitrary number of levels.One quickly realizes that the optimal processor for im-ages with, for example, 128 levels will, apart from requir-ing very long calculations, be computationally demand-ing, because 127 correlation products are needed. For asmaller number of levels, however, the optimal processorsare more within reach when it comes to computational de-mand and will also offer good performance, as was shownin Fig. 1. Fortunately, when we are dealing with imageswith a large number of levels, we have processors thatwere derived by use of continuous probability distribu-tions that approximate reality well enough to yield good,or even very good, performance.
D. Maximum-Likelihood Processor for Five-Level InputImagesIn this subsection we illustrate the performance of the op-timal processors in more detail. We have chosen to workwith the five-level optimal processor because five is a suf-ficiently large number of levels with which to displaygray-scale images at a reasonable gray-scale resolution.The full mathematical expression of the processor is givenin Appendix A. We will for simplicity and comprehen-sion point out only the general form of the five-level pro-cessor as a function of four correlation products:
l~ j ! 5 f5@w * s, w * s2, w * s3, w * s4#. (17)
1. Arbitrary Gray-Level DistributionWe will start by having a look at the processor when it isapplied to images with a noise distribution that cannot bewell approximated by any well-known continuous distri-bution. We generated the images by using the followingparameters: a1 5 a3 5 a4 5 0.2, m1 5 0.1, m2 5 0.05,m4 5 0.4, and m4 5 0.2. a2 was then varied between 0and 0.4 (meaning that a0 is at the same time varying be-tween 0.4 and 0). In Fig. 4 we show two examples of im-ages from the simulation: one with a2 5 0.05 and onewith a2 5 0.35. For each value of a2 we generate 500images and then calculate the probability of correct loca-tion.
In Fig. 5 we show four curves that we now explain anddiscuss. The processor called MLML is the five-level pro-cessor described above. The parameters a0 ,..., a4 andm0 ,..., m4 are estimated with ML estimation. The esti-mated parameter values are used in the ML estimation ofthe target position. We can see that the processor has anoverall high probability of correct location with a smalldip for a2 5 0.1. In every realistic case it will be neces-
sary to estimate the parameters just as we have done. Itis uncommon, not to say rare, to know the gray-level dis-tribution beforehand. In fact, if we did, it would be muchsimpler to use a classical linear filter to detect the target.
The case in which the parameters are known a priori isunrealistic, but we will nevertheless have a look at thiscase, as it will give some more insight into how the pro-cessor is working. In the general case in which we esti-mate the parameters, the parameters themselves will befunctions of the four correlation products. On the other
Fig. 4. (a) Scene with airplane. a1 5 a3 5 a4 5 0.2, m15 0.1, m2 5 0.05, m4 5 0.4, and m4 5 0.2. a2 5 0.05. (b)Same as in (a) but with a2 5 0.35.
Fig. 5. Performance of different versions of the optimal proces-sor on images with discrete arbitrary distribution. The proces-sor labeled MLML is the ML processor with ML estimates of theparameters. The processor labeled 4-factors is the ML processorwith known values of the parameters, and 2-factors is the pro-cessor when only the lower orders of correlation are included.The parameter values are known. The curve labeled 2-factorsest. is the ML processor when the parameters are estimated withonly lower orders of correlation. a1 5 a3 5 a4 5 0.2, m15 0.1, m2 5 0.05, m3 5 0.4, and m4 5 0.2.
Table 1. Influence of the Number of Levels in theInput Image on the Form of the Optimal
Processora
Number of Levels Processor
2 f2@w * s#
3 f3@w * s, w * s2#
4 f4@w * s, w * s2, w * s3#
] ]
N fN@w * s,..., w * sN21#
a Functions fi are different for different values of i.
1986 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 H. Sjoberg and B. Noharet
hand, if the parameters are known, the general expres-sion to be minimized can be simplified, and we are leftwith only four terms (see Appendix A), each one with onecorrelation product. As the correlation products arew * s, w * s2, w * s3, and w * s4, we will call them first- tofourth-order correlation products according to the expo-nent of the scene s. The first two correlation productswill be called lower-order correlation products and thelast two, higher-order correlation products. (Note thatthese correlation orders are not equivalent to the statisti-cal moments of the scene even though, as we will show inSubsection 3.E, there is a strong connection.) The ques-tion is which (or whether perhaps all?) of these correla-tion products must be included in the calculations if opti-mal performance of the processor is to be achieved. Itwould of course be beneficial in terms of speed if we couldobtain the optimal performance with only a limited num-ber of correlation products. This would greatly simplifythe optimal processor when it is applied to images with alarger number of levels. Unfortunately, as is shown inFig. 5, it is necessary to include all four terms in this caseto achieve the optimal performance (we will later have alook at an example where this is not the case). The curvein Fig. 5 labeled 4-factors shows the performance of theprocessor when the parameters are known. As can beseen, the performance of this processor with known pa-rameters is better than that for the processor where theparameters are estimated. This is of course not surpris-ing, as it is always better to use known values than to es-timate them. The curve labeled 2-factors shows the per-formance of the processor consisting of only the two termsthat include the first- and second- (lower-) order correla-
tions. The parameters are known in this case as well,but it is obvious that the two higher-order correlationproducts are of vital importance to the overall processorperformance. The curve labeled 2-factors est. shows theperformance of the processor when the parameters are es-timated but only the two first orders (instead of all fourorders) of correlations are used to estimate the param-eters. As can be seen, the performance is extremely poor(in fact equivalent to zero), and it is clear that the higherorders of correlation play an important role in the param-eter estimation.
2. Discrete Gaussian Gray-Level DistributionIn this Subsection we will look at the performance of theprocessor when it is applied to images with gray-level dis-tributions that, though discrete, are Gaussian. We syn-thesized the images by first generating five numbers yiaccording to
yi 5 expF2~xi 2 m !2
s2 G , (18)
where xi 5 @0, 1, 2, 3, 4#. The five numbers yi are thenrescaled so as to sum to 1. If the mean value m is chosento be 2, this will mean that a0 5 a4 (m0 5 m4) and a15 a3 (m1 5 m3). Examples of images are shown inFig. 6.
In Fig. 7 we show the results from the same simula-tions as for an arbitrary distribution but now applied toimages with the discrete Gaussian distribution. The re-sults are quite different from those obtained in Subsec-tion 3.D.2. As before, the curve labeled MLML shows the
Fig. 6. (a) Scene with airplane. mt 5 2, st 5 sb 5 2, and mb 5 0.5. (b) mt 5 2, st 5 sb 5 2, and mb 5 1.5. (c) mt 5 2, st 5 sb5 2, and mb 5 3.5. (d) mt 5 mb 5 2, st 5 1, sb 5 0.5. (e) mt 5 mb 5 2, st 5 1, and sb 5 0.9. (f) mt 5 mb 5 2, st 5 1, andsb 5 1.3.
H. Sjoberg and B. Noharet Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 1987
performance of the processor for which ML estimates ofthe parameter values are used. We can see a dip in per-formance when the value of the parameter that is varying(the mean of the background, mb) is equal to the mean ofthe target (mt). When this happens we have the samestatistics on the target and on the background, as sb5 st 5 2, and it is impossible to tell the two regionsapart.
The curve 4-factors shows the performance of the pro-cessor when the parameters are known. As above, theperformance is better in this case than when estimatesare used for the parameter values, just as we would ex-pect. The interesting result is shown in the curve labeled2-factors. It shows that the performance of the processorthat consists of only the two lower orders of correlationproducts is better than that of the optimal MLML proces-sor (it is important to remember that the parameters areknown in the 2-factor case, though). It is thus clear thatthe two terms of lower orders of correlation in themselvescarry enough information and that the addition of theterms of higher-order correlation does not more than mar-ginally improve the performance. In fact, in the simula-tions made above it would be possible to use only the termwith the lowest correlation order w * s (as the gray-leveldistributions of the target and the background differ fromeach other because they have different mean values) andstill see a smaller degradation than when the two lowestterms are used. When only the two lower-correlation-order terms are used in the estimates of the parameters,the resulting processor fails to detect the targets whenthe mean of the background is higher than the mean ofthe target (we can also notice a general decrease in theperformance of this processor). If we study the correla-tion planes more carefully we can see that for mb . mtwe obtain correlation planes that are inverted from theposition in which we would like to have them. Thismeans that there is a distinct peak in the correlationplane but that it is inverted, i.e., the peak has a valueclose to zero whereas the rest of the correlation plane hasa value much larger than zero. Because of the error inthe parameter estimates (which arises from the fact that
Fig. 7. Performance of different versions of the optimal proces-sor on images with discrete Gaussian distribution. The proces-sor labeled MLML is the ML processor with ML estimates of theparameters. The processor labeled 4-factors is the ML processorwith known values of the parameters, and 2-factors is the pro-cessor when only the lower orders of correlation are included.The parameter values are known. The curve labeled 2-factorsest. is the ML processor when the parameters are estimated withonly lower orders of correlation. mt 5 2, sb 5 2, and st 5 2.
we use only the two lower-correlation-order terms in theestimates), we do not get the desired correlation plane in-version that the parameters normally automatically en-sure takes place.
3. Optimal Processor for a Continuous GaussianDistributionGoudail and Refregier11 presented a processor optimal forwhite noise with random gray levels. In that paper theyderived an expression for a Gaussian gray-level distribu-tion. This processor is given by
FjGauss 5 ~st
2 2 sb2!@sc
2 * w# j
2 2~mbst2 2 mtsb
2!@sc * w# j , (19)
where sc is the zero-mean scene. The processor is similarto the approximation (including the two lower correlationorders) that was presented above, in that both processorsconsists of two correlation terms, w * s and w * s2 multi-plied by some parameters. Because of their similarity wewould expect similar behavior on images with Gaussiandistributions. The question is only how much the dis-crete nature of the five-level distribution will influencethe results. In Fig. 8 we show the results from simula-tions in which we compared the optimal Gaussian proces-sor with the MLML processor for five discrete levels. Theparameter that was varied is the mean value of the back-ground. The other parameters were mt 5 2 and st5 sb 5 2. As can be seen, the optimal Gaussian proces-sor (labeled Opt. Gauss) is better than the MLML proces-sor. One important point to remember, though, is thatthe optimal Gaussian processor, in the form used here, re-quires knowledge of the parameters st , sb , mt , and mb ,whereas the MLML processor estimates the required pa-rameter values. If we use known parameters in theMLML processor we will obtain the same performance asfor the optimal Gaussian processor. As we noted in Sub-section 3.D.2, it is possible to obtain the same perfor-mance with only the term that contains the first-ordercorrelation (w * s). This result is well in line with thatof Ref. 11, where it was shown that for Gaussian distri-butions in which the mean values differ between targetand background, it is possible to use only the first corre-lation term. In Ref. 11 it was shown that it is possible todetect targets with the same mean value as the back-ground if the variances differ, simply by using the secondcorrelation term w * s2 if the distributions are continuousGaussian.
If the gray-level distribution is not Gaussian but ratherarbitrary, we would expect the MLML processor to per-form better than the optimal Gaussian processor. This isshown in Fig. 9, where we indeed can see the better per-formance of the MLML processor. The parameters in thesimulations were a1 5 a3 5 a4 5 0.2, m1 5 0.1, m25 0.05, m3 5 0.4, and m4 5 0.2.
E. Explanation of the ResultsIn this subsection we will explain the difference in perfor-mance between the optimal MLML and the optimalGaussian processors. It will become clear under whichconditions each of the two processors will perform ad-equately.
1988 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 H. Sjoberg and B. Noharet
We start by having a look at the first correlation prod-uct, w * s. As we are correlating the scene s with a ref-erence that is binary (equal to one inside the contours ofthe target and equal to zero outside), the correlation valuein a pixel will be a measure (estimate) of the local meanvalue. It is local in the sense that it is the mean value ofthe pixels inside the reference centered on the pixel thatwe are observing. Actually, it is not the mean value thatis estimated but rather the sum of the pixels inside thereference. However, as we are dealing only with the dis-tribution of the sums rather than with their absolute val-ues, we may consider the correlation values (sums) to beestimates of the local mean value (the only difference is ascaling factor, Nr , equal to the number of pixels in thereference). Now, if we are considering the correlationvalue of the true position of the target in the input scene,it will be equal to the mean value of the target. Equally,if we are considering a position in the correlation planefar away from the target position, the correlation value inthat point will be an estimate of the mean value of thebackground about that point (by ‘‘far away’’ we mean apoint that is sufficiently far away from the target positionthat a target centered on this point does not overlap a tar-get centered on the true target position). This meansthat a single correlation plane such as w * s is simply theresult of a comparison of the mean of the target and themean of the background. If the mean of the target islower than the mean of the background we will obtain a
Fig. 8. Performance of the optimal Gaussian processor (Opt.Gauss) compared with the optimal MLML processor. The gray-level distributions on the images were discrete Gaussian, and theparameters were mt 5 2 and st 5 sb 5 2.
Fig. 9. Performance of the optimal Gaussian processor (Opt.Gauss) compared with the optimal MLML processor. The gray-level distributions on the images were arbitrary, with the follow-ing parameters: a1 5 a3 5 a4 5 0.2, m1 5 0.1, m2 5 0.05, m35 0.4, and m4 5 0.2.
correlation plane that is inverted, as described in Subsec-tion 3.D.2. This is the reason for the parameters that ap-pear in front of the correlation product w * s in the ex-pressions for the optimal MLML processor and theoptimal Gaussian processor. They ensure that the corre-lation plane is inverted, if needed, so that the peak can bedetected. This line of reasoning can now be extended tothe correlation products w * s2, w * s3 and w * s4, wherethe mean values are now calculated after the pixel valueshave been raised to the appropriate power.
In summary, what we obtain from the correlation prod-ucts are
w * s → E@sz#, (20)
w * s2 → E@sz2#, (21)
w * s3 → E@sz3#, (22)
w * s4 → E@sz4#, (23)
where E@ # is the expectation operator and z P @t, b#,where t and b indicate target and background regions, re-spectively.
E@szi #, i P @1...4#, can be expressed in the form of the
first four statistical moments of the distribution of thescene through
E@sz# 5 v1 , (24)
E@sz2# 5 v2 1 v1
2, (25)
E@sz3# 5 v3 1 3v2v1 1 v1
3, (26)
E@sz4# 5 v4 1 4v3v1 1 6v2v1
2 1 v14, (27)
where v1 5 E@sz# is the mean value of sz , vz 5 E@(sz2 E@sz#)
2# is the variance, and v3 5 E@(sz 2 E@sz#)3#
and v4 5 E@(sz 2 E@sz#)4# are the third- and fourth-
order moments, respectively.This means that the lowest-order correlation product
corresponds to a comparison of mean values of the targetand of the background. The second-order correlationcompares both the mean and the variance of the targetand the background. The third-order one compares thefirst three moments, and the fourth-order one comparesthe first four. Now, if the mean values of the target andthe background are equal, it is necessary to add a second-order correlation so that the variances also are compared.If both mean and variance are equal in the two regions wemust add a third-order correlation, and so on for thehigher moments.
We will now have a look at two of the cases discussedabove, the first when we consider a discrete Gaussian dis-tribution and the second when we consider an arbitrarydistribution and the optimal Gaussian processor fails.
In Fig. 8 we see the performance of the two processorswhen the mean value of the background is varying.When the mean value of the background, mb , is equal to2, the distribution on the target is equal to the distribu-tion on the background. When this happens, the prob-ability of correct location of course drops to zero as can beseen from Fig. 8. When the mean value of the back-ground is equal to 3.5, we see that both processors per-form well as the mean values of the target and back-ground are different. This is well in line with the
H. Sjoberg and B. Noharet Vol. 17, No. 11 /November 2000 /J. Opt. Soc. Am. A 1989
Table 2. Four First Moments of the Target and Background Regions for Different Valuesof the Parametersa
explanation above stating that only a first order correla-tion is needed when the mean values are different (only afirst-order correlation is used in the optimum Gaussianprocessor, as we have sb 5 st 5 2). In Table 2 we showthe first four moments for two values of the mean of thebackground. The moments were calculated from theprobability distributions and thus are not estimates fromany particular realization. As we can see the four mo-ments are identical for mb 5 2, and it is thus also impos-sible for the MLML processor to detect the targets eventhough it uses more higher-order correlations than doesthe optimal Gaussian processor. When mb 5 3.5 we seethat all four moments are different for the target and thebackground, in agreement with the explanations put for-ward above.
Also listed in Table 2 are the first four moments for theimages used in the simulation presented in Fig. 9. Inthis simulation it is shown that the performance of the op-timal Gaussian processor drops for certain values of theparameter a2 , whereas the performance of the MLML re-mains high for the same values. This indicates that thetwo lower-order moments are roughly the same in the tar-get and background regions for these values of a but thatthere is a difference in the third- or fourth-order mo-ments, making it possible for the MLML processor to lo-cate the targets correctly. That this is in fact the case isshown in Table 2, where we can clearly see the differencein the third-order moments for a2 5 0.15. There is asmall difference for the mean and the variance as well,explaining why the probability of correct location does notdrop to zero for the optimal Gaussian processor. If a25 0.4 we can see that the variances of the target andbackground regions have been separated, and, conse-quently, as shown in Fig. 9, the performance of the opti-mal Gaussian processor increases again.
The performance of processors that use lower- orhigher-order correlations, such as the optimal MLML pro-cessor and the optimal Gaussian processor, can thus beexplained with a moment analysis approach. We canalso draw the conclusion that the optimal MLML proces-sor will detect targets as soon as there is a difference inany of the moments included in the processor.
4. DISCUSSIONIn this paper we have presented a new group of proces-sors that are optimal for an arbitrary number of discrete
gray levels. As the gray-level distribution is allowed tobe discrete, it is possible to model any distribution (a con-tinuous distribution can be modeled by using a very largenumber of discrete levels). The processors are thus notlimited to any of the smooth, continuous distributionsthat have so far been used. Gaussian distributionsmodel thermal noise in detectors well, but in many realsituations a Gaussian distribution will not be sufficient tomodel the target or the background. Also, in many casesthe information that we have about the gray-level distri-bution of the target is of minor use because the internalstructure of the target has been distorted by reflectionsfrom the Sun or by shadows. The processors presentedin this paper do not need any a priori information aboutthe gray levels of the target or the background, becausethey estimate them (and thus their distribution) in a MLsense. The only information needed is the binary shapeof the target.
Some image models15 include additive noise coming, forexample, from detectors or electronics. It is interestingto note that our model with discrete distributions takesadditive noise into account, though not explicitly. As weestimate the probability of each gray level within the tar-get and the background, additive noise will be included inthese estimates. The proposed processors are thereforerobust to additive noise.
As the processors are based on a number of correlationproducts—the number is equal to the number of gray lev-els minus 1—it is clear that they will be computationallyextensive for input images with a high number of levels.For a smaller number of levels, on the other hand, use ofthe processors will be more than feasible in terms of com-putational requirements. Images with a small number ofgray levels will probably be more common in the futurewhen the bottleneck in processing systems will be infor-mation transfer rather than processing speed. In opticalprocessing, for example, the limitation of the new, fastmultiple-quantum-well spatial light modulators16 lies ingetting the information to the spatial light modulatorrather than in the switching speed.
For the sake of simplicity and comprehensiveness, inthis paper we derived the processors by using a ML ap-proach. This means that the processors described in thispaper are optimized for images with homogeneous back-grounds. If they were applied to images with nonhomo-geneous backgrounds they would fail if the structural sizeof the background objects were of approximately the same
1990 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 H. Sjoberg and B. Noharet
size as or larger than that of the target. However, theapproach presented here could easily be extended to theML ratio test technique14 to produce processors thatwould be adapted to nonhomogeneous images. This is anarea that is currently under study in our lab.
It is interesting to note the similarities in the basicform of the optimal processors derived here and the pro-cessors presented in Refs. 11 (optimal Gaussian proces-sor) and 17 (polynomial correlation filters). We are cur-rently investigating in what respect, if any, these filtersare related and whether they can be seen as special casesof the optimal processor for an arbitrary number of graylevels.
5. CONCLUSIONIn this paper we have presented a group of processorsthat are optimal in a maximum likelihood sense for homo-geneous images with an arbitrary number of gray levelswith arbitrary discrete distributions. The processors are
based on a number of correlation products, the numberdepending on the number of gray levels in the input im-ages. We derived the processors for two, three, and fivelevels and illustrated the performance of the three- andfive-level processors on different types of image. Weshowed that the processors for three and five levels showgood performance on images with arbitrary gray-level dis-tributions (we are thus not limited to the well-known con-tinuous distributions from the exponential family). Wealso compared the processor for five levels with the opti-mal Gaussian processor11 on images that have a discreteGaussian distribution. The performance of the two pro-cessors on these images is almost the same, showing thatthe discretization of the Gaussian distribution will not de-grade the performance of the optimal Gaussian processor.However, if the gray-level distribution is not Gaussian,the processor MLML for five levels will show superior per-formance. We also give an explanation, based on mo-ment analysis, of the difference in performance betweenthe processors.
APPENDIX AIn this appendix we derive the full expression of the five-level MLML processor. Starting with the following expressionfor the logarithm of the likelihood function,
l~ j ! 5 (iPwj
F ~1 2 si!~2 2 si!~3 2 si!~4 2 si!
24ln~1 2 a1 2 a2 2 a3 2 a4! 1
si~2 2 si!~3 2 si!~4 2 si!
6ln a1
1si~1 2 si!~3 2 si!~4 2 si!
24ln a2 1
si~1 2 si!~2 2 si!4 2 si)
6ln a3 1
si~1 2 si!~2 2 si!~3 2 si!
224ln a4G
1 (i¹wj
F ~1 2 si!~2 2 si!~3 2 si!~4 2 si!
24ln~1 2 m1 2 m2 2 m3m4! 1
si~2 2 si!~3 2 si!~4 2 si!
6ln m1
1si~1 2 si!~3 2 si!~4 2 si!
24ln m2 1
si~1 2 si!~2 2 si!~4 2 si!
6ln m3 1
si~1 2 si!~2 2 si!~3 2 si!
224ln m4G ,
(A1)we will after some simple but lengthy calculations arrive at the following expression, l( j), that should be maximized:
If instead of estimating the parameter values we use values that are known, we can simplify the expression considerably.The only terms in l( j) that will depend on j are the first four, and we are left to maximize the following four terms:
1992 J. Opt. Soc. Am. A/Vol. 17, No. 11 /November 2000 H. Sjoberg and B. Noharet
l~ j ! 5 t1~w * s !j 1 t2~w * s2!j 1 t3~w * s3!j
1 t4~w * s4!j . (A6)
*Also with the Department of Physics–Optics, RoyalInstitute of Technology, Stockholm, Sweden.
REFERENCES1. P. Refregier, ‘‘Filter design for optical pattern recognition:
multicriteria optimization approach,’’ Opt. Lett. 15, 854–856 (1990).
2. P. Refregier, ‘‘Optimal trade-off filters for noise robustness,sharpness of the correlation peak, and Horner efficiency,’’Opt. Lett. 16, 829–831 (1991).
3. A. Mahalanobis, B. V. K. Kumar, and D. Casasent, ‘‘Mini-mum average correlation energy filters,’’ Appl. Opt. 26,3633–3640 (1987).
4. B. V. K. Kumar, ‘‘Minimum-variance synthetic discrimi-nant functions,’’ J. Opt. Soc. Am. A 3, 1579–1584 (1986).
5. B. Javidi and J. Wang, ‘‘Optimum filter for detecting a tar-get in multiplicative noise and additive noise,’’ J. Opt. Soc.Am. A 14, 836–844 (1997).
6. B. Javidi and J. Wang, ‘‘Optimum distortion-invariant filterfor detecting a noisy distorted target in nonoverlappingbackground noise,’’ J. Opt. Soc. Am. A 12, 2604–2614(1995).
7. B. Javidi, A. H. Fazlollahi, P. Willet, and P. Refregier, ‘‘Per-formance of an optimum receiver designed for pattern rec-ognition with nonoverlapping target and scene noise,’’ Appl.Opt. 34, 3858–3868 (1995).
8. B. Javidi, P. Refregier, and P. Willet, ‘‘Optimum receiverdesign for pattern recognition with nonoverlapping targetand scene noise,’’ Opt. Lett. 18, 1660–1662 (1993).
9. F. Guerault and P. Refregier, ‘‘Unified statistically indepen-dent region processor for deterministic and fluctuating tar-gets in nonoverlapping background,’’ Opt. Lett. 23, 412–414(1998).
10. F. Goudail and P. Refregier, ‘‘Optimal detection of a targetwith random gray levels on a spatially disjoint noise,’’ Opt.Lett. 21, 495–497 (1996).
11. F. Goudail and P. Refregier, ‘‘Optimal and suboptimal de-tection of a target with random gray levels imbedded innon-overlapping noise,’’ Opt. Commun. 125, 211–216(1996).
12. F. Guerault and P. Refregier, ‘‘Optimal x2 filtering methodand application to targets and backgrounds with randomcorrelated gray levels,’’ Opt. Lett. 22, 630–632 (1997).
13. P. Refregier, O. Germain, and T. Gaidon, ‘‘Optimal snakesegmentation of target and background with independentgamma density probabilities, application to speckled andpreprocessed images,’’ Opt. Commun. 137, 382–388 (1997).
14. H. Sjoberg, F. Goudail, and P. Refregier, ‘‘Optimal algo-rithms for target location in nonhomogeneous images,’’ J.Opt. Soc. Am. A 15, 1–100 (1998).
15. N. Towghi, B. Javidi, and J. Li, ‘‘Generalized optimum re-ceiver for pattern recognition with multiplicative, additive,and nonoverlapping background noise,’’ J. Opt. Soc. Am. A15, 1557–1565 (1998).
16. K. Kang, J. S. Powell, R. D. Stack, C. G. Garvin, J. A.Trezza, J. C. Kirsch, and B. K. Jones, ‘‘Optical image corre-lation using high-speed multiple quantum well spatial lightmodulators,’’ in Optical Pattern Recognition X, D. P.Casasent and T.-H. Chao, eds., Proc. SPIE 3715, 97–107(1999).
17. A. Mahalanobis and B. V. K. Kumar, ‘‘Polynomial filters forhigher order correlation and multi-input information fu-sion,’’ in Optoelectronic Information Processing, PressMonograph PM54, P. Refregier and K. Javidi, eds. (SPIEPress, Bellingham, Wash., 1997), pp. 221–232.
