Goudail et al. Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1231
Bhattacharyya distance as a contrast parameterfor statistical
processing of noisy optical images
Francois Goudail, Philippe Refregier, and Guillaume Delyon
Physics and Image Processing Group, Fresnel Institute, Unite Mixte de Recherche 6133, and Ecole Generalisted’Ingenieurs de Marseille, Domaine Universitaire de Saint-Jerome, 13397 Marseille Cedex 20, France
Received November 7, 2003; revised manuscript received January 29, 2004; accepted February 11, 2004
1. INTRODUCTION AND POSING OF THEPROBLEMA. IntroductionThe development of new imaging systems has been a topicof growing interest during the past few years. Since themeasured optical images can be perturbed by strong fluc-tuations or noise, they may require specific processing al-gorithms that are in general strongly nonlinear. To char-acterize the performance of the algorithms as well as thequality of the processed optical images, it is very useful tobe able to define a contrast parameter between differentregions in images. However, since the noise may not beadditive Gaussian, the definition of a contrast can beproblematic. For example, active-coherent imaging sys-tems lead to images degraded by speckle noise.1 Fur-thermore, when the intensity level of the reflected light isvery low, the quantum nature of the light leads to an ad-ditional source of fluctuations of the detected signal.Combined with fully developed speckle effect, it producesthe so-called geometrical noise distributions that also re-sult in large fluctuations.2,3
In Ref. 4 a rigorous method for determining the mini-mal set of parameters that characterize the difference be-tween two regions with arbitrary noise distributions hasbeen proposed. In that approach, it was imposed thatdifferent noise configurations with the same contrast pa-rameters necessarily lead to the same performance for de-tection or localization of a signal. This is clearly a desir-able property for a contrast measure. However, thisconstraint does not guarantee the existence of a scalarcontrast measure. Indeed, in the case of coherent-polarization imaging, this approach shows that the per-formance is a function of two parameters, but it does notlead to a single contrast parameter.4 Furthermore, thisapproach is based on statistical group invariance and
thus requires that the probability density function (pdf) ofthe fluctuations possess such an invariance group. Thisis not the case, for example, of the Poisson or geometricalprobability laws that are useful to describe low-flux im-ages.
For imaging system optimization, it is in general moreconvenient to define a scalar contrast parameter that canbe more easily optimized than a set of scalar values. Theprice to pay can be that the contrast parameter only ap-proximately satisfies the property that different imageconfigurations with the same contrast parameter lead toequivalent performance.
A standard approach to defining a contrast parameterconsists of considering the Fisher ratio between two re-gions. We will show that this approach is far from satis-factory. On the other hand, statisticians developed veryearly different approaches to defining a measure of diffi-culty for discrimination problems.5–8 One can cite, forexample, the Stein and Chernoff bounds andapproximations9 of the asymptotic behavior of detectionand discrimination performance.
More recently, an approximation of the area under thereceiver operating characteristic (ROC) as a function ofthe Bhattacharyya distance has been proposed for detec-tion applications.10–13 This approach has been developedfor the ideal observer and consists of detection tasks whenthe target and background statistical parameters areknown. This situation is of great theoretical interestsince it provides upper bounds for the performance, but itdoes not correspond to many practical applications.
In this paper, we propose to demonstrate, with numeri-cal simulations and for noncorrelated noise, that theBhattacharyya distance is an efficient contrast parameterbetween two regions in the image that have fluctuationsthat belong to the same family of probability laws, but
2004 Optical Society of America
1232 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Goudail et al.
that have different and unknown parameters. More pre-cisely, we show that for detection, location, or segmenta-tion algorithms based on the maximum likelihood or onthe minimum-description length (MDL) principle,14 andfor the realistic cases where the target and backgroundstatistical parameters are unknown, the Bhattacharyyadistance characterizes the image difficulty independentlyof the family of probability laws as long as they have thesame number of unknown parameters. This result canbe viewed as an extension of the use of the Bhattacharyyadistance to more realistic scenarios than the ideal ob-server, and it opens new perspectives for imaging systemoptimization. Indeed, this result is a first step towarddefining a general contrast parameter since, in general,more complex situations than the ones discussed here canappear such as spatially correlated noise with a large va-riety of pdf ’s.
B. Posing of the ProblemWe will assume in this paper that we want to perform anautomatic processing task on an image, such as detection,localization, or segmentation. Let us first consider thesimple example of detection, in a classical intensity im-age, of a target with known shape on a homogeneousbackground. Suppose that the image xiui P @1, NT# iscomposed of NT pixels, the target region a is composed ofNa pixels, and its gray levels form a random field with pdfPa(x). Similarly, the background region b is composed ofNb pixels (with NT 5 Na 1 Nb) and its gray levels aredistributed with pdf Pb(x). We will assume that the graylevels of each pixel are statistically independent.
We define a performance measure as being the result ofan averaging over all the possible realizations of the ran-dom field that defines the image model [i.e., over Pa(x)and Pb(x)] and the optimal performance as being the bestperformance that can be achieved.
Let us assume that a bijective transformation T is ap-plied to xi such that the gray-level pdf becomes Pa8(x) inthe target region and Pb8(x) in the background region. Itcan be shown (see, for example, Ref. 4) that the optimalprocessing performance obtained on average over all therealizations of the transformed image Txi is the same asthat obtained with the realizations of the original imagexi . A direct consequence of this property is that anygood contrast parameter should be invariant by bijectivetransformation of the gray levels. Indeed, let us considerthe two situations @Pa(x), Pb(x)# and @Pa8(x), Pb8(x)#,which can be related by a bijective transformation: Sincethey correspond to the same optimal processing perfor-mance, they should correspond to the same contrast pa-rameter.
This discussion raises different interesting questions:Is it possible to obtain a contrast parameter that is, evenapproximately, valid for pure detection tasks, target loca-tion, boundary localization, or segmentation of differentregions? Does this contrast parameter allow one to com-pare different imaging systems with different noise statis-tics such as, for example, a system perturbed by additiveGaussian noise with another one perturbed by specklenoise? Is this contrast parameter still valid for suboptimalalgorithms when the pdf parameters are unknown? Wewill show in the following, with numerical simulations
and for spatially noncorrelated noise, that positive an-swers to these questions can be obtained at least partially.
2. BASIC CONTRAST DEFINITIONS ANDPROPERTIESIn this section, we first summarize the general require-ments for a scalar contrast parameter, and we then pro-pose to analyze three different information-theoretic mea-sures: the Fisher ratio, the Kullback divergence, and theBhattacharyya distance.6–8,15,16
A. General Requirements for a Contrast MeasureThe image processing problems we will consider consist ofdetecting or segmenting a known object in a noisy image.We will assume that the noise present in the system isdue to the optical system and that the pdf of both the tar-get and background regions belong to the same family.In other words, we assume that they can differ only bytheir parameter values. We thus do not consider thecase—which could be useful in some applications—inwhich the target and the background pdf belong to differ-ent families.
The discussion of Subsection 1B shows that differentnoisy optical systems with the same contrast parametermust correspond to the same performance measure. Inother words, the relation between the performance mea-sure and the contrast parameter should be bijective andshould be the same for different families of gray-level pdf.
In that case, two different imaging systems perturbedby noises described with different pdf ’s can be comparedby using this contrast parameter. On the other hand, ifthe relation between the performance measure and thecontrast parameter were different for each family of noisepdf, then looking at the contrast parameter would not besufficient to compare the different imaging systems. Inthat case, the precise relation between the performancemeasure and the contrast parameter should be deter-mined for each pdf and for each processing task to be per-formed, which may not always be easy. A direct conse-quence of this requirement is that the contrast parametershould be invariant under bijective transformations of thegray levels as discussed in Subsection 1.B.
B. Scalar Candidates for Contrast ParametersMany different information-theoretic measures can beconsidered for contrast parameter definition.9,6 Here wewill analyze the Fisher ratio, which is often considered asignal-to-noise ratio, and the Kullback divergence and theBhattacharyya distance, both of which are involved insimple results of information theory. Let us first recallthe definition of these measures and some of their prop-erties.
Fisher ratio. The Fisher ratio is simply the ratio of thesquared difference of the means ma and mb of the twopdf ’s to the sum of their variances sa
2 and sb2:
F 5~ma 2 mb!2
sa2 1 sb
2 (1)
The Fisher ratio represents a rigorous signal-to-noise ra-tio when the pdf ’s of the two regions are Gaussian with
Goudail et al. Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1233
identical variances.4 For other cases, it still representsan empirical evaluation of the ‘‘separability’’ between thetwo distributions and is often used as an empirical esti-mate of the signal-to-noise ratio. One of the main draw-backs of the Fisher ratio is that it is not invariant underbijective transformations of the pixel gray levels.
Kullback divergence. The Kullback–Leibler measurebetween continuous-valued statistical distributions is de-fined as
D@PpaiPpb# 5 E Ppa~x !lnFPpa~x !
Ppb~x !Gdx, (2)
Table 1. Pdf’s Used in This Paperand Their Corresponding Parametersa
Law PdfFree
Parameters
FixedParameters
(if any)
Bernoulli pd (x) 1 (1 2 p)d (1 2 x) p None
Poisson (nPN
d ~x 2 n !exp~2l!ln
n!l None
Gamma SL
mDL xL21
G~L!expF2L
mxG m L
Weibullmxm21
mmexpF2S x
mDmG m m
Gaussian1
A2psexpF2~x 2 m!2
2s 2 G m, s None
Geometric (nPN
d ~x 2 n !p~1 2 p !n p None
a d (x) is the Dirac distribution and N is the set of integers. ForWeibull and Gamma laws m and L are assumed to be known and equal inboth target and background regions for obtaining the reduced parameter.
where we have introduced the notations Ppa and Ppb forthe pdf ’s in the target and background regions to empha-size their dependency on the pdf parameters pa and pb.In the case of discrete, random variables defined withprobability laws, a summation has to be considered in-stead of the integral. The Kullback–Leibler measure isinvolved, for example, in the Stein’s Lemma9 for detectionperformance, which provides an asymptotic behavior (forlarge N) of the probability of detection or of false alarm.However, the distributions Ppa(x) and Ppb(x) do not playsymmetric roles in the Kullback–Leibler measure, whichmay be an undesirable property for a contrast parameter.We will thus consider the Kullback divergence, which issymmetric and defined as:
K 5 D@PpaiPpb# 1 D@PpbiPpa#. (3)
Bhattacharyya distance. Let us first introduce theChernoff measure,
C~s ! 5 2lnF E Ppas
~x !Ppb12s
~x !dx G , (4)
where 0 , s , 1. Let C* be the maximum value of C(s).Here again, in the case of probability laws, a summationhas to be considered instead of the integral. This mea-sure is of interest in detection theory because of the Cher-noff bound, which states that exp(2NC* ) defines an upperbound and constitutes an asymptotic exponent on theprobability of error in discrimination problems.9,17 Themain drawback of the Chernoff measure is that it de-pends on the parameter s* , which maximizes C(s).However, it can be shown that in many cases, one hass* . 0.5, so that C* . C(1/2).8 C(1/2) defines the Bhat-tacharyya distance B:
B 5 2lnF E @Ppa~x !Ppb~x !#1/2dx G . (5)
Table 2. Expression of the Fisher Ratio F, the Kullback Divergence K, and theBhattacharyya Distance B for Different Gray-Level Pdf’sa
Law F K B
Bernoulli~ pa 2 pb!2
paqa 1 pbqb~ pa 2 pb!lnFpaqb
pbqaG 2ln@Apapb 1 Aqaqb#
Poisson~la 2 lb!2
la 1 lb~la 2 lb!lnFla
lbG 1
2~Ala 2 Alb!2
Gamma L~r 2 1 !2
r2 1 1 LS Ar 21
ArD 2
L lnF12 SAr 11
ArD G
Weibull A~r 2 1 !2
r2 1 1 F ~Ar !m 2 S 1
ArD mG 2
lnH1
2 F~Ar !m 1 S 1
ArD mG J
Gaussianb
1 1 a
1
2
b~1 1 a!
a1
1
2 S Aa 21
AaD 2
1
4
b
1 1 a1
1
2lnF12 SAa 1
1
AaD G
Geometric
~ pa 2 pb!2
qapb2 1 qbpa
2
pb 2 pa
papblnSqa
qbD lnF1 2 Aqaqb
ApapbG
a The signification of the pdf parameters is defined in Table 1. A 5 @G(1/a)#2/$2aG(2/a) 2 @G(1/a)#2%, G(u) 5 *O1`xu exp(2x)dx, qu 5 1 2 pu ,
r 5 ma /mb , a 5 sa2/sb
2, and b 5 @(ma 2 mb)2#/sb2.
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It is interesting to note that both the Kullback–Leiblermeasure and the Chernoff measure belong to the family off-divergences,6,18 which can be written
E HFPpa~x !
Ppb~x !GPpa~x !dx. (6)
F-divergences are invariant under bijective transforma-tions of the pixel gray levels.9 This is also the case of theKullback divergence and the Bhattacharyya distance,which clearly corresponds to a property required in Sub-section 2.A.
For the pdf ’s listed in Table 1, we have listed in Table 2the expressions of the three considered measures.
3. EVALUATION OF CONTRASTPARAMETERSWe propose in this section to determine whether, amongthe three listed candidates, there exists a contrast param-eter definition that approximately satisfies the require-ments of Subsection 2.A. For that purpose, we succes-sively consider detection, location, and segmentationtasks.
A. Detection CapabilitiesWe propose to study detection of a target whose gray lev-els are distributed with pdf parameter pa over a back-ground with the same type of pdf but parameter pb. Ef-ficient detection algorithms can be designed by using themaximum-likelihood-ratio test.17,19 For this purpose asubwindow defined by a binary mask W is scanned overthe image. This subwindow W has to correspond to theshape of the target. One can note that if W is larger thanthe target shape, the assumption that the target pixels’gray levels are identically distributed will no longer bevalid. On the other hand, if W is smaller than the targetshape, the background pixels should not be considered asidentically distributed. For each position of W, we take adecision between the two hypotheses that there is a targetor not at this position.
When the parameters pa and pb are known, the deter-mination of the maximum-likelihood-ratio test simplyconsists in computing for each location of W the expres-sion
L 5 (iPW
log@Pp~a !~xi!# 2 (iPW
log@Ppb~xi!#. (7)
This strategy is optimal in the Neyman–Pearson sensesince, for a given false alarm rate, it optimizes the detec-tion probability. The detection test defined in Eq. (7) isoften denominated the ideal observer. A simple and clas-sical way to represent the detection performance is thusto draw the detection probability as a function of the falsealarm probability, which is known as the receiver operat-ing characteristic (ROC). Although sufficient to charac-terize fully the detection performance, the ROC does notprovide a simple scalar figure of merit. A common way toobtain such a scalar figure of merit is to compute the areaunder the curve (AUC) of the ROC. However, its explicitmathematical expression is in general not easy to obtain,
even for the ideal observer. Recently, the following ap-proximation of the AUC of the ideal observer has beenproposed10:
AUC . 12 1
12 erf@~2NaB!1/2#, (8)
where erf@.# is the error function and B is the Bhatta-charyya distance.
We have plotted in Fig. 1 the AUC values against theFisher ratio, the Kullback divergence, and the Bhatta-charyya distance for the different noise pdf ’s listed inTable 1. Each point on the curves corresponds to a givenparameter configuration for the target and the back-ground and to a given pdf type. One can see that theAUC is approximately represented by a bijective functionfor both the Bhattacharyya distance B and Kullback di-vergence K but not for the Fisher ratio. Furthermore,one can see that the relation between AUC and B is wellapproximated by Eq. (8). This is a first indication thatthe Bhattacharyya distance constitutes a good candidatefor defining a contrast parameter.
As mentioned above, the ideal observer is not represen-tative of most practical situations in which the param-eters pa and pb are, in general, unknown. In that case,one can apply a suboptimal strategy such as thegeneralized-likelihood-ratio test.20 This approach con-sists of substituting for the unknown target and back-ground parameter values their maximum-likelihood esti-mates (MLEs). This is a simple and efficient estimationtechnique that possesses interesting optimal properties inthe exponential family.21 To estimate the background pdfparameters, we need to consider pixels from this region.For this purpose a subwindow defined by a binary mask Fis scanned over the image. The subwindow F is com-posed of two disjoint regions W and W so that F5 W ø W, and where W still defines the shape of the tar-get. If pa and pb are the MLEs of pa and pb in W and W,and if p(c) is the MLE of pb in F, one obtains thegeneralized-likelihood-ratio test:
L 5 (iPW
log@P pa~xi!# 1 (iPW
log@P pb~xi!#
2 (iPF
log@P p~c !~xi!#. (9)
We have plotted in Fig. 2 the AUC values against theFisher ratio, the Kullback divergence, and the Bhatta-charyya distance for the pdf and parameter configura-tions considered in Fig. 1. One can see that the AUC ofthe generalized-likelihood-ratio test is still well repre-sented by a bijective function of B and K, but not of theFisher ratio. A more careful analysis shows that this ap-proximate bijective relation is slightly better between theAUC and the Bhattacharyya distance than with the Kull-back divergence, and that this property holds for givenvalues of Na and Nb (see Ref. 3) and for a fixed number ofunknown parameters for the pdf (see Fig. 2). More pre-cisely, the relation between AUC and B is bijective forGaussian pdf with identical variances on the target andon the background; for Gamma and Weibull pdf ’s; and forPoisson, Bernoulli, and geometric probability laws. Onthe other hand, another bijective relation exists for
Goudail et al. Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1235
Gaussian pdf ’s with different variances on the target andon the background. One can thus conjecture that theAUC is well approximated by a bijective function of theBhattacharyya distance for constant (Na , Nb , g) values(where g is the number of scalar unknown parameters ofthe pdf), and thus the Bhattacharyya distance again con-stitutes a good candidate for defining a contrast param-eter.
B. Location CapabilitiesLet us now evaluate the efficiency of the Bhattacharyyadistance as a contrast parameter for edge location in aone-dimensional signal. More precisely, let us consider asignal with Na contiguous pixels, spanning from abscissa1 to abscissa Na , distributed with PDF Ppa(x), and fol-lowed by Nb contiguous pixels, spanning from abscissa
Fig. 1. AUC obtained with the ideal observer as a function ofdifferent statistical distances: Fisher ratio, Kullback diver-gence, and Bhattacharyya distance. Different types of noise sta-tistics and parameter combinations have been considered. Theconsidered task was detection of a target with Na 5 4 pixels.Each ROC has been estimated from 104 random experiments.
Na 1 1 to abscissa N, distributed with pdf Ppb(x). Thetask is to determine the position of the edge in this signal,that is, the abscissa t 5 Na 1 1/2 at which the statisticaldistribution of the pixels changes from pa to pb. For thatpurpose, we will consider the series of hypotheses Ht thatassume that the data ranging from abscissa 1 to t [regiondefined by the mask W(t)] are distributed with param-eter pa and the data ranging from abscissa t 1 1 to N [re-gion defined by the mask W(t)] are distributed with pb.As in Subsection 3.A, we will use the pseudolikelihoodL(t) of hypothesis Ht defined from the MLEs pa and pb ofthe parameters pa and pb estimated respectively in W(t)and in W(t). The expression of this function is
L~t! 5 (iPW~t!
log@P p~a !~xi!# 1 (iPW~t!
log@P pb~xi!#.
(10)
The estimate of the edge position is determined as t
5 arg maxt@L(t)#.
Fig. 2. AUC obtained with the generalized-likelihood-ratio testas a function of different statistical distances: Fisher ratio,Kullback divergence, and Bhattacharyya distance. The consid-ered noise configurations are the same as in Fig. 1. The size ofthe background region is Nb 5 20.
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As an indicator of the edge location quality, we will usethe standard deviation of the edge location estimate: st
5 @^(t 2 ^t&)2/2, where ^•& denotes ensemble averag-ing. In Fig. 3 we have plotted the value of st as a func-tion of the Fisher ratio, the Kullback divergence, and theBhattacharyya distance for the different types of pdf de-fined in Table 1. These experiments have been per-formed with Na 5 Nb 5 50. One can see in the graphcorresponding to Bhattacharyya distance (lower graph)that the points gather on one main curve. From these re-sults, it is clear that for edge location, the Bhattacharyyadistance is a relatively good contrast parameter.
C. Segmentation CapabilitiesTo show that the Bhattacharyya distance constitutes agood candidate to define a contrast parameter for segmen-tation purposes, we propose to analyze segmentation per-
Fig. 3. Standard deviation of the location estimate obtainedwith L(t) as a function of different statistical distances: Fisherratio, Kullback divergence, and Bhattacharyya distance. Differ-ent types of noise statistics and parameter combinations havebeen considered. The considered task is location of an edge in aone-dimensional signal with Na 5 50 and Nb 5 50 pixels.Each ROC has been estimated from 104 random experiments.
formance with the minimum-description length (MDL)snake algorithm,22 which has the advantage of being aparameter-free algorithm.
Let us first briefly review this algorithm. Consider ascene x 5 $xiui P @1, N#% composed of two regions, an ob-ject and a background. Let w 5 $wiui P @1, N#% denote abinary window function that defines a certain shape forthe object, so that wi 5 1 within this shape and wi 5 0elsewhere. The image is thus divided into two regions:Va 5 $i P @1, N#uwi 5 1% and Vb 5 $i P @1, N#uwi5 0%. The gray levels of the object and of the back-ground are considered as independent random variablesrespectively distributed with pdf Ppa(x) for the object re-gion and Ppb(x) for the background region. The purposeof segmentation is therefore to determine the shape wwhich best matches the real shape of the object in thescene. This shape will be modeled with a k-node polygon,and w is thus a polygon-bounded support function, one-valued on and within the snake and zero-valued else-where.
To estimate the number of nodes k of the object, aMDL23 approach can be used. It consists of minimizingthe number of bits D that are necessary for a descriptionof the image;22 that is, the sum of the number of bits Dafor the description of the target gray levels, the number ofbits Db for the description of the background-gray levels,and the number of bits Dw for the description of the poly-gon w. It can be shown that the MDL principle leads tothe minimization of 22
D8 5 J~x, w! 1 k ln~N !. (11)
We have reported in Table 3 the expression of J(x, w) forthe different gray-level pdf ’s that we consider in this pa-per. Segmentation is obtained by minimizing D8 in Eq.11, which is a function of both k and w, by using the two-step strategy proposed in Ref. 22. We show in Fig. 4 re-sults of the MDL segmentation of an object for differenttypes of pdf.
We have plotted in Fig. 5 the average number of mis-classified pixels (ANMP) obtained for the segmentation ofthe airplane of Fig. 4 as a function of the Fisher ratio, the
Table 3. Expression of the Criterion J(x, w)Involved in the MDL Snake for Some of the Pdf’s
Gaussian withidentical variancesc Nf(s 2) 1/2 ln z
a mu 5 1/Nu( iPVuxi , su
2 5 1/Nu( iPVu(xi 2 mu)2 with u 5 a or b and
s 2 5 (Nasa2 1 Nbsb
2)ˆ /(Na 1 Nb).b Corresponds to Gaussian noise with possible different variances in the
two regions.c Corresponds to Gaussian noise with the same variance in the two re-
gions.
Goudail et al. Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1237
Fig. 4. Examples of segmentation results of the MDL snake for different types of noise PDF. (a) Normal with identical variances, (b)normal with identical means, (c) Gamma with order 5, (d) Poisson. All configurations correspond to B . 0.9.
Kullback divergence, and the Bhattacharyya distance fordifferent pdf ’s and parameter configurations. It can beseen that the ANMP is well represented by a bijectivefunction of the Bhattacharyya distance. Here again, B isan efficient contrast parameter, while this is not the case
Fig. 5. ANMP obtained with the snake MDL for the segmenta-tion of the airplane in Fig. 4 (size, 575 pixels) as a function of dif-ferent statistical distances: Fisher ratio, Kullback divergence,and Bhattacharyya distance. Different types of noise statisticsand parameter combinations have been considered. EachANMP has been estimated from 500 random experiments.
for the Fisher ratio. Furthermore, one can remark thatnow the relation between the ANMP and the Bhatta-charyya distance is approximately bijective for all theconsidered pdf ’s. There is no longer any visible differ-ence of behavior between pdf ’s with one or two unknownparameters. This may be explained by the fact that,compared with the detection examples presented in Fig.2, the target and background regions now possess morepixels: The influence of the estimation of the unknownparameters is thus less important than for detection.
It is interesting to remark with regard to Figs. 2, 3, and5 that the difference between the behavior of the Bhatta-charyya distance and of the Kullback divergence is largerwhen these measures are large and when Na is small.Indeed, when the sample size becomes large, only smallvalues of the Kullback divergence and the Bhattacharyyadistance will lead to nonperfect performance measure andthus have to be considered. Since the Bhattacharyya dis-tance and the Kullback divergence become proportional atlow values (i.e., when pa . pb), their behaviors as con-trast parameters become equivalent when Na → `. Wehave seen, however, that they are not totally equivalentfor small size samples.
4. DISCUSSION OF DIFFERENT NOISESITUATIONSWe have seen in Section 3 that the Bhattacharyya dis-tance is a good candidate for contrast definition in thecase of noncorrelated noise. Indeed, for a fixed task, suchas detection, location, or segmentation, one can observeapproximately the same bijective relation between differ-ent performance measures and the Bhattacharyya dis-tance for a large set of probability laws and parametervalues. With this result, it is thus possible to comparedifferent noise situations. We consider in the following afew examples of its practical application.
A. Poisson and Gaussian NoiseLet us assume that one has to compare two imaging sys-tems for a given task (for example, detection or segmen-tation of biological cells), the first one being limited byphoton noise while the second one is perturbed by elec-tronic noise. For low-flux incoherent images, photonnoise is a consequence of the quantum nature of the light,and these fluctuations can be described by Poisson laws.One will also assume that the electronic noise is well de-scribed by an additive Gaussian noise and thus by aGaussian pdf with the same variance s 2 on the targetand background regions. To compare these two imaging
1238 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Goudail et al.
systems, one can compare the Bhattacharyya distancesfor Gaussian and Poisson pdf ’s.
The Bhattacharyya distance between Poisson laws isgiven in Table 2. One can determine the variance sP
2 ofthe additive Gaussian noise that leads to the same pro-cessing performance as in the presence of Poisson noise.For that purpose, let us set equal the Bhattacharyya dis-tance between Poisson laws to the Bhattacharyya dis-tance between Gaussian pdf ’s with the same variances P
2 on the target and background regions:
~la 2 lb!2
8sP2 5
1
2~Ala 2 Alb!2, (12)
which leads simply to sP 5 (Ala 1 Alb)/2.It is well known that the standard deviation of a Pois-
son noise with mean flux l is equal to Al. sP 5 (Ala
1 Alb)/2 is thus the mean of the standard deviations ofthe gray levels in the target and background regions ofthe photon-noise-limited image.
One can conclude from this simple analysis that if thevariance s 2 of the electronic noise is lower than sP
2
5 (Ala 1 Alb)2/4, the imaging system perturbed by theelectronic noise is better. Of course, if (Ala 1 Alb)2
, 4s 2, the imaging system limited by photon noise isbetter.
B. Photon Counting and Event DetectionThere exist imaging systems for which only an event isdetected and not the number of photons. In other words,at each location, instead of measuring the number of pho-tons that can be detected, the known information is onlythat at least one photon has been detected. From amathematical point of view, this means that the underly-ing process is a Poisson noise of parameter la on the tar-get region and lb on the background region. On the
other hand, the measured event is a binary variable dis-tributed with a Bernoulli law of parameter pa5 exp(2la) on the target region and pb 5 exp(2lb) onthe background region:
The Bhattacharyya distance for Poisson laws and Ber-noulli laws are provided in Table 2. For the Bernoulliprocess, one gets:
BB 5 2ln(@exp~2la!exp~2lb!#1/2
1 $@1 2 exp~2la!#@1 2 exp~2lb!#%1/2), (14)
which is lower than BP 512 (Ala 2 Alb)2. For example,
with la 5 0.5 and lb 5 2, one gets BP . 0.25, while BB. 0.14. This result means, for example, that the ANMPis roughly multiplied by less than a factor of 2 if an eventdetection sensor is used instead of measuring the numberof photons in a segmentation experiment analogous to theone considered in Fig. 5.
C. Speckle NoiseLet us now consider speckle images. The Bhattacharyyadistance between gamma PDFs is given in Table 2. Onecan compare this noisy situation with a Gaussian additivenoise situation with difference of means equal to ma2mb .Let us determine the variance of the equivalent additiveGaussian noise. For that purpose, let us set equal theBhattacharyya distances,
~ma 2 mb!2
8sS2 5 L lnF1
2 S Ar 11
ArD G , (15)
which leads simply to
Fig. 6. Left, SAR image of an agricultural region provided by the French Space Agency and delivered by the European Space Agency.Right, three extracts of the image and images with equivalent additive Gaussian noise. B stands for the Bhattacharyya distance.
Goudail et al. Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1239
sS2 5
~ma 2 mb!2
8L ln@12 ~Ama /mb 1 Amb /ma!#
. (16)
This value was not easy to predict directly by looking atthe variances in the target and in the background regions,which are equal respectively to ma
2 and mb2.
To illustrate this result, let us consider the image inFig. 6. It is a single-look view of an agricultural region inUkraine acquired by the spaceborne ERS-1 SAR satellite.It has been verified that the pdf of the gray levels in ho-mogeneous regions is exponential to a very good approxi-mation. We have selected three extracts of this imagethat represent border regions between two agriculturalfields. For each extract, we have estimated the Bhatta-charyya distance between the two parts, and we have de-termined the equivalent noise variance of a Gaussian ad-ditive noise by using Eq. (16). We have generatedGaussian data with this value of the variance in order tovisualize the amount of equivalent additive noise thatwould lead to the same processing performance (e.g., foredge detection) as the original speckle data.
D. Contrast in DecibelsFor additive Gaussian noise, the signal-to-noise ratio isclassically described in decibels as follows:
SNR 5 10 log10F ~ma 2 mb!2
s 2 G , (17)
where log10 is the base 10 logarithm. One can write thisrelation with the Bhattacharyya distance: SNR5 10 log10 (8 B). We have seen that for an image with agiven noise probability law, the image perturbed with ad-ditive Gaussian noise that will lead to the same perfor-mance is the one that has the same Bhattacharyya dis-tance between the two regions. One can thus define anequivalent signal-to-noise ratio in decibels:
SNR 5 10 log10~8 B!. (18)
For example, for the SAR image of Fig. 6, the Bhatta-charyya distances were equal to 0.13, 0.068, and 0.033,which leads, respectively, to a SNR equal to 0.17, 22.64,and 25.78 dB.
5. CONCLUSIONIn conclusion, we have analyzed the capability of theBhattacharyya distance to define a contrast parameter inthe presence of spatially uncorrelated noise for a large setof probability laws when one uses statistical algorithmsfor detection, location, or segmentation. We have also as-sumed that the noise is due to the optical system and thatthe pdf ’s of both the target and background regions be-long to the same family. The pdf of the target and back-ground gray levels differ only by the different parametervalues they may have. We have considered realistic de-tection, location, and segmentation applications wherethe values of the pdf parameters are unknown. A quasi-bijective relation has been obtained between the perfor-mance criterion and the Bhattacharyya distance for dif-ferent performance criteria and probability laws with
different parameter values, such as Poisson, Bernoulli,Gaussian, gamma, geometric, and Weibull distributions.The Bhattacharyya distance thus allows one to comparedifferent spatially uncorrelated noise situations since therelation between the performance criterion and the Bhat-tacharyya distance is the same for a large number of dif-ferent pdf ’s.
This contrast measure can thus be used to compare dif-ferent noise situations, and to simplify the analysis andthe specification of optical imaging systems perturbed bysuch noise as speckle, additive Gaussian, Poisson, or oth-ers as discussed in the paper. This result may also mo-tivate investigations to generalize the theoretical resultsobtained for the ideal observer.10
This study is nevertheless only a first step toward ageneral contrast parameter definition for practical appli-cations. First, it will be interesting to determine forwhich pdf this bijective relation does or does not hold, andfor which number of target pixels. Furthermore, morecomplex situations than the ones discussed in this papercan appear in general, such as spatially correlated noisewith a large variety of pdf ’s or cases where the pdf ’s of thetarget and of the background gray levels belong to differ-ent families. It will be interesting to determine if a con-trast parameter can still be defined for spatially corre-lated noise and to analyze the relevance of theBhattacharyya distance for this purpose.
REFERENCES1. J. W. Goodman, ‘‘The speckle effect in coherence imaging,’’
in Statistical Optics (Wiley, New York, 1985), pp. 347–356.2. J. W. Goodman, ‘‘Some effects of target-induced scintillation
on optical radar performance,’’ Proc. IEEE 53, 1688–1700(1965).
3. F. Goudail, N. Roux, and Ph. Refregier, ‘‘Performance pa-rameters for detection in low-flux coherent images,’’ Opt.Lett. 28, 81–83 (2003).
4. Ph. Refregier and F. Goudail, ‘‘Invariant polarimetric con-trast parameters for coherent light,’’ J. Opt. Soc. Am. A 19,1223–1233 (2002).
5. A. O. Hero and C. Guillouet, ‘‘Robust detection of SAR/IRtargets via invariance,’’ in Proceedings of the Sixth IEEE In-ternational Conference on Image Processing and its Appli-cations (Institute of Electrical and Electronics Engineers,New York, 1997), pp. 472–475.
6. J. A. O’Sullivan, R. E. Blahut, and D. L. Snyder,‘‘Information-theoretic image formation,’’ IEEE Trans. Inf.Theory 44, 2094–2123 (1998).
7. A. D. Lanterman, A. J. O’Sullivan, and M. I. Miller,‘‘Kullback-Leibler distances for quantifying clutter andmodels,’’ Opt. Eng. (Bellingham) 38, 2134–2146 (1999).
8. A. Jain, P. Moulin, M. I. Miller, and K. Ramchandran,‘‘Information-theoretic bounds on target recognition perfor-mance based on degraded image data,’’ IEEE Trans. Pat-tern Anal. Mach. Intell. 24, 1153–1166 (2002).
9. T. M. Cover and J. A. Thomas, Elements of InformationTheory (Wiley, New York, 1991).
10. H. H. Barrett, C. K. Abbey, and E. Clarkson, ‘‘Objective as-sessment of image quality. III. ROC metrics, ideal ob-servers and likelihood generating functions,’’ J. Opt. Soc.Am. A 15, 1520–1535 (1998).
11. J. Shapiro, ‘‘Bounds on the area of the roc curve,’’ J. Opt.Soc. Am. A 16, 53–57 (1999).
1240 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Goudail et al.
12. E. Clarkson and H. H. Barrett, ‘‘Approximations to ideal ob-server performance on signal detection tasks,’’ Appl. Opt.39, 1783–1793 (2000).
13. E. Clarkson, ‘‘Bounds on the area under the receiver oper-ating characteristic curve for the ideal observer,’’ J. Opt.Soc. Am. A 19, 1963–1968 (2002).
14. J. Rissanen, ‘‘Modeling by shortest data description,’’ Auto-matica 14, 465–471 (1978).
15. S. A. Kassam, ‘‘Optimal quantization for signal detection,’’IEEE Trans. Commun. 25, 479–484 (1977).
16. H. V. Poor and J. B. Thomas, ‘‘Application of Ali-Silvey dis-tance measures in the design of general quantizers for bi-nary decision systems,’’ IEEE Trans. Commun. 25, 893–900(1977).
17. H. V. Poor, ‘‘Elements of hypothesis testing,’’ in An Introduc-tion to Signal Detection and Estimation (Springer-Verlag,New York, 1994), pp. 5–39.
18. M. Basseville, ‘‘Distance measures for signal processing
and pattern recognition,’’ Signal Process. 18, 349–369(1989).
19. V. Page, F. Goudail, and Ph. Refregier, ‘‘Improved robust-ness of target location in nonhomogeneous backgrounds byuse of the maximum likelihood ratio test location algo-rithm,’’ Opt. Lett. 24, 1383–1385 (1999).
20. S. M. Kay, ‘‘Statistical decision theory II,’’ in Fundamentalsof Statistical Signal Processing, Vol. II: Detection Theory(Prentice Hall, Upper Saddle River, N.J., 1998), pp. 186–247.
21. T. S. Ferguson, ‘‘Exponential families of distributions,’’ inMathematical Statistics, a Decision Theoretic Approach(Academic, New York, 1967), pp. 125–132.
22. O. Ruch and Ph. Refregier, ‘‘Minimal-complexity segmenta-tion with a polygonal snake adapted to different opticalnoise models,’’ Opt. Lett. 41, 977–979 (2001).
23. J. Rissanen, Stochastic Complexity in Statistical Inquiry(World Scientific, Singapore, 1989).
