Photon-noise effect on detection in coherent active images
Philippe Réfrégier, François Goudail, and Guillaume Delyon
Physics and Image Processing Group/Fresnel Institute, Unité Mixte de Recherche (UMR 6133), Centre National de la RechercheScientifique, École Nationale Supérieure de Physique de Marseille, Domaine Universitaire de Saint-Jérôme, 13397 Marseille Cedex 20,
Active coherent imaging consists of forming an im-age of a scene that has been illuminated with laserlight. This technique allows one to combine night vi-sion capability and improved image resolution for agiven aperture size (since the laser wavelength can beshorter than for ordinary thermal IR bands). How-ever, it leads to the drawback that images are de-graded with speckle noise because of the coherence ofthe light.1 Furthermore, there exist many practicalcases for which the intensity level of the ref lected lightcorresponds to a small number of photons, which leadsto an additional source of f luctuations in the detectedsignal. This situation can arise when the target isfar away, with low-power laser sources, and in multi-spectral and hyperspectral imagery systems. In thiscase the quantum nature of the light leads to Poisson-distributed noise that can become important relativeto the mean value of the signal at a low photon level(i.e., low f lux). It was shown in Ref. 2 with numeri-cal simulations that the Bhattacharyya distance is anefficient figure of merit for characterizing detection al-gorithms based on the generalized likelihood ratio test(GLRT) for the realistic case in which the target andbackground mean values are unknown.
In this Letter we analyze the inf luence of Poissonnoise on coherent images perturbed by speckle noise.In other words, we determine the mean received f luxfor which the photon noise begins to have an inf lu-ence on detection performance as a function of the ratioof target and background ref lectivities and for differ-ent configurations. We also analyze the robustness ofthe classical detection technique for pure speckle im-ages, and we demonstrate that photon noise should betaken into account in the design of eff icient detectionalgorithms.
Low photon f lux in active coherent imaging leadsto a combination of speckle and Poisson f luctuationswhose probability law is3
pl�n� �1
1 1 l
µl
1 1 l
∂2, (1)
where l is the mean value of random number n of de-tected photons. In Eq. (1) we assume that the speckleis fully developed, which is the case that will be con-sidered throughout this Letter.
0146-9592/04/020162-03$15.00/0
Let us consider a target with the mean value of theref lected light equal to la over a background whoselocal mean value is lb. Let us also assume that thegray levels of the target and background regionsare random, spatially uncorrelated, and distributedwith laws given by Eq. (1). When parameters laand lb are known, an eff icient detection algorithmcan be designed by use of the maximum likelihoodratio test4,5 (also known as the ideal observer). Forthis purpose a window defined by a binary mask Wwith Na pixels, which defines the shape of the target,is scanned over the image, and the expression L �P
i[W log�pla �ni�� 2P
i[W log�plb �ni�� is compared witha threshold. This strategy is optimal in the Neyman–Pearson sense, since, for a given false alarm rate, itoptimizes the detection probability. A simple andclassical way to represent the detection performanceis to draw the detection probability as a function of thefalse alarm probability, which is known as the receiveroperating characteristic (ROC). A common way toobtain a scalar measure of the detection performanceis to consider the area under the curve (AUC) ofthe ROC. A theoretical analysis for the case of theideal observer was proposed6 and led to the followingapproximation of the AUC:
AUC � HIO�dB � �12
112
erf�p2dB � , (2)
where erf�.� is the error function and dB is theBhattacharyya distance, dB � 2Na ln�
P`n�0�pla �n� 3
plb �n��1�2�. In the case of the probability law ofEq. (1), one gets
dB � Na ln�p1 1 la
p1 1 lb 2
plalb � . (3)
We plot in Fig. 1 the AUC values versus dB for severalparameter configurations and the curve defined by ex-pression (2). One can see that the plot in Fig. 1 con-stitutes a good approximation of the relation betweenthe AUC and dB .
In practice, la and lb are generally unknown. Aclassical approach thus consists of substituting theirvalues with their maximum likelihood estimates to ob-tain a GLRT.4 For this purpose a subwindow defined
Fig. 1. AUC as a function of Bhattacharyya distance forthe ideal observer and the GLRT �lPHOT � in the presenceof speckle and photon noise. The curves correspond to ex-pression (2) for the ideal observer and Eq. (5) for the GLRT(for two values of Na�Nb). The symbols fitted to eachcurve correspond to Monte Carlo simulations with threedifferent target sizes �Na � 3, 10, 50�, where each ROCwas estimated from 104 random experiments.
by binary mask F is scanned over the image. Sub-window F is composed of two disjoint regions: targetshape W and background region W (with Nb pixels)so that F � W < W . For each position of F we makea decision between the two following hypotheses:(i) There is a target in the middle of F, and thesamples in W and W have different probabilisticparameters (i.e., la fi lb), or (ii) there is no target(only background) in F, and the samples in W and Whave the same probabilistic parameters (i.e., la � lb).
i[W ni�Nb, clc �Pi[F ni��Na 1 Nb� and f �x� � x lnx 2 �1 1 x�ln�1 1 x�.We plot in Fig. 1 the AUC values obtained with the
GLRT versus dB for different parameter configura-tions. One can see that the algorithm performance isstill a bijective function of dB , but its relation with theAUC is no longer described by expression (2). It hasbeen shown2 that, for the GLRT, there nevertheless ex-ists a bijective relation of the form AUC � Hu�dB�, withu � Na�Nb. This relation has been demonstrated bynumerical simulations for a large number of differentparameter conf igurations �la, lb, Na, and Nb� thatinclude fully developed speckle as limit cases. It isshown in Fig. 1 that the relation
Hu�dB � � 1 212
exp�2bu�dB �bu � (5)
is a good approximation of the curves in Fig. 1 if b1 �1.2 and b1 � 0.95 when u � 1 and b1/3 � 2.46 andb1/3 � 0.67 when u � 1�3. Although this relation ispurely empirical, we show below that it is useful forour purpose.
Using the relations between dB and AUC defined byexpression (2) and Eq. (5), we can analyze the inf lu-ence of photon noise on detection performance. More
precisely, let us introduce I � �la 1 lb��2, r � la�lb,and let AUCI denote the AUC for mean f lux I . FromEq. (3) one can show that dB �I � � Na ln��1��1 1
r�� ��1 1 r 1 2rI �1�2�1 1 r 1 2I �1�2 2 2pr I �� and, from
expression (2) and Eq. (5), one has AUCI � H �dB�I ��,where H stands for HIO if one considers the ideal ob-server or for HNa/Nb if one considers the GLRT. Thesecurves AUCI are shown in Fig. 2 for different valuesof r compared with Monte Carlo simulations. Onecan see that there is good agreement between thenumerical simulations and the predicted relation.
Let AUC` denote the value of the AUC when thereis no photon noise (i.e., for pure speckle noise). It isinteresting to determine the value of Ic for which oneobserves a decrease of the AUC from AUC` to a AUC`
(with 0 , a , 1). Since they are approximately linkedby a bijective relation, a decrease of the AUCI corre-sponds to a variation of dB�I �. Using Eq. (3) we caneasily show that
Ic �4r 2 r2
a, r�1 1 r�2
8r�ra, r 2 1�, (6)
where ra, r � exp��d�a�B 2 d̃B��Na� and where d�a�
B�d̃B� is the value of the Bhattacharyya distancethat leads to a AUC` �AUC`�. One thus has d̃B �Na log�1/2�1�
pr 1
pr ��. Using the approximation
of expression (2), we get for the ideal observer d�a�B �
1/2�erf21�2a AUC` 2 1��2. Using the approximation
Fig. 2. AUC as a function of the mean f lux I for the idealobserver and the GLRT for Na�Nb � 1. The curves cor-respond to AUC obtained with expression (2) for the idealobserver and with Eq. (5) for the GLRT. The symbols fit-ted to each curve correspond to Monte Carlo simulationsfrom 104 random experiments: r � la�lb � 2 �3,�� and5 �1, ��.
Fig. 3. Mean f lux Ic required for an AUC equal to a timesthe AUC without the photon-noise effect for two values of a,0.999 and 0.99, as a function of ref lectivity ratio r � la�lb.
Fig. 4. AUC as a function of mean f lux I for the detectoradapted to pure speckle lSPEC, with Na�Nb � 1, r � 2 �1�,and r � 5 �3�. Each ROC was estimated from 104 randomexperiments. The solid and dashed curves were obtainedwith Eq. (5) and correspond to the performance obtainedwith the detector adapted to speckle with photon noiselPHOT (see Fig. 2). The dashed–dotted (dashed) lines rep-resent the AUC obtained when lSPEC �lPHOT � is applied topure fully developed speckle noise.
of Eq. (5), we get for the GLRT d�a�B � 1�b1�bu 3
�2log�2�1 2 a AUC`���1�bu .We report in Fig. 3 the value of Ic for two different
values of a as a function of r for both the ideal observerand the more realistic case of the GLRT. One can seethat Ic is greatly dependent on two factors. The firstis the ref lectivity ratio r, since photon-noise inf luenceappears earlier for low values of r. The second is thea priori knowledge one has of the target and back-ground probability parameters (i.e., la and lb).Indeed, Ic will be different if one considers the idealobserver or the GLRT detector with different valuesof the ratio Na�Nb. As shown in Ref. 5, the choice ofthe ratio Na�Nb can depend on the a priori confidencethat we have in the homogeneous character of thebackground.
The GLRT adapted to pure and fully developedspeckled noise,7 whose mathematical expression is
is often used in remote-sensing applications. Tostudy its robustness to photon noise, we report in
Fig. 4 its detection performance on the same noiseconfigurations as in Fig. 2 (i.e., in the presence ofspeckle with photon noise). One can see that, althoughit is quite robust, the performance of lSPEC is lowerthan that of the detector lPHOT [see Eq. (4)], which isadapted to speckle with photon noise. On the otherhand, it can be seen in Fig. 4 that lPHOT appliedto pure speckle gives a performance that is similarto lSPEC. In other words, in the presence of purefully developed speckle, lPHOT gives approximatelythe same performance as lSPEC, and when photonnoise appears, that is, when I decreases, its perfor-mance becomes better than that of lSPEC. Thus, forfully developed speckle, using the GLRT adapted tolow-f lux speckle noise, i.e., lPHOT , can only improveperformance.
In summary, when photon noise is expected, the per-formance of classical detection techniques designed forpure speckle images can be improved with no increasein the algorithm complexity. Furthermore, the meanphoton number under which photon noise becomes sen-sitive is higher when the target and background meanvalues are unknown than in an ideal case, where theyare assumed to be known, and when the ref lectivityratio between the target and the background is low.
This work was performed in the course of theEuropean Cooperation for the Long Term in Defensecontract 02/EF 08.13/019 provided by the WesternEuropean Union. We gratefully acknowledge this sup-port. F. Goudail’s e-mail address is [email protected].
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