644 OPTICS LETTERS / Vol. 26, No. 9 / May 1, 2001

Statistical techniques for target detection in polarizationdiversity images

F. Goudail and Ph. Réfrégier

Physics and Image Processing Group/Fresnel Institute, École Nationale Supérieure de Physique de Marseille,Domaine Universitaire de Saint-Jérôme, 13397 Marseille Cedex 20, France

Received November 21, 2000

We address the problem of target detection in active polarimetric images. This technique, which has theappealing feature of revealing contrasts that do not appear in conventional intensity images, provides severalimages of the same scene. However, because of the presence of nonhomogeneity in the ref lected intensity, itis preferable to perform target detection on the orthogonal-state contrast image, which is a measure of thedegree of polarization of the ref lected light when the coherency matrix is diagonal. We show that one candetermine a simple nonlinear transformation of this orthogonal-state contrast image that leads to additivenoise, and we then propose a simple and efficient technique for detecting targets in these images. © 2001Optical Society of America

OCIS codes: 040.1880, 100.0100, 100.5010.

For a few years, active imaging has been a topic ofgrowing interest. This technique consists of grabbingan image scene that has been illuminated with laserlight, and one of its main advantages is its capabilityto combine night-vision capability and improved imageresolution for a given aperture size (because the laserwavelength can be shorter than for normal thermal IRbands). However, this technique has the drawbackthat it produces images that are degraded with specklenoise and are spatially nonuniform owing to non-homogeneity in the ref lected intensity. Speckle noise,which is the consequence of the use of coherent light,1

makes the images grainy and causes deterioration inthe detection performance. The spatial nonunifor-mity generally results from interference between thelaser sources and from wave-front distortions that arecaused by atmospheric turbulence. This phenomenoncan perturb the detection algorithms, particularly bycreating false targetlike structures in the images.

With polarization diversity imaging techniques,2,3

the scene is illuminated with laser light that hasa well-defined polarization state, and the imagingsystem contains a polarization analysis module toprovide several images of the scene. In its simplestversion, two images are measured. The first one, de-noted s1, corresponds to ref lected light with the samepolarization state as the incident light; whereas thesecond image, denoted s2, corresponds to the ref lectedlight with orthogonal polarization.3 Let �i, j� be thecoordinates of a given pixel, such that s1�i, j� �s2�i, j��is the intensity at pixel �i, j� in s1 �s2�.

The joint observation of both images s1 and s2 allowsone to estimate how the scene depolarizes the incidentlight. Let G be the coherency matrix.1 The degreeof polarization is defined by �1 2 4 det�G���Tr�G��2�

1/2,where det�G� and Tr�G� are, respectively, the deter-minant and the trace of G. Inasmuch as, in naturaltextures, diattenuation and retardance effects are neg-ligible,3 G is in general a diagonal matrix. In this case,at each pixel �i, j � the degree of polarization is equalto the absolute value of the orthogonal-state contrastimage (OSCI):

0146-9592/01/090644-03$15.00/0 ©

r�i, j � �s1�i, j � 2 s2�i, j �s1�i, j � 1 s2�i, j �

. (1)

The degree of polarization of the ref lected light, andthus the OSCI, can then reveal targets that do not ap-pear with good contrast in intensity images.3 For ex-ample, a metallic object will appear with high contrastagainst a natural background, because the depolariza-tion capabilities of the two are widely different.

It has long been known that the OSCI is invariant toillumination nonhomogeneity because it is normalizedby the total intensity s1�i, j � 1 s2�i, j �. We illustratethis homogeneity property in Fig. 1 with real polariza-tion diversity images. From the two top figures, onecan see that the two images s1 and s2 are highly ho-mogeneous, whereas one can observe from the bottompart of the f igure that the OSCI is homogeneous. TheOSCI has already been shown to be eff icient for visual-ization of polarimetric data4 and contrast enhancement

Fig. 1. Real polarimetric image with channels s1 and s2and the corresponding OSCI.

2001 Optical Society of America

May 1, 2001 / Vol. 26, No. 9 / OPTICS LETTERS 645

in imaging through scattering media.5 Our aim is toconsider target detection on the OSCI in the rigorousframework of Neyman–Pearson theory and to providea rapid and effective algorithm for solving this task.

Let us now analyze some general properties of theprobability-density function (PDF) of the OSCI. LetPa�x� be the PDF of a random variable, x, of meana. The PDF family defined by Pa�x� is said to be in-variant for the group of multiplications with positivenumbers if there exists a function F�x� that is inde-pendent of a such that Pa�x� � �1�a�F�x�a�. For ex-ample, the gamma PDF family satisfies this propertybecause, if x is gamma distributed, ax is also gammadistributed for any a . 0. Because of the dominantspeckle noise, it is natural to assume that s1�i, j� ands2�i, j � are distributed with a PDF invariant for thegroup of multiplication with positive numbers. Letus define s̃n�i, j� � sn�i, j��mn, with n � 1 or n � 2and where mn denotes the mean of sn�i, j�. The ran-dom variable h�i, j � � �s̃1�i, j� 2 s̃2�i, j����s̃1�i, j� 1

s̃2�i, j �� has thus a PDF Ch�h� that is independent ofmn�n � 1, 2�. However, it’s easy to see that the OSCIcan be written r�i, j � � �h�i, j � 1 u���1 1 uh�i, j ��,where u � �m1 2 m2���m1 1 m2�. One thus obtains

P �r�u �r� �

1 2 u2

�1 2 ur�2Ch

µr 2 u1 2 ur

∂. (2)

Furthermore, Ch�h� is independent of mn because h

is independent of mn. u is thus the unique scalar pa-rameter of the OSCI PDF P �r�

u �r�. In the target regionthe PDF is thus P

�r�ua �r� with ua � �ma

1 2 ma2���ma

1 1 ma2�,

and in the background region it is Prub�r� with ub �

�mb1 2 m

b2���mb

1 1 mb2�. In particular, when s1�i, j� and

s2�i, j � are distributed with gamma laws, which arethe standard model for speckle distributions,1 it can beshown that the PDF of the OSCI is

P �r�u �r� �

�2L 2 1�!22L21��L 2 1�!�2

�1 2 u2�L�1 2 r2�L21

�1 2 ur�2L. (3)

Let us now consider target detection on theOSCI. Eff icient detection algorithms can be de-signed by use of the maximum-likelihood ratiotest.6,7 For this purpose a subwindow defined by a bi-nary mask F is scanned over the image. SubwindowF is composed to two disjoint regions, W and W̄ , suchthat F � W < W̄ . W defines the shape of the target.For small targets, it is chosen as a small square. Foreach position of F we make a decision between thetwo following hypotheses: In hypothesis H1 thereis a target in the middle of F , and the samples inW and W̄ have different statistical parameters. Inhypothesis H0 there is no target (only background)in F , and the samples in W and W̄ have the samestatistical parameters.

We assume that the gray levels of the target andthe background regions are random, spatially uncor-related, and distributed with laws given by Eq. (3).When the parameters ua and ub are known, the de-termination of the maximum likelihood ratio test sim-ply consists in computing for each location of F theexpression

L �≥≥≥ X

�i, j �[W log�P �r�ua

�r�i, j ���¥¥¥

1

≥≥≥ X�i, j �[W̄ log�P �r�

ub�r�i, j���

¥¥¥

2

≥≥≥ X�i, j �[F log�P �r�

ub�r�i, j���

¥¥¥. (4)

In practice, the parameters ua, ub, and L areunknown, and we have to deal with this situation.A classic approach consists in substituting someestimates for their values to obtain a generalized like-lihood ratio.6 Because the PDF given by Eq. (3) doesnot belong to the exponential family,8 it is not easy todefine a simple and efficient estimation technique forthese parameters. Furthermore, we can also remarkthat the PDF family of the OSCI defined by Eq. (2)is quite complex inasmuch as the whole shape of thePDF varies with u, as do all the statistical moments.

We propose to show in what follows that there existsa nonlinear transformation of r�i, j� that allows oneto obtain a representation of the degree of polarizationimage in which the noise is additive. We then demon-strate that an algorithm designed for a Gaussian PDFconstitutes an eff icient technique for detection in thisimage.

For that purpose, let us introduce a � tanh21�h�,b � tanh21�r�, and n � tanh21�u�, where tanh21 isthe inverse of the hyperbolic tangent function. Thush � �r 2 u���1 2 ur� becomes tanh�a� � �tanh�b� 2

tanh�n����1 2 tanh�n�tanh�b��, but, as �tanh�b� 2tanh�n����1 2 tanh�n�tanh�b�� � tanh�b 2 n�, it isclear that h � �r 2 u���1 2 ur� is equivalent to a �b 2 n. Using the standard method of probabil-ity theory, it is easy to verify that the PDF ofb�i, j � � tanh21�r�i, j�� is thus

P �b�n �b� � �1 2 tanh2�b 2 n��Ch�tanh�b 2 n�� . (5)

Because Ch�tanh�j�� is a function that is inde-pendent of mn �n � 1, 2� and then of n, the PDFof b is invariant in the simple additive group [i.e.,P �b�

n �b� � P �b�0 �b 2 n�]. Furthermore, because the

function Ch�h� is symmetric, such is also the casefor P �b�

n �b�. In other words, it is possible to con-sider that the image b�i, j � has been corrupted withsymmetric additive noise. We can also remark thatb�i, j � � log�s1�i, j ���2 2 log�s2�i, j ���2, and our resultis analogous, but not identical, to the homomorphicrepresentation techniques that have been introducedfor speckle noise images.

Let us consider the generalized likelihood ratio forGaussian fields with unknown means and with thesame variance in both the target and the backgroundregions. We can show that the ratio is simply

R � A�IW 2 I W̄ �2, (6)

where A is a normalization constant and I l ��1�Nl�

P�i,j �[l b�i, j �, with l � W or W̄ . This tech-

nique is analogous to matched filtering in the b�i, j �image.

646 OPTICS LETTERS / Vol. 26, No. 9 / May 1, 2001

Fig. 2. ROCs of different types of detection test: dia-monds, L ; pluses, R; squares, N . The PDFs of chan-nels s1 and s2 are gamma laws of order L � 50 with meansm1

a � 7.25, m1a � 1.1, m2

b � 5, and m2b � 1. Inset, gamma

laws of order L � 1 with means m1a � 60, m2

a � 3, m1b � 5,

and m2b � 1. The target size is 9 pixels, and the scanning

window size is 25 pixels. The curves have been estimatedwith Monte Carlo simulations with 106 realizations.

We characterize the algorithm performances bytheir receiver operating characteristics (ROCs), whichrepresent the probability of detection as a function ofthe probability of a false alarm when the test (L orR) is compared with a given threshold. The ROCsfor tests L [see Eq. (4)] and R are shown in Fig. 2.We have considered two types of speckle noise forchannels s1 and s2: gamma law of order 1, i.e., anexponential, which models fully developed speckle1

(Fig. 2, inset), and a gamma law of order 50, whichcan model averaged speckle (Fig. 2). We also show inthis figure the results obtained with the generalizedlikelihood ratio adapted to Gaussian statistics withunknown mean and variance7 when it is directlyapplied to r�i, j �, and we denoted it N [note that Ris not appropriate for r�i, j � because both the meanand the variance of P

�r�u �r� vary with parameter u].

We can observe from Fig. 2 that the most efficientalgorithm is L . However, the ROC of L is only a ref-erence because it represents the result that one canexpect in the unrealistic situation when ua and ub areknown. Among the algorithms that estimate the pa-rameters, R leads to better performance than N for

both values of L, which shows that it is preferable towork with b�i, j � than with r�i, j�. Note that test Ris suboptimal when it is applied to b�i, j � because thePDF of b�i, j� is not Gaussian. However, we have ver-ified that the ROCs obtained with R applied to b�i, j�and to Gaussian data with the same means and vari-ances are similar.

These results show that R is efficient when it isapplied to the image b�i, j �, but there of course maystill be a better algorithm. Moreover, it is importantto note that R is equivalent to a matched filter. It canthus be implemented rapidly by a digital Fast Fouriertransform or optical correlators. However, the maxi-mum-likelihood algorithm for OSCI, L , cannot be putinto correlation form, and there is no simple and effi-cient algorithm for estimating PDF parameter u whenit is unknown. This algorithm is thus more computa-tionally intensive than R.

In summary, because of nonhomogeneity in the re-f lected intensity, it is preferable to perform target de-tection on the OSCI. We have demonstrated that asimple nonlinear transformation enables one to trans-form the f luctuations of an OSCI into symmetric ad-ditive noise. This makes it possible to obtain highdetection eff iciency in the transformed image with al-gorithms of low complexity.

The authors thank Sébastien Breugnot of Thomson-CSF Optrosys and Vincent Pagé for fruitful discus-sions. This research has been partially supportedby the French Defense Ministry (grant DGA/DSP/STTC). F. Goudail’s e-mail address is goudail@lsi.u-3mrs.fr.

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