Page 1

Polarimetric target detection in the presence of spatially

fluctuating Mueller matrices

Guillaume Anna, Fran¸ cois Goudail, Daniel Dolfi

To cite this version:

Guillaume Anna, Fran¸ cois Goudail, Daniel Dolfi. Polarimetric target detection in the presence

of spatially fluctuating Mueller matrices. Optics Letters, Optical Society of America, 2011, 36

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Polarimetric target detection in the presence

of spatially fluctuating Mueller matrices

Guillaume Anna,1François Goudail,1,* and Daniel Dolfi2

1Laboratoire Charles Fabry, UMR 8501, Institut d’Optique, CNRS, Université Paris Sud 11, 91127 Palaiseau, France

2Thales Research and Technology—France, RD128, 91767 Palaiseau Cedex, France

*Corresponding author: francois.goudail@institutoptique.fr

Received September 9, 2011; revised October 20, 2011; accepted October 23, 2011;

posted October 24, 2011 (Doc. ID 154458); published November 28, 2011

In polarimetric imaging systems, the main source of perturbations may not be detection noise but fluctuations of the

Mueller matrices in the scene. In this case, we propose a method for determining the illumination and analysis

polarization states that allow reaching the highest target detection performance. We show with simulations and

real-world images that, in practical applications, the statistics of Mueller matrix fluctuations have to be taken into

account to optimize polarimetric imagery.© 2011 Optical Society of America

OCIS codes:110.5405, 100.0100.

Polarimetric images are useful for gathering information

that is not visible on intensity images and can be useful in

such domains as machine vision, remote sensing, biome-

dical imaging, and industrial control [1,2]. In many appli-

cations, the objective is to discriminate a target from its

background, and a lot of efforts have been done to per-

form this task in an optimal way. The first results have

been obtained in the radar community, where images

are mainly perturbed by multiplicative speckle noise

[3,4]. More recently, this problem has been addressed

in the optics community, and the cases of additive detec-

tor noise [5] and of Poisson shot noise [6] have been in-

vestigated. However, in many situations such as target

detection in foliage or in biological tissues, the dominant

source of perturbations is not due to detector, speckle, or

shot noise but to the fluctuations of the Mueller matrices

in the scene. For the first time to our knowledge in the

optical domain, we address the theoretical aspects of this

issue and demonstrate on simulated and real-world

images that, in practical applications, the statistics of

spatial Mueller matrix fluctuations have to be taken into

account to optimize polarimetric imagery.

We consider an active polarimetric imaging system

that illuminates the scene with light whose polarization

state is defined by a Stokes vector S and is produced by a

polarization state generator (PSG) (see Fig. 1). The po-

larimetric properties of a region of the scene correspond-

ing to a pixel in the image is characterized by its Mueller

matrix M. The Stokes vector of the light scattered by this

region is S0¼ MS. It is analyzed by a polarization state

analyzer (PSA), which is a generalized polarizer whose

eigenstate is the Stokes vector T. The number of photo-

electrons measured at a pixel of the sensor is

i ¼ηI0

2TTMS;

ð1Þ

where the superscript T denotes matrix transposition. In

this equation, S and T are unit intensity, purely polarized

Stokes vectors, I0is a number of photons, and η is the

conversion efficiency between photons and electrons.

We will consider that the image is disturbed by two dif-

ferent types of noise. The first one is additive sensor

noise, which will be assumed of zero mean and variance

σ2. The second one is due to the fact that, from one pixel

to the next, the Mueller matrix randomly fluctuates

around its average value. This spatial fluctuation can lead

to a significant noise disturbing the intensity image and

thus has to be taken into account. For the sake of sim-

plicity, we will assume that the scene is composed of two

regions: a target characterized by an average Mueller ma-

trix hMai and a background characterized by an average

Mueller matrix hMbi. Each pixel belonging to region u ∈

fa;bg has a Mueller matrix M that deviates from the aver-

age matrix hMui of this region. These spatial fluctuations

are characterized by their correlation matrices defined as

Gu¼ hðVM− VhMuiÞðVM− VhMuiÞTi;

with VMthe 16-component vector obtained by reading

the Mueller matrix M in the lexicographic order. Using

this notation and taking into account additive noise,

Eq. (1) can be written as

ð2Þ

i ¼ηI0

2½T ⊗ S?TVMþ n;

ð3Þ

where ⊗ denotes the Kronecker product and n is a ran-

dom variable of zero mean and variance σ2. Notice that i

is now a random variable whose statistical properties de-

pend on the region where the pixel is located. If we as-

sume that the fluctuations of VMand n are independent,

the mean and variance of i in region u ∈ fa;bg are given

by

hiiu¼ ηI0=2 × ½T ⊗ S?TVhMui;

ð4Þ

Fig. 1.(Color online) Polarimetric imaging setup.

4590 OPTICS LETTERS / Vol. 36, No. 23 / December 1, 2011

0146-9592/11/234590-03$15.00/0© 2011 Optical Society of America

Page 3

var½i?u¼ ðηI0=2Þ2× ½T ⊗ S?TGu½T ⊗ S? þ σ2:

Our objective is to optimize the discrimination be-

tween the target and the background region in the inten-

sity image. To quantify the quality of this discrimination,

we will use the Fisher ratio [7]. Using the notation de-

fined above, it is defined as

ð5Þ

FðS;TÞ ¼

½hiia− hiib?2

var½i?aþ var½i?b

:

ð6Þ

By using Eqs. (3)–(5), it can be put in the following form:

FðS;TÞ ¼

½T ⊗ S?TGtarget½T ⊗ S?

½T ⊗ S?TGfluct½T ⊗ S? þ 8=SNR;

ð7Þ

where Gfluct¼ Gaþ Gbis the average covariance matrix

of the Mueller matrix fluctuations over the scene,

Gtarget¼ ðhMai − hMbiÞðhMai − hMbiÞTis the interclass

covariance matrix, which represents the difference be-

tween the average Mueller matrices of the two regions,

and SNR ¼ ðη2I2

the presence of additive noise.

Our goal will be to determine the optimal couple of il-

lumination and analysis states (S, T) that maximizes the

function FðS;TÞ defined in Eq. (7). In order to show the

importance of taking into account the actual statistics of

Mueller matrix fluctuations in this optimization, we will

take one example based on simulation and another one

based on real-world polarimetric images.

In order to simulate random Mueller matrices, let us

use the Lu–Chipman decomposition [8], which consists

in describing each Mueller matrix as a product of three

components: retarder (MR), diattenuator (MD), and de-

polarizer (MΔ). The resulting Mueller matrix M is then

given by M ¼ MΔMRMD. We will assume that the Mueller

matrices of the diattenuator andthe retarder are constant

in each region and only the diagonal coefficients of the

depolarizer matrices Mu

dard deviations of these fluctuations in the two regions

are defined in Table 1. We generate two sets of N ¼ 5000

Mueller matrices according to this model. The average

Mueller matrices of regions a and b are estimated as

0Þ=σ2is the signal-to-noise ratio due to

Δrandomly fluctuate. The stan-

hMaihMbi

0:64

0:00

0:00 −0:03

0:00 −0:01 −0:02 0:16

0:00

0:31

0:01

0:03

0:24

0:00

0:03

0:02

2

664

3

775

0:60 0:01 0:00

0:00 0:19 0:00 −0:02

0:00 0:00 0:15

0:00 0:02 0:00

0:00

0:00

0:22

2

664

3

775

:

ð8Þ

They are mainly depolarizing but with depolarization

coefficients that are anisotropic and region dependent.

Let us first assume that the fluctuations of Mueller ma-

trices are neglected in the optimization process; that

is, Gfluctis assumed to be zero. The SNR of additive noise

defined in Eq. (7) is assumed equal to ðηI0=σÞ2¼ 152. The

optimal states maximizing the Fisher ratio are presented

in the first row of Table 2 (α denotes the azimuth of a

polarization state and ε its ellipticity). The value of the

Fisher ratio obtained with this couple of optimal PSG

and PSA states is F ¼ 2:3, and the image obtained ap-

pears in Fig. 2(a). Let us now take into account the fact

that the Mueller matrices fluctuate inside each region.

The covariance matrices Gaand Gbof these fluctuations

are estimated from the generated data sets, and these es-

timates are used in Eq. (7). The obtained optimal states

are presented in the second row of Table 2: it is seen that

they are different from those obtained without taking

into account the Mueller matrix fluctuations. They lead

to a better value of the Fisher ratio F ¼ 4:1, as can be

seen on the scalar image obtained using this couple of

optimal states in Fig. 2(b). In order to interpret this result,

let us look at the matrices in Eq. (8). The difference be-

tween the two average Mueller matrices is higher for the

coefficient M11, and thus the optimal states maximizing

the contrast when taking into account only the additive

noise are roughly linear with azimuth 90°. However, we

can notice in Table 1 that it is also for coefficient M11that

the Mueller matrix fluctuations are the highest. Thus,

when these fluctuations are taken into account, the op-

timal states correspond to the best compromise between

maximization of the separability of the average intensi-

ties and minimization of the fluctuations. It is seen in

Table 2 that these states are elliptic with nonparallel

azimuth.

Let us now consider a real-world polarimetric imaging

scenario. The observed scene is a piece of translucent

birefringent plastic with spatial fluctuations, on the back-

side of which two pieces of translucent adhesive tape

have been stuck on a polarizer. The target regions are

constituted by the pieces of adhesive [see Fig. 3(a)].

The standard intensity image of this scene is shown in

Fig. 3(b): the adhesive tapes are not visible on it. The

scene is now observed with a polarimetric imager of the

type represented in Fig. 1. The average and covariance

matrices of the Mueller matrices of the region a (target)

Fig. 2.

noise. (b) Optimal image taking into account Mueller matrix

fluctuations.

(a) Optimal image taking into account only additive

Table 1.

the Diagonal Coefficients of the Depolarizer

Matrices Mu

Standard Deviation of the Fluctuations of

Δ

RegionMΔ;00

0.05

0.05

MΔ;11

0.1

0.03

MΔ;22

0.05

0.02

MΔ;33

0.01

0.02

a

b

Table 2.Optimal States Obtained by Taking into

Account or Not the Fluctuations of the

Polarimetric Properties of the Scene

εS

Gfluct¼ 0

Gfluct≠ 0

−30°

Algorithm

αS

αT

85°

20°

εT

10°

−19°

−90°

−10°

−21°

December 1, 2011 / Vol. 36, No. 23 / OPTICS LETTERS4591

Page 4

and the region b (background) are estimated thanks to a

database collected previously and containing sets of

Mueller matrices associated with the two regions that

have to be discriminated. These Mueller matrices are ac-

quired by generating 16 different configurations of the

PSG and PSA states [9].

Based on these estimations, we compute the PSA and

PSG states that maximize the contrast under two noise

hypotheses: additive noise only, which leads to ðαS;ϵSÞ ¼

ð15°;−30°Þ and ðαT;ϵTÞ ¼ ð−90°;−35°Þ, and Mueller ma-

trix fluctuations, which leads to ðαS;ϵSÞ ¼ ð−35°;−15°Þ

and ðαT;ϵTÞ ¼ ð−50°;−10°Þ. It is noticeable that these

two pairs of states are different. By implementing them

on the polarimetric imager, we obtain the scalar polari-

metric images represented in Fig. 4. It is seen that the

optimal states taking into account only additive noise re-

veals adhesive pieces by maximizing the difference be-

tween of average intensity coming from the target and

the background (Δ ¼ 1100). However, the presence of

high fluctuations decreases the contrast. On the other

hand, using the optimal states that take into account

the specific statistics of Mueller matrix fluctuations, the

difference between the average intensity is only about

(Δ ¼ 500), but, as the fluctuations are much lower, the

contrast is better. This is a clear illustration that correctly

optimized polarimetric imagers can help to enhance the

contrast in the presence of spatial fluctuations.

Let us now briefly address the computational issues for

estimation of the optimal states. Since there is no closed-

form solution to optimization of Eq. (7), one has to use a

numerical algorithm. According to our experience, the

function FðS;TÞ may have more than one local maxi-

mum but always has smooth variations. We have thus de-

signed an iterative code with a progressive mesh. For a

given value of the analysis state T (the same reasoning

can be done by fixing the illumination state), the illumi-

nation state S maximizing the contrast can be found as

follows. First, the whole Poincaré sphere is swept using

a sampling of 10° for the azimuth and the ellipticity of the

polarization state. The two highest local maxima that are

separated of minimum 20° are kept. Then around each of

these two maxima, a 20° × 20° wide area in azimuth and

ellipticity is swept with a sampling step of 5°, and the two

states leading to the highest contrast are kept. Finally,

around these two states, a 10° × 10° wide area is swept

with a sampling step of 1°, and the state leading to the

highest contrast is kept. This sequence is alternatively

performed on S and T in order to obtain a result in about

2s with a precision of 1°, which is sufficiently close to the

globally optimal performance.

In conclusion, the fluctuations of the polarimetric

properties of the scene induce a noise that can be more

detrimental to polarimetric target detection than additive

sensor noise. We have proposed a methodology that ad-

dresses successfully this issue and makes it possible to

optimize polarimetric imaging systems in the presence

of such heavy fluctuations. It opens up perspectives for

optimal adaptation of polarimetric imaging systems to

difficult applications such has imaging through diffusing

media or decamouflaging. Interesting perspectives in-

clude generalization to the discrimination of more than

two regions.

G. Anna’s Ph.D. dissertation is supported by the Délé-

gation Générale pour l’Armement (DGA), Mission pour la

Recherche et l’Innovation Scientifique (MRIS).

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Fig. 4.

account only additive noise. (b) Optimal image taking into

account Mueller matrix fluctuations.

(a) Optimal image of the scene in Fig. 3 taking into

Fig. 3.

(b) Intensity image of the scene.

(Color online) (a) Scheme of the observed scene.

4592OPTICS LETTERS / Vol. 36, No. 23 / December 1, 2011