Full analytical solution of adapted polarisation state contrast imaging

Debajyoti Upadhyay; Sugata Mondal; Eric Lacot; Xavier Orlik

doi:10.1364/OE.19.025188

1. Introduction

Taking into account the vectorial nature of the electromagnetic fields, polarimetric imaging contains obviously additional informations compared to classical imaging. The Mueller formalism describes the full polarimetric response of an object and thus is able to describe completely its properties of birefringence, dichroism, depolarization and polarisance [1–4]. However, these above mentioned properties exhibit respectively 3, 4 and 9 degrees of freedom and only a few of them remain pertinent for a given polarimetric scene. Moreover, there could be some situations when the object and the background would differ very weakly from 2 or more polarimetric degrees of freedom and that they could become distinguishable only by cumulating the global polarimetric information. Thus, we have proposed in earlier works [5, 6] a specific method to maximize the polarimetric contrast between an object and its background that takes into account the total polarimetric information of the scene [7–9]. This method consists of finding the fully polarised incident states that excite the object and background in such a way that they scatter light with polarimetric states located as far as possible on the Poincaré sphere [10]. Also optimized with adapted detection states, this method has provided in the presence of shot noise, gains in contrasts quantified by Bhattacharyya distances [11, 12] that could reach up to some factors of the order of 10 in some situations [6]. It has also been shown that the performances were not noticeably degraded by additional high contrast circular Gaussian speckle noise [13]. However, the ensemble of solutions describing the incident adapted polarimetric states were determined numerically by an iterative optimization process and no explanations were given concerning the nature of these solutions (circle or point on the Poincaré sphere) in function of the scene under investigation.

In this paper, we provide the full analytical solution of the adapted polarimetric states and explain the underlying physics through introducing the concept of 3-Distance Eigen Space. By mapping the mathematical theory with physical imaging technique we address the following issues :

What are the ensembles of solutions for the specific excitations states depending on the polarimetric scenes considered.
What is the effect of the shot noise on these solutions and
for a given polarimetric scene, what is the theoretical maximal distance in the Poincaré sphere between the two Stokes vectors describing the field scattered by the object and background region.

In a first part, we propose a mathematical modeling of the APSCI method and introduce the 3-Distance Eigen Space(3DES) to solve the problem. Considering the Stokes vectors describing the object and background scattering after an adapted excitation, we deduce their maximal theoretical distance in the Poincaré sphere. In the result section, for some generic polarimetric scenes, we show different possible ensembles of solutions of these adapted states. We show the effect of shot noise in 3-Distance Eigen Space and thus in APSCI method. We then evaluate the improvement in terms of contrast and reliability due to the use of the exact solution for the adapted state of illumination by comparing with the well-known existing techniques such as Lu-Chipman Decomposition [2] and our last APSCI random search technique [6, 13].

2. The 3-Distance Eigen Space and Adapted Polarisation State Contrast Imaging

2.1. The Mathematical Modeling of the Problem

We consider a scene with a polarimetrically homogeneous circular object surrounded by a homogeneous background. The Mueller matrices M_O and M _B fully describes the polarimetric properties of the two disjoint regions 𝒪 and ℬ respectively for the object and the background. Here M_O _ij and M_B _ij belongs to R ∀ (i, j) ∈ [0, 3]. Considering experimental situation, the scene is considered to be a priori unknown and hence we need an initial estimation of the Mueller matrices of the object M̃_O and of the background M̃_B by Classical Mueller Imaging (CMI) before using APSCI method. Here similarly, from the physicality criterion M̃_O and M̃_B belongs to R ⁴XR ⁴. During CMI, we consider that each pixel of the detector indexed by (u, v) receive an intensity I(u, v) perturbed by a shot noise of Poisson distribution. Then from noisy intensity matrices Ĩ(u, v), the Mueller matrices M̃ (u, v) are retrieved for each pixel. The noise at any two pixels of the detector are also considered to be mutually independent.

Let us consider a Stokes vector S used to illuminate the scene after the initial evaluation of Mueller matrices for 𝒪 and ℬ. Here S ∈ B ⁴. B ⁴ is the closed ball in 4 dimensions. Which implies, S ∈ R ⁴ : ||S||² ≤ 2. The estimations of the Stokes vectors of the field scattered by the object S̃_O and background S̃_B can be expressed as :

{\tilde{S}}_{O} = {[S_{O_{0}}, S_{O_{1}}, S_{O_{2}}, S_{O_{3}}]}^{T} = {\tilde{M}}_{O} S, {\tilde{S}}_{B} = {[S_{B_{0}}, S_{B_{1}}, S_{B_{2}}, S_{B_{3}}]}^{T} = {\tilde{M}}_{B} S

In APSCI, we only use fully polarised illumination states and hence, ||S||² = 2 i.e. S ∈ 𝒫, where 𝒫 is the set of fully polarised Stokes vectors corresponding to all the possible states on the surface of Poincaré sphere.

We define the 3-Distance between S̃_O and S̃_B in the Poincaré sphere by the Euclidean distance D considering their last three parameters in following way:

D (S) = \sqrt{\sum_{k = 1}^{3} {(S_{O_{k}} - S_{B_{k}})}^{2}}

The goal is to find the specific Stokes vector S_max for which,

D (S_{m a x}) = max_{S \in 𝒫} (D (S))

2.2. The Solution

Let us consider the following equations,

Δ M = {\tilde{M}}_{O} - {\tilde{M}}_{B} = [\begin{array}{l} Δ m_{00} & Δ m_{01} & Δ m_{02} & Δ m_{03} \\ Δ m_{10} & Δ m_{11} & Δ m_{12} & Δ m_{13} \\ Δ m_{20} & Δ m_{21} & Δ m_{22} & Δ m_{23} \\ Δ m_{30} & Δ m_{31} & Δ m_{32} & Δ m_{33} \end{array}]

Then, under illumination of the scene with the Stokes vector S = (s ₀, s ₁, s ₂, s ₃)^T, the difference vector is

Δ M . S = ({\tilde{M}}_{O} - {\tilde{M}}_{B}) . S = [\begin{array}{l} Δ m_{00} s_{0} + Δ m_{01} s_{1} + Δ m_{02} s_{2} + Δ m_{03} s_{3} \\ Δ m_{10} s_{0} + Δ m_{11} s_{1} + Δ m_{12} s_{2} + Δ m_{13} s_{3} \\ Δ m_{20} s_{0} + Δ m_{21} s_{1} + Δ m_{22} s_{2} + Δ m_{23} s_{3} \\ Δ m_{30} s_{0} + Δ m_{31} s_{1} + Δ m_{32} s_{2} + Δ m_{33} s_{3} \end{array}]

Hence, from Eq. (3), we have to maximise the sum of the squares of the last three components of the right hand side of Eq. (5). Now this term is the 3-Distance quadratic function of s_i’s, ∀i ∈ 0, 1, 2, 3. Let us denote this quadratic function by 𝒬 (s ₀, s ₁, s ₂, s ₃). Then we get,

𝒬 (s_{0}, s_{1}, s_{2}, s_{3}) = \sum_{j = 1}^{3} {(\sum_{i = 0}^{3} Δ m_{j i} s_{i})}^{2} = \sum_{i, j = 0}^{3} a_{i j} s_{i} s_{j}

Here, we want to specify that we would like to have a symmetric solution for a_ij and which is unique. It can be calculated from Eq. (6) that

a_{i j} = \sum_{k = 1}^{3} Δ m_{k i} Δ m_{k j}, \forall i, j \in 0, 1, 2, 3

From Eq. (7), we can easily verify that a_ij’s ∀i, j ∈ 0, 1, 2, 3 are symmetric. Hence,

A (i, j) = a_{i j} = a_{j i}, \forall i, j \in 0, 1, 2, 3

Here A is a real symmetric 4X4 matrix, which implies that all its eigen values and eigen vectors are real and can be diagonalised to A^d by an orthogonal matrix transformation of form O^T AO, where O is orthogonal matrix (columns of which are eigen vectors of A) and A^d is real and diagonal (having the corresponding eigenvalues of A on the diagonal).

Thus,

m a x (𝒬 (s_{0}, s_{1}, s_{2}, s_{3})) = m a x (\sum_{i, j = 0}^{3} a_{i j} s_{i} s_{j}) = m a x (S^{T} A S)

From linear algebra we know that 𝒬 will be maximum for S_in = S_max, where S_max is the eigen vector of A corresponding to maximum eigen value λ_max. Hence we get,

m a x (𝒬) = S_{m a x}^{T} A S_{m a x} = λ_{m a x} {‖ S_{m a x} ‖}^{2}

2.3. 3-Distance Eigen Space

An orthogonal transformation in 2D is similar to an operation of rotation. Thus, the transformation of S ₀, S ₁, S ₂, S ₃ space to the eigen space of matrix corresponding to 3-Distance Quadratic Function is equivalent to a rotation in 4D space. Moreover, from a set of orthogonal basis (here it is S ₀, S ₁, S ₂, S ₃) if we move to the eigen bases of a symmetric matrix (in this case it is A), the new bases will also remain orthogonal. The nomenclature for this new rotated space is from now on will be 3-Distance Eigen Space (3DES). 3DES has two intrinsic properties. Firstly, it depends on the quadratic form we work with, which can be very general and irrespective of the imaging problem we are addressing here (in our case the Cartesian distance function on Poincaré space of Stokes vectors) Eq. (6) and can be broadened further and secondly it depends on the exact Mueller matrices of the scene we work with and this is what makes the 3DES adaptive to our imaging. Now if we concentrate on the trace of A, we can see that:

T r (A) = \sum_{i = 0}^{3} a_{i i} = \sum_{i = 0}^{3} \sum_{k = 1}^{3} Δ m_{k i} Δ m_{k i} = \sum_{i, k = 0}^{3} {(Δ m_{k i})}^{2} - \sum_{j = 0}^{3} {(Δ m_{0 j})}^{2} = \sum_{i = 0}^{3} λ_{i}

Here λ_i’s (∀i ∈ (0, 1, 2, 3)) are the eigen values of A. As we know under similarity transformation the trace is conserved and hence Eq. (11). In the result section we analyse some properties of 3DES and it becomes immensely helpful to understand underlying physics of APSCI.

3. The Illumination Technique of APSCI

Here we finally determine the exact polarimetric illumination state of APSCI by taking into account the above analytical solution along with physical realisation criterions. Firstly, as we have seen in the last section λ_max and S_max are real, hence we can reach a physically realisable solution. Though due to the solution space (i.e. ||S||² ≤ 2) we work with, we may come up with solutions where input Stokes vectors are not fully polarised. For the sake of APSCI, only fully polarised solutions are selected. As we can readily see from Eq. (10), the maximum 3-Distance is proportional to $‖ S_{m a x}^{2} ‖$ , which is maximum and equal to 2 for only fully polarised illumination. However, the upper limit of the 3-Distance is 2λ_max, but it might not be achievable in most of the imaging situations. In fact here, we are obtaining the direction of the illumination state using mathematical algorithm and then we are stretching out Stokes vector norm to unity in order to achieve specific fully polarised incident light. So, the algorithm we use to get $S_{m a x}^{n e w}$ after initial evaluation of S_max is as follow :

Normalisation by the last 3 parameters: $S_{m a x}^{n e w} = \frac{S_{m a x}}{\sqrt{\sum_{k = 1}^{3} {(S_{{m a x}_{k}})}^{2}}}$
Selection of the Physical solution by multiplying by −1 the entire Stokes vector in case its component S_o describing the intenty is negative.

Concerning step 2 as we can see from Eq. (10), S_max and –S_max both will yield the maximum of 3-Distance Quadratic Function, and hence we choose the physically possible solution so that S _max₀ is positive.

We would also like to mention here that though the input Stokes vector are fully polarised, the output Stokes vectors S̃_O and S̃_B are not normalized and hence they can exhibit different rates of depolarization. As a consequence, the maximization of the 3-Distance mentioned above takes into account both the physical entities: the polarization state and the intensity of the polarized part of the scattered field.

4. The Detection Technique of APSCI

In the detection part we project the scattered fields $S_{o u t}^{O}$ and $S_{o u t}^{B}$ respectively from 𝒪 and ℬ into some state S_d on the surface of Poincaré sphere. Then to maximise the imaging contrast we have to maximise the difference of projection. Hence we have to maximize the following function F(S_d)

F (S_{d}) = S_{d} . S_{o u t}^{O} - S_{d} . S_{o u t}^{B}

where ||S_d||² = 2

In order to optimize F(S_d) we differentiate both side of Eq. (13) and set it equal to 0 and get,

d F (S_{d}) = d S_{d} . (S_{o u t}^{O} - S_{o u t}^{B}) = 0

Hence we can say from Eq. (14), that dS_d is orthogonal to $(S_{o u t}^{O} - S_{o u t}^{B})$ . Again differentiating ||S_d||² = 2 we get,

2 d S_{d} . S_{d} = 0

So by comparing Eqs. (14) and (15), |F| will be maximum for S_d parallel and antiparallel to $(S_{o u t}^{O} - S_{o u t}^{B})$ . Now, we use a Two Channel Imaging (TCI) system that projects $S_{o u t}^{O}$ and $S_{o u t}^{B}$ , into the following 2 orthogonal polarisation states of analyzer:

S_{o u t 1} = {[1, Δ {\vec{S}}^{T} / ‖ Δ \vec{S} ‖]}^{T}, S_{o u t 2} = {[1, - Δ {\vec{S}}^{T} / ‖ Δ \vec{S} ‖]}^{T}

Where $Δ {\vec{S}}_{i} = S_{o u t_{i}}^{O} - S_{o u t_{i}}^{B}$ , ∀i ∈ (1, 2, 3). Thus, S_out ₁ and S_out ₂ will increase the detected intensities corresponding respectively to the object and background pixels while imaging.

Now we define the APSCI parameter for each pixel of the detector indexed by corresponding coordinates (u, v) as :

A P S C I (u, v) = \frac{I_{1} (u, v) - I_{2} (u, v)}{I_{1} (u, v) + I_{2} (u, v)},

Where I ₁(u, v) and I ₂(u, v) are the detected intensities after projection respectively on the 2 states of polarization S_out ₁ and S_out ₂.

Finally, in order to quantify the imaging contrast of the scene under investigation, we calculate the Bhattacharyya Distance, defined by the measurement of the amount of overlap between the two statistical populations that are the APSCI parameter over the background and object region.

5. Results and Analysis

In the following analysis we choose to illustrate the results obtained from previous sections by considering scenes with various polarimetric properties. In the first case the object and background have a difference of 10% in scalar birefringence, while in the second situation, the difference is of 10% only in the azimuth of diattenuation or dichroic vector. In the last case, we investigate a more complex scene where object and background differ by 10% in scalar dichroism, scalar birefringence and scalar depolarisation simultaneously. In first two generic cases, we address different possibilities of the ensemble of solutions of specific excitations from the analytical as well as graphical point of view. We investigate further the 3DES and effect of noise over it and hence on measurement. At last in the complex scene we study both the performances of APSCI using analytical method and simplex search iterative method and compare the contrast level with respect to what is achievable from Classical Mueller Imaging (CMI).

We can see in Fig. 1(a), that under several evaluations of S_max for different realizations of shot noise, the ensemble of solutions is on and around a great circle over Poincaré sphere. From theoretical solutions, we can see in this case if we consider pure Mueller matrices (not affected by shot noise), then the eigen space corresponding to λ_max is degenerate. In this kind of cases, two eigen vectors, let’s say $S_{m a x}^{1}$ , $S_{m a x}^{2}$ , both corresponds to the maximum eigen value. Hence we get the solution ensemble as the plane defined by $S_{m a x}^{1}$ , $S_{m a x}^{2}$ . In presence of shot, we consider the evaluated Mueller matrices and we see that the degeneracy in 3DES corresponding to maximal eigen value breaks and hence one of the eigen directions becomes preferred over the others. In this figure we have plotted the degenerate eigen vectors ([1, 0, 1, 0] using the red arrow and [1, 0, 0,1] using the pink arrow, both correspond to same eigen value λ_max = 0.0979) considering pure Mueller matrices and modified eigen directions ([1, 0.0184, 0.2636, −0.9644] using the dark green arrow corresponds to eigen value = 0.0887 and [1, 0.0107, 0.9645, 0.2639] using the light green arrow corresponds to eigen value = 0.0991) considering Mueller matrices perturbed by a single instance of shot noise. We can easily verify from the Fig. 1(a) that the two eigen planes have a small angle between them in (S ₁, S ₂, S ₃) co-ordinate system and this statistics of the angular shift between the eigen planes in 3DES can provide the effect of noise during measurements. To provide an intuitive understanding we investigate the case graphically in Fig. 1(b). In case of birefringence, the direction of the birefringence vector in Poincaré sphere is defined by corresponding azimuth and ellipticity, while the scalar birefringence is defined by the rotation angle around it. Now as in our scene the object and the background only differ in scalar birefringence, hence the angles of rotation (for the object denoted by dark blue, for the background denoted by bluish green) of a polarimetric state around the same birefringence axis (denoted by the red arrow) are different. The distance between the final states (heads of arrows) will in turn denote the 3-Distance (denoted by the pink arrow) in the Poincaré sphere. For any point S_in not lying in the equatorial plane perpendicular to the birefringence vector, the same angular rotation will always yield lesser length of the pink arrow compared to any point $S_{m a x}^{n e w}$ chosen over the great circle lying on this plane and for all the points $S_{m a x}^{n e w}$ ’s on that great circle we expect same length of the pink arrow. From this graphical explanation, then we can further confirm the reason of circular cluster for ensemble of solutions.

Fig. 1 (a) Ensemble of solutions of the incident adapted states for a scene with difference in 10% in scalar birefringence. The arrows defines the eign vectors using pure Mueller matrices(red and pink) and using Mueller matrices pertubed by a generic shot noise(light and dark green) (b) Graphical representation of the birefringence scene and at upper right corner: the ensemble of solutions over Poincaré sphere for different realisations of shot noise. (c) Ensemble of solutions for a scene with difference in 10% in azimuth of dichroic vector. The arrows show the average direction of incident adapted states from iterative simplex search method (blue) and from analytical method (green) (d) Graphical representation of the dichroic scene and at the upper right corner: the ensemble of solutions over Poincaré sphere for different realisations of shot noise. The colour mapping on the spheres at the upper right corners for (b) and (d) maps linearly the 3-Distance for all the polarimetric states on Poincaré sphere considering corresponding pure Mueller matrices respectively.

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In the upper right corner image of Fig. 1(b), we show the distribution of evaluated states on Poincaré sphere. The colourmap of the surface of sphere maps the 3-Distance considering pure Mueller matrices. The black lines defines the boundary of the region where 3-Distance is 95% of maximum or above.

To analyse the second case, let us consider Fig. 1(c). In this scene, the object and the background differ by 10% only in azimuth of their dichroic vectors. This time our 3DES is non-degenarate with respect to the maximum eigen value and hence we get the solution ensemble as a point cluster. The blue points are the ensemble of solutions using simplex search algorithm and the green points are using the analytical method, while the average of the eigen vectors under several number of trials using different Poissonian shot noise are plotted ([1, 0.5315, 0.7955, −0.0039] using the blue colour arrow for simplex search method) and ([1, 0.533, 0.8042, −0.0035] using the green colour arrow for analytical method which is superimposed with the blue arrow in the figure). At Signal to Noise Ratio(SNR) = 4.3, though the average of specific excitations from both the method are more or less close, but still we get more robust contrast by the analytical method. Here the average contrast = 0.96, standard deviation of the contrast = 0.04 using analytical method and average contrast = 0.94, standard deviation of the contrast = 0.05 using simplex search method. The reason can easily be understood from the fact that the cluster of specific excitations (green points) of analytical methods are more compact on Poincaré sphere compared to distribution of states (blue points) by the simplex search method for different realisations of shot noise. We would like to point out that, due to their different dichroism, the energy scattered by the object and background and focalized by imaging elements towards the detector can be different, and hence their corresponding classical SNRs. Thus, we choose to define here a global SNR by considering the shot noise generated by the amount of energy received by the detector without the use of any polarizer and after the back-scattering on a virtual perfectly lambertian and non-absorbing object, whose size and position are similar to that of the scene under investigation.

From Fig. 1(d), using graphical analysis we explain intuitively the ensemble of solutions in this case. For a dichroic object we define the direction dichroic vector by its azimuth and ellipticity. Under some polarimetric illumination of a dichroic object the Stokes vector turns towards the dichroic vector and the decrement of its magnitude is dictated by the scalar dichroism.

In the case concerned, our object and background differ only in the azimuth of their dichroic vectors. Object dichroic vector is denoted by D⃗_O using the blue arrow, while the background dichroic vector is by D⃗_B using the greenish blue arrow in Fig. 1(d). Any incident Stokes vector S_in on Poincaré sphere will provide more or less similar 3-Distance. Only the Stokes vector, that is bisecting the angle between D⃗_B and D⃗_O, will provoke a scattering in almost opposite direction for object and background and hence will yield the maximum 3-Distance(denoted by the pink arrow) and that’s the reason why we get the ensemble of solutions as a cluster of points.

From Fig. 1(d), using graphical analysis we explain intuitively the ensemble of solutions in this case. For a dichroic object we define the direction dichroic vector by its azimuth and ellipticity. Under some polarimetric illumination of a dichroic object the Stokes vector turns towards the dichroic vector and the decrement of its magnitude is dictated by the scalar dichroism. In the case concerned, our object and background differ only in the azimuth of their dichroic vectors. Object dichroic vector is denoted by D⃗_O using the blue arrow, while the background dichroic vector is by D⃗_B using the greenish blue arrow in Fig. 1(d). Any incident Stokes vector S_in on Poincaré sphere will provide more or less similar 3-Distance. Only the Stokes vector, that is bisecting the angle between D⃗_B and D⃗_O, will provoke a scattering in almost opposite direction for object and background and hence will yield the maximum 3-Distance(denoted by the pink arrow) and that’s the reason why we get the ensemble of solutions as a cluster of points.

In order to understand the standard deviation of imaging contrast, which is higher in the dichroic case than the earlier birefringence case, in our last work [13] we have defined a new parameter, called the Cartesian distance between Mueller matrices of the object and the background. This matrix distance is calculated by sum of the squares of position-wise differences of the elements of two matrices. Intuitively this parameter seems to be very pertinent and explains some results [13]. Here from Eq. (11), we confirm that this Cartesian distance is proportional to the sum of the eigen values of 3DES. The more will be the Cartesian distance, the more will be the maximum eigen value. Hence from Eq. (10) we can verify that the scatterings from object and background will the further in Poincaré sphere and in turn we will reach higher contrast in imaging. However, in some cases we have observed that the same Cartesian distance might yield quite different image contrast as well as different standard deviation of the contrast. This can be explained by the distribution of eigen values over R. Though in these case, the sum of the eigen values are same, the more will be the maximum eigen value with respect to the others, the more will be our achieved contrast. Hence we can say that the maximum eigen value and the ratios of maximum eigen value with respect to other eigen values can be the more pertinent parameters for characterization of a polarimetric scene in these sorts of imaging. Now, if we again concentrate on Fig. 1(d), in the corner image we show the distribution of evaluated states on Poincaré sphere, which spreads due to the small value of Cartesian distance between object and background Mueller matrices. Again the colourmap is linearly related to the 3-Distance for each state using pure Mueller matrices. The black lines defines the boundary of the region with 95% of maximum 3-Distance or above like the birefringence case. The colourmap goes upto 0 in this case, and which can be justified from the fact that here we have a null eigen value corresponding to 3DES.

In Fig. 2 we analyze a complex scene with a difference of 10% between object and background in scalar birefringence, scalar dichroism and scalar depolarisation simultaneously. In this case the real Mueller matrices of the object and background regions are the following,

M_{O} = [\begin{matrix} 0.500 & 0.300 & 0 & 0 \\ 0.180 & 0.300 & 0 & 0 \\ 0 & 0 & - 0.112 & 0.212 \\ 0 & 0 & - 0.176 & - 0.094 \end{matrix}] and M_{B} = [\begin{matrix} 0.500 & 0.250 & 0 & 0 \\ 0.125 & 0.250 & 0 & 0 \\ 0 & 0 & - 0.037 & 0.213 \\ 0 & 0 & - 0.213 & - 0.037 \end{matrix}]

Fig. 2 (a) Comparisons of contrast from different pertinent parameters by Bhattacharyya distance vs. SNR curves. (b) at SNR = 3.4, (i.) cluster of solutions from analytic (green points) and simplex search (blue points) method on Poincaré sphere, and Visual comparison of the complex scene obtained from (ii.) using best parameter of Mueller Matrix, (iii.) using APSCI by simplex search method and (iv.) using APSCI by analytical method.

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Under the effect of the shot noise the average of the retrieved Mueller matrices using CMI for the object and the background are as follows,

{\tilde{M}}_{O} = [\begin{matrix} 0.501 & 0.296 & 0.001 & - 0.002 \\ 0.182 & 0.297 & 0.002 & 0.003 \\ - 0.001 & 0.001 & - 0.112 & 0.209 \\ - 0.002 & 0.001 & - 0.171 & - 0.091 \end{matrix}] and {\tilde{M}}_{B} = [\begin{matrix} 0.499 & 0.249 & 0.001 & - 0.006 \\ 0.123 & 0.245 & 0.002 & - 0.000 \\ 0.000 & - 0.002 & - 0.040 & 0.213 \\ 0.000 & 0.008 & - 0.210 & - 0.042 \end{matrix}]

In Fig. 2(a) we have compared the relative performances of APSCI using simplex search method and analytical method with respect to the relevant parameters of CMI, by evaluating the Bhattacharyya distance for different levels of SNR. For comparison purposes, the Bhattacharyya distances are plotted versus SNR for a same number of intensity acquisitions. Here B(M), B(M _Δ), B(Δ), B(APSCI) and B(MAPSCI) are the Bhattacharyya distance available from best element of Mueller Matrix (the one element out of the 16 elements of total Mueller matrices that provides maximum contrast), from the best element of Dichroic matrix after Lu-Chipman forward decomposition [2, 14] (the one element out of the 16 elements of dichroic parts of the total Mueller matrices that provides the maximum contrast), from scalar dichroism parameter, from APSCI using simplex search method [13], using the above analytical method respectively. We can see that the performance of APSCI is quite higher in terms of achieved contrast than that of the other pertinent parameters of current CMI technique, even at low SNR, which is particularly important for noninvasive biological imaging. To provide a visual comparison, at a low SNR = 3.4, we have provided the corresponding image of the scene in Fig. 2(b). The image contrast level is 0.05 and is shown in Fig. 2(b)ii., using the best parameter of the total Mueller matrices. In Fig. 2(b)iii., using the simplex search method the average imaging contrast we reach in APSCI is 0.19, while in Fig. 2(b)iv., from the analytical method in APSCI we reach an average contrast 0.22, which is around 15% higher than what is achievable by iterative simplex search algorithm [6]. At SNR = 3.4, we also plot the ensemble of solution for different realisations of shot noise in Fig. 2(b)i. Here we can see that the cluster of solution of APSCI by simplex search tool (using blue points) and by analytical method (using green points) are having an angular separation of around 10° on the Poincaré sphere. This explains the difference in contrast level from these two methods. Moreover we can also observe that the standard deviation of contrast is lower using the analytical method. Keeping in mind, a single shot imaging, which is important for in-vivo applications, using the analytical method we significantly improve our last APSCI technique in terms of reliability and contrast. The faster and precise analytical method also make the imaging method more robust from the point of view of possible movements in object during in-situ imaging compared to exponential order simplex search method.

6. Conclusion

The APSCI method takes into account the global cumulative polarimetric differences between the object and background and such technique is expected to increase noticeably our ability to detect and characterize objects. In this work, the underlying physics driving the APSCI method has been explained, using linear algebraic structure of the problem. In the general case, we have provided the full analytical solution of the ensemble of adapted illumination polarimetric states needed to optimize the contrast between an object and its background in function of their respective polarimetric properties. Given a scene with polarimetric object and background, we have provided the maximum value of contrast achievable using the APSCI method. We have also shown that the presence of shot noise broke the degeneracy in the ensemble of adapted states. Moreover, at low SNR condition, the analytical method has significantly improved the value and reliability of contrast in imaging, which is an upshot from experimental point of view. Such specific polarimetric imaging technique opens the vast field to promising applications concerning biological tissues that exhibit often several polarimetric properties simultaneously.

Acknowledgments

We would like to thank the Région Midi-Pyrénées for providing us the financial support for this work.

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