1. Introduction

Polarimetric imaging for object detection has attracted much interest in the past few years especially concerning topics such as submersible object detection [1] [2], target detection in scattering media [3] [4] and detection of inhomogeneities in biological tissues [5].

The various polarimetric systems can be divided into three main categories: Mueller matrix polarimeters [6][7][8], Stokes polarimeters [9][10] and two-channel imaging systems (TCI) [3][11][12][13][14][15]. A Mueller system is composed of a polarization-state generator (PSG) and a polarization-state analyser (PSA) that respectively generates and analyses the light using four linearly independent states. Such a system is able to provide a complete polarimetric characterization of a scattering object. A Stokes polarimeter is composed only of a PSA and measures the polarization state of the scattered or reflected light. Concerning the TCI systems, the scene is illuminated with a single polarization state and the scattered light is analysed into the polarization states parallel and orthogonal to the one of the incident light. These two corresponding intensity images can be combined to form a polarization-difference image [11] [12]. It has been shown recently that in the hypothesis of a purely depolarizing material, the difference of these two intensity images divided by their sum represents an efficient estimator of the degree of polarization according to its close proximity to the Cramer-Rao bound [16]. Such technique is able to reveal contrasts that does not appear in conventional reflectance images [3] [2] [11] [15]. Let us notice that this method can be generalized to multiple wavelengths as pointed out by Demos and al. [13] [14] who have performed deep subsurface imaging in tissues using spectral polarization difference images.

We would like to emphasize that the polarization states used in Stokes and Mueller polarimeters are generally chosen to minimize the noise induced correlations between successive intensity measurements in order to minimize the propagation of the corresponding errors in the calculation of the polarimetric data (the Stokes vector or the Mueller matrix) from the raw data [10] [17] [18] [7] [8]. Such optimizations imply the use of precise elliptic polarization states. Very recently, an optimized illumination in active Stokes polarimeters has been proposed for partially depolarizing scene in the presence of additive Gaussian detection noise [19] and confirms the interest of a polarimetric excitation adapted to the scene under investigation [20]. Concerning the TCI method, only linear or circular polarization states have been used until now. We propose in this paper the study of a general method for obtaining an optimum polarimetric contrast between an object and its background for a TCI system. The proposed method is based on the determination of the optimized incident and detection polarization states specific to the couple object/background considered, and can be used whatever are their polarimetric properties. More precisely, it consists in finding the specific incident polarization that will provide scattered polarimetric states of the object and background as opposite as possible on the Poincare sphere [21] and then, to detect suitable projections of these scattered states via the PSA. Let us notice that this method implies a first evaluation of the polarimetric properties of the object and background that can be performed by a Mueller imaging system [8].

After a brief review of the Stokes-Mueller Formalism and the polar decomposition in section 2, we describe the proposed Adapted Polarization State Contrast Image (APSCI) method in section 3. In Section 4, we quantify the corresponding enhancement of polarimetric contrast for various object/background couples with different polarimetric properties (diattenuation, re-tardance, and depolarization) using Monte Carlo simulations in the case of low flux coherent imagery. Taking into account the shot noise of the detector and the numerical propagation of errors in the calculation of the polarimetric data from the raw data, the contrast obtained by the APSCI method is quantified with the Bhattacharyya distance [22], and compared to the various polarimetric contrasts we can achieve using a classical Mueller Imaging system [8] with or without the polar decomposition [23].

2. Brief review of the formalism

2.1. The Stokes-Mueller formalism

The Stokes-Mueller formalism [24] is particularly adapted to describe the polarimetric properties of a field scattered by a partially depolarizing object. The Stokes vector S⃗ can be used to describe the state of polarization of such a field:

where for Φ = 0°,90°, and ±45°, I _Φ represents the intensity transmitted through a linear polarizer set at the azimuthal angle Φ, and I_RC and I_LC represent the intensity transmitted through a right and a left circular polarizer respectively. Brackets mean ensemble average of the intensity that is generally performed by space and/or time averaging. The Stokes parameter S ₀ is the total intensity of the light, S ₁ and S ₂ describe the partial linear polarization, and S ₃ describes the partial circular polarization. These four parameters verify the following Stokes-Verdet inequality [24]

that becomes an equality in the case of a completely polarized field. A powerful representation to illustrate and to handle graphically polarimetric entities is the Poincaré sphere where each point of its surface represents a unique state of polarization.

A Mueller matrix M of a sample relates the Stokes vector S⃗_out of an outcoming light with the Stokes vector S⃗_in of an incoming light. It characterizes the linear interaction between the sample and the light, and is a function of the wavelength, and of the incident and scattered directions of the field. This relation is written as:

where M = (m_ij) is a 4 × 4 real matrix with 0 ≤ i,j ≤ 3; m ₀₀ is the sample transmission for unpolarized incident light; m ₀₁, m ₀₂, and m ₀₃ describe the dichroism (diattenuation); and m ₁₀, m ₂₀ and m ₃₀ represent the components of the polarizance vector. This latter vector characterizes the capability of the sample to polarize an unpolarized incident light. The Mueller Matrix also contains information about retardance and depolarization, however such physical properties need a mathematical decomposition of the Mueller Matrix to become accessible.

2.2. Polar decomposition

S. Y. Lu and R. A. Chipman proposed in 1996 [23], a decomposition of a Mueller matrix into a product of three matrix factors: a diattenuator M _D, a retarder M _R, and a depolarizer M _Δ. M _D has 4 degrees of freedom that are: the transmission T of the sample for an unpolarized incoming light, the scalar diattenuation D, and the azimuth λ _D⃗ and the ellipticity ε _D⃗ of the diattenuation vector D⃗. The matrix M _R has 3 degrees of freedom: the retardance R, the azimuth λ_R⃗ and the ellipticity ε_R⃗ of the retardance vector R⃗. The matrix M _Δ has 9 degrees of freedom. Six of them are devoted to the depolarization property described by its three orthogonal principle axes (v⃗₁,v⃗₂,V⃗₃) associated to their eigenvalues (Δ₁,Δ₂,Δ₂). These axes can be conveniently described by an axial rotation Rot_[V⃗,ϕ] from the orthonormal basis (S⃗₁,S⃗₁,S⃗₃) of the Poincaré sphere. As such rotation is not unique, we choose in the calculations the one that minimizes the rotation angle ϕ. The last three degrees of freedom of the depolarizer matrix are the polarizance P _Δ, the azimuth λ_P⃗Δ and the ellipticity ε_P⃗Δ of the polarizance vector P⃗_Δ.

As the matrix multiplication is not commutative, the decomposition of a Mueller matrix into a product of three matrix factors can give rise to 6 possible arrangements that are not equivalent. J. Morio and F. Goudail [25] have shown that these arrangements can be split into two families, which depend on the order in which the diattenuator and the depolarizer matrices are multiplied:

Lu and Chipman have demonstrated that the three possible arrangements of the family 𝓕 _ΔD give rise to the same diattenuation, retardance and depolarization power. Moreover, J. Morio and F. Goudail have shown that only this family always leads to physical Mueller matrices when using the generalized forward decomposition. R. Ossikovski et al. [26] have then given a mathematical procedure to perform a dual decomposition (also called reverse decomposition) able to generate physical Mueller Matrices for the family 𝓕 _DΔ inside which, as in the forward decomposition, the three possible arrangements exhibit the same polarimetric properties. They have shown moreover that the choice of the decomposition to use should be driven by the relative position of the depolarizing and diattenuator components: in case depolarization precedes diattenuation, the experimental Mueller matrix should be factorized by using the reverse decomposition and vice versa. However, when nothing is known a priori about an experimental Mueller matrix, questions still remain concerning the most suitable decomposition to use. As most of the depolarizing media can be expressed by diagonal depolarizing matrices, Ossikovski and al. have proposed to retain the decomposition that leads to the depolarizer Mueller matrix the closest to a diagonal matrix form [26].

3. The APSCI method

Let consider an homogeneous object of interest surrounded by an homogeneous background, both being possibly textured. The image of the scene can be divided into two disjoint regions 𝒪 and 𝓑 for respectively the object and background. The polarimetric responses of these regions are described by their corresponding Mueller matrices M_O and M _B.

As the scene is a priori unknown, in order to adapt the TCI system to the particular couple object/background under investigation, the method begins by an estimation of the object and background Mueller matrices, called respectively $\widetilde{{\mathbf{M}}_{\mathbf{O}}}$ and $\widetilde{{\mathbf{M}}_{\mathbf{B}}}$, using a classical Mueller imaging system. Let an illumination Stokes vector S⃗∈ 𝒫, 𝒫 being the set of polarization states totally polarized, the estimation of the Stokes vector of the field scattered by the object $\widetilde{{\overrightarrow{S}}_O}$ and background $\widetilde{{\overrightarrow{S}}_B}$ are obtained from $\widetilde{{\mathbf{M}}_{\mathbf{O}}}$ and $\widetilde{{\mathbf{M}}_{\mathbf{B}}}$ as follow:

We define the Euclidean distance D between the last three parameters of $\widetilde{{\overrightarrow{S}}_O}$ and $\widetilde{{\overrightarrow{S}}_B}$ as:

Then, we determine numerically the specific incident Stokes vector S⃗_in that will maximize this Euclidean distance by using a simplex search algorithm performed from several initial polarimetric states uniformly distributed on the Poincaré spherein order to avoid local maxima. We would like to emphasize that the Stokes vectors $\widetilde{{\overrightarrow{S}}_O}$ and $\widetilde{{\overrightarrow{S}}_B}$ are not normalized. As a consequence, we maximize the difference between the totally polarized parts of the fields scattered by the object and background by taking into account two physical entities: the polarization state and the intensity of the polarized part of the scattered field.

Thus, we calculate the optimized incident Stokes vector S⃗_in so that:

Finally, by the use of the PSA, the scattered field resulting from the selective excitation S⃗_in, is projected on two specific polarization states S⃗_out1 and S⃗_out2 chosen to maximize, between the object and background regions, the contrast of the polarimetric data APSCI(u, v) that is defined as:

where I ₁(u,v) and I ₂(u,v) describe the two intensity images obtained by the projection of the scattered fields respectively into S⃗_out1 and S⃗_out2.

S⃗_out1 and S⃗_out2 are defined as the Stokes vectors that maximize respectively $\left(\widetilde{I_O}-\widetilde{I_B}\right)$ and $\left(\widetilde{I_B}-\widetilde{I_O}\right)$ where $\widetilde{I_B}$ and $\widetilde{I_O}$ are the evaluations of the mean intensity detected respectively from the object and background scattering. Thus, if we define the vector ΔS⃗ in the following way:

a straightforward calculation shows that the Stokes vectors S⃗_out1 and S⃗_out2 can be written simply as:

where ∥ … ∥ is the Euclidean distance.

On the Poincaré sphere, S⃗_out1 and S⃗_out2 are in fact respectively parallel and anti-parallel to the vector [1, ΔS⃗]_T. Let us notice that in these 2 latter equations, the first component of the Stokes vectors have been set to 1, as our optimization is performed only on the polarized part of the field. Thus, on the ideal case (when the maximum value of D corresponds to a selective excitation S⃗_in leading to polarimetric scattered fields of background and object diametrically opposed on the Poincaré sphere), the average value of the APSCI parameter in the region 𝒪 (resp.𝓑) is equal to 1 (resp. -1).

4. Evaluation of the polarimetric contrast of the APSCI method compared to Mueller imaging

This section is devoted to the evaluation of the polarimetric contrast that can be obtained from three types of polarimetric data: the raw data of a Mueller matrix, the data obtained from the polar decomposition of this Mueller matrix and the data obtained from the proposed APSCI method. All the following calculations are performed for an uniform monochromatic illumination and for a low-flux detection. We use the Bhattacharyya distance as a contrast parameter for each physical quantity under investigation.

4.1. The contrast parameter: the Bhattacharyya distance

For experimental situations where the object and background mean values are unknown, it has been demonstrated that the Bhattacharyya distance is a efficient figure of merit when we use detection algorithms based on the generalized likelihood ratio test. Especially, the area under the curve (AUC) of the receiver operating characteristic (ROC) of the ideal observer is an increasing function of the Bhattacharyya distance [27] [28].

Let P_O (x) and P_B (x) be the probability density functions of a random variable x describing respectively the object and background regions. The Bhattacharyya distance B(x) is then defined as:

with ln being the neperian logarithm.

This equation is used to perform a quantitative comparison of the detection capabilities of the 3 types of polarimetric data cited above.

We calculate the Bhattacharyya distance B_APSCI of the APSCI parameter from the equations 10 and 13. Regarding the Bhattacharyya distance obtained from the Mueller imaging alone, we take the maximum Bhattacharyya distance value among the 16 elements:

where B(M _ij) is the Bhattacharyya distance corresponding to the element (i,j) of the Mueller matrices M.

From the Mueller imaging associated to the polar decomposition, we get a set of matrices, vectors and scalars, and we define Bhattacharyya distances for each of these polarimetric entities. As defined in eq. (14) for the Mueller matrix, we calculate the contrast parameters B(M _D), B(M _R) and B(M _Δ) corresponding respectively to the three elementary matrices of diattenua-tion, retardance and depolarization obtained from the polar decomposition. Then we determine the Bhattacharyya distances corresponding to the 16 degrees of freedom coming from the polar decomposition of the Mueller matrix (see §2.2). The last 2 contrast parameters that can be estimated are the Bhattacharyya distance of the average linear and circular depolarization power respectively written as B(D_opL) and B(D_opC). All the polarimetric parameters obtained from the polar decomposition are summarized in Fig. 1. The polarimetric entities on which we calculate the Bhattacharyya distance are encircled.

 

Fig. 1. From the Mueller matrix, we have access directly to the average linear and circular depolarization power (respectively D_opL and D_opC). Performing the polar decomposition, we have access to the 3 elementary Mueller Matrices of Retardance M _R, Depolarization M _Δ and Diattenuation M _D, from which we calculate the 16 degrees of freedom represented at the bottom of the diagram. Bhattacharyya distances will be performed on all encircled entities in order to evaluate their detection capabilities.

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4.2. Modelling by Monte Carlo simulations

In order to evaluate the detection capabilities of the 3 following methods: the Mueller imaging, the Mueller imaging associated to the polar decomposition and the APSCI method, we compute by Monte Carlo simulations the detected intensity images from the polarimetric response of various scenes exhibiting some retardance, diattenuation and depolarization effects. For each of the three methods, we compare the Bhattacharyya distances of some of the most relevant physical parameters mentioned above.

The Mueller matrices of the object M_O and background M_B characterize the scene under investigation. From the Mueller imaging, at each pixel indexed by (u,v), we get the intensity matrix I(u,v) that is calculated from the following relation:

where M(u, v) is the Mueller matrix M_O (respectively M_B) in the region 𝒪 (respectively 𝓑), W the matrix of the PSG that contains the 4 incident Stokes vectors, and A the matrix of the PSA that contains the 4 states of analysis. In order to take into account the shot noise, we perturb the intensity matrix I(u,v) with a Poisson distribution and thus generate the corresponding noisy matrix called Ĩ(u, v) from which we get the Mueller matrix M̃(u, v) at each pixel:

$\widetilde{{\mathbf{M}}_{\mathbf{O}}}$ and $\widetilde{{\mathbf{M}}_{\mathbf{B}}}$ are then estimated by taking the most probable values of the Mueller matrices M(u, v) among all the pixels imaging respectively the object and the background region.

Moreover, all the calculations are performed in the framework of the following assumptions:

Let notice that the APSCI method uses the Matrices M̃_O and M̃_B obtained from a first Mueller imaging and as a consequence, has to deal with the corresponding initial shot noise. For each configuration of scene, the polarimetric properties of the object and background are chosen to differ only from a single degree of freedom with a difference equal to 10% of the total dynamic of the polarimetric property under investigation.

We propose, as a first step, to compare the contrast of the APSCI method to the one of the Mueller Imaging for an example of scene that is composed of 2 purely birefringent elements (a square shaped one mentioned as the background, with a hole at the center accepting a round shaped one mentioned as the object) that are different of 18° in the azimuth of their retardance vector (background: R = 100°, λ_R⃗ = 18°, ε_R⃗ = 0°; object: R = 100°, λ_R⃗ = 36°, ε_R⃗ = 0°). Both methods are compared for a chosen mean SNR value (without any polarizer) of 0.55. On Fig. 2, we see in (a) that only a few elements of the Mueller matrices allow a weak distinction between the 2 birefringent elements, the best Bhattacharyya distance being of 0.026 for their M ₁₁ element that is magnified on (b). On (c), the APSCI parameter benefits from a Bhattacharyya distance nearly 10 times larger (0.249). We notice that limiting the scale bar of Fig. 2(b) to [-1;+1] for comparison purposes with Fig. 2(c) has not been chosen here because near half of the pixels of Fig. 2(b) are out of this range and as a consequence the corresponding image is strongly saturated and moreover doesn’t exhibit any appreciable visual enhancement. Furthermore, we point out, as will be shown after, that the association of the polar decomposition to the Mueller imaging doesn’t enhance the polarimetric contrast for such low SNR levels.

 

Fig. 2. The scene under investigation is composed of 2 birefringent elements with a difference of 18° in the azimuth λ_R⃗ of their retardance vector. (a) Image of the Mueller matrices (b) a magnification of their M ₁₁ element that exhibits the best Bhattacharyya distance (0.026) among the 16 elements and (c) the APSCI parameter that exhibits a Bhattacharyya distance of 0.249. The range of the scale bars is set by the minimum and maximum value of the pixels inside each image.

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In order to generalize such results, in Fig. 3, Bhattacharyya distances of Mueller matrix elements and of several polarimetric parameters have been plotted in function of the SNR for various object/background couples: (a) describes the case of 2 birefringent elements with different scalar retardances, and (b) with different retardance vector azimuths; (c) describes 2 dichroic elements with different scalar diattenuations and (d) with different diattenuation vector azimuths; (e) describes 2 depolarizing elements with different linear degrees of polarization, and (f) with different angles ϕ of the axial rotation Rot[V⃗, ϕ] describing the principal depolarization axes of the materials. For each situation, B_APSCI is compared to B(M), to one of the three following contrast parameters depending on the polarimetric property under investigation: B(M _D), B(M _R) or B(M _Δ), and to the Bhattacharyya distances of some relevant parameters specific to each situation: the scalar retardance R for (a) and the azimuth of the retardance vector λ_R⃗ for (b); the scalar diattenuation D for (c) and the azimuth of the diattenuation vector λ_D⃗ for (d); the linear degree of polarization D_opL and the depolarization power Δ for (e) and the angle ϕ of the axial rotation Rot[V⃗, ϕ] for (f).

We first notice that for the case (c), the B(M _D) and B(M) curves are superimposed, as for the B(M _Δ) and B(M) curves for the case (e).

We can observe that for a sufficient SNR, in all the situations studied, the physical parameters calculated from the polar decomposition give rise to better Bhattacharyya distances compared to the ones obtained from the Mueller matrix of the scene. This can be explained by the fact that the polar decomposition isolates the investigated polarimetric property from the others, and thus the effect may appear more clearly. However, this product decomposition necessarily adds some noise in each of the three components with respect to the raw data leading to a rapid degradation of its performance at low SNR levels.

Moreover, for all the situations under study, the Bhattacharyya distance of the APSCI parameter exhibits some noticeable increases compared to the larger one achievable with other parameters: in the range of SNR studied, the maximum gain of the APSCI method evolves from a ratio near 1.6 for the situation (d), to a ratio near 12 for the situation (f) (both values being obtained for a SNR around 3.9).

 

Fig. 3. Comparison of the Bhattacharyya distances of some Mueller matrix elements and of several polarimetric parameters for 6 different situations where the object and background are: (a) 2 birefringent elements with different scalar retardances, and (b) with different retardance vector azimuths; (c) 2 dichroic elements with different scalar diattenuations and (d) with different diattenuation vector azimuths; (e) 2 depolarizing elements with different linear degrees of polarization, and (f) with different angles ϕ of axial rotations Rot[V⃗, ϕ]. On the associated Poincaré spheres (that have been drawn for the maximum value of SNR of each situation) are plotted with white dots the distribution of the optimized states S_in used by the APSCI method. In the hypothesis of a perfect evaluation of the Mueller matrices of the object and background, the colors of the sphere represent the Euclidean distance D between their scattered states in function of all the totally polarized incident states.

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Moreover, we observe that even in the case of very low SNR, the APSCI method can give rise to important gains in the Bhattacharyya distance: for example, a factor of 9.4 is obtained with a SNR around 0.9 for the case (b). This latter observation demonstrates that the APSCI method can be of particular interest for increasing the detection capabilities when we have to deal with difficult experimental conditions related to very low SNR levels.

If now we compare the detection performance of the APSCI method between the 6 situations, we notice that the evolution of the Bhattacharyya distance in function of the SNR is strongly correlated to the uncertainty bars of this distance: for example, a SNR of 3 is sufficient to reach a Bhattacharyya distance near 1 in situation (a) whereas a SNR of 12 is needed to reach a similar value in situation (c) where uncertainty bars are much larger. In fact, a low Euclidean distance between the Mueller matrices of the object and background implies a low distance D (as defined in equation 9), and as a consequence, a low mean value of the Bhattacharyya distance of the APSCI parameter. Moreover, such low Euclidean distance between the object and background matrices implies a strong uncertainty on the optimized illumination states, and as a consequence, provokes the corresponding uncertainty bars in the Bhattacharyya distance.

In order to precise this latter observation, for each of the 6 situations under study, we have drawn on a Poincaré sphere the distribution of the optimized states S_in (in white dots) used for the evaluation of the APSCI parameter (Fig. 3). The color scale represents the evolution of the Euclidean distance D(S⃗) (defined previously in equation 8) calculated in the hypothesis of perfectly evaluated Mueller matrices of the object and background. The black lines on the sphere (called 95% D_max) define a domain inside which an error around the ideal optimized incident state(s) gives rise to a reduction in the Euclidean distance D(S⃗) of less that 5% compared to its optimum value. In situations (a) and (b), the states S_in exhibit a distribution around the circle of the optimum states that appears thin compared to the domain delimitated by the 95% D_max curves. On the other hand, the distribution of the states S⃗_in appears clearly wider compared to this domain on situations (c) and (d) that exhibit, as a consequence, higher uncertainty bars in the Bhattacharyya distance.

As pointed out previously, for each configuration of scene, the polarimetric properties of the object and background have been chosen to differ only from a single degree of freedom with a difference equal to 10% of the total dynamic of the polarimetric property under investigation. However, such variation of 10% applied on different polarimetric properties does not result in an equivalent perturbation in the Euclidean distance between the matrices of the object and background. In order to deepen this aspect, On figure 4, for each situation (a)-(f) of Fig. 3, we have plotted the Bhattacharyya distances of the APSCI parameter in function of the SNR and of the Euclidean distance between the object and background matrices. We observe some noticeable differences in this latter distance for the 6 situations investigated (plotted with black lines): its minimum occurs for situation (c) with a value of 0.005 and its maximum occurs for the situation (b) with a value of 0.92. Moreover, we observe that for a wanted value of the Bhattacharyya distance, the SNR needed is strongly dependant of this Euclidean distance that appears to be a relevant parameter to estimate the performance of the APSCI method regarding the detection capabilities.

5. Conclusion

We have proposed a method to optimize the polarimetric contrast between homogeneous objects and backgrounds, that can be performed whatever are their polarimetric properties. Using the Bhattacharyya distance as a contrast parameter, we have compared by Monte Carlo simulations, the detection capabilities of the proposed APSCI method with classical Mueller imaging associated or not with the polar decomposition, in the case of low flux imaging. In all the simulations performed, the polarimetric contrast of the APSCI method has exhibited noticeable increases in the Bhattacharyya distance. In consequence, this method seems particularly well suited to increase the detection capabilities of any Mueller system. We would like to point out that the APSCI simulations have been performed by taking into account the detection noise for the estimation of the Mueller matrices which results in an uncertainty concerning the determination of the optimized polarization states to use. As a consequence, the results can be greatly enhanced if the polarimetric response of the object and/or background is precisely known a priori. In the case of coherent imaging, additional studies taking into account the optical speckle noise on the performance of the APSCI method would be needed. Moreover, robustness of this method in function of calibration errors of the PSA and PSG would deserve an entire study. Experimental validations are planned. Applications can be expected in many domains in optics where detection capabilities need to be enhanced, including medical imaging where for example an optimization of the polarimetric contrast between healthy and cancerous tissue can be of great interest for delimiting surgical ablation areas.

 

Fig. 4. The black lines represent the Bhattacharyya distances of the APSCI parameter in function of the SNR and of the Euclidean distance between the object and background matrices for the 6 situations (a)-(f) depicted in Fig. 3 (filled colored parts are just a guide for eyes). We observe some noticeable differences in Euclidean distances for the various situations, and can appreciate the corresponding consequences on the value of B_APSCI.

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