1. Introduction

As a label-free method to obtain structural and optical information of media, nowadays polarization imaging has been more and more prevalently used in various fields such as remote sensing [1], materials characterization [2], underwater geolocalization [3], and biomedicine [4–6]. Specifically, for biomedical studies and clinical applications, Mueller matrix (MM) imaging is gaining increasing attention because it can comprehensively characterize the sample’s polarization-related vectorial properties, which are crucial for microstructural identification and evaluation [7]. Recently, MM microscopy has shown great potential in pathological detection of abnormal tissue areas [8–13]. On the other hand, backscattering MM polarimetry provides a promising tool for quantitative in-vivo and endoscopic tissue measurement [14–19]. To more efficiently acquire the structural and optical information of complex tissue sample contained in a MM, in the past decades several MM analyzing methods have been proposed [20–26]. For instance, the MM polar decomposition (MMPD) decomposes the MM into a product of three matrices, namely diattenuation, retardance and generalized depolarization, to derive a series of polarization feature parameters with clear physical significances [20]. The MM Cloude decomposition (MMCD) maps the MM to the Pauli matrix to extract different degrees of depolarization effect [24,25]. Besides, we put forward the Mueller matrix transformation (MMT) based on phantom experiments and Monte Carlo simulations to extract more rotation invariant parameters with clear relationship with certain structures from the MM [26]. All the parameters obtained via MM analyzing methods mentioned above lay the foundation for tissue MM polarimetry.

For backscattering MM polarimetry there are two photon coordinate selection conventions for the light. According to the relationship between the photon coordinate systems of the emergent and incident light, these two conventions are: (i) both the photon coordinate systems of emergent and incident light conform to the right-hand rule. However, their x-axes are not unified, so we call it right-handed-nonunitary (RR) coordinate system (adopted in numerous optics books including Principles of Optics [27] and Light scattering by small particles [28]); (ii) the x-axes of the photon coordinate systems of the emergent and incident light are unified, so we call it non-right-handed-unitary (RL) coordinate system (widely used in tissue polarimetric studies such as Monte Carlo simulation of polarization photons propagation in turbid media [26,29,30]). It should be noted that, here “nonunitary” is used to emphasize that in the RR system, the x-axes of the emergent and incident light are not consistent. “Unitary” is used to emphasize that in the RL system, the x-axes of the emergent and incident light are consistent. Basically, both the conventions are acceptable for backscattering MM imaging. For instance, several experimental and theoretical studies chose the RR coordinate system [31–39], while other researches adopted the RL coordinate system for their measurements [40–48]. However, it should be noted that the following MM parameters calculation and information acquisition are obviously influenced by the photon coordinate selections.

In this study, we analyze the influence of photon coordinate system selection on the backscattering MM polarimetry. We first compare the MM elements in the RR and RL coordinate systems, and specifically deduce the changes of MMPD, MMCD and MMT parameters widely used in backscattering MM imaging as the photon coordinate system varied. Then based on the analysis, a group of photon coordinate system transformation invariants (PCSIs) for backscattering MM polarimetry is provided. The phantom experiments confirm that some MM parameters can be prominently affected by photon coordinate system selection. The effectiveness of the proposed invariant MM parameters is also demonstrated. The findings presented in this study give a crucial criterion of parameters selection for backscattering Mueller matrix imaging under different photon coordinate systems.

2. Mueller matrix elements in different photon coordinate systems

Here, we firstly analyze the influence of different photon coordinate systems on the MM. Figure 1 shows the schematics of two photon coordinates usually used for backscattering polarimetry. Note that for transmission and grazing incidence measurement systems, the emergent light and incident light have the same coordinate as the red arrows show in Fig. 1, where the xyz-axes follow the laboratory coordinate system, and the incident light as the yellow arrow indicated is in the xoz-plane.

 

Fig. 1. Schematics of two photon coordinate systems. The incident light is indicated by the yellow arrows, the emergent light is mainly divided into: forward transmitted light indicated by the red arrows and backscattering light indicated by the blue arrows. Their corresponding angles are $\theta$, $\theta _t$ and $\theta _r$, respectively. The $s$ wave is indicated by $s$, the $p$ wave is indicated by $p$, and the light wave propagation direction is indicated by $k$. (a) Right-handed-nonunitary (RR) coordinate system. (b) Non-right-handed-unitary (RL) coordinate system. It should be noted that $k_r$ represents the actual light wave propagation direction, and ${k_r^{\prime }}$ represents the light wave propagation direction conformed to the right-hand rule.

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Ideally, for a backscattering setup with normal incidence, the incident angle $\theta$ is 0 deg; the z-axis of the photon coordinate system is consistent with the incident light wave propagation direction; and the y-axis of the photon coordinate system (the positive direction of the $s$ wave) is consistent with the y-axis of the laboratory coordinate system. As for the real backscattering tissue polarimetry such as MM endoscopy, the incident angle $\theta$ is often within 20 deg [49], then the x-axis of the backscattered photon coordinate system (the positive direction of the $p$ wave) can be defined in two ways: (a) approximately opposite to the x-axis of the incident light as Fig. 1(a) shows, where both the emergent light indicated by the blue arrow and the incident light indicated by the yellow arrow conform to the right-hand rule. However, here the x axis of the emergent light is different with that of the incident light. (b) approximately the same as the x-axis of the incident light as Fig. 1(b) shows, where the emergent light no longer conforms to the right-hand rule, but its x axis definition is mutually unified with that of the incident light. Most optical textbooks, such as Principles of Optics [27] and Light scattering by small particles [28], are discussed in the right-hand rule system illustrated as Fig. 1(a). We call it the right-handed-nonunitary (RR) coordinate system. Meanwhile, for tissue polarimetric studies such as the Monte Carlo simulation of polarization photons propagation in turbid media [26,29,30], the non-right-handed-unitary (RL) coordinate system as shown in Fig. 1(b) were often adopted. Based on the above two coordinate system selections, the MM representations can be derived as shown in Eqs. (1)–(4).

According to Eq. (4), apparently the third and fourth rows elements of $\boldsymbol {M}_{RR}$ and $\boldsymbol {M}_{RL}$ have opposite signs, which can further affect the calculation and analysis of the MM derived parameters measured using backscattering polarimetry. Thus, to explore the influence of the RR and RL photon coordinate systems selection on backscattering MM derived parameters images, in the following sections we take Eqs. (1)–(4) as the basis for the derivation and demonstration of the three groups of MM derived parameters frequently used in tissue polarimetry, namely MMPD, MMCD and MMT.

3. Mueller matrix polar decomposition method

As one of the most prevalently used analyzing tools in tissue polarimetry, the MMPD method [20] assumes that the optical behavior in a biomedical sample is continuous and follows the order of dichroism ($D$), retardance ($R$), and generalized depolarization ($\Delta$), as indicated by Eq. (5).

In Eq. (5), $m_{ij}$ represents the unnormalized MM element, and $M_{ij}$ represents the MM element normalized by $m_{11}$. Here, $i$ and $j$ denote the corresponding row and column indices of the MM element, respectively. Based on Eq. (5), the dichroism vector $\boldsymbol {\vec {D}}$ and its modulus $D$ can be calculated directly by using the first row elements of $\boldsymbol {M}$, as shown in Eq. (6) and (7).

On the basis of Eq. (6) and (7), we can further obtain the dichroism matrix $\boldsymbol {M}_{D}$ and its lower-right $3\times 3$ submatrix $\boldsymbol {m}_{D}$, as shown in Eq. (8) and Eq. (9).

According to Eqs. (6)–(9), since $\boldsymbol {\vec {D}}$, ${D}$, $\boldsymbol {M}_{D}$ and $\boldsymbol {m}_{D}$ can all be calculated directly from the first row elements of $\boldsymbol {M}$ without involving the elements from third and fourth rows, the RR and RL photon coordinate systems selection does not affect the dichroism parameters, which are all PCSIs.

Next, we can separate the dichroism matrix $\boldsymbol {M}_{D}$ from $\boldsymbol {M}$ to get the product matrix ${\boldsymbol {M}^{\prime }}$ of the generalized depolarization matrix ${\boldsymbol {M}_{\Delta }}$ and the retardance matrix ${\boldsymbol {M}_R}$ and its lower-right $3\times 3$ submatrix ${\boldsymbol {m}^{\prime }}$, as shown in Eq. (10). Further, the generalized depolarization matrix ${\boldsymbol {M}_{\Delta }}$ and its lower-right $3\times 3$ submatrix ${\boldsymbol {m}_{\Delta }}$ can be constructed as shown in Eqs. (11) and (12). Finally, the scalar parameter ${\Delta }$ used to characterize the depolarization effect can be obtained by calculating the rank of ${\boldsymbol {m}_{\Delta }}$, as shown in Eq. (13). Here, ${\boldsymbol {\vec {P}}_\Delta }$ denotes the polarizance vector and $\mathrm {eig}\left (\cdot \right )$ denotes the eigenvalue of the matrix.

As ${\boldsymbol {M}_D}$ is a PCSI, combined with Eq. (10), we can deduce that ${\boldsymbol {M}^{\prime }}$, ${\boldsymbol {m}^{\prime }}$ and ${\boldsymbol {\vec {P}}_{\Delta }}$ satisfy the following relations in the RR and RL coordinate systems:

Substituting Eq. (16) into Eq. (12), it can be observed that $\boldsymbol {m}_{\Delta RR}^{c}$ and $\boldsymbol {m}_{\Delta RL}^{c}$ exhibit a “Fish-shaped” relationship, which are symmetric matrices in common but the elements along their upper edge, left edge, upper-right corner, and lower-left corner, have opposite signs. Here, we denote this “Fish-shaped” transformation operation as $Fish\left (\cdot \right )$, as indicated in Eqs. (17) and (18). Because that the determinant expansions of the two matrices satisfying the “Fish-shaped” relationship are the same with respect to their eigenvalues, it means the eigenvalues of $\boldsymbol {m}_{\Delta RR}^{c}$ and $\boldsymbol {m}_{\Delta RL}^{c}$ are equal, as shown in Eq. (19).

Based on Eqs. (12) and (19), since the “Fish-shaped” $\boldsymbol {m}_{\Delta RR}^{c}$ and $\boldsymbol {m}_{\Delta RL}^{c}$ have the equal eigenvalues, $\boldsymbol {m}_{\Delta }^{a}$ and $\boldsymbol {m}_{\Delta }^{b}$ also exhibit a similar “Fish-shaped” relationship in the RR and RL coordinate systems. Then, by substituting Eqs. (17) and (18) into Eq. (12) and taking the inverse of $\boldsymbol {m}_{\Delta RR}^{a}$ and $\boldsymbol {m}_{\Delta RL}^{a}$, we can derive the following relationship:

Considering that the determinants of $\boldsymbol {m}_{\Delta RR}^{a}$ and $\boldsymbol {m}_{\Delta RL}^{a}$ are equal, and the adjoint matrices of $\boldsymbol {m}_{\Delta RR}^{a}$ and $\boldsymbol {m}_{\Delta RL}^{a}$ satisfy the “Fish-shaped” relationship, it can be concluded that the inverse matrices of $\boldsymbol {m}_{\Delta RR}^{a}$ and $\boldsymbol {m}_{\Delta RL}^{a}$ also maintain the “Fish-shaped” relationship. In other words, the process of taking inverses does not change the “Fish-shaped” relationship. To ultimately obtain the depolarization parameter $\Delta$, Eq. (20) should be substituted into Eq. (12), to obtain the specific expressions of ${\boldsymbol {m}_{\Delta }}$ in both RR and RL coordinate systems:

From Eq. (21), it can be observed that ${\boldsymbol {m}_{\Delta }}$ satisfies the “Fish-shaped” relationship in both photon coordinate systems. Similar to Eq. (19), the eigenvalues of ${\boldsymbol {m}_{\Delta RR}}$ and ${\boldsymbol {m}_{\Delta RL}}$ are equal. Therefore, we can conclude that the depolarization parameter ${\Delta }$ is a PCSI, as shown in Eq. (22).

With this, we can separate the retardance matrix ${\boldsymbol {M}_{R}}$ from ${\boldsymbol {M}^{\prime }}$ and proceed the discussion of linear retardance, optical rotation, and other polarization parameters, as shown in Eqs. (23)–(27).

It is evident from Eq. (28) that the elements in the third and fourth rows of ${\boldsymbol {M}_R}$ have the opposite signs in RR and RL coordinate systems. Therefore, the parameters ${R}$, ${\delta }$, ${\alpha }$ computed from Eqs. (25)–(27) relying on the photon coordinate system are all photon coordinate system transformation variants.

4. Mueller matrix Cloude decomposition method

The MMCD method [24,25] is another widely used tool in MM tissue polarimetry. The general steps of a MMCD decomposition are as follows:

Firstly, the MM is mapped to a Hermite matrix $\boldsymbol {H}\left (\boldsymbol {M}\right )$ using the Pauli matrix $\boldsymbol {\sigma }$ as the basis, as shown in Eq. (29).

Then, the four real eigenvalues of $\boldsymbol {H}\left (\boldsymbol {M}\right )$ are arranged in descending order as $\lambda _1\geq \lambda _2\geq \lambda _3\geq \lambda _4\geq 0$, which contains crucial information related to depolarization. When the depolarization exists, the eigenvalues are non-zero, and we can establish the following relationship:

As a series of MMCD parameters are all related to the eigenvalues of ${\boldsymbol {H}\left (\boldsymbol {M}\right )}$, we can compute the eigenvalues of ${\boldsymbol {H}\left (\boldsymbol {M}\right )_{RR}}$ and ${\boldsymbol {H}\left (\boldsymbol {M}\right )_{RL}}$, which confirm that their determinants have the same expansion, thus satisfying the relationship of equal eigenvalues:

Several studies have performed combination operations on the eigenvalues of ${\boldsymbol {H}\left (\boldsymbol {M}\right )}$, thereby obtaining parameters such as polarization entropy [50,51], indices of polarimetric purity [52,53], depolarization index [54], and overall purity index [55], as shown in Eqs. (34)–(37).

Apparently, since the calculations of MMCD parameters above only involve the eigenvalue $\lambda$ and matrix element ${m_{11}}$, which do not depend on the photon coordinate system selection, it can be concluded that the MMCD parameters are all PCSIs.

5. Mueller matrix transformation method

The MMT method [26] mainly focuses on directly mapping polarization parameters with clear physical significance through combination operations on MM elements, offering high computational efficiency for in vivo tissue MM polarimetry. Here in this section, we discuss and analyze the frequently used MMT parameters in backscattering MM imaging, as presented in Eqs. (38)–(43).

In summary, we have successfully derived and demonstrated the PCSIs of MMPD, MMCD, and MMT, as presented in Table 1.

Table 1. Photon coordinate system transformation invariants in the Mueller matrix derived parameters. ($T$ indicates the invariant and $F$ indicates the variant under the photon coordinate system transformation)

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6. Experimental results and analysis

To further validate the influence of the RR and RL coordinate systems selection on the MM derived parameters, we experimentally measured the MM of an anisotropic phantom composed of concentrically well aligned silks, whose effectiveness for revealing the relationship between fibrous structures and backscattering polarimetric parameters has been demonstrated previously [26,43]. The experimental results of several MM derived parameters are illustrated in Fig. 2.

 

Fig. 2. (a) Experimental results of MM derived parameters for the silk phantom. For each class of MM derived parameters, the upper row and lower row represent the measurement results under RR and RL coordinate systems, respectively. (b) Azimuthal curves of photon coordinate system transformation invariants (the units of ${P_3}$ and ${PI}$ are ${1\times 10^3}$). (c) Azimuthal curves of the photon coordinate system transformation variants. Additionally, in (b) and (c), the horizontal and vertical axes represent the azimuth angle and parameter value, respectively.

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We can observe from Fig. 2(a) that for the PCSIs ${D}$, ${\Delta }$, ${P_3}$, ${PI}$, ${S\left (\boldsymbol {H}\right )}$ and ${\left \|\boldsymbol {B}\right \|}$, the measurement results under RR (upper row) and RL (lower row) coordinate systems are the same. However, the variants ${R}$, ${b}$ and ${A}$ images exhibit prominent differences between the RR and RL coordinate systems. Especially, the periodical intensity variations of MMT parameters ${b}$ and ${A}$ are apparently errors induced by coordinate transformation. To more clearly compare the measurement results in Fig. 2(a), we obtain the azimuthal curves for each parameter within the range of $\left [0,2\pi \right ]$ as shown in Fig. 2(b) and (c). Furthermore, the differences between the parameters in the two coordinate systems are quantitatively evaluated by using indicators of mean absolute error (MAE), dynamic time warping (DTW) and peak signal-to-noise ratio (PSNR). For the PCSIs as shown in Fig. 2(b), the MAE, DTW, and PSNR values in both coordinate systems exhibit ideal values. When it comes to the variant parameter ${R}$, its azimuthal curves exhibit the most significant difference between the two coordinate systems, with each indicator also deviating the most from the ideal values. For the variants ${b}$ and ${A}$, their curves display characteristics of rotation invariant [56] in the RL system. However, in the RR system, their curves show periodical intensity variations of false negative values within the range of $\left [0,2\pi \right ]$, which contradicts their true physical significance. It is evident that for backscattering MM polarimetry such as MM endoscopy, the choice of photon coordinate system is crucial to acquire correct MM derived parameters analyzing results.

7. Conclusions

In this study, we analyzed the influence of photon coordinate system selection on the backscattering MM polarimetry. We first compared the MM elements in the RR and RL coordinate systems. It showed that the third and fourth rows elements of $\boldsymbol {M}_{RR}$ and $\boldsymbol {M}_{RL}$ have opposite signs, which can further affect the calculation and analysis of the MM derived parameters measured using backscattering polarimetry. Then we specifically deduced the changes of MMPD, MMCD and MMT parameters widely used in backscattering MM imaging as the photon coordinate system varied. Based on the analysis, we provided a group of photon coordinate system transformation invariants for backscattering MM polarimetry as MMPD parameters ${D}$ and ${\Delta }$, MMT parameters ${\left \|\boldsymbol {B}\right \|}$, ${t_2}$ and ${t_3}$, and all MMCD parameters. The phantom experiments further confirmed that some MM parameters can be prominently influenced by photon coordinate system selection. The effectiveness of the proposed invariant MM parameters was also demonstrated by evaluation indicators as MAE, DTW, and PSNR. Evidently, for backscattering MM polarimetry such as MM endoscopy, the choice of photon coordinate system should be clarified to acquire correct MM derived parameters analyzing results. The findings presented in this study give a crucial criterion of parameters selection for backscattering Mueller matrix imaging under different photon coordinate systems.