Isolating individual polarization effects from the Mueller matrix: comparison of two non-decomposition techniques

: The prevailing formalisms for isolating individual polarization effects from the experimental Mueller matrix M can be broadly divided into two categories; decomposition of M to derive the individual optical effects and directly associating the individual optical effects to specific elements of M (i.e., non-decomposition techniques). Mueller matrix transformation (MMT) and direct interpretation of Mueller matrix (DIMM) are two popular techniques of the latter category. In this study, these two non-decomposition techniques (i.e., MMT and DIMM) are compared in a detailed quantitative analysis comprising of tissues (n = 53) and phantom (n = 45) samples. In particular, two commonly investigated polarimetric variables (i.e., depolarization and retardance) were calculated from the experimentally measured M using both the non-decomposition (i.e., MMT and DIMM) techniques. The comparison carried out with scatter plots (integrated with the correlation coefficients), violin plots and Bland and Altman plots revealed better agreement of depolarization-related variables (as compared to the retardance) between the two non-decomposition techniques. The comparative analyses presented here would be beneficial for the interpretation of polarimetric variables and optical characterization of turbid media.


Introduction
Several formalisms have been proposed and implemented towards exploring the interaction of polarized light with turbid substrates like biological tissues. Among them, one of the most popular and vital is the Stokes-Mueller calculus, where the polarized light and the turbid medium are described by the four-dimensional real Stokes vector and 4 × 4 Mueller matrix M, respectively. Since the Stokes-Mueller formalism is capable of handling the problems of partially polarized light sensitively, it has been used in numerous applications [1][2][3][4][5][6][7][8][9].
The Mueller matrix M contains a complete set of optical polarization variables of the medium. Several techniques have been proposed to derive the individual polarization properties from the experimentally measured M of the medium; these techniques can be broadly divided into two categories. First, decomposition techniques where the M is split in multiple base matrices, each base matrix corresponds to a specific set of optical parameters. Examples of the techniques included in this category are polar- [10,11], differential- [12,13], symmetric- [14,15] and root- [16] decompositions; second, non-decomposition techniques where empirical/ analytical relations are used to study the individual polarization effects of the medium. Specifically, the analytical relations express the polarization properties of the medium in terms of specific elements of M. Thus the techniques in this category permit to directly obtain the polarization parameters from measured M without any decomposition. Consequently, these techniques are fast, simple to understand and easy to implement. Mueller matrix transformation (MMT) is a well-known example of the techniques included in this category [17,18]. Recently, a new analytical method has been proposed for directly obtaining the optical metrics from the M, where a comprehensive set of mathematical relations has been developed, which correlate the elements of the measured M and the optical variables. However, no specific nomenclature was adopted for this method. For convenience, we call this new method as Direct Interpretation of Mueller Matrix (DIMM) [19].
The algorithms of the aforementioned methods for studying individual polarization properties have substantial differences, even for the simple non-decomposition techniques. For example, to analyze the polarization properties of a medium, the MMT algorithm defines a group of analytical parameters which link the polarization properties of the sample to the elements of measured M [17]. In particular, the MMT formalism comprises of three analytical variables, describing the depolarization, retardance and normalized anisotropy of the sample [20][21][22][23]. On the other hand, the DIMM algorithm is based on the postulate that only one unique form of the canonical decomposition exists for the experimental M [24,25]. A set of mathematical equations for the individual polarization variable have been derived by expressing the M as the product of three canonical matrices, similar to the one exploited in polar decomposition [19]. Such differences in these algorithms may directly influence the derived individual polarimetric variables, complicating their physical interpretation and a one-to-one comparison among different techniques.
Analysis of the differences among the individual polarimetric variables derived from M using different algorithms is critical to the applications of optical characterization. Several comparative studies have analyzed the polarimetric variables calculated from different techniques [26,27]. However, a comparative study of the non-decomposition techniques has not yet been implemented through thematic analysis and a quantitative argument is still missing. Consequently, a study focused on the quantitative comparison of the two non-decomposition techniques (i.e., MMT and DIMM) with detailed discussion on how such techniques are connected is required. In this context, a comprehensive approach towards this topic is presented here. Specifically, the two non-decomposition techniques of MMT and DIMM are quantitatively compared in terms of their derived individual polarization variables so that the similarities and contrast could be assessed.

Mueller matrix transformation
The concept of a non-decomposition technique to analyze individual polarization parameters encoded in M, makes it worthy to be utilized as a promising tool of Mueller matrix polarimetry (MMP). To this end, as a simple and fast algorithm, the Mueller matrix transformation (MMT) is widely used [22,28,29]. In order to realize the study of the individual polarization properties, the MMT is contingent on three analytical parameters, namely b, t and A; these three parameters of MMT have been derived by analyzing the trends in angular variance of Mueller matrix elements and subsequently fitting them with trigonometric functions [17,21,30]. Moreover, these parameters have been manifested in the form of polarimetric characterization of biological substrates, such as depolarization and anisotropy.
The parameters of MMT have been linked to the elements of the experimental Mueller matrix through the following relations The first parameter of MMT, i.e., 1 − b has been interpreted as the depolarization metric of the sample. The parameter 1 − b, in particular, indicates the structure, size and number density of scattering centers of the medium. The second MMT parameter is t which has been defined in terms of three different groups of Mueller matrix elements. Nevertheless, all three forms of t (viz t 1 , t 2 , t 3 ) have the same physical interpretation and are associated to the retardance of the medium. The final parameter of MMT formalism, A, shows the normalized degree of anisotropy [17,20,22,29]. The permissible ranges for the depolarization parameter b is 0-1 and it is typically expressed in percentages while the retardance parameter t is a unitless quantity, with t ∈ [0, b] where the lower and upper limits represent completely isotropic and anisotropic tissues, respectively.
It has been established that all polarimetric properties, including the retardance, of the given tissue sample are present in multiple elements of the M in a dispersed form. For instance, the overall retardance of the tissue may be contributed by both the scattering from the fibrous structures and from optical birefringence (i.e., light propagation in birefringent media), each presenting different features in the M. In particular, it has been postulated that the tissue retardance anisotropy originating from its optical birefringence is reflected in the M (4, where the negative sign in these relations define the direction of the tissue fibers with respect to the incident beam. Thus, the signs and magnitudes of these elements are helpful in determining the directions of the aligned fibers (i.e., birefringence).

Direct interpretation of Mueller matrix
Direct interpretation of Mueller matrix (DIMM), another non-decomposition technique, has been recently reported for the study of individual polarization properties from the measured M. The fundamental postulate of the DIMM algorithm is that there exists only one unique form of the canonical decomposition for the M, impeding the need for considering multiple determined forms of the decomposition, as in the case of polar decomposition [19,24,25]. Details of the derivation of mathematical relations of the polarimetric variables, such as depolarization and retardance, can be found in reference [19]; these equations are presented in their final form below.
The depolarization ∆ of the sample has been associated to the elements of the M, with the following relations It may be noted that the use of a particular relation (i.e., Eqs. (6) or (7)) for computing the depolarization will depend on the tissue morphology, as dictated by the corresponding boundary conditions. Specifically, if the tissue sample exhibit higher optical birefringence (e.g., due to fibrous structures) compared to depolarization {M(3, 3), M(4, 4)> |M(3, 4)|, |M(4, 3)|}, Eq. (6) may be employed for calculating the depolarization and vice versa. It is important to mention that higher values/ magnitude of M(3, 3) and M(4, 4) are indicative of lower depolarization.
Likewise, the expression for the retardance of the medium is The permissible range the depolarization parameter ∆ and retardance parameter R are 0-1 and 0-180 degree (∼0-3.14 radians), respectively.
It is clear from the above equations that the DIMM formalism offers substantially different relations, contrary to all existing contemporary techniques, for calculating the depolarization and retardance from M. Consequently, a thorough analysis and comparison of Mueller matrix elements are essential before selecting a specific relation for calculating the polarimetric variables from M.

Literature survey
To perform a comprehensive comparison of MMT and DIMM techniques, we searched several online databases, including Medline (PubMed), Web of Science, IEEE Explore, ScienceDirect, Springer, Wiley, SciELO, LILACS and Ovid [31]. A combination of key terms related to the Mueller matrix, polarimetry and decomposition was applied. Moreover, the reference list of the relevant articles were hand-searched to identify additional related studies. All English language articles that contained the full Mueller matrix (i.e., all sixteen elements) data related to biological tissues or tissue phantoms were included herein. Two researchers (MI, IA) independently screened all studies for inclusion, resolving the differences in opinion and reaching consensus through discussion. We omitted the reported M of standard optical components (e.g., polarizer, retarder), as they offer only one polarization effect, contrary to the tissues where several polarization effects occur concurrently, enabling extensive comparison. Ultimately, 41 articles were included in the comparative analysis.
One investigator (SK) initially extracted all necessary data, such as type of sample (i.e., tissues vs. phantom), pathological condition of the analyzed tissues, parameters for optical measurements (i.e., source of the optical field, wavelength, power), experimental geometry (forward vs. backward), and values of sixteen elements of Mueller matrix from the included studies. The extracted data were divided into two groups (tissue and phantom) and independently double-checked by another team member (BG).

Data analyses
All descriptive analyses that are common in the framework of the comparative studies were carried out. As a preparatory phase for these analyses, two polarimetric variables were selected and calculated from the M. Specifically, depolarization and retardance are the two most critical and commonly investigated variables; these variables were calculated using both the nondecomposition (i.e., MMT and DIMM) techniques. This resulted in the two pairs of polarimetric variables, depolarization (1 − b, ∆) and retardance (t, R). The first and second variable in each pair corresponds to the MMT and DIMM methods, respectively. These polarimetric variables were computed from the cohort of M (n = 100) collected from the published literature (Table 1).
For completeness, another related optical parameter called the diattenuation/ dichroism is sometimes used in tissue polarimetry. However, in general, the diattenuation is not investigated/ included optical studies, owing to the fact that the magnitude of diattenuation presented by the tissue samples used in the preclinical studies is very small, thus hindering/ compounding the interpretation of this polarimetric variable in terms of the tissue morphology. What's more, no explicit parameter for the diattenuation has been offered by the MMT formalism. Consequently, the quantitative comparison of MMT and DIMM methods was carried out in terms of depolarization and retardance only.
To carry out sufficient descriptive analyses that would cover all the necessary attributes of this comparative study, graphical tools were coupled with quantitative evaluation metrics. Specifically, the relation between the two non-decomposition techniques was investigated by scatter plots of the corresponding optical variables in tandem with the correlation coefficients. Such graphical analysis was then followed by the Bland and Altman plots that quantified the bias and agreement between the two techniques.

Results and discussions
Although the potential of MMP for optical characterization of turbid media is widely acknowledged, studies aimed at comparison of various algorithms used for extraction of individual polarimetric variables from M are somewhat limited. Nevertheless, such comparative analysis enables us to pursue and contribute, as a second-order effect, to the quality of the optical polarimetric characterization. Accordingly, this study was aimed to systematically analyze, evaluate and elicit the connection between the two non-decomposition methods (MMT and DIMM) in the regime of MMP. The results presented below, alongside the relevant discussion, revealed consistent patterns of depolarization while complex and variable effects of retardance exist across the two non-decomposition schemes.
The attributes of the samples and experiments used for the measurement of Mueller matrices included in this comparative study have been summarized in Table 1. In the published literature, complete data for M was available for n total = 98 turbid samples; the number of tissue samples is fifty three (i.e., n tissue = 53) and the number of phantom samples is forty-five (i.e., n phantom =45). As shown in Table 1, these samples reflect a broad range of pathologies, reflecting the increasing body of evidence supporting the use of MMP as an optical characterization tool.
When comparing and eliciting the two non-decomposition formalisms, the calculation of corresponding polarimetric variables would rationally rest on all their reported definitions (i.e., equations), integrated with the respective boundary conditions and structural inequities as imposed on the elements of M. Accordingly, for depolarization, the single variable of the MMT method (Eq. (1)) was compared with each of the two corresponding variables of the DIMM method (Eqs. (6) and (7) in both groups of tissue and phantom samples. Such analyses for the second polarimetric variable, namely retardance, were relatively complex. Specifically, each of the MMT and DIMM formalism has three relations for computing the retardance, Eqs. (2)(3)(4) and Eqs. (8-10), respectively, rendering a total of nine combinations for comparisons. It is noteworthy that each polarization variable was calculated, using its specific relation as given in the previous section, from only those M whose elements satisfy the corresponding boundary conditions. For example, to compute the depolarization variable of DIMM, the cohort of M was divided into two subsets on the basis of the boundary conditions specified by Eqs. (6) and (7). The same procedure was followed for computing the retardance variable. The findings drawn from such analyses are presented below, highlighting alongside the contrast and similarities in the variables calculated with the two non-decomposition formalisms. Figure 1 presents the comparison of the variables describing depolarization in the two nondecomposition techniques, where the variables of MMT and DIMM formalisms were plotted on the x-and y-axis, respectively. For the DIMM method, depolarization was computed with Eqs. (6) and (7), as represented by the black-hollow squares and red-solid circles, respectively. The depolarization data of the tissue samples are depicted in Fig. 1(A), while that of phantom samples in Fig. 1(B). The red-solid line and black-dotted line shows the linear fitting of the scattered data computed by Eq. (6) and (7), respectively. The correlation between the depolarization variables of the two non-decomposition formalisms was quantified by computing the correlation coefficient R 2 , as summarized in Table 2. Depolarization calculated with Eq. (6) of the DIMM method revealed a better correlation with the corresponding variable of the MMT method as compared to the results of Eq. (7) (R 2 = 0.83 versus 0.40 for tissue and R 2 = 0.97 versus 0.81 for phantoms).
The linear relationships for the depolarization variables, shown in Fig. 1, can also be interpreted exploiting the respective equations of the MMT and DIMM formalisms; such algebraic analysis would also result in the theoretical expressions for the slopes and intercepts with the y-axis. The said algebra can be carried out by substituting Eq. (1) in Eq. (6), which provides the following explicit relation between these metrics: µs (mm −1 ) χ deg /cm g ∆n = ne -no τ Extension (mm) Sucrose concentration (M) Power (mW) λ (nm)     Fig. 1, as compared to equation (1, 7).
Comparison of the two non-decomposition methods was extended by determining the values of retardance for the cohort of collected M (n = 100) and identifying any possible correlation in their optical behaviors. The retardance, calculated from M, provide critical information about the fibrous structures of the sample, and allows to study the micro-structural organization in relation to different pathological conditions. Figure 2 illustrates the comparison of the retardance variables as calculated with the MMT and DIMM methods. Specifically, the three variables (t 1 , t 2 , t 3 ) describing retardance for the MMT method were calculated with Eq. (2-4), while the corresponding variables for the DIMM method were computed by Eqs.  Figure S1 and Figure S2, respectively. The retardance data for the variables (t 1 , t 2 , t 3 ) have been represented by the black hollow-squares, red solid-circles and blue solid-triangles, while the linear fitting of the scattered data has been demonstrated by the corresponding lines, respectively. The retardance-based correlation of the two formalisms for the tissue and phantom samples is depicted in Figs. 2(a) and 2(b), respectively. The summary of correlation coefficients calculated for all possible combinations of Eqs. (2)(3)(4) and Eqs. (8-10) has been given in Table 3. Among all these combinations, the highest value of correlation coefficient was observed for Eq. (4) of MMT and Eq. (9) of DIMM methods; this trend was consistent for both tissue and phantom samples, as summarized in Table 3. The lowest value of correlation coefficient (R 2 ≈ 0) was observed for Eq. (10) of the DIMM method against all corresponding relations of the MMT method for tissue samples; thereby, these results are lacking in Table 2.  Albeit the correlation of retardance between the two formalisms (i.e., MMT and DIMM) has been represented by linear fitting in the graphical method, the quantitative assessment of such correlation revealed poor metrics (i.e., lower R 2 values). The poor linear correlation may also be inferred from the involvement of cosines and square roots in these equations. Moreover, the algebra for solving the system of retardance equations (one each for the MMT and DIMM methods, e.g., Eq. (2) and 8) to obtain the theoretical expressions that correlate these variables was highly complex and the solutions does not converge. Specifically, the substitution method (as performed above for the depolarization variables) was not suitable as each retardance variable/ equation contains different elements of the M (contrary to the case for depolarization variables). On the same grounds, the elimination method also fails to converge at any specific point. Briefly, finding a one-to-one correlation for the six retardance equations (three each in MMT and DIMM methods) would need to separately solve the following nine binary system of equations: (2,8), (2,9), (2,10), (3,8), (3,9), (3,10), (4,8), (4,9), (4,10). Of these, the binary system of equations (4,9) seems the simplest one, because of the involvement of the similar elements of M. However, the solution of even this simplest system does not converge with both the substitution and elimination methods. In summary, although the linear fitting in the graphical method resulted in poor correlation, indicated by lower R 2 values (except for Eqs. (4) and (9)), the substitution and elimination methods fails to provide a converging solution. Figure 3 shows the comparison of depolarization variable for the two non-decomposition formalisms in the form of violin plots-a graphical tool benefiting from a compact size, delivering rich information and a powerful visual presentation. Specifically, a violin plot integrates the probability density function with a box plot to confer the data dispersion and type/ shape of the distribution concurrently. The violin plots for the depolarization variables of tissue (n tissue = 53) and phantom (n phantom = 45) samples are shown in Fig. 3(A, B) and 3(C, D), respectively; the data were computed from Eq. (1) (MMT method: black color) and Eqs. (6) and (7) (DIMM method: red color), as mentioned on the x-axis. The fine structure of the violin plots illustrated a nearly uniform distribution and closer resemblance for the values of depolarization variables computed from Eq. (1) (MMT method) and Eq. (6) (DIMM method), as compared to Eq. (1) and Eq. (7). This trend was consistent for both tissue and phantom samples. Specifically, depolarization values calculated from Eq. (7) were consistently higher than the corresponding values calculated with Eq. (1). The violin plots for the data computed with Eqs. (2)(3)(4) versus Eq. (8) and (10) have been provided as supplementary Figure S3 and Figure S4, respectively.  Figure 4 demonstrates the violin plots for the retardance variables of tissue ( Fig. 4(A)) and phantom ( Fig. 4(B)) samples. The retardance data were computed with/from Eqs. (2-4 (MMT method) and Eqs. (8-10) (DIMM method); presented here, however, is the comparison of data calculated with Eqs. (2)(3)(4) against Eq. (9) only, while the plots for the remaining data can be found in supplementary Fig. 3(S) and 4(S). The distribution and range of the retardance values in the data space of the two non-decomposition formalisms can be explored by analyzing the structural features (viz width/ waist and height) of these violin plots. In the process of inferring the comparison, a nearly unimodal distribution of retardance values can be observed for both formalisms. However, the tissue samples showed relatively more positive skewness compared to phantom samples (Fig. 4(A) vs. 4(B). Moreover, the violin plots indicated a better match of Eq. (9) (DIMM method) with Eq. (2) (MMT method). This matching trend was more prominent for the tissue samples.
The well-established Bland and Altman (BA) method were utilized to assess the limits of agreement and bias between the two non-decomposition formalisms-an integral part of any comparative analysis. A typical BA plot may be constructed by mapping the paired differences against the mean for the two formalisms. It may be pointed out that such analysis is not possible with the scatter and violin plots. The bias quantifies the mean difference in the paired polarimetric variables as computed with the two non-decomposition techniques, while the data range of bias ± 1.96 SD (i.e. Standard deviation) defines the limits of agreement. The BA plots for the depolarization variables as computed with the two non-decomposition techniques for tissue (n tissue = 53) and phantom (n phantom = 45) samples are shown in Fig. 5   6(A) and 6(B) illustrate the BA plots for the paired differences of retardance variables for the tissue and phantom samples as computed from the two non-decomposition formalisms, respectively. In particular, Eqs. (2-4) of the MMT method and Eq. (9) of the DIMM method were used for the calculations of the retardance variable; such analysis for Eqs. (2)(3)(4) with Eq. (8) and (10) are provided as supplementary Fig. 5(S) and Fig. 6(S), respectively.While inferring the comparison from Fig. 6(A) and 6(B), it may be noted that the level of agreement for the retardance variables for the tissue and phantom samples is almost similar, as opposed to the depolarization variable (Fig. 5). Several formalisms have been developed for isolating the individual polarization variables from the experimental M. These can be broadly classified into three domains: first, decomposition formalisms which assume that all polarization effects occur in a stepwise fashion in the medium [10,11,35,71], second, decomposition formalisms where the concurrent occurrence of multiple polarization effects is assumed [10,13,59,62,72], and third, the non-decomposition formalisms [17][18][19]73]. The methodology of formalisms falling in the third group is substantially different from the first two groups. Specifically, the third group contains formalisms where analytical parameters are defined on the basis of elements of M, and correlated to the structural attributes of the turbid samples. Consequently, it is vitally important to note that the third group of formalisms impedes the complex process of decomposition of M, and thereby the complex matrix calculus, contrary to the first two groups of formalisms.
Different formalisms in the aforementioned three groups have been examined and described comparatively. To exemplify, the differential decomposition was used to decode the optical variables contained in M and compared with the corresponding optical variable extracted with polar decomposition in tissue sample of different clinical pathologies such as tuberculosis, Crohn's disease, etc. [23,59,63,72]. Also, several studies have compared the polar decomposition with the MMT method, both qualitatively [21,22], and quantitatively [17]. Moreover, the polarimetric variables have been extracted with polar decomposition from a 3 × 3 and 4 × 4 M and compared [74]. Recently, the polar decomposition has also been compared with MMT in the regime of 3 × 3 M [26]. Altogether, notwithstanding all these comparative studies, the comparison of the non-decomposition methods was not performed yet, and thereby this study was designed to carry out this comparison in a large cohort of experimental M.

Conclusion
Comparison of two non-decomposition methods-MMT and DIMM-used for the isolation of polarimetric variables from the Muller matrix M have been investigated. Using both these methods, the depolarization and retardance were computed from M for tissues (n=53) and phantoms (n=45) samples. Comparative graphical methods showed linear correlation of depolarization variables between the two formalisms; analysis of the depolarization equations in the two formalisms with substitution method resulted in theoretical expressions for the said linear correlation. Comparison and correlation of the retardance variables was more difficult to quantify. Specifically, comparative graphical methods showed poor linear correlation of the retardance variables between the two methods, as indicated by lower R 2 values. Substitution and elimination methods for correlation analysis of the retardance equations failed to converge. Overall, the MMT method seems simple to understand, computationally fast and provide more accurate description of the tissue morphology, particularly in terms of the three different retardance variables.