# Polarimetric Images of Biological Tissues Based on the Arrow Decomposition of Mueller Matrices

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## Abstract

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## 1. Introduction

**M**). Even though the sixteen elements of a given Mueller matrix can be used to build their respective images, each of those elements are related in an intricate manner to the polarimetric properties of the sample at the particular point under consideration. Consequently, the identification of appropriate sets of physical parameters representing, in a separate manner, the fundamental (phenomenological) polarimetric properties of the sample at each point, appears a key aspect to optimize the contrast in imaging polarimetry, while the said properties can be monitored and represented.

**M**that allow a physical interpretation of some characteristics of samples [1,2,3,4,5,6]. In particular, one can mention properties such as [2]:

- depolarization, which can be globally characterized by means of the depolarization index (or degree of polarimetric purity) [7], the polarization entropy [9], the depolarization power [8], the first and second Lorentz depolarization indices [10], the overall purity index [11], etc., while the detailed information on depolarization can be characterized by the indices of polarimetric purity [12,13];
- retardance, whose characterization for general Mueller matrices requires a criterion to define both the entrance and exit retardance vectors.

## 2. Theoretical Background

**s**and ${s}^{\prime}$ are the Stokes vectors that represent the states of polarization of the incident and emerging light beams, respectively, while

**M**is the Mueller matrix associated with this kind of interaction and can always be expressed as [8,59,60].

**M**; the superscript T indicates transpose matrix; ${m}_{00}$ is the mean intensity coefficient (MIC), i.e., the ratio between the intensity of the emerging light and the intensity of incident unpolarized light;

**D**and

**P**are the diattenuation and polarizance vectors, with absolute values D (diattenuation) and P (polarizance); and

**m**is the normalized 3 × 3 submatrix associated with

**M**.

**M**, the Mueller matrix that represents the same linear interaction as

**M**, but with the incident and emergent directions of the propagation of the electromagnetic wave interchanged, is given by [2,61,62]

**M**, showing that D and P share a common essential nature related to the ability of the medium to enpolarize (increase the degree of polarization) unpolarized light incoming in either forward or reverse directions [2]. Since magneto-optic effects only affect the sign of certain elements of

**M**, this does not affect D, P and other quantities considered below (when applied to the reverse Mueller matrix), which are defined from the square averages of some Mueller matrix elements.

**D**and

**P**can be represented in the Poincaré sphere; in fact, they are closely linked to the Stokes vectors $M\hspace{0.17em}{\widehat{s}}_{u}$ and ${M}^{T}\hspace{0.17em}{\widehat{s}}_{u}$, ${\widehat{s}}_{u}={(1,0,0,0)}^{T}$, representing input unpolarized light and parameterized as follows:

**M**to preserve the degree of polarization (DOP) of totally polarized incident light, a proper measure is given by the degree of polarimetric purity of

**M**(also called depolarization index) [7], ${P}_{\Delta}$, which can be expressed as

**m**.

**C**associated with

**M**. The values of the IPP satisfy the nested inequalities $0\le {P}_{1}\le {P}_{2}\le {P}_{3}\le 1$ and the following weighted square average of them equals the degree of polarimetric purity [12]:

**X**(akin to that used for the diattenuation, polarizance, Poincaré entrance retardance and Poincaré exit retardance vectors):

## 3. Arrow-Form-Inspired Parameterization of the Information Contained in a Mueller Matrix

**m**of

**M**[56]

**m**, so that the following orthogonal Mueller matrices (representing respective retarders) can be defined as

**M**is then defined as

**M**is [56] (see Figure 1)

**M**are recovered from those of ${M}_{A}$ through the respective rotations in the Poincaré sphere representation $D=\hspace{0.17em}{m}_{RI}^{T}{D}_{A}$ and $P=mRO{P}_{A}$ (thus preserving the respective absolute values $\left|{D}_{A}\right|=\hspace{0.17em}\left|D\right|=D$, $\left|{P}_{A}\right|=\hspace{0.17em}\left|P\right|=P$), which are directly determined from the entrance and exit retarders ${M}_{RI}$ and ${M}_{RO}$ of

**M**.

**M**shows that

**M**can be interpreted through the serial combination of the entrance retarder ${M}_{RI}$ of

**M**, the arrow form ${M}_{A}$ of

**M**and the exit retarder ${M}_{RO}$ of

**M**. Consequently, the physical information held by

**M**can be parameterized through the following set of sixteen independent parameters [64,65]:

- the three parameters $({\phi}_{I},{\chi}_{I},{R}_{I})$ determining the entrance retarder;
- the three parameters $({\phi}_{O},{\chi}_{O},{R}_{O})$ determining the exit retarder;
- the MIC ${m}_{00}$ of
**M**(which coincides with that of ${M}_{A}$); - the three parameters $({\phi}_{D},{\chi}_{D},D)$ determining the diattenuation vector
**D**of**M**, or, alternatively, the three parameters $({\phi}_{DA},{\chi}_{DA},D)$ determining the diattenuation vector ${D}_{A}=\hspace{0.17em}{m}_{RI}D$ of ${M}_{A}$; - the three parameters $({\phi}_{P},{\chi}_{P},P)$ determining the polarizance vector
**P**of**M**, or, alternatively, the three parameters $({\phi}_{PA},{\chi}_{PA},P)$ determining the polarizance vector ${P}_{A}=\hspace{0.17em}{m}_{RO}^{T}D$ of ${M}_{A}$; - the three indices of polarimetric purity ${P}_{1},{P}_{2},{P}_{3}$ of
**M**(which coincide with those of ${M}_{A}$).

**D**and ${D}_{A}$ and between the polarizance vectors

**P**and ${P}_{A}$, and since the polarimetric images generated from their respective parameters only depend on their variations, for imaging purposes the use of

**D**(

**P**) is entirely equivalent to that of ${D}_{A}$ $\left({P}_{A}\right)$.

## 4. Materials and Methods

#### 4.1. Experimental Setup Description: Complete Image Mueller Matrix Polarimeter

**M**of the analyzed samples is a complete imaging Mueller polarimeter. The polarimeter comprises two main parts: the Polarization State Generator (PSG) and the Polarization State Analyzer (PSA). The PSG and the PSA are composed of the respective series of optical elements (see Figure 2a) and devices, which allow to generate and analyze, respectively, any state of fully polarized light. In the case of the PSG, for being able to generate any state of polarization, it is comprised by a linear polarizer oriented at 0° with respect to the laboratory vertical and two Parallel Aligned Liquid Crystals (PA-LC) retarders oriented at 45° and 0° respectively. The PSA is comprised of the same optical elements as the PSG but located in inverse order (Figure 2a). To obtain the Mueller matrix images of the samples, a CCD camera is placed after the PSA to capture the intensity of the sample correspondent to each pixel. In addition, the PSG is illuminated with a light source which can work at different wavelengths in the visible spectrum (625 nm, 530 nm and 470 nm) allowing us to inspect different characteristics of samples. In particular, larger wavelengths polarimetrically interact with deeper tissues and shorter wavelengths mostly interacts with superficial tissues [66,67]. To reduce the spectrum of the different wavelengths of the LED source and to prevent artificial depolarization originated by the PA-LC’s performance dependence on said parameter, 10 nm filters for blue and green illumination are used. Note that, for an accurate experimental determination of the Mueller matrix of the samples, it is important to control external light sources, so that the only light source interacting with the sample is that present in the PSG. For this reason, experimental measurements of the Mueller matrix have been conducted in dark conditions in the laboratory.

**M**. This has been done in terms of condition numbers and equally weighted variance metrics, by using the six based polarizing basis described in [68]. In this sense, the polarimeter provides an accuracy of 2% in the measurements [69].

^{®}high resolution objective followed by an Allied Vision Manta G-504B CCD camera, with 5 Megapixel GigE Vision and Sony ICX655 CCD sensor, 2452(H) × 2056(V) resolution, and cell size of 3.45 µm × 3.45 µm, so that a spatial resolution of 22 µm is achieved.

#### 4.2. Sample Description and Preparation

## 5. Application of the Mueller Matrix Parameterization to Polarimetric Imaging of Biological Tissues

_{I}, ${\phi}_{D}$ and ${\phi}_{I}$; see Section 3). To highlight the potential of this set of observables calculated for each one of the studied samples, we provide the best observables-based images results in terms of tissue visualization in this section. Note that common biological samples present different polarimetric features and specific observables will focus on a particular characteristic inspecting such features. This limits the situation of retrieving all possible biological structures at the same time with a single polarimetric channel and, thus, multiple observables should be considered for a complete analysis of the sample under inspection. In turn, the sixteen arrow-decomposition-derived observables discussed in this work present a set of metrics studying the main polarimetric characteristics of samples (retardance, diattenuation and depolarization). In the following examples, from all calculated metrics (see Section 2), we choose to present the channels providing the best visualization of particular biological structures of interest.

_{I}) (Figure 3b,c). We clearly observe the visualization enhancement between different structures of the plant associated with the polarimetric channel R

_{I}when comparing Figure 3a,b. To benefit from the visual improvement related to colormaps, an image based on the entrance retardance parameter (R

_{I}) is represented in Figure 3c in a different colormap than the grayscale in Figure 3b. For the following discussion, we will compare images (a) and (c).

_{I}. For instance, the pathological areas of the plant (see yellow arrows) present different retardance values than the rest of the healthy leaf lamina. That is, the structural changes produced in these necrotic areas of the leaf (see description in Section 4) produce a very different entrance retardance, R

_{I}, behavior which allows us to have a great contrast between the not infected part of the plant and the necrotic stage of the pathology. Moreover, we also can differentiate the vascular structure of the plant, especially the highlighted primary veins (see yellow dashed rectangles). Note how the stated visualization improvement can be of interest for characterization as well as the pathological analysis of plants.

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Arrow decomposition of a Mueller matrix. For any incident polarization state, with Stokes vector

**S**, the effect of any given Mueller matrix

**M**is equivalent to that of a serial combination of an entrance retarder ${M}_{RI}$, the arrow form ${M}_{A}$ associated with M and an exit retarder ${M}_{RO}$.

**Figure 2.**Three-dimensional representation of (

**a**) the PSG and PSA optical components; the complete image Mueller polarimeter configurations used in this work; (

**b**) reflection configuration; and (

**c**) transmission configuration. Image reproduced from Ref. [15].

**Figure 3.**Images of the pathological grapevine sample for the 470 nm illumination wavelength. (

**a**) Intensity image; (

**b**,

**c**) images of the R

_{I}polarimetric channel with different colormaps (indicated on the right of the image). The yellow arrows indicate the necrotic lesions of the leaf and, the yellow dashed rectangles indicate some of the primary veins in the leaf.

**Figure 4.**Images of the tendon sample for the 470 nm illumination wavelength. (

**a**) Intensity image, (

**b**,

**c**) azimuth of the diattenuation, ${\phi}_{D}$, images with different colormaps (indicated on the right of each image). The tendon is partially (*) enveloped by fascia and areolar fatty tissue corresponding to paratenon. Red arrows in (

**c**) indicate boundaries between different fascicles inside the same tendon.

**Figure 5.**Images of a coronal section of a cow brain sample for the 470 nm illumination wavelength. The sample corresponds to a section taken in the crossroad between the posterior parietal and occipital lobes with cortical grey matter (GM) and subcortical white matter (WM) fibers correspondent to different types of radiation; parietal radiations of the corona radiata (PCR) and optical radiations (OR). (

**a**) Macroscopic plain view of the full cut of the sample, (

**b**) intensity image, (

**c**,

**d**) entrance retarder azimuth images, ${\phi}_{I}$, with different colormaps (indicated on the right of each image).

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## Share and Cite

**MDPI and ACS Style**

Gil, J.J.; San José, I.; Canabal-Carbia, M.; Estévez, I.; González-Arnay, E.; Luque, J.; Garnatje, T.; Campos, J.; Lizana, A.
Polarimetric Images of Biological Tissues Based on the Arrow Decomposition of Mueller Matrices. *Photonics* **2023**, *10*, 669.
https://doi.org/10.3390/photonics10060669

**AMA Style**

Gil JJ, San José I, Canabal-Carbia M, Estévez I, González-Arnay E, Luque J, Garnatje T, Campos J, Lizana A.
Polarimetric Images of Biological Tissues Based on the Arrow Decomposition of Mueller Matrices. *Photonics*. 2023; 10(6):669.
https://doi.org/10.3390/photonics10060669

**Chicago/Turabian Style**

Gil, José J., Ignacio San José, Mónica Canabal-Carbia, Irene Estévez, Emilio González-Arnay, Jordi Luque, Teresa Garnatje, Juan Campos, and Angel Lizana.
2023. "Polarimetric Images of Biological Tissues Based on the Arrow Decomposition of Mueller Matrices" *Photonics* 10, no. 6: 669.
https://doi.org/10.3390/photonics10060669