Fig. 1. (a) Lu–Chipman decomposition of a forward-scattering Mueller matrix. (b) Reciprocal polar decomposition of a backscattering Mueller matrix in reciprocal polarization imaging. The diattenuation and retardance matrices ${\mathit{M}}_{\mathit{D}}^{\#}$ and ${\mathit{M}}_{\mathit{R}}^{\#}$ in the backward path are specified by the reciprocal of their counterparts ${\mathit{M}}_{\mathit{D}}$ and ${\mathit{M}}_{\mathit{R}}$ in the forward path.

Fig. 2. Polarization imaging system. P, polarizer; QW, quarter-wave plate; L, lens; BS, beam splitter; M, mirror; PCCD, polarization camera; $\text{S}1$, sample position for backscattering Mueller matrix measurement; $\text{S}2$, sample position for forward-scattering Mueller matrix measurement.

Fig. 3. Polarization imaging of a birefringence resolution target in forward and backward geometries. The orientation angle, linear retardance, and depolarization from (a1), (b1), (c1) Lu–Chipman decomposition in forward geometry; (a2), (b2), (c2) Lu–Chipman decomposition; and (a3), (b3), (c3) reciprocal polar decomposition of the Mueller matrix measured in backward geometry. Profiles along the white arrow are displayed for (a4) the target orientation angle, (b4) linear retardance, and (c4) depolarization from Lu–Chipman decomposition of the Mueller matrix measured in forward geometry; Lu–Chipman and reciprocal polar decompositions of the Mueller matrix measured in backward geometry. Scale bar: 0.5 mm.

Fig. 4. Polarization imaging of fresh beef tissue sections in serial cuts with thicknesses of 100 and $300\mu \mathrm{m}$. The orientation angle, linear retardance, and depolarization: (a1), (b1), (c1) Lu–Chipman decomposition of the Mueller matrix for the $100\text{-}\mu \mathrm{m}$ section measured in forward geometry; Lu–Chipman decomposition of (a2), (b2), (c2) the $100\text{-}\mu \mathrm{m}$ section and (a4), (b4), (c4) the $300\text{-}\mu \mathrm{m}$ section; and reciprocal polar decomposition of (a3), (b3), (c3) the $100\text{-}\mu \mathrm{m}$ section and (a5), (b5), (c5) the $300\text{-}\mu \mathrm{m}$ section measured in backward geometry. Boxplots are shown for the orientation angle (a6), linear retardance (b6), and depolarization (c6) of the whole tissue section obtained by Lu–Chipman and reciprocal polar decompositions of the Mueller matrices measured in the forward and backward geometries. Only reciprocal polar decomposition correctly yields consistent orientation angles, approximately three times greater linear retardance, and greater linear retardance for the $300\text{-}\mu \mathrm{m}$ section than for the $100\text{-}\mu \mathrm{m}$ section. Scale bar: 0.5 mm.

Fig. 5. Polarization imaging of a $30\text{-}\mu \mathrm{m}$-thick unstained gastric cancerous tissue section. (a) Original image, (b) H&E stained histological image, and the orientation angle, linear retardance, depolarization, and depolarization anisotropy: (c), (e), (g), (i) Lu–Chipman decomposition of the forward scattering Mueller matrix, and (d), (f), (h), (j) reciprocal polar decomposition of the backscattering Mueller matrix. The red dashed line represents the boundary between the cancer [left (C)] and normal [right (N)] regions. Boxplots are shown for the orientation angle (k), linear retardance (l), depolarization (m), and depolarization anisotropy (n) of the cancerous and normal regions obtained by Lu–Chipman measured in the forward and reciprocal polar decomposition of the Mueller matrices measured in the backward geometry. Triple (linear retardance, depolarization, and depolarization anisotropy): (o) Lu–Chipman decomposition of the forward-scattering Mueller matrix and (p) reciprocal polar decomposition of the backscattering Mueller matrix. The latter better differentiates cancerous from normal gastric tissue. Scale bar: 0.5 mm.

Fig. 6. (a)–(b) Four configurations of the polarization imaging system.

Table 1. Mean and standard deviation of the orientation angle θ, linear retardance δ, and depolarization Δ for the birefringent (subscript “B”) and clear (subscript “C”) regions of the target measured in the forward and backward geometries. Erroneous values are in bold.

Table 2. Reciprocal polar decomposition of the Mueller matrix measured in the backward geometry for a multiple scattering chiral turbid medium and the extracted polarization parameters compared with the Lu–Chipman decomposition of the Mueller matrices measured in the forward and backward geometries for the same medium. Erroneous values are in bold.