Contrast Matrix Construction in JM-OCT

• Contrast matrix construction is a method that unifies multiple optical coherence tomography contrasts, including intensity, polarization, and flow, into a single co-registered dataset.
• The process employs techniques such as adaptive Jones-matrix averaging, eigenanalysis, and phase correction to derive cumulative and local polarization contrasts.
• It enables quantitative tissue analysis by integrating conventional and advanced OCT metrics, enhancing both biomedical imaging research and clinical diagnostics.

Contrast matrix construction is the process by which multiple optical coherence tomography (OCT)-derived contrasts—including intensity, phase retardation, polarization uniformity, attenuation, Doppler flow, angiography, and dynamic properties—are systematically computed and co-registered from the fundamental volumetric Jones-matrix measurement in Jones-matrix optical coherence tomography (JM-OCT). Central to multi-contrast imaging, contrast matrix construction enables the unification of conventional and advanced OCT contrasts into a single, structured data object, thereby facilitating quantitative tissue analysis and visualization across several biological and clinical modalities.

1. Acquisition and Preprocessing of the Jones Matrix

Contrast matrix construction begins with the acquisition of the four complex OCT channels stemming from two independent, known input polarization states and two orthogonal output channels, typically captured by a polarization-diversity detection scheme. The measured Jones matrix at voxel coordinates $(x, y, z)$ and time %%%%1%%%% is represented as:

$$J_{m}(x, y, z, t) = \begin{pmatrix} E_{HH} & E_{HV} \ E_{VH} & E_{VV} \end{pmatrix}$$

where $E_{pq}$ denotes the complex field amplitude for input polarization $p$ and detection polarization $q$. Bulk phase fluctuations are removed prior to further processing. Adaptive Jones-matrix averaging (AJA) is optionally applied to reduce random complex noise while preserving phase, implemented as:

$$J_{m}^{\mathrm{avg}}(r) = \frac{1}{N}\sum_{k=1}^N e^{-i\phi_k} J_{m}^{(k)}(r)$$

with the phase offset estimated by $\phi_k = \arg[\det J_{m}^{(k)}(r)]$.

2. Derivation of Cumulative and Local Jones Matrices

The measured Jones matrices are transformed to derive cumulative polarization interactions across depth and localized, depth-differential polarization effects. The cumulative Jones matrix is defined by:

$J_c(x, y, z) = J_m(x, y, z) J_m(x, y, z=0)^{-1}$

This matrix captures the net round-trip retardation up to depth $z$. The local Jones matrix, isolating the effect in the thin slice between $z$ and $z + \Delta z$, is given by:

$$J_L(x, y; z, z+\Delta z) = J_c(x, y, z+\Delta z) J_c(x, y, z)^{-1} = J_m(x, y, z+\Delta z) J_m(x, y, z)^{-1}$$

3. Extraction of Conventional and Polarization OCT Contrasts

A series of contrasts is then extracted via analytical or statistical processing:

• Polarization-insensitive Intensity (coherent or incoherent sum):

$$I(x, y, z) = \operatorname{Tr}[J_m J_m^{\dagger}] = |E_{HH}|^2 + |E_{HV}|^2 + |E_{VH}|^2 + |E_{VV}|^2$$

• Attenuation Coefficient $\alpha$, assuming single-exponential decay:

$$I(z) \approx I_0 e^{-2\alpha z} \implies \alpha(x, y) = -\frac{1}{2} \frac{d}{dz}\ln I(x, y, z)$$

• Scatterer Density $\rho$ estimated by speckle statistics or machine learning:

$$\rho = \mathrm{SDE}(\{I_i\}) \quad \text{(learned estimator)}$$

• Polarization Contrasts through eigenanalysis of $J$:
  • Eigenvalue Decomposition: $J = A \Lambda A^{-1}$, $$\Lambda = \operatorname{diag}(\lambda_1, \lambda_2)$$
  • Cumulative Phase Retardation:

  $$\Delta\varphi_{\mathrm{cum}} = \arg[\det J_c] = \arg(\lambda_1 \lambda_2)$$ - Local Phase Retardation (Birefringence per slice):

  $$\Delta\varphi_{\mathrm{loc}}(z) = \arg[\lambda_1/\lambda_2],\qquad b(z) = \frac{\Delta\varphi_{\mathrm{loc}}}{2 k_0 \Delta z}$$ - Diattenuation:

  $$d = \frac{|\lambda_1|^2 - |\lambda_2|^2}{|\lambda_1|^2 + |\lambda_2|^2}$$ - Polar Decomposition: $J = UP$ with $U$ unitary (pure retardance) and $P$ Hermitian positive-definite (diattenuation + isotropic loss).

4. Temporal Contrasts: Flow, Angiography, and Dynamics

Sequential Jones-matrix measurements enable the extraction of flow and dynamic tissue properties:

• Doppler OCT (phase-resolved flow):

$$\Delta\phi(\mathbf{r}; t) = \arg[E(\mathbf{r}; t+\tfrac{\Delta t}{2}) E^*(\mathbf{r}; t-\tfrac{\Delta t}{2})], \quad v \propto \frac{\Delta\phi}{2 k_0 \Delta t}$$

• OCT Angiography (OCTA) via speckle-variance or decorrelation:

$$\mathrm{OCTA}(\mathbf{r}) = \operatorname{Var}_t[I(\mathbf{r}, t)] \quad \text{or} \quad 1 - |\langle \hat{E}(\mathbf{r}, t) \hat{E}^*(\mathbf{r}, t + \Delta t) \rangle_t |$$

• Dynamic OCT:
  • Log-Intensity Variance (LIV): $\mathrm{LIV} = \mathrm{Var}_t[10 \log_{10} I(t)]$
  • OCT Correlation-Decay Speed (OCDS): slope of temporal autocorrelation $\operatorname{Corr}_t[I]$ vs delay

5. Degree of Polarization Uniformity and Advanced Polarization Metrics

Degree-of-polarization uniformity (DOPU) quantifies the uniformity of backscattered polarization states within a spatial kernel:

$$\mathbf{S} = \langle I, Q, U, V \rangle_{\text{kernel}}, \quad \mathrm{DOPU} = \frac{\sqrt{Q^2 + U^2 + V^2}}{I}$$

A practical implementation uses a noise-corrected form to enhance robustness under low SNR conditions.

6. Noise Correction, Quantity Estimation, and Self-Calibration

Random complex noise is suppressed via adaptive Jones-matrix averaging. Maximum-a-posteriori (MAP) estimation, informed by Monte-Carlo likelihoods $p(\mathrm{meas}|\mathrm{true}, \mathrm{SNR})$, is used to correct bias in derived contrasts including birefringence, phase retardation, and intensity. The system’s self-calibrating nature follows from the observation that imperfections, such as non-ideal polarizing beam splitter (PBS), detector gain $\eta$, and reference birefringence $R$, factor out in the cumulative Jones calculation:

$$J_c = (\eta X' R \cdots) J_{r0i} (\cdots R^{-1} {X'}^{-1} \eta^{-1}) \implies \text{eigenvalues unaffected}$$

7. Construction and Organization of the Multi-Contrast Matrix

For each voxel $(x, y, z)$ (optionally at each time $t$), scalar and vector contrasts are collected into a column vector:

$$C(x, y, z) = [I; \alpha; \rho; \Delta\varphi_{\mathrm{cum}}; \Delta\varphi_{\mathrm{loc}}; d; \mathrm{DOPU}; v_{\mathrm{Doppler}}; \mathrm{OCTA}; \mathrm{LIV}; \mathrm{OCDS}; \ldots ]^{T}$$

These data can be structured as a 5-D array $(x, y, z,\, \text{contrast index}, t)$ or as a collection of co-registered 3-D volumes, one per contrast. Practical considerations include pre-filtering to improve SNR, choosing a depth kernel $\Delta z$ in the $20$–$50~\mu$m range to balance depth resolution with sensitivity for local Jones-matrix derivation, GPU acceleration for computationally intensive steps (eigenanalysis, MAP estimation, speckle variance), and sample motion registration and correction prior to temporal analyses.

The resulting contrast matrix provides a comprehensive, co-registered dataset capturing the multi-parametric optical properties of biological tissue, directly traceable to the measured Jones-matrix sequence. This systematic construction supports advanced quantitative imaging in both research and clinical contexts, integrating conventional OCT, polarization-sensitive, flow, angiographic, and dynamic contrasts within a unified analytical and computational framework.