Polarization-Aided Adaptive Region Growing (PARG)

• The paper introduces PARG, which segments the polarized object mask into locally convex subregions to resolve π-periodicity in azimuth angles and enhance 3D reconstruction fidelity.
• It employs a 4D polarization feature tensor with adaptive weighting based on local variance to robustly guide the region growing process.
• The strategy integrates into the SMSfP framework by decomposing complex surfaces into locally well-posed convex subproblems, ensuring consistent surface normal and height recovery.

Polarization-Aided Adaptive Region Growing (PARG) is a segmentation strategy introduced within the segmentation-driven monocular shape-from-polarization (SMSfP) framework for enhanced 3D surface reconstruction from single-view polarization images. PARG addresses the intrinsic azimuth ambiguity found in conventional monocular shape-from-polarization (SfP) methods by partitioning the polarized object mask into locally convex subregions and adapting the convexity prior separately in each. This decomposition enables independent, locally well-posed convexity-prior optimizations, effectively suppressing global azimuth ambiguities while preserving overall surface coherence (Zhang et al., 8 Jan 2026).

1. Conceptual Foundations and Motivation

The central challenge in monocular SfP arises from azimuth angle ambiguity ($\phi \leftrightarrow \phi + \pi$), which leads to instability in global surface normal and height recovery. Traditional global optimization methods suffer from poor disambiguation, especially for complex geometries. PARG is specifically designed to segment the object mask into a set of labeled, locally convex regions. Within each region, the multi-scale fusion convexity prior (MFCP) can be applied independently, so the $\pi$-periodic azimuth ambiguity is consistently resolved per region and not globally, drastically improving geometric fidelity and reconstruction accuracy.

2. Mathematical Formulation

PARG’s mathematical structure is built around polarization-based features and local adaptation:

• Polarization Feature Tensor: At each pixel $(x,y)$, a 4D tensor is constructed:

$$F(x, y) = \begin{bmatrix} \rho(x, y) \ \cos 2\phi(x, y) \ \sin 2\phi(x, y) \ |\nabla \phi(x, y)| \end{bmatrix}$$

Here, $\rho$ is the degree of polarization (DOP), $\phi$ the angle of polarization (AOP), and $|\nabla \phi|$ the modulus of the AOP gradient, revealing orientation discontinuities. Wrapping $\phi$ into $\cos 2\phi$ and $\sin 2\phi$ removes $\pi$-periodicity.

• Local Variance and Reliability: For each pixel $q$, a $5 \times 5$ centered window $W_5$ is defined. Compute local variances:

$$\sigma^2_\phi(q) = \mathrm{Var}\{\phi(u) \mid u \in W_5(q)\}$$

$$\sigma^2_\rho(q) = \mathrm{Var}\{\rho(u) \mid u \in W_5(q)\}$$

Reliability scores normalize these via

$$R_\rho(q) = \exp\left(-\frac{\sigma^2_\rho(q)}{\max_{\text{all pixels}}(\sigma^2_\rho)}\right)$$

$$R_\phi(q) = \exp\left(-\frac{\sigma^2_\phi(q)}{\max_{\text{all pixels}}(\sigma^2_\phi)}\right)$$

• Adaptive Channel Weighting: With user-chosen hyperparameters $\lambda_\rho$, $\lambda_\phi$, the pixelwise weights are

$$W(q) = \begin{bmatrix} 1 + \lambda_\rho R_\rho(q) \ 1 + \lambda_\phi R_\phi(q) \ 1 + \lambda_\phi R_\phi(q) \ 1 \end{bmatrix}$$

• Feature Distance and Region Growing: For a region seed with current mean feature $F_\mathrm{seed}$, a neighbor $q$ is assessed with:

$$d_\mathrm{feature}(q) = \| W(q) \odot (F(q) - F_\mathrm{seed}) \|_2$$

where $\odot$ indicates element-wise multiplication. $q$ is added if $d_\mathrm{feature}(q) < \tau$, with $\tau$ a global threshold.

3. Algorithmic Workflow and Pseudocode

PARG is implemented as a seeded, weighted adaptive region-growing algorithm. The pipeline follows these sequential steps:

1. Feature Computation: Generate $F(x, y)$ for all mask pixels.
2. Seed Initialization: Choose seeds (object boundary, skeleton, or uniform grid).
3. Region Growing: For each seed:
   • Assign new region label $l$.
   • Initialize mean feature $F_\mathrm{seed}[l]$ with the seed’s features.
   • Iteratively expand region, adding unlabeled neighbors whose weighted feature distance to the current mean is below $\tau$; update $F_\mathrm{seed}[l]$ incrementally.
4. Post-processing: Apply morphological hole filling and $3 \times 3$ Gaussian smoothing (with $\sigma \approx 1.0$).
5. Output: Segmentation label map $L(x, y) \in \{0, 1, 2, \ldots, K\}$.

Implementation parameters include window size ($5 \times 5$), recommended $\lambda_\rho = \lambda_\phi = 2$, and empirical $\tau$ in $[0.1, 0.3]$ normalized feature-distance units. Post-processing operations further refine region boundaries (Zhang et al., 8 Jan 2026).

Component	Function	Notes
Feature tensor $F$	Encodes DOP, unwrapped AOP, gradient	4D, combines intensity and geometric info
Reliability scores $R$	Modulate adaptive weighting	Based on local variance
Seeds	Initialize region-growing locations	Chosen at boundaries, skeleton, or grid of mask
Threshold $\tau$	Controls inclusion via feature similarity	Empirically chosen in implementation
Post-processing	Fills holes, smooths boundaries	Morphological and Gaussian filtering

4. Segmentation-Driven Disambiguation Mechanisms

PARG’s core advantage is in mitigating the $\phi \leftrightarrow \phi + \pi$ global ambiguity. By restricting each growing region to pixels with homogeneous polarization signatures, local azimuth direction variation remains smooth and consistent. Consequently, when the multi-scale fusion convexity prior is enforced on each segmented region, ambiguity collapses to a single, internally consistent solution per segment rather than across the whole mask, preserving both geometric detail and continuity. The global height-from-polarization recovery task becomes a union of locally well-posed convex problems.

A plausible implication is that this approach could generalize to other segmentation-driven physical inversion contexts where local homogeneity improves global tractability, subject to suitable prior design and feature selection.

5. Inputs, Hyperparameters, and Practical Considerations

PARG depends on inputs derived from four polarization-state images to compute:

• Degree of Polarization $\rho$: Sensitive to surface geometry; boundaries correspond to high local $\rho$ changes.
• Angle of Polarization $\phi$: Encoded as $\cos 2\phi$ and $\sin 2\phi$ to remove periodic ambiguity.
• Gradient Magnitude $|\nabla \phi|$: Emphasizes azimuthal boundaries and discontinuities.
• Foreground Mask $M(x,y)$: Binary mask limiting segmentation to the object region.

Hyperparameters include:

• Variance window size: $5 \times 5$.
• Adaptive weight strengths: $\lambda_\rho = \lambda_\phi = 2$ in reported experiments.
• Similarity threshold $\tau$: $0.1$–$0.3$ (empirically determined; normalized units).
• Seed strategy: mask boundary, skeleton, or grid sampling.
• Morphological and Gaussian post-filtering.

These design choices are necessary for accurate region partitioning, ensuring each subregion is “locally convex” for subsequent MFCP surface height recovery.

6. Workflow Integration within SMSfP

The role of PARG is situated between raw polarization inference and local convexity-regularized optimization. The complete SMSfP framework follows:

1. Compute DOP $\rho$ and AOP $\phi$ from raw polarization images.
2. Encode pixelwise 4D feature vectors, $F$.
3. Segregate the masked object into locally convex regions using PARG.
4. Apply region-level convexity prior (MFCP) optimization independently in each segment.
5. Fuse local surface reconstructions into a global 3D solution.

By structuring global reconstruction as a composition of locally optimized subproblems, SMSfP with PARG demonstrates substantial improvement in disambiguation accuracy and surface quality compared with prior monocular physics-based SfP techniques (Zhang et al., 8 Jan 2026).

7. Limitations and Scope of Applicability

PARG’s efficacy is tightly linked to the correctness of input polarization data, the reliability of the adaptive weighting scheme, and the empirical setting of $\tau$ and other hyperparameters. The method assumes foreground masks are available and that object surfaces can be decomposed into regions sufficiently convex to suppress azimuth ambiguities with a locally applied convexity prior. Topologically complex or highly concave shapes may challenge the region-growing process and subsequent optimization unless subregion convexity hypotheses are valid.

While the SMSfP framework with PARG is validated on both synthetic and real-world datasets, further investigation is warranted to establish generalization across diverse material classes and observation configurations.