Objective The polarimetric bidirectional reflectance distribution function (pBRDF) is a powerful tool for accurately describing the polarized scattering properties of light interacting with material surfaces, enabling highly realistic rendering and 3D modeling of objects. However, previous metal modeling and rendering methods, while widely simulating the polarization effects of specular reflection, often overlooked the polarization of diffuse reflection to avoid the complexity of polarized reflection modeling. In addition, few studies have explored inverse rendering methods specifically for metal pBRDF models. To address these challenges, we propose a hybrid polarized reflection model for metal surfaces which combines specular reflection and interreflection components. Based on this metal pBRDF model, we further develop an inverse rendering method capable of recovering the 3D shape, surface normals, and polarimetric SVBRDF information of metal surfaces. Methods In this paper, we propose a metal surface pBRDF model that integrates specular reflection and interreflection. The specular reflection component is modeled based on microfacet theory and the complex refractive index of metals, while the interreflection component is approximated using the phase Mueller matrix to address partial light energy loss due to occlusion and shadowing effects. Initially, a coarse point cloud of the object is obtained using methods such as structure- from- motion. By constructing a loss function that minimizes mixed reflection components, diffuse reflection components, and surface normals, the point cloud is optimized through sequential quadratic programming to estimate the SVBRDF and normals. The refined geometry is further enhanced using selective Poisson reconstruction methods to update the point cloud. Iteration continues until the final point cloud and normals are obtained, followed by a separate optimization of the SVBRDF to achieve accurate results. The final object undergoes relighting using a renderer for visualization. Results and Discussions Based on our model, we conduct forward rendering simulations and practical inverse rendering experiments. In simulations, we used the KAIST pBRDF model as a benchmark and compared our results with the C- T model. Figure 4 shows the Mueller matrix images of a rendered chromium sphere, while Fig. 5 shows the same for a rendered golden rabbit. Figures 5(a), 5(b), and 5(d) correspond to the Mueller matrix images generated by the KAIST model, our model, and the C- T model, respectively. Figs. 5(c) and 5(e) highlight the differences between our model and the C- T model in comparison to the KAIST model. The results demonstrate that both our model and the C- T model capture the polarization variations in specular reflection, with diagonal elements closely matching actual values. However, our model also accounts for the polarization effects of diffuse reflection, leading to non- diagonal elements that align more consistently with the KAIST model. To assess rendering quality, we render images from various angles and analyze each model's performance, as shown in Fig. 6. The C- T model exhibits blurriness in specular highlights, causing detail loss. In contrast, our model preserves highlight details and suppresses blurring, resulting in more realistic rendering. In practical applications, our inverse rendering method successfully reconstructs the 3D shape, surface normals, and pBRDF information of objects. Figure 15 presents the reconstruction results. Our experiments demonstrate that the proposed model and rendering method accurately characterize the polarized reflection properties of metals and enable effective rendering. Conclusions Polarization plays a crucial role in the scattering and reflection of light on material surfaces, showing significant variations across different materials. Introducing a pBRDF model improves the ability to effectively model surface properties, thereby enhancing the realism of image rendering. In this paper, we address the limitations of existing metal pBRDF models, particularly the neglect of diffuse polarization, by proposing a hybrid polarization model. This model captures metal surface reflection as a mixture of specular and diffuse polarization, with the diffuse component modeled as a linear retarder with a 45 degrees phase delay. A Mueller matrix model is derived to accurately represent depolarization and phase delay in multiple reflections, providing a detailed characterization of polarized reflection in metals. In addition, we propose an inverse rendering method based on the metal pBRDF model, capable of recovering surface attributes and normals, effectively reproducing the appearance of metals under different lighting conditions. Experimental results validate the effectiveness of the proposed model and reconstruction method, offering new approaches for metal material simulation and rendering. However, the current model does not consider light scattering at the shallow surface of metals or diffraction effects caused by periodic structures, making it unsuitable for thin- coated metals and metals with such structures. In addition, since this study employs a structure- from- motion method, it has high requirements for the texture of the objects being measured. Furthermore, the Poisson reconstruction results in point clouds that lose fine details, which limits the method's ability to recover objects with intricate structures. Future research will focus on refining the model to accommodate different metal structures, exploring other methods to obtain more detailed initial point clouds, and further optimizing the iterative algorithm to enhance computational efficiency and practicality.
