Implicit Neural hpBRDF Modeling

• Implicit neural hpBRDF modeling is a technique that uses neural networks to represent the continuous mapping from spectral and angular inputs to polarization-resolved reflectance.
• It employs a fully connected MLP to compress gigabyte-scale hpBRDF data into a compact model, providing smooth interpolation and accurate prediction of 4×4 Mueller matrices.
• The approach is applicable to physically-based rendering, inverse rendering, and remote sensing, offering practical benefits in efficient material simulation and detailed spectral-polarimetric analysis.

Implicit neural hyperspectral polarimetric BRDF (hpBRDF) modeling is the use of neural networks to directly and compactly represent the high-dimensional, continuous mapping from spectral and angular configurations to polarization-resolved reflectance (encoded as Mueller matrices) of real-world materials. This approach enables efficient storage, continuous interpolation, and seamless integration of hpBRDF data into rendering and analysis pipelines, surpassing the limitations of discrete tabulated datasets, especially when handling the joint spectral, angular, and polarimetric dependencies inherent in real-world BRDFs (Moon et al., 17 Sep 2025).

1. Foundations and Motivation

Traditional BRDF measurements capture the reflectance of materials under varying illumination and observation directions, typically for RGB wavelengths and scalar reflectance only. In contrast, hpBRDFs extend this by densely sampling:

• Wavelength (typically 68 bands, 414–950 nm, covering visible and near-IR)
• Illumination and viewing geometry (often parameterized in Rusinkiewicz angles: $\lambda$, $\varphi_d$, $\theta_d$, $\theta_h$)
• Full polarization state at each configuration, recorded as a $4 \times 4$ Mueller matrix.

Tabulated hpBRDF datasets may require up to 13 GB for a single material due to the concomitant angular, spectral, and polarimetric dimensionality. This data volume precludes direct storage and use in simulation and motivates compact, continuous, and differentiable representations. Implicit neural representations provide a solution by training a function $f_\theta$ (an MLP) to approximate the mapping:

$$M(\lambda, \varphi_d, \theta_d, \theta_h) \simeq f_\theta(\lambda, \varphi_d, \theta_d, \theta_h)$$

where $M$ is the measured Mueller matrix.

2. Model Architecture and Training

The typical implicit neural hpBRDF model comprises a fully connected MLP with several hidden layers (e.g., four layers with 256 hidden units each), accepting as input the concatenated spectral and angular parameters. The output is a vector representing the flattened $4 \times 4$ Mueller matrix. The network is trained with supervised learning to minimize the mean-squared error loss over the measured table:

$$\mathcal{L}(\theta) = \sum_i \| f_\theta(x_i) - M(x_i) \|^2$$

where $x_i=(\lambda, \varphi_d, \theta_d, \theta_h)$ are the sampled input configurations during training.

To address missing samples and measurement noise, data is inpainted and optionally pre-processed for physical admissibility before being used as ground truth. Experimental evaluation demonstrates that neural models smoothly interpolate between measured points and accurately capture fine structure in the angular and spectral/polarimetric domains, including rapid variations near specularities.

A notable advantage is the dramatic storage compression: a network of only $\sim$146 kB can replace a 13 GB table, representing a $10^5 \times$ reduction in footprint.

3. Representation of Spectral and Polarimetric Variations

The input domain combines three angular variables (using Rusinkiewicz or equivalent) and the wavelength, with the network tasked with capturing the associated spectral and polarimetric effects. The output, a $4 \times 4$ Mueller matrix, fully encodes the change of the Stokes vector, supporting arbitrary combinations of input/output polarization.

Component-wise, the model learns:

• Spectral features: e.g., material-specific absorption edges, dispersion, or fluorescence
• Polarimetric effects: depolarization, diattenuation, retardance as a function of geometry and $\lambda$
• High-frequency angular dependencies: notably, specular peaks and orientation-dependent polarization signatures

This joint modeling enables unified prediction at arbitrary $(\lambda, \varphi_d, \theta_d, \theta_h)$, facilitating dense or continuous rendering and analysis.

4. Practical Efficiency and Interpolation

Comparison with discrete table-based representations shows that neural models eliminate discretization artifacts, especially near highly directional features. Table lookups with nearest-neighbor or linear interpolation can produce visible discontinuities due to coarse angular/spectral resolution, whereas the neural approach yields smooth transitions through the domain. Storage and inference costs are minimized: gigabyte-sized data is replaced by a compact parametric model, suitable for integration with real-time or batch rendering systems and interactive material design workflows.

Empirical results indicate that, when trained over the post-processed measurement data, neural models attain low mean-squared error and maintain close physical fidelity between predicted and measured Mueller matrices, both for local variations (e.g., edge cases near specularities) and global spectral trends.

Representation	Storage Size	Artifacts	Continuous?
Discrete Table	~13 GB	Yes	No
Implicit Neural MLP	~146 kB	No	Yes

5. Integration and Applications

Implicit neural hpBRDF modeling is immediately applicable to:

• Physically-based rendering: Allows direct construction of spectral-polarimetric shaders, as shown by integration with Mitsuba 3, enabling realistic simulation of light transport with joint spectral and polarization effects.
• Inverse rendering and material recovery: Facilitates differentiable or optimization-based recovery of material properties and geometry from hyperspectral-polarimetric image data, leveraging the network’s continuity and differentiability.
• Remote sensing and scientific imaging: Compact models support rapid matching and simulation of complex spectral-polarimetric signatures, critical in object identification and surface characterization.
• Material analysis and visualization: Enables the paper of dependence on wavelength, angle, and material category within a single continuous framework.

6. Key Challenges and Limitations

High-dimensionality and Generalization

The hpBRDF domain combines four variables and outputs a matrix-valued function. While cost-effective, neural models are sensitive to training data coverage—undersampling in certain domains or inadequate physical regularization can cause poor generalization or physical inconsistency (e.g., nonphysical Mueller matrices). The chosen architecture must balance depth (capacity) with overfitting risk. Physical constraints (e.g., enforcing that output matrices map admissible Stokes inputs to physical outputs) are not automatically satisfied and may require further loss engineering.

Physical Validity

Ensuring strict physical admissibility (e.g., positive semi-definiteness, energy conservation, reciprocity) is nontrivial. In the referenced work, some of these concerns are addressed by post-processing or inpainting measured data prior to training; explicit physics-informed constraints during training remain a topic for further research.

7. Implications and Outlook

Implicit neural hpBRDF modeling provides a scalable solution to the challenges of high-dimensional, polarization- and spectrum-aware material representation. By enabling continuous, data-driven approximation of complex table data, it overcomes the limitations of legacy approaches in both storage and accuracy. The approach is poised for impact across rendering, remote sensing, and scientific imaging, with ongoing efforts directed toward physical regularization (e.g., physics-informed loss functions), extension to anisotropic or spatially-varying materials, and integration with learning-based inverse problems.

In summary, neural hpBRDF modeling enables the practical use of dense hyperspectral-polarimetric BRDF measurement in rendering and analysis workflows by providing an efficient, continuous, and differentiable representation that faithfully captures the joint angular, spectral, and polarimetric structure of real-world materials (Moon et al., 17 Sep 2025).