Sample-Based Volumetric Models

• Sample-based volumetric models are techniques that represent 3D functions or fields using a discrete set of samples to encode continuous spatial data.
• They employ methods such as basis function expansions and analytic kernel mixtures, balancing representation power and computational efficiency.
• Key applications include scientific visualization, tomography, robotics, and point cloud compression, with tradeoffs in accuracy and complexity.

Sample-based volumetric models are a unifying framework for representing three-dimensional functions, densities, or fields using a finite set of samples. These models underpin diverse tasks in scientific visualization, computer graphics, tomography, robotics, and point cloud compression. Unlike surface-based or purely point-based representations, sample-based volumetric models encode spatial data throughout a region, enabling interpolation, integration, and rendering operations that capture both continuous variation and localized structure.

1. Definitions and Mathematical Foundations

Sample-based volumetric models represent a function $f: \mathbb{R}^3 \to \mathbb{R}$ (or $\mathbb{C}$, $\mathbb{R}^d$) as a sum or composition of primitives determined by a discrete set of samples. Canonical formulations include expansions in continuous bases fitted to discrete data, mixtures of analytic kernels (e.g., Gaussians), or direct transformations of regular/sparse grids.

Functional Expansion Approaches

Given samples $\{ (x_n, f_n) \}_{n=1}^N$, a volumetric function is expressed in a finite basis $\{ \varphi_k \}$:

$f(x) = \sum_k c_k \varphi_k(x)$

The basis may consist of B-spline volumes (Krivokuća et al., 2018), prolate spheroidal wavefunctions (Katz et al., 2018), or Fourier modes (Kwon et al., 14 Nov 2025). For point clouds or grid samples, coefficients are obtained by least-squares projection or orthogonalization.

Analytic Kernel Mixtures

Volumetric densities are approximated by a sum of parameterized primitives (Editor’s term: kernel mixture models):

$\rho(x) = \sum_{i=1}^M c_i \mathcal{K}_i(x)$

where $\mathcal{K}_i$ is typically a 3D Gaussian, each with center $\mu_i$, covariance $\Sigma_i$, and amplitude $c_i$ (Sharma et al., 14 Sep 2025). Primitives are adapted to data locality and can be fit to regular, adaptive, or unstructured samples.

2. Basis Function Choices and Multiresolution Construction

The choice of basis or primitive is crucial in balancing representation power, locality, and computational efficiency.

• B-spline Wavelet Basis: Compactly supports piecewise polynomial representations with global $C^{p-2}$ continuity. Multiresolution structure enables sparse, adaptive encoding and direct generalization of Haar (RAHT) transforms for point cloud attributes (Krivokuća et al., 2018).
• Generalized Prolate Spheroidal Wavefunctions (GPSWFs): Optimal for reconstructing essentially bandlimited, spatially localized functions. GPSWFs offer explicit sample-based expansions and error bounds in high-dimension volumetric settings (Katz et al., 2018).
• Fourier Basis: Used in ergodic control to aggregate the coverage statistics of volumetric "footprints" for sampled physical volumes or sensor models (Kwon et al., 14 Nov 2025).

Multiresolution decomposition enables flexible analysis/synthesis—e.g., encoding geometry and attributes at variable fidelity and supporting pruning or quantization schemes adapted to signal complexity (Krivokuća et al., 2018).

3. Sample Grouping, Compression, and Primitives Fitting

Compression of large volumetric datasets demands grouping samples into optimal support regions for each primitive.

• Voxel Grouping to Gaussians: In sparse grid data formats like OpenVDB, voxels are partitioned into blocks, and for each block, a Gaussian is fit via centroid and block-variance statistics (Sharma et al., 14 Sep 2025). Smart grouping algorithms further adapt block sizes in sparsely populated regions.
• OcTree and AMR Integration: Hierarchies such as octrees or adaptive mesh refinement (AMR) are naturally exploited to guide grouping, enabling unified handling of regular, adaptive, and particle data (Sharma et al., 14 Sep 2025).
• Wavelet/Transform Coding: Coefficient thresholding and quantization on wavelet trees or multi-level splines permit significant bit-rate reductions for geometry and attribute data, outperforming discrete point coding in rate-distortion performance (Krivokuća et al., 2018).

4. Rendering, Volume Integration, and Reconstruction

Sample-based volumetric models admit efficient analytic and numerical procedures for rendering and integration.

• Ray Marching with Analytic Kernels: For mixtures of 3D Gaussians, the line integral of density along a ray is computed in closed form using error function evaluations after finding segment-ray/ellipsoid intersections via Mahalanobis distance bounds (Sharma et al., 14 Sep 2025). This enables interactive rendering rates and flexible transfer-function selection.
• Volume Rendering for Physical Tomography: For physically-grounded models of attenuation and scattering, the radiative transfer equation is solved using Monte Carlo integration (ratio tracking, phase function importance sampling), with parameters directly linked to physical quantities in each sampled voxel (Nakath et al., 2023).
• Sample-based Bandlimited Reconstruction: GPSWF expansions reconstruct bandlimited volumetric signals with control over error via explicit truncation and sampling rules, allowing precise error budgeting in functional approximations (Katz et al., 2018).
• Implicit Surface Extraction: For B-spline coefficient representations of an SDF, surfaces are rendered by recursive subdivision or ray-casting of the implicit zero-level set, leveraging spline smoothness (Krivokuća et al., 2018).

5. Applications Across Domains

Sample-based volumetric models support a wide array of applications:

• Scientific Visualization: Compression and real-time rendering of large-scale 3D fields (e.g., smoke, fluid flow, astrophysical volumes) via grouped Gaussian primitives and GPU-accelerated optics (Sharma et al., 14 Sep 2025).
• Point Cloud Compression: Unified representation and coding of both geometry and attributes for sparse point sets, enabling superior rate-distortion properties compared to MPEG G-PCC (Krivokuća et al., 2018).
• Tomographic Inversion: Physically-faithful recovery of scattering and absorption properties in translucent media from sets of photographic projections, coupling sample-based grids with Monte Carlo optimization (Nakath et al., 2023).
• Optimal Coverage in Robotics: Ergodic control using volumetric footprints, generalized to arbitrary sample-based robot or sensor models, maximizing spatial coverage efficiency over tasks such as search and mechanical manipulation (Kwon et al., 14 Nov 2025).

6. Quantitative Performance and Limitations

The models achieve substantial gains in computational and memory efficiency, continuous representation, and flexibility across irregular data structures, but face specific tradeoffs.

Table: Example Compression and Rendering Results (from (Sharma et al., 14 Sep 2025))

Dataset	PSNR	Gaussians	FPS
explosion	24.0	1.5 M	125
smoke2	24.1	2.6 M	266
smoke	24.1	0.9 M	235

Notable limitations:

• Axis-aligned bounding boxes over-bound anisotropic primitives, increasing false positives in BVH traversal (Sharma et al., 14 Sep 2025).
• Compression by block-clustering may saturate or underfit in high-contrast regions unless further global clustering or variance-aware fitting is included (Sharma et al., 14 Sep 2025).
• GPSWF-based and spline-based approaches require substantial precomputation of basis functions and careful parameter selection to balance accuracy and computational load (Katz et al., 2018).
• Many frameworks emphasize absorption-only models; incorporating scattering (for physical realism) increases computational overhead (Nakath et al., 2023).

7. Future Directions and Extensions

Research is extending sample-based volumetric models via:

• Variance-aware clustering and global mixture model fitting to better capture sharp interfaces and overlapping heterogeneous features (Sharma et al., 14 Sep 2025).
• Incorporation of learned neural implicit representations (e.g., NeRF, SIREN) to replace or hybridize with analytic/transform bases, leveraging initialization or regularization benefits (Nakath et al., 2023, Krivokuća et al., 2018).
• Real-time, adaptive recomputation for robot control and active sensing, with scalable runtime complexity and coverage guarantees (Kwon et al., 14 Nov 2025).
• Expanding to support efficient encoding and interoperability across surface, voxel grid, AMR, and unstructured samples, enhancing cross-domain applicability (Sharma et al., 14 Sep 2025).

Sample-based volumetric modeling continues to provide a mathematically rigorous, adaptable foundation for 3D data acquisition, compression, manipulation, and rendering across the scientific, engineering, and graphics communities.