Fixed-Budget Gaussian Volume Encoding with Structure-Aware Allocation

Michael R. Martin; Joseph Insley; Victor A. Mateevitsi; Silvio Rizzi; Kwan-Liu Ma

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Fixed-Budget Gaussian Volume Encoding with Structure-Aware Allocation

Michael R. Martin, Joseph Insley, Victor A. Mateevitsi, Silvio Rizzi, Kwan-Liu Ma

arXiv:2608.14112v1Aug 14, 2026cs.CVcs.AIcs.CEcs.GRcs.LG

TL;DR

Scientific scalar volumes are costly to store, transfer, and process under constrained in situ budgets. This paper analytically allocates a fixed set of anisotropic Gaussians from field structure, refines them without changing count, and demonstrates useful rate–fidelity trade-offs across five datasets.

Problem
Existing encoders determine capacity through architecture choices or error-driven growth, leaving allocation and encoded size unpredictable before optimization.
Method
The method allocates a complete anisotropic Gaussian ensemble analytically from field structure, then performs camera-free field-space refinement while preserving the exact primitive count.
Results
Across five datasets, fixed-count representations provide useful rate–fidelity trade-offs; Bubble Plume reaches 38.7 dB at 200,000 primitives.
Takeaways & Limitations
The approach makes encoded footprint a deterministic constraint aligned with fixed memory budgets, while rendering cost scales with the selected budget rather than source resolution.
Takeaways & Limitations
One-shot allocation can leave room for further spatial refinement when field complexity is concentrated within narrow interfacial regions.

Abstract

from arXiv · show

Scientific simulations often produce scalar volumes faster than they can be stored, transferred, and loaded, while in situ reduction must use only a limited share of simulation resources. This work encodes scalar fields as anisotropic Gaussian primitives under a fixed budget. The complete primitive set is allocated analytically from local field structure, including position, orientation, and shape, then refined directly against the scalar field without densification, pruning, or count changes. The selected budget determines encoded storage before refinement and, together with the iteration schedule, provides a controllable refinement-time budget. In a controlled benchmark, truncation-aware field evaluation reduces encoding time by up to 51x; 1.4 million Gaussians encode a billion-voxel volume in at most four minutes on one desktop GPU, with reduced-iteration refinement completing in under one minute. Across five datasets spanning 2.1 million to 1.1 billion evaluated voxels, compression-useful configurations achieve 15.0-38.7 dB PSNR at compression ratios from 2.2x to over 40,000x. Pre-encoding structure statistics characterize fields for which one-shot allocation yields limited gains from additional capacity. Because primitives retain scalar attributes rather than baked appearance, a single compact model serves every subsequent visualization state - supporting post-hoc transfer-function, colormap, lighting, and viewpoint changes without re-encoding.

I. INTRODUCTION … C. Gaussian Splatting and Explicit Primitive Representations

Scientific visualization needs compact scalar-volume representations because dense fields are costly to store, transfer, and load, especially under strict in situ budgets. Prior neural and Gaussian approaches provide compact or explicit representations but differ in accessibility, supervision, and primitive-set control.

I. INTRODUCTION: Spatially distributed, multiscale structures make scalar fields difficult to characterize from slices or isolated viewpoints.These fields occur across fluid simulation, combustion, medical imaging, and computational science.
I. INTRODUCTION: A common in situ rule of thumb limits visualization and reduction to 5% of the simulation run’s cost.A fixed bound requires an encoder whose cost is known in advance.
II. RELATED WORK: Dense-grid encoding addresses the storage and movement problem that remains after hardware acceleration reduces direct-volume-rendering cost.Direct volume rendering remains the reference method for inspecting scalar volumes.
A. In Situ Data Reduction and Visualization: In situ processing reduces data while it remains in simulation memory, but the same compact encoding also supports post hoc analysis after direct data dumps.Both settings require effective reduction under practical storage and computation constraints.
C. Gaussian Splatting and Explicit Primitive Representations: 3D Gaussian Splatting uses anisotropic Gaussian primitives with differentiable rendering and density control that splits and prunes primitives during optimization.VEG adapts Gaussians to scientific volumes, storing scalar values while assigning color and opacity through a transfer function at render time.
C. Gaussian Splatting and Explicit Primitive Representations: W-VEG moves supervision from image space to direct world-space field sampling because image-space-trained models do not transfer reliably to direct sampling.Its primitive set grows through error-guided densification until a target compression ratio is reached.

D. Gaussian Count and Capacity Allocation … A. Overview

The paper fixes the Gaussian primitive count before training, allocating the full budget analytically from field structure rather than letting optimization determine capacity. Its pipeline separates pre-training allocation from refinement and preserves scalar attributes for render-time transfer-function control.

D. Gaussian Count and Capacity Allocation: Prior densification pipelines split, insert, or prune primitives during optimization, paying the cost of density-control machinery at every iteration.Their final count emerges from the growth process and its steering machinery.
E. Positioning of This Work: Analytical allocation from field structure, held fixed throughout training, is presented as an unexplored position for scientific scalar volumes.The contrast is with capacity learned through architecture choice or grown in response to reconstruction error.
D. Gaussian Count and Capacity Allocation: Fixed primitive count is treated as a budget chosen before training, unlike prior approaches that grow or prune capacity toward reconstruction targets.This makes encoded size a consequence of the preset budget rather than an emergent optimization outcome.
III. METHODOLOGY: The pipeline allocates the full primitive budget in one analytic pass before considering the error signal.The overview identifies this as the first of three pipeline stages.
A. Overview: Primitive positions are drawn from a mixture of gradient-weighted and uniform distributions over the occupied domain.This is part of the structure-analysis allocation stage described in the overview.
A. Overview: The active transfer function assigns color and opacity at render time in the 8-bit variant.The scalar attributes remain available for rendering rather than being permanently baked into appearance.

B. Volume Preprocessing · C. Scalar-Aware Gaussian Primitives

The method normalizes structured scalar volumes into a fixed coordinate and value representation, then encodes them with N anisotropic Gaussian primitives whose scalar attributes remain available for later transfer-function-based rendering.

B. Volume Preprocessing: Input scalar values are converted to float32 and linearly normalized to [0, 1] using the field minimum and maximum.Seeding and refinement use normalized values and voxel-index coordinates, so reconstruction errors are reported for the normalized field.
B. Volume Preprocessing: Compression ratios are measured against the source byte count at the evaluated volume.The supplied passage defines this comparison basis but truncates the remainder of the sentence.
C. Scalar-Aware Gaussian Primitives: The encoded volume is represented by a set of N Gaussian primitives.This provides the primitive-count basis for the scalar-volume representation.
C. Scalar-Aware Gaussian Primitives: Each primitive uses anisotropic scales, a unit-quaternion rotation, a scalar attribute, and a weight.The primitives are optimized using the standard unconstrained parameterization of 3DGS-family encoders, including log-scales and normalized quaternions.
C. Scalar-Aware Gaussian Primitives: Scalar attributes remain data attributes rather than being converted directly into RGB appearance.At render time, the active transfer function maps the reconstructed scalar field to color and opacity, while wi modulates field contribution.
C. Scalar-Aware Gaussian Primitives: The primitive contribution combines its weight, scalar attribute, and Gaussian term.The supplied expression fragment contains wi, ai, and Gi(p), while the full equation is truncated.
C. Scalar-Aware Gaussian Primitives: Gaussian kernels are truncated at 5σ, corresponding to squared Mahalanobis distance below 25.The covariance is induced by each primitive’s scales and rotation.
C. Scalar-Aware Gaussian Primitives: The density floor ρ0 = 0.05 prevents unstable normalization in sparsely covered regions.This applies where the reconstructed volume approaches zero.

D. Direct 3D Supervision · E. Structure-Aware Fixed-Budget Allocation

The method supervises Gaussian encoding directly against voxel-space scalar values, producing a self-contained continuous model reusable across visualization configurations. It allocates the full primitive budget analytically from field structure before refinement, using detail and coverage sampling plus locally derived geometry.

D. Direct 3D Supervision: Voxel-space scalar supervision avoids cameras, rendered images, and transfer functions during optimization, with each volume producing a self-contained primitive set.Allocation and training operate directly on sampled scalar values in voxel coordinates.
D. Direct 3D Supervision: Analytic allocation completes in seconds even at billion-voxel scale, with stages S1a–S1e separately timed and memory-profiled.These stages together constitute allocation before refinement.
D. Direct 3D Supervision: One trained model supports every transfer function, colormap, and lighting configuration without re-encoding.Transfer functions are evaluated at sample positions, while the continuous reconstructed field also provides shading gradients.
D. Direct 3D Supervision: Continuous field evaluation makes classification granularity follow sampling rate rather than primitive extent and exposes derived quantities such as ∇Ṽ for shading.The transfer function is applied to reconstructed field values at sample positions.
E. Structure-Aware Fixed-Budget Allocation: 55% of the primitive budget samples gradient magnitude over the occupied domain, while 45% samples coverage uniformly.The occupied domain is defined by V > 0.02, and both draws enforce distinct positions so the realized budget matches specification.
E. Structure-Aware Fixed-Budget Allocation: Distinct seed positions decouple primitive centers from grid alignment, allowing the representation to express off-grid structure.Sparse fields lacking sufficient seed positions receive additional encoder handling.
E. Structure-Aware Fixed-Budget Allocation: 73 structure tensors determine local orientation, aligning each principal axis with the tangent eigenvector associated with the smallest eigenvalue.This orientation is derived independently for each seed.
E. Structure-Aware Fixed-Budget Allocation: Axis scales use within-group seed spacing, remain tighter for detail and broader for coverage, then elongate along the tangent according to gradient magnitude.The resulting scales are clamped by the allocation procedure.

F. Training Objective · G. Truncation-Aware Training at Gigavoxel Scale

Training jointly optimizes all Gaussian primitive parameters against the scalar field using a composite objective and stratified, jittered samples. Truncation-aware spatial binning and memory-bounded analysis reduce evaluation overhead and enable billion-voxel volumes on a single 24 GB GPU with budgets up to at least 1.4 million primitives.

F. Training Objective: All primitive parameters are jointly refined with Adam under direct supervision from the scalar field.The optimized parameters are µ_i, s_i, q_i, a_i, and w_i.
F. Training Objective: 35% of samples come from high-value voxels, 40% from occupied mid-range voxels, and 25% uniformly across the domain.All sample points are jittered off-grid.
F. Training Objective: The objective combines value, density, smoothness, anchor, anisotropy, and scale-capacity terms.Its stated form is L = Lval+λρ Lρ+λsm Lsm+λanc Lanc+λa Laniso+λs Lcap.
F. Training Objective: The density term targets uniform unit density in occupied space, while smoothness suppresses speckles and anneals from strong early coverage to weaker late detail recovery.Bidirectional density matching addresses both spatial gaps and overlap accumulations; opacity remains governed by the transfer function.
G. Truncation-Aware Training at Gigavoxel Scale: 2.0 × 10^9 candidate pairs shrink to 0.034% contributing under 5σ truncation at N = 650,000 primitives and 3,072 samples per iteration.Primitives are binned by truncation radius, and samples are bucketed into grids matched to each bin’s maximum radius.
G. Truncation-Aware Training at Gigavoxel Scale: A two-level budget draw and slab-wise or block-wise processing support volumes exceeding a billion voxels on one 24 GB commodity GPU.The approach avoids full-volume index tensors and supports budgets up to at least 1.4 million primitives.

H. Implementation · I. Checkpointing, Export, and Rendering · J. Compression and Storage Cost

The implementation provides instrumented direct and memory-bounded execution paths with shared evaluation and export components, while checkpointing supports resumable runs. Storage is fixed analytically by the selected primitive budget: full-precision encodings use 48N + H bytes, and an optional 8-bit format reduces this to approximately 12 bytes per primitive.

H. Implementation: PyTorch/CUDA execution is divided into independently timed and memory-profiled stages, with direct and memory-bounded paths sharing the field evaluator, objective, and export pipeline.The shared architecture is intended to ensure path-invariant execution.
I. Checkpointing, Export, and Rendering: Periodic checkpoints preserve the full primitive state and accumulated timing, allowing interrupted runs to resume without repeating allocation.The resumed run retains its reported cost.
I. Checkpointing, Export, and Rendering: 48 bytes per primitive are stored in the final full-precision representation, alongside a small fixed header containing volume metadata and default lookup tables.Each primitive uses twelve float32 values for position, logscale, quaternion, scalar, and weight; the representation remains distinct from the underlying representation.
J. Compression and Storage Cost: Senc(N) = 48 N + H bytes, where H is a fixed header containing volume metadata and default transfer-function lookup tables.The source volume occupies Ssrc = Mb bytes when it has M voxels stored at b bytes per voxel.
J. Compression and Storage Cost: Compression ratio is determined a priori by the selected primitive count N, so targeting a specific ratio reduces directly to solving Eq. 8 for N.Unlike optimization-driven encoders with variable output sizes, the ratio depends exclusively on the source volume and chosen budget.
J. Compression and Storage Cost: A 200,000-primitive encoding occupies an identical byte budget regardless of field complexity, making the storage footprint content-agnostic.The fixed-budget property follows from H being independent of N.
J. Compression and Storage Cost: Approximately 12 bytes per primitive are required by the optional 8-bit format, increasing the compression ratio by approximately 4×.Both full-precision and 8-bit formats remain independent of transfer-function state.

K. Relation to Prior Approaches … A. Datasets

The method distinguishes itself from prior scalar-aware Gaussian encodings by analytically allocating and preserving the complete primitive budget before optimization. Evaluation covers five scientific scalar datasets across 2.1 million to 1.1 billion voxels, with optional render-time density mapping that leaves encoded primitives unchanged.

K. Relation to Prior Approaches: The complete Gaussian set is analytically allocated before optimization and preserved at the exact count throughout training.This contrasts with W-VEG, which grows and prunes a partial set toward a ratio-derived target count.
K. Relation to Prior Approaches: Capacity is committed in one analytic pass from field structure alone before refinement.The method uses field-space supervision by querying scalar values at sampled 3D positions.
L. Scope and Design Trade-offs: Render-time density mapping can emphasize user-selected visual density features without modifying encoded primitives.The mapping may be tuned against reference renders from selected viewpoints while preserving field fidelity.
L. Scope and Design Trade-offs: The optional render-time mapping requires only a small fraction of the initial encoding cost.Experimental boundaries and core-method evaluation parameters are addressed separately in Section VI-B.
A. Datasets: Compression ratios and PSNR values are measured against the distributed source form at the evaluated resolution, including stride-2 cases where marked.Occupancy is defined as the fraction of voxels with normalized value above 0.02.
A. Datasets: Five scientific scalar datasets span evaluated grids from 2.1 million to 1.1 billion voxels.The set includes Vortex, Bubble Plume, Miranda, Chameleon, and Richtmyer–Meshkov, with native resolutions reaching 2048×2048×1920.
A. Datasets: The datasets cover structured flow, sparse multiphase flow, hydrodynamic instability, CT imaging, and turbulence.Bubble Plume has 2.4% occupancy, while Chameleon has 8.7% occupancy.
A. Datasets: Chameleon’s 8.7%-occupancy field gains 16.5 dB across a budget ladder and continues improving at the largest budget.The comparison uses identical transfer function, camera, and lighting in both panels.

B. Training Configuration … A. Reconstruction Quality Across Budgets

The evaluation uses common training settings and volumetric PSNR to examine reconstruction quality across primitive budgets, encoding costs, and field-structure indicators. Across five datasets, budget efficiency varies with spatial structure, with sparse concentrated fields converting small allocations into substantial quality.

B. Training Configuration: Budgets span 10^3 to 1.4×10^6 primitives, with a 48byte format losing compression once its encoded size exceeds the source volume.Budget ladders and costs were measured by default on one NVIDIA RTX 4090 GPU with 24 GB.
C. Evaluation Metrics: Volumetric PSNR is defined as 10 log10(1/MSE) over evaluation positions normalized to Vmax = 1.It is computed at 500,000 uniformly sampled voxel centers using a fixed evaluation seed.
A. Reconstruction Quality Across Budgets: Miranda’s nearly uniform complexity distribution allows one-shot allocation to convert budget into quality steadily.The figure uses identical transfer function and camera in both panels.
C. Evaluation Metrics: PSNR evaluation uses the pipeline’s native evaluator, with sampled estimates matching full-grid evaluations within 0.01 dB.Because it evaluates scalar values directly, the metric is independent of transfer functions and renderers and weights every position equally.
V. RESULTS: The evaluation covers reconstruction quality across budgets, admissible budget ranges, encoding cost, reduced-iteration time budgets, and structure indicators for limited gains from added capacity.These experiments assess both quality and the relationship between allocation capacity, runtime, and field structure.
A. Reconstruction Quality Across Budgets: Bubble Plume reaches 27.2 dB at 204× compression from 2,000 primitives and 38.7 dB at 200,000 primitives.The reported pattern identifies sparse, structure-concentrated fields as converting budget most efficiently.
A. Reconstruction Quality Across Budgets: Richtmyer–Meshkov concentrates 77.7% of gradient mass in the top decile of occupied voxels at its thin mixing interface.This region is identified in advance by the paper’s structure analysis.

B. Budget Limits Imposed by Field Structure · C. Training Cost and Memory

Feasible and compression-beneficial budgets are constrained by per-primitive storage and field structure, so requested budgets are validated against structure-dependent ceilings. Training remains practical through deterministic output sizing, bounded memory, seconds-to-minutes preprocessing and refinement, and truncation-aware sparsity.

B. Budget Limits Imposed by Field Structure: At 48 bytes per primitive, compression ends above approximately 175,000 primitives for Vortex and 437,000 for Bubble Plume.These storage thresholds determine part of each field’s feasible, compression-beneficial budget range.
B. Budget Limits Imposed by Field Structure: Bubble Plume’s 125,787 eligible voxels support approximately 2.3 × 10^5 primitives, whereas Vortex’s 99.6% occupancy supports 1.4 × 10^6 primitives.The budget ceiling therefore depends on the distribution of field structure rather than source size.
B. Budget Limits Imposed by Field Structure: Primitive budgets determine encoded footprints before optimization, and upfront validation returns the requested complete model or an immediate status signal.The encoded size follows from Eq. 7 alone and is identical across datasets for a given budget.
C. Training Cost and Memory: Representative encodings complete in at most 4.0 minutes on one RTX 4090, with peak refinement memory from 1.7 to 9.5 GiB.One-time analysis and seeding complete in seconds, including for gigavoxel inputs.
C. Training Cost and Memory: Only 0.034% of 2.0 × 10^9 candidate Gaussian–point pairs are nonzero per iteration under the 5σ cutoff.This truncation-aware sparsity enables efficient field evaluation.
C. Training Cost and Memory: 51× sparsity-aware evaluation reduced Miranda encoding time from 218.8 to 4.25 minutes and peak memory from 20.5 to 8.4 GiB.The comparison is against dense evaluation at 650,000 primitives.

D. Cross-GPU Cost and a Time-Budget Mode · E. Rendering the Encoded Model

Cross-GPU reconstruction quality remains consistent across implementations, while runtime varies by GPU architecture and can be controlled through iteration scheduling. Encoded rendering scales with the selected primitive budget, which determines model size and interactive performance independently of source resolution or training engine.

D. Cross-GPU Cost and a Time-Budget Mode: 0.17 dB was the maximum reconstruction difference across 21 cross-GPU and implementation configurations, with a mean difference of 0.08 dB.The configurations covered three gigavoxel datasets and nine budgets.
D. Cross-GPU Cost and a Time-Budget Mode: 280.5 s versus 201.5 s was the refinement time for Miranda on an A100 versus an RTX 4090 at 1.4 million primitives and 1,500 iterations.Runtime varied by GPU architecture despite consistent reconstruction results.
E. Rendering the Encoded Model: Encoded models render interactively through both Section III-H rendering paths and VTK-based dense-grid ray casting.The exported format and primitive count are identical across training engines, so closed-form-gradient and reference-engine models render equivalently.
D. Cross-GPU Cost and a Time-Budget Mode: The reduced-iteration time-budget mode provides a faster refinement option while preserving the selected primitive budget.Figure 5 identifies the last panel as the reduced-iteration time-budget mode.
E. Rendering the Encoded Model: 16.5 dB was the overall quality gain across Chameleon’s budget ladder under one transfer function and density setting.Each panel reports primitives, RTX 4090 refinement seconds, and PSNR.
E. Rendering the Encoded Model: Gradient concentration tracks ∆PSNR, and Richtmyer–Meshkov is identified in advance as the field with interface-concentrated gradient mass.Table VI defines ∆PSNR as the dB gain across each dataset’s compression-useful budget range.
E. Rendering the Encoded Model: Rendering cost scales with the chosen primitive budget rather than source resolution or the training engine, using the same knob that sets encoded size.A controlled rendering-performance comparison under matched settings is left to future work.
E. Rendering the Encoded Model: 6.9–16.5 dB was the quality improvement converted from budget by four datasets, while Richtmyer–Meshkov reached +3.6 dB.The improvement curves are compared over each dataset’s compression-useful range; absolute PSNR levels differ across fields.

F. Predicting Allocation Sufficiency from Field Structure … VII. CONCLUSION

The paper presents fixed-budget, structure-aware Gaussian encoding with deterministic primitive allocation and scalar-preserving representations, while identifying one-shot allocation as the main limitation when complexity is spatially concentrated. Results also show reproducible reconstruction quality across hardware and a controllable runtime–quality trade-off through reduced refinement iterations.

F. Predicting Allocation Sufficiency from Field Structure: 0.17 dB maximum and 0.08 dB mean differences across 21 configurations show effectively hardware- and implementation-independent reconstruction quality.The comparison used three gigavoxel datasets and nine budgets on an RTX 5090 against Table IV.
VI. DISCUSSION: At 300 iterations, 1.4-million-primitive models refined in 52.8 s for Miranda, 42.6 s for Chameleon, and 44.2 s for Richtmyer–Meshkov, with 1.3–2.7 dB lower quality than 1,500 iterations.Reduced iteration count therefore provides a direct runtime–quality trade-off.
A. Representation Behavior: Scalar attributes remain attached to individually addressable primitives, enabling primitive-level selection and post-hoc visualization changes without re-encoding.The scalar-field representation supports transfer-function, colormap, lighting, and viewpoint changes after encoding.
B. Limitations: One-shot allocation can under-refine narrow, high-complexity interfaces because capacity is committed before reconstruction error is available.Capacity experiments indicate allocation, rather than total budget or initial primitive size, is the likely bottleneck.
F. Predicting Allocation Sufficiency from Field Structure: Pre-training field-structure analysis identifies cases where additional capacity yields limited gains because the initial primitive distribution, rather than total budget, constrains refinement.This condition is associated with complexity concentrated in narrow interfacial regions.
VII. CONCLUSION: The method allocates the complete anisotropic Gaussian ensemble analytically before optimization, preserving an exact primitive count and deterministic encoded footprint.Primitive count and encoded size remain fixed specifications rather than optimization outcomes, supporting in situ data reduction.