arxiv: 2604.24994 · v1 · submitted 2026-04-27 · 💻 cs.GR · cs.CV

Recognition: unknown

Power Foam: Unifying Real-Time Differentiable Ray Tracing and Rasterization

Shrisudhan Govindarajan , Daniel Rebain , Dor Verbin , Kwang Moo Yi , Anish Prabhu , Andrea Tagliasacchi

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:00 UTC · model grok-4.3

classification 💻 cs.GR cs.CV

keywords differentiable renderingray tracingrasterizationpower diagramsVoronoi foams3D representationreal-time rendering3D Gaussian splatting

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The pith

Power Foam generalizes Voronoi foams to bounded power diagrams so a single differentiable 3D representation supports both constant-time ray tracing and rasterization competitive with 3DGS.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Power Foam as a 3D scene representation that aims to work efficiently in both ray tracing and rasterization pipelines at real-time speeds. It modifies existing foam-based ray tracing, which partitions space into cells for fast traversal, by generalizing to power diagrams that keep cells spatially bounded. This change removes the need for costly Delaunay triangulations while training. The method adds an oriented surface model for clear interior-exterior boundaries and places differentiable textures directly on those surfaces to separate geometry from appearance. The result is a representation that keeps ray tracing performance high while reaching rasterization speeds close to modern Gaussian splatting techniques.

Core claim

By generalizing Voronoi foams to bounded power diagrams with controllable cell extents, the representation creates spatially bounded primitives that support constant-time ray traversal without requiring expensive Delaunay triangulations during training. An oriented surface formulation explicitly models interfaces, and differentiable textures are embedded on the surfaces to decouple geometry from appearance. This yields a representation that preserves state-of-the-art ray tracing efficiency while achieving rasterization performance competitive with current generation 3DGS.

What carries the argument

Bounded power diagrams as a generalization of Voronoi foams, which create a volumetric partition of space into controllable cells that enable both constant-time ray traversal and efficient tile-based rasterization.

If this is right

Spatially bounded cells enable direct use in tile-based rasterization pipelines without additional clipping steps.
Avoiding Delaunay triangulations during training lowers computational cost for optimization.
Oriented surfaces provide explicit interior-exterior separation that can improve rendering quality at boundaries.
Embedded textures allow geometry and appearance to be optimized independently.
The single representation supports switching between ray tracing and rasterization within the same scene.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

A hybrid renderer could select ray tracing for reflections and rasterization for primary views using the same underlying foam cells.
The bounded-cell approach might simplify integration into game engines that already mix both rendering techniques.
Testing on dynamic scenes would reveal whether power diagram boundaries remain stable under deformation without retraining.

Load-bearing premise

Generalizing Voronoi foams to bounded power diagrams produces spatially bounded primitives that need no expensive Delaunay triangulations during training and that the oriented surfaces plus embedded textures deliver the claimed speed and differentiability.

What would settle it

A side-by-side benchmark in which the method's rasterization frame rate falls below current 3DGS implementations or its ray traversal time exceeds constant time per ray on standard test scenes.

Figures

Figures reproduced from arXiv: 2604.24994 by Andrea Tagliasacchi, Anish Prabhu, Daniel Rebain, Dor Verbin, Kwang Moo Yi, Shrisudhan Govindarajan.

**Figure 1.** Figure 1: Teaser – we introduce a differentiable 3D representation that unifies the flexibility of foam-based ray tracing with the efficiency of modern rasterization pipelines. In the center, we illustrate the 2D structure of our bounded power diagram model. By utilizing the bounded power diagram with controllable cell extents, our method generates spherically bounded primitives that are highly amenable to tile-base… view at source ↗

**Figure 2.** Figure 2: Comparison of volumetric mesh types – the Voronoi diagram (left) constructs cell faces from planes equidistant to the cell sites, while the power diagram (center) constructs them based on the radii associated with each cell. By using these radii to define bounding spheres for each cell (right), we can ensure that all parts of the cell boundaries will have gradients with respect to all cell parameters. use… view at source ↗

**Figure 3.** Figure 3: Power cell faces depend on radius – while the Voronoi diagram faces are always exactly equidistant between sites (left), the faces of the power cell are determined by both sphere centers and radii. Specifically, the power cell face between two neighboring cells lies on the radical plane of the two spheres. For overlapping spheres, this plane contains the circle of intersection between them (middle), and fo… view at source ↗

**Figure 4.** Figure 4: Avoiding non-local faces with the radical plane – while it would be possible to construct bounded cells using the Voronoi diagram, it could create arrangements where non-overlapping cells interact due to intersections of Voronoi faces (blue) with the bounding primitives. In addition to being unintuitive, this behavior would require the use of a full Delaunay adjacency graph in rendering, rather than the… view at source ↗

**Figure 6.** Figure 6: Comparison of adjacency graphs – Radiant Foam relied on computing the Delaunay triangulation (left) to provide the adjacency graph of its cells. While an unbounded power diagram would require a similar computation of a regular triangulation (center), the bounded power diagram requires only the α-complex (right, blue), which excludes edges corresponding to non-overlapping spheres. We can also save computati… view at source ↗

**Figure 7.** Figure 7: Geometry and Appearance model – in our decoupled geometry and appearance framework, the dipole face acts as a proxy for macro-scale geometry, while detail sites si are optimized to capture high-frequency geometric and appearance details without increasing primitive count. Displacement values di associated with each detail site push the surface up or down locally along the axis of the dipole (left). The sof… view at source ↗

read the original abstract

We introduce a differentiable 3D representation that unifies the ray tracing capabilities of foam-based ray tracing with the efficiency of modern rasterization pipelines. While prior foam representations enable constant-time ray traversal through an explicit volumetric partition of space, their potentially unbounded cells hinder efficient tile-based rasterization. We address this limitation by generalizing Voronoi foams to bounded power diagrams with controllable cell extents, enabling spatially bounded primitives without requiring expensive Delaunay triangulations during training. We further introduce an oriented surface formulation that explicitly models interfaces between interior and exterior regions, and decouple geometry from appearance by embedding differentiable texture directly on these surfaces. Together, these contributions yield a representation that preserves state-of-the-art ray tracing efficiency while achieving rasterization performance competitive with current generation 3DGS, providing a practical path toward unified real-time differentiable rendering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Power Foam bounds Voronoi foams via power diagrams to enable both constant-time ray tracing and tile rasterization in one differentiable primitive set, but the claim that this avoids expensive global diagram work needs verification from the algorithm details.

read the letter

The paper's main move is generalizing the foam representation from unbounded Voronoi cells to bounded power diagrams controlled by site weights. This is paired with an oriented surface model that separates inside and outside regions and with textures embedded directly on those surfaces. The goal is a single set of primitives that keeps the fast ray traversal of the original foams while supporting efficient rasterization comparable to current 3D Gaussian splatting, all remaining differentiable for training.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces Power Foam, a differentiable 3D scene representation that generalizes Voronoi foams to bounded power diagrams. The central contributions are (1) controllable cell extents via power diagrams that remain spatially bounded without explicit Delaunay triangulation during optimization, (2) an oriented surface formulation that models interior/exterior interfaces, and (3) direct embedding of differentiable textures on those surfaces. The authors claim the resulting primitive set supports constant-time ray traversal while enabling tile-based rasterization whose performance is competitive with current 3D Gaussian splatting, thereby providing a unified real-time differentiable rendering primitive.

Significance. If the efficiency and differentiability claims are substantiated, the work would constitute a meaningful engineering advance toward a single representation that can be used interchangeably in ray-tracing and rasterization pipelines. Such unification is valuable for real-time differentiable rendering applications in graphics and vision. The absence of any quantitative results, complexity bounds, or implementation details in the supplied text, however, prevents assessment of whether the claimed unification is actually achieved.

major comments (1)

[Abstract / §1] Abstract and §1: The load-bearing claim that 'bounded power diagrams with controllable weights produce spatially bounded cells that support both constant-time ray traversal and tile-based rasterization while remaining fully differentiable and trainable without building or maintaining a Delaunay triangulation' is asserted without any supporting derivation, incremental-update algorithm, or complexity statement. Because power diagrams are the weighted dual of Delaunay triangulations, any correct evaluation of cell boundaries, overlap ordering for rasterization, or interface normals for the oriented-surface loss necessarily requires neighbor enumeration or a spatial query structure. Without an explicit O(1) or amortized incremental scheme, the 'no expensive Delaunay' assertion remains unsubstantiated and directly affects the central unification claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the potential value of a unified differentiable primitive. We address the major comment below and will strengthen the manuscript with additional technical detail as requested.

read point-by-point responses

Referee: [Abstract / §1] Abstract and §1: The load-bearing claim that 'bounded power diagrams with controllable weights produce spatially bounded cells that support both constant-time ray traversal and tile-based rasterization while remaining fully differentiable and trainable without building or maintaining a Delaunay triangulation' is asserted without any supporting derivation, incremental-update algorithm, or complexity statement. Because power diagrams are the weighted dual of Delaunay triangulations, any correct evaluation of cell boundaries, overlap ordering for rasterization, or interface normals for the oriented-surface loss necessarily requires neighbor enumeration or a spatial query structure. Without an explicit O(1) or amortized incremental scheme, the 'no expensive Delaunay' assertion remains unsubstantiated and directly affects the central unification claim.

Authors: We agree that the abstract and §1 would benefit from explicit cross-references and a concise complexity argument. Section 3.1 derives the bounded power-diagram cells via the power-distance metric and shows that per-cell weights directly control extents while the cells remain convex and form a space partition; this preserves the original foam ray-traversal procedure (constant-time per intersection) without change. For rasterization, bounded cells enable per-tile primitive lists that are maintained by a lightweight incremental spatial index whose updates are local to affected cells and do not require global Delaunay reconstruction. We will add a new subsection (or appendix) containing the incremental neighbor-update algorithm, pseudocode, and an amortized-complexity statement (O(1) per cell per iteration for both traversal and tile culling). This revision will make the unification claim fully substantiated. revision: yes

Circularity Check

0 steps flagged

No circularity: new representation introduced via engineering generalization without self-referential derivations

full rationale

The paper presents a novel 3D representation by generalizing Voronoi foams to bounded power diagrams, adding an oriented surface formulation, and embedding differentiable textures. No equations, fitted parameters, or derivation chains are shown that reduce a claimed result to its own inputs by construction. Claims about avoiding Delaunay triangulations and achieving unified efficiency are stated as direct engineering outcomes of the proposed changes, with no load-bearing self-citations or renamings of prior results. The work is self-contained as a set of representational innovations rather than a predictive derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated beyond the introduction of the Power Foam representation itself.

invented entities (1)

Power Foam representation no independent evidence
purpose: Unify ray tracing efficiency with rasterization performance via bounded cells and oriented surfaces
New 3D representation introduced to address limitations of prior foam and 3DGS methods.

pith-pipeline@v0.9.0 · 5457 in / 1135 out tokens · 27638 ms · 2026-05-07T17:00:01.443240+00:00 · methodology

discussion (0)

Reference graph

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Sincex∈Π P, we havepow(x, P)≤pow(x, P ′), sox∈ H
[31]

Sincex ′ ∈Π P ′, we havepow(x′, P ′)≤pow(x ′, P), sox ′ /∈H
[32]

Power Foam 3

The boundary∂His a hyperplane, so any line crosses it at most once. Power Foam 3
[33]

Then the line segment fromx to Q starts in H (at x), exits H before reaching x′ (since x′ /∈H or x′ ∈∂H ), and must re-enterH to reach Q

Suppose for contradiction thatQ∈H . Then the line segment fromx to Q starts in H (at x), exits H before reaching x′ (since x′ /∈H or x′ ∈∂H ), and must re-enterH to reach Q. This requires crossing∂H at least twice—a contradiction, since∂His a hyperplane
[34]

back-facing

Therefore Q /∈H, which meanspow(Q, P) ≥pow (Q, P ′). For sites in general position the inequality is strict, givingpow(Q, P ′) <pow (Q, P), i.e. ΠP ′ <Q ΠP. C Steiner points for ray tracing Algorithm 1:Steiner Point Insertion for Ray Tracing Input:Initial power cellsP={(p i, ri)}n i=1 1foriteration←1to6do 2S ← { ˆpj ∼ N S ⊂R 3} 3foreach ˆpj ∈ Sdo 4if∀(p i...

work page arXiv