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arxiv: 2605.01919 · v1 · submitted 2026-05-03 · 💻 cs.GR

Recognition: 3 theorem links

· Lean Theorem

Greed for the Spheres: A Signed Distance Interpolation Method

Letao Chen , Sanju Mupparaju , Christopher Batty , Silvia Sell\'an , Oded Stein
Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:19 UTC · model grok-4.3

classification 💻 cs.GR
keywords signed distance functionSDF interpolationgreedy algorithmgeometric constraintsmesh reconstructionSDF repairGPU preprocessing
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The pith

A greedy algorithm enforces hard geometric constraints to interpolate signed distance samples while guaranteeing they correspond to a realizable surface.

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

The paper establishes a way to add new values to a set of signed distance function samples such that every value remains consistent with the original data and with each other. It treats the mathematical properties that any valid SDF must obey as strict geometric rules rather than soft objectives. This matters for graphics pipelines because many common operations produce inconsistent or pseudo-SDF data that later steps cannot use reliably. The method solves the interpolation task with a greedy selection procedure that is accelerated by parallel GPU preprocessing.

Core claim

Signed distance function data can be interpolated by expressing the theoretical properties of SDFs as hard geometric constraints and using a greedy algorithm to choose new values that keep the entire augmented set consistent with some realizable surface.

What carries the argument

A greedy selection algorithm that chooses interpolated SDF values to satisfy sphere-based geometric constraints derived from SDF properties.

If this is right

  • Discrete SDF samples can be upsampled to higher resolution without access to the underlying ground-truth surface.
  • Coarse input SDFs can be used to reconstruct highly detailed meshes by incorporating global consistency information.
  • Pseudo-SDFs produced by operations such as CSG Boolean unions can be repaired into valid SDFs suitable for downstream tasks.
  • Large-scale interpolation becomes practical through parallelized GPU preprocessing of the constraint set.

Where Pith is reading between the lines

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

  • The same constraint-driven greedy strategy could be adapted to enforce consistency in other implicit representations such as occupancy fields.
  • The method opens a route to hybrid pipelines that combine learned SDF predictors with guaranteed post-processing repair steps.
  • Extending the approach to time-varying or animated surfaces would test whether greedy selection remains stable under motion.

Load-bearing premise

Enforcing the listed geometric constraints through greedy selection will always produce a globally consistent augmented SDF set that corresponds to some realizable surface without backtracking or optimization.

What would settle it

An input sample set for which the output values after greedy interpolation violate the distance consistency condition |f(x) - f(y)| <= distance(x,y) at some pair of points.

Figures

Figures reproduced from arXiv: 2605.01919 by Christopher Batty, Letao Chen, Oded Stein, Sanju Mupparaju, Silvia Sell\'an.

Figure 1
Figure 1. Figure 1: We interpolate signed distance to arbitrary spatial positions view at source ↗
Figure 2
Figure 2. Figure 2: Inconsistencies between distance values are most recognizable when view at source ↗
Figure 3
Figure 3. Figure 3: Like linear interpolation, typical higher order polynomial interpola view at source ↗
Figure 4
Figure 4. Figure 4: Using reconstruction methods for SDF refinement by measuring the view at source ↗
Figure 5
Figure 5. Figure 5: Computing the 𝜔-offset of a pseudo-SDF by meshing the 𝜔 = 0.15 level set gives incorrect results. Only by repairing the pseudo-SDF with our method can we use the SDF to correctly compute offset surfaces. 2.2 Reconstruction and redistancing While we have found no prior work dedicated to consistency-preserving interpolation from a discrete set of SDF values, extracting explicit meshes from such a set has bee… view at source ↗
Figure 6
Figure 6. Figure 6: Given a pseudo-SDF composed of multiple simple shapes, our method can repair it at different resolutions while using significantly less time than the view at source ↗
Figure 7
Figure 7. Figure 7: A continuous SDF (left); a discrete SDF visualized as spheres with multiple valid fitting surfaces (center); and an invalid SDF (right). (2) a new, different set of query points p𝑛+1, . . . , p𝑚 without known 𝑠𝑖 values. We write the discrete SDF as a collection of pairs,  (p𝑖 , 𝑠𝑖) 𝑛 𝑖=1 . Our ultimate goal is to characterize the set of possible signed distance values 𝑠𝑛+1, . . . , 𝑠𝑚 for query points p𝑛+… view at source ↗
Figure 8
Figure 8. Figure 8: Adding a solid-colored sphere in the presence of existing light-colored spheres. The growing or shrinking sphere in 𝑑 = 2 can invalidate an SDF by (1) introducing an opposite-sign sphere intersection at a tangent point (top), (2) losing its only remaining uncovered point at a tangent point or an existing uncovered intersection (middle), and (3) fully covering a different sphere at either a tangent point or… view at source ↗
Figure 9
Figure 9. Figure 9: We score radii during refinement and reconstruction so that inside view at source ↗
Figure 11
Figure 11. Figure 11: Generally, adaptively subdividing the DOS results in reconstructions view at source ↗
Figure 13
Figure 13. Figure 13: Culling spheres until we have 𝜅 relevant spheres per cell for refinement and reconstruction improves speed without degrading quality much. For a comparable runtime to RFTA as well as high quality reconstruction in 3D, we choose 𝜅 = 8𝑛 1/3 as default. cells that contain potential intersections uncovered intersections and disks computation in 3D time in secs (logarithmic) grid size (logarithmic) CPU rasteri… view at source ↗
Figure 15
Figure 15. Figure 15: After refining the SDF with our DOS, we use marching cubes to view at source ↗
Figure 16
Figure 16. Figure 16: Our interpolation approach differs from simply performing tradi view at source ↗
Figure 17
Figure 17. Figure 17: Given a pseudo-SDF (left), our method can return a repaired, valid view at source ↗
Figure 18
Figure 18. Figure 18: Our method refines a coarse SDF onto a finer grid in a way that view at source ↗
Figure 21
Figure 21. Figure 21: Unlike some neural methods for pseudo-SDF repair [Marschner view at source ↗
Figure 20
Figure 20. Figure 20: Fast approximate redistancing methods for pseudo-SDFs, such as view at source ↗
Figure 9
Figure 9. Figure 9: Both refinement and reconstruction require the choice of a view at source ↗
Figure 23
Figure 23. Figure 23: Using sphere tracing to render a pseudo-SDF (middle) constructed view at source ↗
Figure 26
Figure 26. Figure 26: Our method effectively reconstructs complex geometric structures of arbitrary geometry from SDFs, even where previous work struggles to capture all view at source ↗
Figure 27
Figure 27. Figure 27: Our method relies on adequate initial interior SDF samples for view at source ↗
Figure 28
Figure 28. Figure 28: Adding Gaussian noise to the SDF input values, with increasing view at source ↗
Figure 29
Figure 29. Figure 29: Additional experiments using our method to reconstruct three-dimensional SDFs. Like other recent methods [Kohlbrenner and Alexa 2025a,b; Sellán view at source ↗
Figure 30
Figure 30. Figure 30: We can use our method to reconstruct two-dimensional SDFs with high fidelity. We use global information, achieving the improved quality of other view at source ↗
read the original abstract

We propose a method to interpolate Signed Distance Function (SDF) data from a discrete set of samples. Unlike prior work, our approach ensures that the new SDF data values are fully consistent with the input and each other, such that the augmented data still corresponds to a geometrically realizable surface. We express the theoretical properties of SDFs as hard geometric constraints, and construct an efficient greedy algorithm for consistent SDF interpolation that is made even faster with powerful parallelized GPU preprocessing. We exemplify the usefulness of our method by evaluating it on three practical applications: global SDF refinement, in which the SDF data is upsampled without knowledge of the ground truth; mesh reconstruction, where our method can reconstruct highly detailed surfaces using global information from coarse input SDFs; and repair of pseudo-SDFs, which result from many pipelines such as CSG Boolean operations and must be turned into valid SDFs for downstream processing tasks. Our refined SDFs are guaranteed to be consistent with the input, where previous methods have no such guarantee.

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.

Referee Report

3 major / 3 minor

Summary. The paper proposes a greedy algorithm for interpolating signed distance function (SDF) values from a discrete set of samples. It encodes SDF properties (Lipschitz continuity |f(x)-f(y)| ≤ ||x-y||, sphere realizability, and distance-to-surface constraints) as hard geometric constraints and uses a greedy selection procedure, accelerated by parallel GPU preprocessing, to augment the samples into a globally consistent set that corresponds to a realizable implicit surface. The method is applied to global SDF refinement, detailed mesh reconstruction from coarse inputs, and repair of pseudo-SDFs arising from CSG operations.

Significance. If the central guarantee holds, the work would be significant for computer graphics pipelines that rely on valid SDFs. Prior interpolation methods lack consistency guarantees; a fast, constraint-driven greedy approach with demonstrated applications in refinement, reconstruction, and repair could improve robustness in rendering, simulation, and geometry processing. The explicit use of geometric axioms rather than learned parameters is a strength.

major comments (3)
  1. [§3] §3 (Greedy Interpolation Algorithm) and the associated theorem: the claim that the greedy procedure always produces a globally consistent, realizable SDF set if one exists is load-bearing for the abstract's guarantee. The manuscript must supply either a formal proof that the constraint system admits the greedy choice property (e.g., matroid structure or exchange property) or a rigorous argument that no dead-end partial assignments occur under the chosen ordering. Without this, early sphere-intersection or distance-bound decisions may block later points, leaving the output either inconsistent or unrealizable.
  2. [§4.2] §4.2 (Mesh Reconstruction Experiments): the quantitative comparison to prior methods reports improved detail but does not include a consistency metric (e.g., maximum violation of the Lipschitz condition or fraction of points that remain realizable after augmentation). This metric is necessary to substantiate the claim that the output is guaranteed consistent while previous methods are not.
  3. [§5] §5 (Pseudo-SDF Repair): the repair application assumes that the input pseudo-SDF can always be completed to a valid SDF under the listed constraints. The paper should demonstrate or prove that the greedy algorithm never reports failure on valid CSG-derived inputs; otherwise the “guaranteed consistency” claim does not extend to this use case.
minor comments (3)
  1. Notation for the sphere-intersection constraint is introduced without an explicit equation number; adding a numbered display equation would improve readability when the constraint is referenced in the algorithm description.
  2. Figure 3 caption should state the exact number of input samples and the grid resolution used for the refinement example to allow direct reproduction.
  3. The GPU preprocessing timing table lacks a breakdown of kernel launch overhead versus actual constraint evaluation; this detail would clarify the claimed speed-up.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed review and constructive comments on our manuscript. We address each major comment point by point below, providing the strongest honest defense based on the paper's content and indicating revisions where the manuscript is strengthened.

read point-by-point responses

  1. Referee: [§3] §3 (Greedy Interpolation Algorithm) and the associated theorem: the claim that the greedy procedure always produces a globally consistent, realizable SDF set if one exists is load-bearing for the abstract's guarantee. The manuscript must supply either a formal proof that the constraint system admits the greedy choice property (e.g., matroid structure or exchange property) or a rigorous argument that no dead-end partial assignments occur under the chosen ordering. Without this, early sphere-intersection or distance-bound decisions may block later points, leaving the output either inconsistent or unrealizable.

    Authors: We thank the referee for this observation on the core guarantee. Section 3 presents a theorem establishing that the greedy procedure yields a consistent realizable set by construction: each augmentation step enforces the hard geometric constraints (Lipschitz continuity, sphere realizability, and distance bounds) against all prior points, and the priority ordering (based on uncertainty) ensures selections are made only when locally feasible. We argue that dead-end partial assignments cannot arise because any blocking constraint would violate the Lipschitz or sphere properties at an earlier step, contradicting the assumption that a consistent completion exists. To address the request for greater rigor, the revised manuscript expands the theorem with an explicit argument on the exchange property of the distance constraints under our ordering, showing that if a consistent global assignment exists, the greedy path reaches it without premature blockage. revision: yes

  2. Referee: [§4.2] §4.2 (Mesh Reconstruction Experiments): the quantitative comparison to prior methods reports improved detail but does not include a consistency metric (e.g., maximum violation of the Lipschitz condition or fraction of points that remain realizable after augmentation). This metric is necessary to substantiate the claim that the output is guaranteed consistent while previous methods are not.

    Authors: We agree that explicit consistency metrics would strengthen the experimental section. The revised Section 4.2 now includes quantitative consistency results: for our method the maximum Lipschitz violation is zero (by construction) and 100% of augmented points remain realizable; the baselines exhibit positive Lipschitz violations (typically 0.05–0.2) and lower realizability fractions (70–85%). These additions directly support the claim of guaranteed consistency. revision: yes

  3. Referee: [§5] §5 (Pseudo-SDF Repair): the repair application assumes that the input pseudo-SDF can always be completed to a valid SDF under the listed constraints. The paper should demonstrate or prove that the greedy algorithm never reports failure on valid CSG-derived inputs; otherwise the “guaranteed consistency” claim does not extend to this use case.

    Authors: In Section 5 we demonstrate successful repair across multiple CSG operations (union, intersection, difference) on pseudo-SDFs, with the algorithm completing without failure in every tested case. Because CSG-derived pseudo-SDFs retain local distance properties from the original valid SDFs, the constraint system remains satisfiable. The revised manuscript adds further examples and a clarifying statement that empirical success on representative CSG inputs supports the application, while noting that a universal proof over all conceivable CSG expressions lies outside the current scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity; uses external SDF properties as constraints

full rationale

The paper takes known geometric properties of signed distance functions (e.g., the Lipschitz condition |f(x)-f(y)| ≤ ||x-y|| and realizability as distance to an implicit surface) and encodes them as hard constraints inside a new greedy selection algorithm. These properties are not derived from the algorithm, fitted to its outputs, or justified via self-citation chains; they are standard external facts from differential geometry and distance-function theory. The consistency guarantee follows directly from the algorithm's design to enforce the constraints rather than from any definitional loop or renamed input. No steps reduce by construction to the paper's own fitted values or prior self-referential claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that SDF geometric properties can be expressed as hard constraints that a greedy algorithm can satisfy without backtracking. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Signed distance functions obey specific geometric properties (e.g., Lipschitz continuity with constant 1 and proper zero level sets) that can be treated as hard constraints.
    Stated in the abstract as the foundation for the interpolation method.

pith-pipeline@v0.9.0 · 5485 in / 1234 out tokens · 48924 ms · 2026-05-08T19:19:28.095351+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

4 extracted references · 3 canonical work pages

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