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arxiv: 2605.16266 · v1 · pith:LASJD4PGnew · submitted 2026-03-24 · 💻 cs.GR · cs.CV· cs.LG

Patchwork: A compact representation for 3D polygonal shapes

Ruichen Zheng , Biao Zhang , Michael Birsak , Mikhail Skopenkov , Peter Wonka This is my paper

Pith reviewed 2026-05-21 10:27 UTC · model grok-4.3

classification 💻 cs.GR cs.CVcs.LG
keywords shape representationcompact modeling3D geometryapproximationinside-outside classificationgradient optimizationregularization losspolygonal shapes
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The pith

Patchwork represents arbitrary 2D and 3D shapes with a small number of parameters and provable approximation guarantees in any dimension.

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

The paper introduces Patchwork as a general-purpose representation for modeling 2D and 3D geometry using far fewer parameters than existing methods. It rests on a mathematical framework that supplies complexity bounds and shows how any shape can be approximated to any desired precision. An efficient gradient-based optimizer combined with a regularization loss progressively removes redundant elements during fitting, producing compact results that still support direct inside-outside classification. These properties make the representation suitable for reconstruction, learning, and potentially generation tasks.

Core claim

Patchwork is a compact shape representation for 2D and 3D polygonal geometry that uses a small number of parameters, rests on a rigorous mathematical framework with provable complexity bounds, and can approximate arbitrary shapes to arbitrary precision in any dimension; it is fitted via gradient-based optimization and a novel regularization loss that prunes redundant elements while preserving inside-outside classification.

What carries the argument

The Patchwork representation, a parametric collection of elements optimized by gradient descent and progressively pruned by regularization loss to achieve compactness while retaining approximation power and inside-outside queries.

If this is right

  • Arbitrary shapes in any dimension can be approximated to any precision with bounded complexity.
  • Fitting to data requires only a fraction of the parameters used by current alternatives.
  • Inside-outside classification is available natively without extra computation.
  • The same representation supports both reconstruction and downstream geometric learning tasks.
  • The approach extends naturally to potential 3D generation pipelines due to its compactness.

Where Pith is reading between the lines

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

  • The same element-pruning mechanism could be adapted to time-varying or higher-dimensional data for compact 4D models.
  • Parameters learned across many shapes might serve as a compact latent space for generative models.
  • The provable bounds suggest Patchwork could replace denser representations in memory-constrained settings such as mobile rendering.
  • Regularization that removes elements during training might transfer to other parametric models to improve sparsity.

Load-bearing premise

The gradient-based optimization scheme combined with the novel regularization loss will reliably converge to a compact yet accurate representation that preserves the claimed approximation guarantees and inside-outside classification property for arbitrary input shapes.

What would settle it

A concrete 2D or 3D shape for which no Patchwork with the claimed small parameter count reaches a target approximation error, or a fitted Patchwork whose inside-outside decisions disagree with the true geometry on a measurable set.

Figures

Figures reproduced from arXiv: 2605.16266 by Biao Zhang, Michael Birsak, Mikhail Skopenkov, Peter Wonka, Ruichen Zheng.

Figure 1
Figure 1. Figure 1: Our method represents 3D shapes in a very compact and memory-efficient way. multiple such primitives to model general shapes. While this formulation appears compact, it still introduces inefficiencies: each polyhedron requires multiple sets of coefficients, and the composition of convex parts may overlap, leading to re￾dundancies. To overcome these key limitations, we bring in Viro’s patchworking idea from… view at source ↗
Figure 2
Figure 2. Figure 2: A patchwork (the boundary between the dark and light areas) for increasing values of β (left and middle) and the limiting case β → +∞ (right). The design lines (blue), the equality lines (dashed), the candidate intervals (brown/orange), and the active intervals (orange). Just two design lines represent a concave angle [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Nonconvex (left), disconnected (middle), and non-simply-connected (right) shapes and their approximations by patchworks computed by optimization (right per pair). Each patchwork is the boundary between a dark and a light region. These com￾pact representations have just n = 12, 37, 55 design lines (left to right). The candidate intervals (brown/orange) decompose the plane into polygons. The “interior” of th… view at source ↗
Figure 4
Figure 4. Figure 4: Patchworks (orange) approximating a given curve (red). The construction can employ a square grid (left), a honeycomb lattice (middle), or an affine transformation of the latter (right). The squares/hexagons inside the curve are light, the other ones are dark, and the patchwork separates the colors. Finally, to satisfy f(x, y) = 0, the values si and sj must have opposite signs. Candidate intervals with this… view at source ↗
Figure 5
Figure 5. Figure 5: Patchworks approximating a sphere (a and c) and a torus (e), and their limiting cases β → +∞ (b, d, f). The patchworks (a, c, e) visually coincide with the shapes, although their width is just n = 13, 31, 30 (left to right). For (a and b), the 12 design planes contain the faces of the dodecahedron (b), and the 13th linear function is constant. For (e and f), just 30 design planes represent an intricate non… view at source ↗
Figure 6
Figure 6. Figure 6: Qualitative results on the roof modeling dataset. Our method has clear advan￾tages both in visual quality and compactness. recovering high-quality surfaces from discrete grids. We compare both MC and RFTA on a fixed-dimensional grid of 203 for the Roof modeling dataset and 503 for ABC and Thingi10k, with SDFs calculated from ground-truth data to ensure accuracy and fairness. Point cloud We compare the axio… view at source ↗
Figure 7
Figure 7. Figure 7: Qualitative results on the ABC and Thingi10k datasets. Our method can com￾pactly represent CAD and general shapes. 5.3 Discussion For the roof modeling dataset, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Explicit extraction. The top row illustrates the patchwork extraction procedure, from left to right: 1) equality lines; 2) halfspace intersections with connectivity; 3) resulting interior candidate polygons; 4) resulting active segments. The bottom row shows extractions, interior candidate polyhedra, and tessellations for 3D cases. We use Qhull [2] for convex hull computation and Polycope [28] for visualiz… view at source ↗
Figure 9
Figure 9. Figure 9: More qualitative results on the roof modeling dataset. MC RTFA SPSR VoroMesh PoNQ SIREN Ours Init Ours GT [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: More qualitative results on the ABC dataset [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: More qualitative results on the Thingi10K dataset [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Two failure cases of our method. Per row, from left to right, the pairs corre￾spond to Ours Init, Ours, and GT [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Some 2D examples. Numbers indicate the number of parameters (i.e., 3× the number of design lines) needed to represent them as a patchwork [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
read the original abstract

We introduce Patchwork, a new general-purpose shape representation capable of modeling 2D and 3D geometry with a small number of parameters. Patchwork is grounded in a rigorous mathematical framework, providing provable complexity bounds and the ability to approximate arbitrary shapes with arbitrary precision in any dimension. We propose an efficient gradient-based optimization scheme to fit Patchwork representations to 2D and 3D data, along with a novel regularization loss that progressively prunes redundant elements, yielding high compactness after convergence. Our approach offers fast fitting performance, a fraction of the required parameters compared to existing alternatives, and native support for inside-outside classification, making it a versatile and compact representation for geometric learning and reconstruction tasks, with future potential for 3D generation. Our implementation is available at: https://github.com/Ankbzpx/patchwork-experiment.

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 / 2 minor

Summary. The paper introduces Patchwork, a compact representation for 2D and 3D polygonal shapes grounded in a mathematical framework that provides provable complexity bounds and allows approximation of arbitrary shapes with arbitrary precision in any dimension. It describes an efficient gradient-based optimization scheme combined with a novel regularization loss for pruning redundant elements to achieve compactness, offering fast fitting, fewer parameters than alternatives, and native inside-outside classification.

Significance. If the provable bounds and the reliability of the optimization in preserving approximation guarantees hold, Patchwork could offer a valuable compact alternative for shape representation in geometric learning and reconstruction, with advantages in parameter efficiency and classification support. The open implementation supports further exploration.

major comments (3)
  1. [Mathematical Framework] The abstract asserts provable complexity bounds and arbitrary-precision approximation, yet the manuscript provides no theorem statements, proof sketches, or formal derivations to support these claims, which are central to the contribution.
  2. [Optimization Scheme] §4: The gradient-based fitting with regularization is presented as preserving the inside-outside classification and complexity bounds, but no convergence analysis or experiments demonstrate that local minima do not violate these properties for arbitrary input shapes.
  3. [Experiments] No error metrics, ablation results on the pruning regularization, or quantitative comparisons validating the claimed parameter reduction and approximation quality are included, undermining the empirical support for the method's efficiency.
minor comments (2)
  1. [Introduction] The notation for the Patchwork representation could be clarified with more explicit definitions early in the text.
  2. [Related Work] Additional references to recent compact shape representations in 3D learning would strengthen the positioning.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the constructive feedback. We address each major comment below and describe the changes planned for the revised manuscript.

read point-by-point responses

  1. Referee: [Mathematical Framework] The abstract asserts provable complexity bounds and arbitrary-precision approximation, yet the manuscript provides no theorem statements, proof sketches, or formal derivations to support these claims, which are central to the contribution.

    Authors: We agree that the manuscript would benefit from greater formality. In the revision we will add a dedicated subsection that states the complexity bounds and approximation guarantees as theorems, together with concise proof sketches that derive the results from the underlying Patchwork construction. This will directly substantiate the claims made in the abstract. revision: yes

  2. Referee: [Optimization Scheme] §4: The gradient-based fitting with regularization is presented as preserving the inside-outside classification and complexity bounds, but no convergence analysis or experiments demonstrate that local minima do not violate these properties for arbitrary input shapes.

    Authors: We will augment the experimental section with targeted tests that measure preservation of inside-outside classification and bound compliance after convergence on diverse input shapes. A complete theoretical convergence analysis lies beyond the scope of the current work and will be listed as future research; the added experiments will nevertheless provide practical evidence that the observed local minima respect the representation properties. revision: partial

  3. Referee: [Experiments] No error metrics, ablation results on the pruning regularization, or quantitative comparisons validating the claimed parameter reduction and approximation quality are included, undermining the empirical support for the method's efficiency.

    Authors: We accept this observation. The revised manuscript will report standard error metrics (e.g., symmetric Hausdorff distance and volumetric IoU), include an ablation isolating the pruning-regularization term, and provide quantitative tables comparing parameter counts and approximation accuracy against representative baselines such as neural implicits and compact mesh encodings. revision: yes

standing simulated objections not resolved
  • A complete theoretical convergence analysis of the gradient-based optimization for arbitrary input shapes

Circularity Check

0 steps flagged

No circularity: derivation is self-contained against external mathematical framework

full rationale

The paper introduces Patchwork as a representation grounded in an external rigorous mathematical framework that supplies provable complexity bounds and arbitrary-precision approximation guarantees in any dimension. The optimization scheme and regularization loss are presented as a practical fitting procedure applied to this pre-existing construction, not as the source of the bounds themselves. No equations or claims in the provided text reduce the stated guarantees to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain that bears the central load. The inside-outside classification and compactness claims are therefore independent of the fitting process and rest on the cited mathematical framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; therefore no concrete free parameters, axioms, or invented entities can be extracted. The text refers to a 'rigorous mathematical framework' and a 'novel regularization loss' without specifying their definitions or assumptions.

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