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UCSG-Net -- Unsupervised Discovering of Constructive Solid Geometry Tree

Kacper Kania, Maciej Zięba, Tomasz Kajdanowicz

arXiv:2006.09102v3cs.CVcs.LG

TL;DR

Existing CSG-tree methods require parse-tree supervision, while UCSG-NET learns tree structure and primitive composition without it. The authors report state-of-the-art 2D reconstruction compared with supervised CSG-NET, and describe inferred trees as interpretable and useful for CAD prototyping.

  • Problem

    Existing interpretable CSG approaches require an assigned parse tree for each training example, making their supervision costly.

    Source: §1 · p. 1

  • Method

    UCSG-NET predicts primitive parameters and their CSG composition, converting SDF primitives to occupancy values and dynamically selecting Boolean operations.

    Source: §2 · p. 2, §2.1 · p. 4

  • Results

    The authors report state-of-the-art 2D reconstruction compared with supervised CSG-NET and reasonable reconstruction accuracy with inferred CSG trees.

    Source: §1 · p. 2, §5 · p. 8

  • Takeaways & Limitations

    Inferred trees can be visualized step by step and used for quick prototyping of 3D objects in CAD applications.

    Source: §5 · p. 8, Broader Impact · p. 9

    The authors recommend using the method mainly as a prototyping device because reconstructed CSG trees can produce erroneous 3D-printed elements.

    Source: Broader Impact · p. 9

Abstract

from arXiv · show

Signed distance field (SDF) is a prominent implicit representation of 3D meshes. Methods that are based on such representation achieved state-of-the-art 3D shape reconstruction quality. However, these methods struggle to reconstruct non-convex shapes. One remedy is to incorporate a constructive solid geometry framework (CSG) that represents a shape as a decomposition into primitives. It allows to embody a 3D shape of high complexity and non-convexity with a simple tree representation of Boolean operations. Nevertheless, existing approaches are supervised and require the entire CSG parse tree that is given upfront during the training process. On the contrary, we propose a model that extracts a CSG parse tree without any supervision - UCSG-Net. Our model predicts parameters of primitives and binarizes their SDF representation through differentiable indicator function. It is achieved jointly with discovering the structure of a Boolean operators tree. The model selects dynamically which operator combination over primitives leads to the reconstruction of high fidelity. We evaluate our method on 2D and 3D autoencoding tasks. We show that the predicted parse tree representation is interpretable and can be used in CAD software.

1 Introduction

Interpretable CSG trees represent shape construction, but existing CSG approaches require costly tree supervision, while convex decompositions can need many primitives for concave shapes. UCSG-NET learns CSG trees without supervision and introduces CSG layers for explainable Boolean operations.

1 Introduction

  • Existing CSG-tree methods require an assigned parse tree for each training example, while convex decompositions can require many primitives for concave shapes.The paper also notes that convex-decomposition decoding is difficult to explain and modify with external expert knowledge.Source: §1 · p. 1
  • UCSG-NET learns CSG parse trees without supervision by introducing CSG layers that learn explainable Boolean operations between primitives.These layers form an interpretable network of geometric operations for constructing complex shapes from a limited number of simple primitives.Source: §1 · p. 2
  • The method is reported to achieve state-of-the-art 2D reconstruction results compared with supervised CSG-NET, with interpretable predictions that can aid CAD applications.The paper also proposes a novel formulation of CSG operations for occupancy-value representations in 2D and 3D.Source: §1 · p. 2

2 Method

UCSG-NET predicts primitive parameters and composes primitives into a CSG tree without tree supervision. It uses occupancy-valued Boolean operations and a two-stage training procedure to produce interpretable reconstructions.

2 Method

  • UCSG-NET predicts primitive parameters and their CSG composition end-to-end, learning parse trees without supervision for mesh reconstruction.The predicted tree and primitives can be passed to external rendering software.Source: §2 · p. 2

2.1 Constructive Solid Geometry Network

  • Primitive parameters define analytic SDFs, which a learnable α-clipping function converts into occupancy values for Boolean operations.As α approaches zero, occupancy approaches binary values; gradual learning initially distributes gradients across shapes.Source: §2.1 · p. 3
  • Each CSG layer selects left and right operands from input shapes using latent-code queries and Gumbel-Softmax, with temperature annealed toward single-shape choices.The layer applies Boolean operations to selected operands, allowing the model to choose a composition during learning.Source: §2.1 · pp. 3–4
  • When α and layer temperatures approach zero, selected operands become one-hot and the occupancy operations correspond to explicit mesh CSG operations, enabling tree retrieval and pruning.The final layer outputs a union, which is described as non-empty in most cases.Source: §2.1 · pp. 4–5
  • Layers pass prior outputs alongside the original primitives, allowing primitives to remain available for later compositions such as nested rings.The latent code is also updated using information about what remains to reconstruct.Source: §2.1 · p. 4

2.2 Training

  • Training first minimizes occupancy reconstruction error and regularizes primitive translations and α, then lowers layer temperatures after α ≤ 0.05 to encourage one-hot CSG relations.The second stage targets interpretable relations by making operand masks resemble one-hot selections.Source: §2.2 · p. 5

3 Related Works

Prior work spans explicit mesh and implicit shape representations, interpretable primitive models, and neural CSG methods. The cited CSG parse-tree approaches rely on supervision, while other decompositions impose fixed structures or use convex primitives.

3 Related Works

  • Mesh reconstruction methods include genus-0 refinement and sequential vertex prediction, while Mesh-RCNN first predicts voxels before mesh conversion.Implicit methods also face representation-specific conversion or complexity issues: voxels can incur cubic prediction complexity, and point clouds lack vertex connectivity.Source: §3 · p. 6
  • DeepSDF and DualSDF use variational autodecoders, while OccNet and IM-NET predict inside-versus-outside occupancy.BSP-NET and CVXNET decompose shapes into unions of convexes; their fixed structure and low interpretability are noted as limitations for CAD use.Source: §3 · p. 6
  • Visual Primitives and Superquadrics use learnable unions of defined primitives and offer interpretable results, but Superquadrics can become unstable outside closed parameter domains.Source: §3 · p. 6
  • Neural CSG approaches such as CSG-NET and DeepPrimitive integrate parse-tree discovery with networks but rely heavily on supervision at each tree step.CSG-NET outputs a program with a defined rendering grammar.Source: §3 · p. 6

4 Experiments

UCSG-NET is evaluated on 2D CAD reconstruction and 3D ShapeNet autoencoding. It outperforms CSG-NET variants on the 2D task, while reference approaches outperform it on 3D Chamfer Distance; its reconstructions and parse trees expose how shapes are composed.

4 Experiments

  • The experiments compare 2D reconstructions with CSG-NET and 3D reconstructions with VP, SQ, BAE, and BSP-NET.The 3D task uses 5 CSG layers and 64 sphere and box parameters.Source: §4 · p. 6, §4.2 · pp. 7–8

4.1 2D Reconstruction

  • Figure 4 shows the primitives used for CAD reconstructions alongside their resulting CSG parse trees; rectangles are often combined by overlapping unions.The paper attributes this rectangle-heavy pattern to the dataset and says overlapping shapes can pass a primitive through unchanged.Source: Figure 4 · p. 7, §4.1 · p. 7
  • UCSG-NET is significantly better than the best CSG-NET variants without output refinement on the 2D CAD reconstruction task.The comparison uses the same validation split and Chamfer Distance; the method uses k = 1 and i = 0.Source: Table 1 · p. 7, §4.1 · p. 7

4.2 3D Autoencoding

  • Reference approaches outperformed UCSG-NET on 3D Chamfer Distance, partly because missed details such as airplane engines are metric-sensitive.The 3D evaluation uses 4096 points sampled from reconstructed surfaces.Source: Table 2 · p. 8, §4.2 · p. 8
  • On 3D autoencoding, UCSG-NET accurately reconstructs main shape components, with hulls and wings represented as separate semantic parts before a final union.The evaluation trains on 643 voxelized ShapeNet volumes and presents surface reconstructions with ground truth above predictions.Source: §4.2 · pp. 7–8, Figure 5 · p. 8

5 Conclusions

UCSG-NET discovers CSG parse trees without supervision and composes primitives with Boolean operations to reconstruct shapes. The trees can be visualized step by step and used for quick CAD prototyping; future work includes weak supervision and changes to primitive initialization and layer design.

5 Conclusions

  • UCSG-NET discovers CSG parse trees without supervision, composes primitives with Boolean operations, and forms meshes directly from inferred trees.The trees can be visualized step by step, supporting interpretability and quick prototyping of 3D objects in CAD applications.Source: §5 · p. 8
  • Figure 6 illustrates an airplane parse tree; the model approximates airplanes with similar CSG trees.The conclusion describes the inferred trees as interpretable reconstructions that can be visualized step by step.Source: §5 · p. 8, Figure 6 · p. 9
  • Future directions include weak supervision, RANSAC-based primitive initialization, Set Transformer layers, and regularization to increase tree diversity.The authors note that many CSG trees can reconstruct the same object and the solution space is vast; weak supervision may improve final results.Source: §5 · pp. 8–9

Broader Impact

The authors position recovered CSG trees as a way to bring sculpted objects into CAD prototyping, while cautioning against treating reconstructions as production-ready. They also identify tree non-uniqueness as an unresolved research constraint.

Broader Impact

  • UCSG-NET can approximate a sculpted object with primitives and operations, integrate the reconstruction into existing CAD models, and speed up 3D modeling prototyping.The authors describe retrieving a CSG parse tree for an object of interest.Source: Broader Impact · p. 9
  • Use the method mainly as a prototyping device: inexperienced CAD users may rely heavily on assumptions, and reconstructed prints may be erroneous enough to break an item.Source: Broader Impact · p. 9
  • A single object can have infinitely many CSG trees, motivating approximations that recover trees with short inference times.Source: Broader Impact · p. 9

A Mathematical Formulation of Primitives in the Signed Distance Field Representation

The section describes primitives through signed distance fields evaluated at points, focusing on simple shapes for 2D and 3D data. It also specifies the formulation conventions and assumptions used for these primitives.

A Mathematical Formulation of Primitives in the Signed Distance Field Representation

B Implementation details

The implementation uses convolutional encoders and an MLP decoder to predict primitive parameters, with parameter encoders and GRU units sized around a 256-dimensional latent code. Experiments specify the layer settings, optimizer, and task-specific CSG layer counts.

B Implementation details

  • The encoder uses convolutional layers that halve spatial dimensions, while an MLP decoder with leaky ReLU outputs primitive parameters from a latent code of size dz=256.The parameter encoder h(l) has one hidden fully connected layer of size dz, and GRU units also use latent dimension dz.Source: B Implementation details · p. 12
  • In 2D and 3D autoencoding tasks, experiments use Adam with learning rate 10^-4, beta parameters (0.5, 0.99), batch size 16, and 2 CSG layers in 2D versus 5 in 3D.Learning starts with initial values α=1 and τ(l)=2.Source: B Implementation details · p. 12
  • Each convolution uses a 4×4 kernel, stride 2, leaky ReLU slope 0.01, and a bias term.The architecture table summarizes the encoder, decoder, and h(l) layers.Source: Table 4 · p. 12

C Constructive Solid Geometry Tree Visualizations

The visualizations show reconstructed objects as CSG trees and indicate that training learns tree structures. They also show similar structures for semantically similar objects.

C Constructive Solid Geometry Tree Visualizations

  • The reconstructed-object visualizations display CSG trees using union, intersection, and difference operations, with similar structures for chair and table legs.Source: Figure 7 · p. 13
  • Training with a new set of weights from scratch learns CSG structures again.Source: Figure 8 · p. 14

D Diversity of CSG structures - Discussion

UCSG-NET initially exhibits diverse CSG structures during training, but converges to a single structure after the second stage; enforcing diversity across samples remains future work.

D Diversity of CSG structures - Discussion

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