PaperlayerView exact revision on arXiv

Source-linked AI summary

Superquadrics Revisited: Learning 3D Shape Parsing beyond Cuboids

Despoina Paschalidou, Ali Osman Ulusoy, Andreas Geiger

arXiv:1904.09970v1cs.CV

TL;DR

The paper addresses the challenge of representing complex 3D shapes with compact, interpretable parts beyond limited cuboid abstractions. It learns superquadrics directly from 2D or 3D inputs without supervision and introduces a tractable analytical Chamfer loss. Across ShapeNet and SURREAL, the model produces consistent parses that capture structure and fine details, while future work targets global deformations and large-scale scenes.

  • Problem

    Existing 3D reconstruction methods are not sufficiently parsimonious or interpretable for higher-level scene understanding, while cuboids lack expressiveness for many shapes.

  • Method

    An unsupervised neural network predicts superquadric primitives with variable existence, using a linear-time closed-form Chamfer loss and a parsimony regularizer.

  • Results

    The model learns consistent superquadric parses on ShapeNet and SURREAL, converges faster than [37], and produces more accurate reconstructions while capturing structure and fine details.

  • Takeaways & Limitations

    Superquadrics provide expressive, compact part-based abstractions that capture details and variable shape complexity beyond cuboid representations.

  • Takeaways & Limitations

    The current model does not include global deformation parameters such as tapering and bending and is not yet extended to large-scale scenes.

Abstract

from arXiv · show

Abstracting complex 3D shapes with parsimonious part-based representations has been a long standing goal in computer vision. This paper presents a learning-based solution to this problem which goes beyond the traditional 3D cuboid representation by exploiting superquadrics as atomic elements. We demonstrate that superquadrics lead to more expressive 3D scene parses while being easier to learn than 3D cuboid representations. Moreover, we provide an analytical solution to the Chamfer loss which avoids the need for computational expensive reinforcement learning or iterative prediction. Our model learns to parse 3D objects into consistent superquadric representations without supervision. Results on various ShapeNet categories as well as the SURREAL human body dataset demonstrate the flexibility of our model in capturing fine details and complex poses that could not have been modelled using cuboids.

1. Introduction

The paper revisits compact 3D shape parsing with an unsupervised neural model based on superquadrics rather than cuboids. Superquadrics provide more expressive representations, while an analytical Chamfer loss enables efficient learning.

  • Compact part-based 3D representations could support recognition, detection, shape manipulation, and physical reasoning.
  • Cuboids capture overall structure but lack the expressiveness needed for many natural and man-made shapes.
  • Existing 2.5D, volumetric, mesh, and point-cloud reconstructions are not sufficiently parsimonious or interpretable for higher-level 3D scene understanding.
  • The proposed unsupervised neural network extracts parsimonious 3D representations conditioned on a 3D shape or 2D image.
  • Superquadrics represent diverse shapes in a continuous parameter space, enabling smooth variation, faster optimization, and more stable training.
  • The model uses a closed-form Chamfer distance evaluated in linear time, avoiding computationally expensive reinforcement learning and supporting standard backpropagation.

2. Related Work

Related work spans general 3D reconstruction, compact shape modeling, learning-based cuboid parsing, and classical superquadric fitting. The paper extends unsupervised cuboid abstraction with superquadrics and a tractable Chamfer-based learning objective.

  • 3D Reconstruction: Depth maps, volumetric grids, point sets, and meshes address different reconstruction limitations but generally lack parsimonious semantic interpretations.
  • Compact Shape Modeling: Constructive solid geometry methods learn sequences of Boolean operations or drawing commands over simple primitives.
  • Learning-based Scene Parsing: Recent learning-based scene parsers predict cuboid primitives and relationships, but supervision may require manual annotations or greedy heuristics.
  • Learning-based Scene Parsing: The proposed approach is unsupervised and avoids ambiguities from different prediction sequences producing the same cuboid assembly.
  • Superquadrics: The paper extends prior unsupervised cuboid abstraction with superquadrics, a linear-time analytical Chamfer solution, backpropagation, and a parsimony loss.
  • Superquadrics: Superquadrics provide a broader shape vocabulary than cuboids, including cubes, cylinders, spheres, octahedra, and ellipsoids.

3. Method

The model predicts variable-sized sets of superquadrics from 3D inputs and trains them without primitive annotations using bidirectional reconstruction and parsimony losses. An analytical Chamfer computation reduces the existence-variable expectation from exponential to linear evaluations, while superquadrics provide flexible shape representations.

  • The network maps an input and oriented target point cloud to M primitive parameters, each describing shape, size, position, and orientation.
  • Existence variables allow the model to predict a variable number of primitives from an upper bound M.Each primitive has an existence probability γm, and the final representation samples zm from a Bernoulli distribution.
  • Without primitive annotations, training combines bidirectional reconstruction with a Minimum Description Length prior that favors parsimonious parses.The reconstruction loss measures shape discrepancy, while the prior favors a small number of primitives.
  • 3.1. Reconstruction Loss: The bidirectional Chamfer objective measures both primitive-to-point-cloud and point-cloud-to-primitive distances.The two terms are weighted by 1.2 and 0.8, respectively, based on empirical performance.
  • 3.1. Reconstruction Loss: A closed-form computation reduces the point-cloud-to-primitive expectation from 2^M to M evaluations by sorting primitive distances and using closer-primitive existence probabilities.This enables standard backpropagation instead of computationally expensive reinforcement learning.
  • 3.2. Parsimony Loss: The existence-probability regularizer prevents the all-absent trivial solution and encourages sparse parses, with α = 1.0 and β = 10^-3.Its first term expects at least one active primitive, while the second uses a sub-linear penalty to encourage sparsity.
  • 3.3. Superquadric Parametrization: Superquadrics use size parameters α and shape parameters ϵ, with ϵ1 and ϵ2 bounded to [0.1, 1.9] to prevent non-convex shapes.A rigid-body motion parameter extends the canonical superquadric to arbitrary position and orientation.
  • For every tested primitive count, superquadrics achieve lower ShapeNet animal reconstruction loss than cuboids.The comparison uses the proposed model for both representations and averages three random initializations.

4. Experimental Evaluation

Experiments on ShapeNet and SURREAL show that unsupervised superquadric parsing produces consistent primitives, captures finer geometric detail, and trains with more stable, efficient gradients than the cuboid baseline.

  • Datasets and Baseline: The evaluation uses ShapeNet animals, aeroplanes, and chairs, plus SURREAL human meshes, comparing against Tulsiani et al.’s unsupervised cuboid parser.ShapeNet models use 32 × 32 × 32 voxel occupancy grids; SURREAL contains 4,500 training and 500 testing meshes.
  • Superquadrics vs. Cuboids: For any given number of primitives, superquadrics achieve lower fitting loss than cuboids when both are optimized with the proposed model.The comparison varies the maximum number of primitives and averages results over three random initializations.
  • Results on ShapeNet: With M = 20, superquadrics consistently capture object structure and fine details that the cuboidal predictions mainly represent as coarse structure.Examples include animal bodies, tails, and heads across ShapeNet categories.
  • Results on ShapeNet: The model captures complex parts such as an open dog mouth and allocates different numbers of primitives according to input-shape complexity.It uses six primitives for one airplane and more for a more complicated jetfighter, while distinguishing vehicle types through their parts.
  • Training Evolution: During training, predictions progress from blob-shaped primitives modeling overall structure to superquadrics that closely match unknown object parts and finer details.Initially, ε1 and ε2 are close to 1.0; later iterations attend to details.
  • Results on SURREAL: On SURREAL, the model represents feet, heads, and bodies with varied superquadrics while consistently assigning primitives to semantic body parts across poses.Correspondences can be mirrored for more complicated poses, which the authors speculate is caused by body symmetries.
  • Analytical Loss Formulation: The analytical loss yields orders-of-magnitude lower gradient variance, lower per-iteration runtime, and smoother convergence than the sampling-based baseline.The method avoids sampling-based gradient approximation, while increasing sample counts raise Monte Carlo runtime linearly.

5. Conclusion and Future Work

The paper concludes that unsupervised superquadric parsing captures both structure and detail with consistent part representations, while identifying global deformations and large-scale scenes as future extensions.

  • Conclusion: The proposed model learns consistent superquadric representations from data without primitive supervision and captures both target-object structure and details.The conclusion describes this as the first learning-based approach for parsing 3D objects into consistent superquadric representations.
  • Future Work: Future work will add global deformation parameters such as tapering and bending and extend the approach to large-scale scenes.The authors identify hierarchical strategies as important for unsupervised parsing at room-, building-, and city-level scales.

Supplementary Material for Superquadrics Revisited: Learning 3D Shape Parsing beyond Cuboids

The supplementary material provides author and affiliation information for the paper.

  • Authors: The listed authors are Despoina Paschalidou, Ali Osman Ulusoy, and Andreas Geiger.The passage also includes a generic author email format.

Abstract

The supplementary material derives the analytical Chamfer-loss solution and adds qualitative results for complex ShapeNet objects, SURREAL bodies, and RGB-image inputs.

  • Supplementary Contributions: The supplement presents a detailed derivation of the analytical Chamfer loss, avoiding computationally expensive reinforcement learning or iterative prediction.It also reports additional qualitative results on cars, motorbikes, and SURREAL human bodies.
  • Superquadric Representation: Superquadrics combine simple continuous parameterization with a diverse shape vocabulary suitable for geometric primitives and deep learning.They are modeled using 11 parameters, with α controlling size and ε controlling global shape.

B. Derivation of Pointcloud-to-Primitive Loss

The pointcloud-to-primitive loss measures how well target points are explained by selected primitive surfaces. Although direct computation is exponential in the number of primitives, sorting point-to-primitive distances yields a linear-time analytical evaluation.

  • Loss construction: The target point cloud is represented as sampled points, while each superquadric surface is approximated by sampled 3D points.The primitive transformation maps points between world and primitive-local coordinates.
  • Loss construction: The pointcloud-to-primitive distance measures each target point’s minimum distance to an existing primitive surface.This ensures every observation is explained by at least one selected primitive.
  • Efficient evaluation: Naïve evaluation has exponential complexity, requiring 2^M evaluations for M primitives.The exponential cost comes from summing over all combinations of binary existence variables.
  • Efficient evaluation: Sorting primitive distances in ascending order identifies the closest existing primitive through the first active primitive in the ordering.This ordering enables the expectation over existence variables to be simplified iteratively.
  • Efficient evaluation: The iterative simplification produces the analytical pointcloud-to-primitive distance used by the model.The formulation assumes at least one primitive exists, but a virtual primitive can relax this assumption.

C. Qualitative Results on SURREAL

On SURREAL humans, the model captures body parts across varied shapes, poses, and articulations while often maintaining semantic primitive correspondences. Some correspondences are lost when the viewing orientation is ambiguous.

  • Human body parsing: The model accurately captures diverse human body parts using superquadric surfaces across varied poses and articulations.The qualitative results cover humans with different body shapes and articulations.
  • Semantic correspondences: The same primitive often represents the same semantic body part across instances, such as the head using a red primitive.For some poses, these correspondences are lost, possibly because front-facing and back-facing humans are ambiguous to the network.

D. Qualitative Results on ShapeNet

On ShapeNet objects, superquadrics provide consistent part correspondences and capture details across animals, chairs, motorbikes, and cars. Their shape vocabulary supports distinctions among object types that cuboids cannot represent as effectively.

  • Vehicles: Superquadrics capture complicated motorbike and car structures, including indicative parts such as wheels and front forks.Motorbike wheels are consistently represented using flattened ellipsoids.
  • Category distinctions: Using up to 20 primitives per object, the model differentiates vehicle types such as scooters, racing bikes, choppers, sedans, convertibles, and coupes.The same expressive abstractions also distinguish animal categories such as dogs and cats.
  • Category distinctions: The learned primitive correspondences remain consistent across object parts while representing rounded surfaces that cuboids cannot capture.This supports distinctions among different vehicle types and detailed parts across ShapeNet categories.
  • Animals and chairs: The model captures structure and fine details consistently for ShapeNet animals and chairs.Rounded chair legs are represented with flattened ellipsoids, which cuboids alone could not model.

E. Network Architecture Details

The architecture encodes an input into a low-dimensional representation and uses separate regressors to predict superquadric parameters and existence probabilities. Image inputs use a ResNet18 encoder, while the loss and primitive heads remain shared in structure.

  • Architecture: The network combines an encoder with five regressors that predict size, shape, translation, rotation, and primitive-existence probability.The predicted parameters are α, ϵ, t, q, and γ.
  • Architecture: The volume-based architecture operates on 3D volumes and K-dimensional features, with M denoting the maximum number of predicted primitives.The figure’s cubes represent volume operations and rectangles represent feature layers.
  • Image-based variant: For RGB-image inputs, the encoder is replaced by ResNet18 without its final fully connected layer.Five independent heads then regress translation, rotation, size, shape, and existence probability for each primitive.
  • Loss design: The parsimony loss avoids the zero-primitive trivial solution while penalizing overly large expected primitive counts.Typical settings are α = 1.0 and β = 10^-3, with β smaller than α to permit multiple primitives.
  • Evaluation: The quantitative comparison reports mean Chamfer distance and mean Volumetric IoU, with cuboids treated as a special case of superquadrics.The comparison therefore expects cuboid fitting to perform worse than superquadric fitting.

H. Empirical Analysis of Reconstruction Loss

The section empirically compares the proposed Chamfer-based reconstruction loss with Tulsiani et al.’s truncated bi-directional loss. Direct primitive-fitting experiments visualize parameter evolution and indicate that the truncated loss is more prone to local-minimum convergence.

  • Experimental setup: The proposed Chamfer-based reconstruction loss is evaluated for training stability against Tulsiani et al.’s truncated bi-directional loss.The comparison is motivated by the claim that Chamfer-based reconstruction produces more stable training.
  • Experimental setup: The experiment directly fits primitive parameters rather than optimizing a neural network’s weights.It uses a 2D toy example and compares the proposed loss with Tulsiani et al.’s truncated distance formulation.
  • Findings: The truncated loss is more likely to converge to local minima during direct primitive optimization.The parameter trajectories are visualized as training progresses in the 2D toy experiment.
  • Findings: Figure 20 shows the evolution of two cuboid abstractions under the Chamfer and truncated bi-directional reconstruction losses across iterations 0–140.The panels report snapshots at iterations 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, and 140.
Loading 1904.09970v1…