SuperFlex: Deformable Superquadrics for Point Cloud Decomposition¶

Conference: ECCV2026
Paper: ECCV Paper
Project: https://superflex3d.github.io/
Area: 3D Vision
Keywords: point cloud decomposition, superquadrics, bending and tapering, volumetric supervision, occlusion completion

TL;DR¶

SuperFlex learns bendable and taperable superquadrics with joint volumetric and surface losses, then supervises occluded inputs with optimized complete decompositions, raising feed-forward ShapeNet IoU from SuperDec's 0.59 to 0.72 while retaining a compact representation of 5.64 primitives on average.

Background & Motivation¶

Point clouds and meshes can record detailed surfaces, but they do not inherently describe which geometric parts compose an object. Superquadrics offer an alternative: a small parameter set describes scale, shape, and pose, approximating a complex object with a collection of manipulable parts. This representation supports structural abstraction, but decomposition must jointly determine which points belong together and which shape fits each part. Per-object optimization repeatedly solves this coupled problem; SuperDec instead uses a set-prediction network to predict primitives and point-to-primitive assignments together. It establishes fast feed-forward decomposition, but its reliance on Chamfer distance can leave gaps between primitives or produce overly rounded edges.

Simply increasing the primitive count does not remove the limitations of the representation itself. The rigid superquadrics considered in the paper have convex, symmetric shape constraints, making a curved armrest or a gradually changing cross-section difficult to represent with one primitive. Splitting such parts into more pieces can approximate their surfaces, but weakens compactness and coherent structure. Moreover, small distances between sampled surface points do not guarantee that the union of primitives occupies the correct three-dimensional volume. The authors therefore modify both the representable shapes and the supervision, rather than merely replacing the point cloud encoder.

Real scans introduce another problem: the visible point set is not the complete object. Fitting only visible points does not directly teach a model to restore occluded parts and can even encourage missing geometry. The authors first obtain high-quality primitive decompositions of complete objects, then train occluded inputs to predict the same complete structures, turning geometric fitting into supervision for completion. This distinction matters: complete-cloud reconstruction by the base model, optional test-time optimization, and additional fine-tuning of the robust variant are separate settings. Core Idea: constrain deformable primitives through both volume and surface geometry, then use refined complete primitive sets as geometric supervision for partial point clouds.

Method¶

Overall Architecture¶

The input is a single object's point cloud; outputs are deformable superquadrics with existence probabilities and soft point-to-primitive assignments used during training. The base network performs "Deformable Set Prediction" and learns complete-object decomposition through "Joint Geometric Supervision"; "Active Primitive Refinement" is optional per-object optimization. For occlusions, "Matched Structural Completion" uses refined results to train a robust predictor that outputs complete primitive structure from partial inputs in a forward pass. The edge leading to completion denotes training supervision, not a requirement to obtain a complete object before every partial-cloud inference.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    Input["Complete Object Point Cloud"] --> Predict["Deformable Set Prediction"]
    Predict --> Joint["Joint Geometric Supervision"]
    Full["Complete Geometry Occupancy<br/>Training or Refinement Only"] -.-> Joint
    Joint --> Refine["Active Primitive Refinement"]
    Refine -->|Complete Decomposition Pseudo-labels| Robust["Matched Structural Completion"]
    Partial["Partial Object Point Cloud"] --> Robust
    Robust --> Output["Complete Primitive Structure"]

Key Designs¶

1. Deformable Set Prediction: explain curved parts and changing cross-sections with one primitive

The model follows the main SuperDec architecture: PVCNN extracts pointwise features, while a separate branch initializes a fixed number of superquadric queries. Queries exchange part information through self-attention, read point features through cross-attention, and pass through a regression head to predict geometry and existence probabilities. The segmentation head matches pointwise features against projected primitive queries and applies a per-point softmax to obtain soft primitive assignments. Thus, the network jointly learns partitioning and geometry rather than performing hard segmentation followed by independent fitting. Existence probabilities let a fixed-size candidate set represent objects of varying complexity without requiring every candidate to become an output part.

The rigid representation contains 3 scales, 2 shape parameters, and 6 pose degrees of freedom, counted as 11 parameters in the paper. SuperFlex adds x- and y-direction tapering that varies along the z-axis, plus bending curvature and orientation angles defined for each of the three axes. This adds 2 tapering and 6 bending parameters, giving 19 geometric parameters per primitive; existence probability is predicted separately. Tapering progressively narrows or widens a cross-section, while bending turns a straight profile into a circular arc, avoiding many rigid fragments for one continuous part. To evaluate a spatial point against a deformed primitive, the method applies the inverse rigid transform, inverse bending around z, x, and y in sequence, and inverse tapering to reach the canonical superquadric. This order comes from pages 4-5; the explicit deformation equations are corrupted in the cached extraction, so reconstructed analytic expressions are not presented as author equations.

2. Joint Geometric Supervision: constrain object volume, local surfaces, and decomposition redundancy together

Chamfer distance between sampled surfaces alone need not penalize a volumetric gap between neighboring primitives. The volumetric term samples spatial points in the enclosing unit cube, obtains per-primitive occupancy probabilities through a soft indicator of radial distance, and aggregates them into global occupancy. A differentiable IoU objective aligns predicted occupancy with complete-object ground-truth occupancy, directly supervising the primitive union rather than isolated surface points. Radial distance measures the distance to the surface along the line through the primitive center and query point; it is a convenient proxy, not an exact Euclidean SDF. Although the paper calls its local objective an SDF loss, it uses this radial distance and a stabilizing activation.

The local surface term penalizes distances from input points to individual primitives, weighted by predicted point-to-primitive assignments. An activation improves stability and robustness to outliers, but its detailed form is deferred to an appendix absent from the supplied cache. The volumetric term handles global occupancy and the surface term preserves detail; neither automatically eliminates multiple primitives explaining the same region. An overlap regularizer therefore penalizes locations where the sum of primitive soft occupancies exceeds 1. Sparsity regularization acts on the 0.5-norm of mean primitive assignment weights, while existence regularization derives targets by thresholding usage and trains the existence probabilities. Together, these constraints discourage the degenerate solution of treating additional overlapping volume as better reconstruction.

Following the textual definitions on pages 6-7, the base training objective is:

\[ \mathcal{L}=\lambda_{\mathrm{IoU}}\mathcal{L}_{\mathrm{IoU}}+\lambda_{\mathrm{SDF}}\mathcal{L}_{\mathrm{SDF}}+\mathcal{L}_{\mathrm{reg}}. \]

The regularization term contains overlap, sparsity, and existence constraints; although manual part labels are unnecessary, complete-geometry occupancy is still required. Consequently, "self-supervised decomposition" does not mean that the same training objective can be applied directly to arbitrary partial scans alone.

3. Active Primitive Refinement: optimize the geometric union after selecting the part count

Feed-forward predictions can be used directly; for greater fidelity, the method retains primitives with existence probability above 0.5 and optimizes their geometry. Refinement adjusts the pose, shape, and deformations of selected primitives rather than optimizing a changing candidate set. Instead of weighting distances by predicted assignments as in training, it treats primitives as one global shape and approximates the hard minimum of distance fields with LogSumExp. This provides smooth gradients near the surfaces without requiring the original pointwise segmentation weights to adjust primitives. The volumetric term also changes: a soft occupancy function is applied directly to the unified distance field, replacing the probabilistic aggregation used in feed-forward training.

Thus, "the same loss" means retaining volumetric and surface objectives, not keeping every computational detail identical. Experiments use Adam for 1000 iterations per object with an optimization temperature of 0.01. In Table 2, the average active primitive count remains 5.64, showing that improvement comes from fitting existing primitives rather than adding parts. The cost rises from 0.0082 s for a forward pass to a reported 5 s per object; refined accuracy and feed-forward latency must not be combined into one result. This stage also generates offline labels, supplying high-quality complete-object decompositions for subsequent robust training.

4. Matched Structural Completion: supervise occluded inputs with complete geometry without requiring parameter equality

The method first generates refined primitives for a complete object, then trains partially visible inputs to recover that set. However, axis permutations, sign flips, and other parameter symmetries can describe the same geometry, causing conflicting supervision under direct parameter regression. The authors formulate set prediction and use Hungarian matching to establish one-to-one correspondences between predicted and target primitives. After matching, Chamfer distance between sampled surfaces is computed only for existing target primitives, substituting geometric agreement for parameter equality. Here Chamfer distance supervises corresponding parts, which differs from relying on a whole-object surface distance alone to fit the base model.

Completion training retains an IoU term based on complete-object occupancy so that visible regions alone do not determine the global shape. An additional rigid geometric term compares matched primitive surfaces with bending and tapering disabled, supplying a prior on coarse part structure. It does not require the final output to remain rigid, but discourages complex deformations from compensating for incorrect coarse structure. Occlusion augmentation uses Hidden Point Removal for viewpoint-dependent visibility and random spherical masking for foreground occlusions. The authors additionally fine-tune on ASE partial point clouds for 100 epochs; this is target-domain adaptation and must be distinguished from ShapeNet occlusion augmentation alone.

Page 8 gives the completion objective as:

\[ \mathcal{L}_{\mathrm{sup}}=\mathcal{L}_{\mathrm{geom}}+\lambda_{\mathrm{IoU}}\mathcal{L}_{\mathrm{IoU}}+\mathcal{L}_{\mathrm{rigid}}. \]

Loss & Training¶

ShapeNet experiments use 13 categories and the train/validation/test splits of Choy et al., with objects pre-aligned to a canonical orientation. Farthest Point Sampling selects 4096 points per object; the network uses 16 candidate primitives, feature dimension 128, and 3 decoder layers. The base model starts from SuperDec's ShapeNet checkpoint and trains for 1000 epochs, so the gains are not all obtained from random initialization. Training uses 4 GPUs and batch size 32; the standard learning rate is \(3\times10^{-4}\) and robust fine-tuning uses \(1\times10^{-4}\). The occupancy temperature is 0.001; IoU and SDF weights are 0.2 and 3.2, while overlap, sparsity, and existence weights are 5, 1.26, and 0.01. The cache extracts the usage threshold as 24, incompatible with the earlier definition as an average probability; it is not presented here as a directly reproducible hyperparameter.

Key Experimental Results¶

Main Results¶

The following selection comes from Table 1, page 10: 13 ShapeNet categories, canonical orientation, and 4096 points per object; both SuperFlex rows are feed-forward results. Higher IoU and F-score are better; lower Chamfer L1/L2 and mean primitive counts are better. L1/L2 are displayed as raw values multiplied by \(10^2\), following the source table.

Method	Bending and Tapering	IoU	F-score	L1	L2	Mean Primitives	Runtime
Marching Primitives	No	0.56	0.18	2.08	0.10	24.10	163.41 s
SuperDec	No	0.59	0.28	1.77	0.050	5.79	0.0075 s
SuperFlex	No	0.70	0.35	1.75	0.049	6.04	0.0075 s
SuperFlex	Yes	0.72	0.37	1.54	0.043	5.64	0.0082 s

Rigid SuperFlex already raises IoU to 0.70, identifying joint supervision as an important source of improvement; deformations further raise it to 0.72. Table 2, page 11, reports refined IoU of 0.87, F-score of 0.49, L1 of 1.32, L2 of 0.036, and runtime of 5 s. The prose claims a 22% relative refinement gain, whereas the rounded table values 0.72 and 0.87 imply approximately 20.8%; the original table is preserved rather than silently reconciling the discrepancy.

Ablation Study¶

The following selection comes from Table 4, page 14; sparsity and existence constraints remain fixed while reconstruction and overlap terms are introduced incrementally. L1/L2 retain the paper's reported values. Overlap is the reported overlap percentage; its precise evaluation denominator is not specified in the cached main text.

Reconstruction and Overlap Configuration	IoU	F-score	L1	L2	Mean Primitives	Overlap
SuperDec reconstruction loss	0.59	0.29	1.75	0.050	5.78	4%
IoU only	0.74	0.40	1.76	0.124	5.94	32%
IoU + SDF	0.72	0.37	1.54	0.042	5.80	14%
IoU + SDF + overlap	0.72	0.37	1.54	0.043	5.64	8%

IoU-only supervision has the highest volumetric score but worse L2 and extensive overlap, so IoU rankings alone do not establish decomposition quality. Adding the surface term lowers L2 from 0.124 to 0.042; adding overlap regularization then reduces Overlap from 14% to 8% while keeping IoU at 0.72. The SuperDec-loss baseline in Table 4 does not exactly match the SuperDec row in Table 1; each set of source values is retained separately. Table 5, pages 14-15, additionally reports IoU of 0.81, 0.83, 0.84, and 0.85 for no deformation, tapering only, bending only, and both, when optimizing from undeformed feed-forward initializations. That deformation ablation evaluates optimization, so its 0.85 must not be reported as the feed-forward performance in Table 1.

Key Findings¶

The following selection comes from Table 3, page 13: inputs are partial object point clouds extracted from ASE depth maps and instance masks, and evaluation targets are complete ABO meshes. "Occlusion augmentation" means fine-tuning on ShapeNet; "ASE adaptation" adds ASE partial clouds. Only IoU, F-score, and mean primitive count are reproduced here.

SuperFlex Configuration	Occlusion Augmentation	ASE Adaptation	IoU	F-score	Mean Primitives
Base model	No	No	0.20	0.16	6.3
Robust variant	Yes	No	0.48	0.16	4.9
Robust variant with target-domain adaptation	Yes	Yes	0.54	0.18	4.3

Occlusion augmentation increases IoU from 0.20 to 0.48 while F-score remains 0.16, suggesting that recovering overall occupancy does not imply simultaneous improvement in local surface detail. Further adaptation raises IoU to 0.54; this result includes target-domain training and is not an unadapted zero-shot result on real scenes. ScanNet++ real-scan demonstrations are qualitative results in Figure 7 and use ground-truth instance masks, rather than evaluating end-to-end scene detection and decomposition.

Highlights & Insights¶

Loss improvements work independently of shape extensions. The rigid variant already gains substantial IoU, so the contribution should not be attributed entirely to 8 additional deformation parameters.
A correct union and reasonable parts are different objectives. IoU, surface distance, and overlap constraints address global occupancy, local fitting, and geometric redundancy; the ablation demonstrates their complementarity.
Supervise in geometry space rather than parameter space. Matching surface sets avoids ambiguities from equivalent parameterizations, a transferable idea for other symmetry-affected structural prediction tasks.
Refinement also produces training data. Expensive per-object optimization can provide offline complete-structure supervision in addition to improving one reconstruction; this is a reusable methodological insight, not an additional experimental finding.

Limitations & Future Work¶

Training geometry requirements are substantial. Base learning and refinement depend on complete occupancy information; the paper does not establish equally strong unsupervised training from arbitrary partial scans alone.
Evidence on real scenes remains limited. ASE supplies quantitative results, while ScanNet++ mainly supplies qualitative examples and object extraction depends on instance masks.
Refinement has explicit latency. Per-object optimization taking 5 s is not a free addition to a 0.0082 s forward pass; applications must choose according to their latency budget.
Expressiveness remains constrained. A 19-parameter primitive is still a restricted shape family; the paper provides no comprehensive guarantee for arbitrary topology, thin structures, or uncertain completions.
Some reproducibility details are unavailable in the extraction. The supplied cache lacks the appendix, contains corrupted equations and a damaged usage threshold, and does not clearly specify the F-score threshold in readable main text; these gaps should not be filled by guessing.
Reader-proposed directions. Robustness to erroneous instance segmentation and uncertainty across multiple plausible completions merit evaluation; neither is an established capability of this paper.

SuperDec supplies the query-based decomposition architecture and compactness constraints; SuperFlex chiefly changes reconstruction supervision, deformable parameterization, and complete-to-partial structural supervision.
Marching Primitives and EMS represent per-object optimization approaches; SuperFlex learns reusable priors and optionally refines them, so Table 1 runtime and primitive count should be considered alongside accuracy.
CvxNet informs differentiable occupancy and union-related designs; this paper applies related geometric optimization ideas to deformable superquadrics rather than switching to arbitrary convex representations.
Barr's global deformations and classical superquadric work provide the geometric foundations for bending and tapering; the contribution integrates these representations with learning, refinement, and occlusion supervision rather than inventing deformation itself.

Rating¶

Novelty: 4/5. The representation and individual components have clear precedents, but joint supervision and complete-to-partial geometric training form a useful integration.
Experimental Thoroughness: 4/5. Complete objects, loss and deformation ablations, and partial-cloud evaluation are covered; quantitative evidence on real scans remains limited.
Writing Quality: 4/5. The main argument and ablation motivations are clear, although the claimed relative gain differs from rounded table values and the cache contains equation extraction errors.
Value: 4/5. Relevant to compact interpretable 3D representations, geometric completion, and combinations of learned prediction with refinement.