Multi-Superquadric Decoding Strategy

• Multi-superquadric decoding strategy is a method that represents complex 3D scenes by decomposing them into multiple parametric superquadric primitives that interpolate between basic shapes like spheres and cuboids.
• It employs transformer-based and region-wise architectures with coarse-to-fine scheduling to efficiently capture local geometric details and semantic nuances.
• The approach achieves computational efficiency and improved semantic occupancy prediction through probabilistic modeling, regularization techniques, and iterative hierarchical refinement.

A multi-superquadric decoding strategy refers to the process of representing complex 3D scenes or objects by decomposing them into a set of superquadric primitives—parametric volumetric shapes whose exponents and axes can smoothly interpolate between spheres, ellipsoids, cuboids, and cylinders. Such strategies aim to achieve a compact, semantically-meaningful, and geometrically expressive approximation using a judiciously chosen number of parameterized primitives. This section provides a comprehensive examination of its mathematical underpinnings, architectural designs, algorithmic pipelines, and evaluation methodologies across several recent state-of-the-art approaches, with a focus on both discriminative and generative paradigms.

1. Motivation and Theoretical Foundations

The impetus for multi-superquadric decoding strategies arises from the inadequacy of single-primitives, particularly ellipsoids, to capture the geometric diversity encountered in real-world scenes. Local geometries—such as sharp corners, composite objects, and adjacent flat and curved surfaces—typically cannot be faithfully represented using a solitary superquadric due to the "one-shape-fits-all" restriction, unless one resorts to a prohibitive number of queries or dense sampling (Yu et al., 22 Jan 2026, &&&1&&&). The multi-primitive approach, in which local regions are modeled by the union (or mixture) of multiple superquadrics, yields both greater geometric expressiveness and computational efficiency by leveraging the sparsity of the representation. This design underpins recent frameworks for semantic occupancy prediction in autonomous driving, robotic grasp synthesis, and 3D shape abstraction (Xu et al., 4 Mar 2025, Fedele et al., 1 Apr 2025).

2. Mathematical Formulation of Superquadrics and Mixtures

A superquadric is defined in its canonical (local) frame by the implicit function

$$\left( \left| \frac{x'}{s_x} \right|^{2/\epsilon_2} + \left| \frac{y'}{s_y} \right|^{2/\epsilon_2} \right)^{\epsilon_2/\epsilon_1} + \left| \frac{z'}{s_z} \right|^{2/\epsilon_1} = 1,$$

where $(x',y',z')$ is the coordinate of a point transformed into the primitive’s frame via rotation and translation, $(s_x,s_y,s_z)$ are positive axis scales, and $(\epsilon_1,\epsilon_2)$ are exponents controlling "squareness".

To model complex shapes, multi-superquadric decoding composes the predicted primitives, typically through a probabilistic mixture model or via union rules. For instance, in probabilistic occupancy prediction, the overall occupancy probability at a point $\mathbf p$ is given by

$$p_o(\mathbf{p}) = 1 - \prod_{i=1}^M \left( 1 - p_o(\mathbf{p}; S^i) \right),$$

where $S^i$ denotes the $i$-th decoded superquadric and $p_o(\mathbf{p}; S^i)$ is its individual occupancy field (Yu et al., 22 Jan 2026, Zuo et al., 12 Jun 2025). Semantic predictions aggregate the class scores across primitives using occupancy and opacity-weighted averages, yielding interpretable, class-aware representations: $$p_s(\mathbf{p}) = \frac{ \sum_{i=1}^{M} p_o(\mathbf{p};S^i)\,\sigma^i\,\mathbf{c}^i } { \sum_{j=1}^{M} p_o(\mathbf{p};S^j)\,\sigma^j },$$ where $\mathbf{c}^i$ are semantic logits and $\sigma^i$ opacity values.

3. Decoder Architectures and Algorithmic Pipelines

Transformer/Attention-Based Decoding

Recent approaches (e.g., SuperOcc (Yu et al., 22 Jan 2026), SuperDec (Fedele et al., 1 Apr 2025), QuadricFormer (Zuo et al., 12 Jun 2025)) employ query-based transformers or attention mechanisms for multi-superquadric decoding. These networks maintain a set of queries (e.g., $\sim600$ in SuperOcc) that, after progressive self- and cross-attention updates and temporal modeling (view-centric/object-centric), each decode into a small cluster of superquadrics per query via specialized heads. The number of primitives per query is commonly scheduled in a coarse-to-fine manner, with increasing multiplicity deeper in the decoder.

Example: SuperOcc Decoder Layer Schedule

Decoder Layer	Primitives per Query ($K_i$)
1	2
2	2
3	4
4	4
5	8
6	8

Early layers produce a rough approximation, while later layers refine structure and detail (Yu et al., 22 Jan 2026).

Region-Wise and Pipeline-Based Strategies

Other approaches, such as RGBSQGrasp (Xu et al., 4 Mar 2025) and SuperDec (Fedele et al., 1 Apr 2025), segment the scene into local regions/instances (using segmentation masks or region proposals), fit a superquadric per region using regression networks or self-supervised matching, and aggregate the union of all fitted primitives as the final multi-superquadric decomposition.

Iterative, Hierarchical, and Coarse-to-Fine Methods

Hierarchical approaches recursively partition the object or scene space, fitting primitives at different levels of a tree (hierarchical binary splits), while iterative frameworks (ISCO (Alaniz et al., 2023)) grow the primitive set adaptively by adding new superquadrics at locations of maximal reconstruction error and refining all parameters jointly.

4. Training Objectives, Supervision, and Regularization

Supervision schemes depend on the task:

• For occupancy prediction and segmentation, only the final composite predictions are supervised via per-voxel cross-entropy (possibly augmented with softmax or Lovász losses), with no explicit fitting of individual primitive parameters (Yu et al., 22 Jan 2026, Zuo et al., 12 Jun 2025).
• For geometric fitting, e.g., in grasp synthesis or point cloud abstraction, losses include symmetrized Chamfer distance between ground truth and predicted superquadric surfaces, often augmented with parameter regularizers (scale positivity, exponent ranges) and normal consistency (Xu et al., 4 Mar 2025, Fedele et al., 1 Apr 2025).
• For probabilistic EM-based fitting (Liu et al., 2021), the objective maximizes data likelihood under a Gaussian-uniform mixture over all primitives, with outlier handling and mixture assignment inferred in latent space.

Regularization is effected through $L_2$ weight decay, explicit parameter range clamping (e.g., $0.1 < \epsilon_{1,2} < 2$), and parsimony losses to suppress redundant primitives (Fedele et al., 1 Apr 2025, Zuo et al., 12 Jun 2025).

5. Implementation Techniques and Computational Considerations

Voxel Splatting and Efficient Occupancy Computation

For occupancy prediction, efficient splatting kernels map each primitive’s contribution onto sparse or dense voxel grids. SuperOcc introduces a tile-level binning strategy, pre-binning primitives per spatial tile and using CUDA shared memory for accumulation—yielding an $\sim80\%$ speedup over naive strategies and up to $20\%$ higher end-to-end FPS (Yu et al., 22 Jan 2026).

Pruning, Splitting, and Hierarchical Management

To improve efficiency and expressive allocation of primitives, advanced strategies perform post-hoc pruning (removing primitives with low volume or occupancy contribution), splitting of coarse primitives with excessive support, and iterative LM- or ICP-based refinement to enhance local fit (especially post-inference) (Zuo et al., 12 Jun 2025, Fedele et al., 1 Apr 2025).

Pipeline Summary

A typical pipeline proceeds as follows:

1. Input scene (RGB-D, depth, or multi-view images).
2. Regional or query-based feature extraction.
3. Decoder predicts clusters of superquadric parameters per region/query: scale, shape, pose, and semantic/logit attributes.
4. Primitives are combined via probabilistic occupancy or geometric union.
5. Losses and gradients are computed over the final prediction or reconstructions, propagated to all primitives.
6. Optional post-processing: pruning, splitting, region assignment, and optimization (LM/ICP/EM).

6. Quantitative Evaluation and Empirical Results

Benchmarks across autonomous driving (SurroundOcc, Occ3D, nuScenes), point cloud datasets (ShapeNet, ScanNet++), and bin-picking environments demonstrate state-of-the-art or competitive performance for multi-superquadric approaches:

Method	mIoU	RayIoU	Inference FPS	Notes
SuperOcc (single quadric)	27.9%	33.3%	33.0	Single per-query primitive (Yu et al., 22 Jan 2026)
SuperOcc (multi-sq, coarse-to-fine)	29.1%	34.9%	31.7	$K=[2,2,4,4,8,8]$ schedule
QuadricFormer (1600 prim.)	20.04	30.71	162 ms (2.55 GB)	nuScenes; outperforms Gaussian/voxel baselines (Zuo et al., 12 Jun 2025)
RGBSQGrasp	92%	—	—	Real-world bin-picking grasp success (Xu et al., 4 Mar 2025)

Performance increases with the number of primitives until saturation (typically at 8 per query in SuperOcc); coarse-to-fine scheduling consistently outperforms uniform allocation.

7. Comparative Strategies and Research Directions

Variants such as hierarchical superquadric decomposition (Šircelj et al., 2022) and EM-based probabilistic recovery (Liu et al., 2021) demonstrate that multi-superquadric strategies are adaptable across learning-based, optimization-based, and hybrid paradigms. Research directions include further integration with object-centric perception, semantic consistency, and exploration of regularization regimes to enhance out-of-distribution robustness (Alaniz et al., 2023).

The multi-superquadric decoding strategy has proven effective in bridging the gap between dense voxel methods and minimal parametric models, providing a balance between efficiency, geometric expressiveness, and scalability, and establishing a generalizable framework for 3D scene abstraction and semantic reasoning (Yu et al., 22 Jan 2026, Zuo et al., 12 Jun 2025, Fedele et al., 1 Apr 2025).