Superquadric Representation

Dense point cloud maps, as a typically used mapping representation, are difficult to deploy on resource-constrained robots because their memory consumption grows rapidly with scene scale. Although compact single-model representations reduce memory cost, their fixed geometric expressiveness is insufficient for structurally diverse environments. Existing multi-model methods improve representational flexibility, yet their feature extraction and model selection are often dominated by local geometry, which can cause overfitting and adhesion between objects. To address these issues, this paper presents M2-SMap, a memory-efficient semantic mapping framework based on hierarchical multi-model representation. First, a hierarchical geometric decomposition partitions RGB-D point clouds into compact Gaussian components. Then, a projection-guided semantic annotation mechanism assigns instance identities to each component. Subsequently, these annotations are incorporated into an object-aware Gaussian fusion strategy. Furthermore, a multi-scale feature extraction strategy separates large planar regions, semantic objects, and complex residual structures, which are respectively represented by bounded planes, object-level superquadrics, and GMM primitives. Experiments on three RGB-D sequences show that M2-SMap runs in real time at no less than 29.37 Hz while achieving the lowest primitive count, with an average reduction of 18.7% over the best baseline. It also reduces the mean per-frame number of measured inter-object adhesion cases from 2.808 to 0, demonstrating efficient and semantically consistent scene representation.

Superquadrics have proven to provide a compact, geometrically meaningful representation for 3D objects. However, existing methods suffer from limited reconstruction accuracy, are restricted to rigid primitives, and lack robustness to partial point clouds. In this work, we present SuperFlex, an enhanced framework that expands the expressive power and applicability of superquadric decompositions. First, we introduce a novel loss formulation which significantly improves reconstruction accuracy. Second, we include bending and tapering deformations, enabling high-fidelity representation of curved and asymmetric geometries. Finally, we leverage these high-quality decompositions as supervision to train a model that is robust to partial real-world point clouds. Experiments demonstrate substantial improvements in reconstruction accuracy over both optimization- and learning-based baselines while maintaining a highly compact primitive representation.

Recent indoor occupancy prediction methods adopt Gaussian primitives as a sparse 3D representation for computational efficiency. However, their training relies on voxel classification, which imposes only local constraints and lacks global supervision on the distribution of the primitives. Therefore, they inevitably predict spurious primitives in empty regions, undermining both representational and computational efficiency. To address this, we propose Feed-forward Likelihood Maximization (FLM), a novel framework that reformulates occupancy prediction as voxel distribution estimation. In FLM, a network is trained to predict a mixture model that maximizes the likelihood over ground-truth occupied voxels in a feed-forward manner. To enable end-to-end training of networks and voxelization of a standard mixture model, we define mixture weights as normalized primitive volumes to implicitly enforce simplex constraints and derive novel voxelization formulas. Based on FLM, our FLM-Occ, a novel method that is capable of relocating randomly initialized primitives over long distances to model a scene. On Occ-ScanNet, FLM-Occ achieves superior accuracy using only 32 superquadrics, 2.7% of the prior SoTA, while running 3.7 times faster.

Generative design of robots requires navigating a vast search-space, encompassing physical configurations and behavioural parameters. Evolutionary Algorithms (EAs) have shown promising results, but often converge prematurely to a small set of sub-optimal designs. Most EAs fail to maintain sufficient diversity in the population that would allow the discovery of distinct functional robots. To counter premature convergence, we introduce a superquadrics-based representation (SQs) for robot bodies. SQs are interpretable, compact and computationally efficient mathematical representations of 3D geometrical shapes that can be tuned to specific design-spaces. To encourage morphological diversity, we combine this representation with a quality-diversity (QD) algorithm (MAP-Elites). We compare SQs and Compositional Pattern Producing Networks representations as generators of morphologies, combining them with standard EAs and MAP-Elites. In two test environments, we find that using SQs to generate morphology in conjunction with the MAP-Elites algorithm reaches the highest QD-score across both environments, maximising diversity of design and functionality of generated robots. The findings highlight the benefits of using a compact and interpretable geometric representation for exploring a complex design-space and suggest that combining SQs with an explicit diversity mechanism increases the quality and number of designs generated.

Diffusion models have advanced 3D shape generation, yet most methods still denoise in high-cardinality spaces (e.g., voxel/SDF grids, meshes, or point clouds), which is computationally and memory intensive and makes it difficult to scale in terms of both higher resolution and stronger controllability. We rethink the diffusion representation and propose to move diffusion from dense geometry to compact geometric primitives, representing each shape as a small set of superquadrics. Instead of operating on thousands to millions of geometric representation values, we leverage 7KB superquadric parameters (pose, size, and shape), drastically reducing diffusion-state dimensionality and per-step compute/memory. Our diffusion-over-superquadrics improves scalability by supporting broader capabilities (e.g., resolution-free point-cloud decoding, part-level editing, and constraint-based design) and achieving competitive surface-fidelity and distributional performance on standard benchmarks after point-cloud decoding, while enabling efficient generation within 0.6s per shape for most conditions.

Hierarchical 3D Scene Graphs (3DSG) have emerged as an actionable and scalable representation for long-term autonomy incorporating metric, semantic, and topological information in the scene. However, the question of geometric representation of objects in 3DSG has been overlooked as most methods use simplified geometric models such as partial point clouds or 3D bounding boxes. In this work, we introduce a hierarchical object representation that can be leveraged for high-fidelity object-level reconstruction, object-based robust re-localization or map alignment, and efficient and analytical collision checking for safe robot navigation planning in dense and cluttered environments. The representation is structurally organized into four distinct layers, progressively abstracting the scene from raw sensor data to dense 3D meshes to analytical primitives such as superquadrics, which provide a sparse and analytical representation for object geometry. We develop a pipeline that builds the hierarchical object representation from RGB-D image stream captured by a robot, and demonstrate its working in real-world open-set object scenes in both indoor and outdoor environments. Extensive experiments across diverse datasets including HOPE, ReplicaCAD, Kimera-Multi, and NUS Campus Dataset collected using Unitree B2 Robot validate our pipeline in both indoor and outdoor environments. We show that our superquadric-based map alignment method outperforms the current state-of-the-art object based map alignment method ROMAN. Our code can be found at https://github.com/perceptica-robotics/Hickory.

This work presents a novel method for fitting superquadrics to point clouds under the contamination of noise and outliers, which has many applications for shape modeling across diverse fields. Unlike prior approaches that either exclusively focus on fitting rigid or deformable superquadrics, or suffer from robustness and numerical instability issues, our method redefines the problem from a new unsupervised clustering perspective, enabling the holistic fitting of both rigid and deformable superquadrics within a unified framework. Central to our approach is a stable optimization function inspired by unsupervised clustering analysis, where we formulate the point cloud data and samples from the potential parametric surface as clustering members and centroids, respectively. Then, the clustering process with dynamic updates to centroid locations serves as a direct proxy for optimizing superquadric parameters, establishing a principled link between geometric fitting and clustering dynamics. We further derive the relationship between pairwise computations of clustering centroids and clustering members to orthogonal distances, effectively eliminating the need for the time-consuming surface sampling process. Moreover, our formulation provides closed-form analytical solutions for both the fuzzy membership degree vector and the covariance matrix, ensuring efficient iteration optimization and enabling more effective handling of geometric deformations. In addition, we provide a theoretical certificate of convergence analysis and demonstrate that the clustering-inspired fitting method can escape local minima by inherently increasing the convexity of the objective function. The implementation is publicly available at https://github.com/zikai1/SuperquadricFitting.

Robotic grasping from single-view observations remains a critical challenge in manipulation. However, existing methods still struggle to generate reliable grasp candidates and stably evaluate grasp feasibility under incomplete geometric information. To address these limitations, we present SuperGrasp, a new two-stage framework for single-view parallel-jaw grasping. In the first stage, we introduce a Similarity Matching Module that efficiently retrieves valid and diverse grasp candidates by matching the input single-view point cloud with a precomputed primitive dataset based on superquadric coefficients. In the second stage, we propose E-RNet, an end-to-end network that expands the grasp-aware region and takes the initial grasp closure region as a local anchor region, capturing the contextual relationship between the local region and its surrounding spatial neighborhood, thereby enabling more accurate and reliable grasp evaluation and introducing small-range local refinement to improve grasp adaptability. To enhance generalization, we construct a primitive dataset containing 1.2k standard geometric primitives for similarity matching and collect a point cloud dataset of 100k samples from 124 objects, annotated with stable grasp labels for network training. Extensive experiments in both simulation and real-world environments demonstrate that our method achieves stable grasping performance and good generalization across novel objects and clutter scenes.

Neural reconstructions often trade structure for fidelity, yielding dense and unstructured meshes with irregular topology and weak part boundaries that hinder editing, animation, and downstream asset reuse. We present DualPrim, a compact and structured 3D reconstruction framework. Unlike additive-only implicit or primitive methods, DualPrim represents shapes with positive and negative superquadrics: the former builds the bases while the latter carves local volumes through a differentiable operator, enabling topology-aware modeling of holes and concavities. This additive-subtractive design increases the representational power without sacrificing compactness or differentiability. We embed DualPrim in a volumetric differentiable renderer, enabling end-to-end learning from multi-view images and seamless mesh export via closed-form boolean difference. Empirically, DualPrim delivers state-of-the-art accuracy and produces compact, structured, and interpretable outputs that better satisfy downstream needs than additive-only alternatives.