This dissertation addresses the methodological foundations of unsupervised three dimensional (3D) shape representation using superquadrics, focusing on challenges of accuracy, computational efficiency, and interpretability in implicit modelling. Superquadrics form a compact analytic family of parametric surfaces capable of approximating a broad range of geometric forms, making them suitable for part-based decom position of complex shapes. Their integration into deep learning frameworks depends on careful methodological design to support stable training, maintain structural fidelity, and enable interpretable outcomes. Building on this foundation, the dissertation is structured into two methodological stages, reflecting a progression from stability-oriented design to structure-aware refinement. The first stage focuses on establishing stability and feasibility in unsupervised learning with superquadrics and comprises three core components: (i) a reformulated objective function that guides part-based decomposition, (ii) a point-based neural encoder that processes input point clouds directly without volumetric discretization, and (iii) a composite loss formulation combining complementary terms to balance coverage and geometric fidelity, supported by adaptive weighting and stable optimization settings. Collectively, these components establish the methodological foundation for integrating analytic shape primitives into deep neural architectures under an unsupervised setting, providing the groundwork for the refinements introduced in the second stage. The second stage advances the framework, emphasizing interpretability, compactness, and systematic evaluation. This stage comprises three refinements: (i) an overlap constraint that limits redundant intersections between primitives, (ii) a curvature-aware uniform sampling strategy to stabilize the training process, and (iii) a structure-aware evaluation framework that extends traditional surface-based metrics toward part-level and global assessment through Structural Accuracy (SA), Primitive Accuracy (PA), and Overlap Percentage (OP). Together, these criteria establish a consistent methodological foundation for comparing primitive-based models in terms of fidelity, compactness, and interpretability within unsupervised settings. Taken together, the two stages define a unified methodological framework for implicit three-dimensional shape representation using superquadrics. The framework encompasses contributions to objective function design, training stability, and structure-aware evaluation methodology, thereby contributing to the methodological foundations of un supervised geometric learning. It also offers a methodological basis for applications in computer vision, robotics, and graphics, where accuracy and interpretability are essential for reliable three-dimensional modelling.
Mahmoud Eltaher (Thu,) studied this question.