CvxNet: Learnable Convex Decomposition (1909.05736v4)

Abstract: Any solid object can be decomposed into a collection of convex polytopes (in short, convexes). When a small number of convexes are used, such a decomposition can be thought of as a piece-wise approximation of the geometry. This decomposition is fundamental in computer graphics, where it provides one of the most common ways to approximate geometry, for example, in real-time physics simulation. A convex object also has the property of being simultaneously an explicit and implicit representation: one can interpret it explicitly as a mesh derived by computing the vertices of a convex hull, or implicitly as the collection of half-space constraints or support functions. Their implicit representation makes them particularly well suited for neural network training, as they abstract away from the topology of the geometry they need to represent. However, at testing time, convexes can also generate explicit representations -- polygonal meshes -- which can then be used in any downstream application. We introduce a network architecture to represent a low dimensional family of convexes. This family is automatically derived via an auto-encoding process. We investigate the applications of this architecture including automatic convex decomposition, image to 3D reconstruction, and part-based shape retrieval.

• The paper presents an autoencoder that learns convex decompositions of 3D shapes through differentiable convex indicator functions.
• It achieves state-of-the-art reconstruction performance on the ShapeNet dataset using metrics like IoU and Chamfer-L1.
• The method offers significant computational benefits by efficiently generating explicit polygonal meshes without expensive iso-surface extraction.

An Overview of CvxNet: Learnable Convex Decomposition

CvxNet introduces a novel approach to representing 3D geometries through learnable convex decompositions. The core idea centers on decomposing any solid object into a set of convex polytopes, which serve as an efficient approximation method, particularly beneficial for applications in computer graphics and physics simulations. This paper presents a network architecture designed to encode a low-dimensional family of convex shapes, enabling tasks such as automatic convex decomposition, image-to-3D reconstruction, and part-based shape retrieval.

Methodology and Implementation

The CvxNet framework applies an autoencoding architecture to derive convex decompositions without human supervision. The architecture comprises an encoder that processes an input, such as a 2D image or 3D point cloud, to generate a latent representation. This representation is then decoded into the parameters defining the convex components of the object, specifically through the use of half-space constraints. A significant aspect of this method is the differentiable nature of the convex indicator functions, enabling the network to learn shape representations effectively.

Unlike traditional methods requiring computationally expensive procedures like Marching Cubes for iso-surfaces, CvxNet generates explicit polygonal meshes from convex decompositions efficiently. The process involves duality transformations and convex hull computations, which are independent of resolution parameters, offering computational advantages.

Experimental Results and Comparative Analysis

CvxNet demonstrates superior performance over several state-of-the-art methods in self-supervised 3D reconstruction tasks. The experiments conducted on the ShapeNet dataset reveal that CvxNet not only matches but often exceeds the accuracy of other leading techniques, such as Occupancy Networks (OccNet), in terms of reconstruction quality and interpretability. Key metrics used in evaluations include Volumetric Intersection over Union (IoU), Chamfer-L1 distance, and F-score, with CvxNet showing robustness across various categories, including single and multi-view reconstruction tasks.

The paper also includes ablation experiments that explore the influence of model parameters such as number of convexes and hyperplanes, as well as different loss terms. It shows that each component of the loss function contributes positively to the network's performance.

Implications and Future Work

The introduction of CvxNet offers significant implications for computational efficiency and scalability in 3D geometric modeling. Its ability to efficiently decompose complex shapes into convex components provides a promising direction for real-time applications in graphics, simulation, and robotics. The semantic correspondence of parts in CvxNet's decomposition also suggests potential enhancements in part-based retrieval and scene understanding.

Future research could extend CvxNet to handle variable numbers of parts, model rotations, and explore the use of permutation-invariant architectures. The approach's implicit ordering of hyperplanes may also inspire more sophisticated encoding techniques to capitalize on the rich geometric information captured by the convex decomposition.

Overall, CvxNet establishes itself as a valuable tool for developing geometric understanding in AI, providing a flexible yet powerful framework for shape representation and analysis.