Many computer graphics applications use simpler yet faithful approximations of complex shapes to conduct reliably part of their computations. Some tasks, such as physical simulation, collision detection, occlusion queries or free-form deformation, require the simpler proxy to strictly enclose the input shape. While there are algorithms that can output such bounding proxies on simple input shapes, most of them fail at generating a proper coarse approximant on real-world complex shapes, which may contain multiple components and have a high genus. We advocate that, before reducing the number of primitives to describe a shape, one needs to regularize it while maintaining the strict enclosing property, to avoid any geometric aliasing that makes the decimation unreliable. Depending on the scale of the desired approximation, the topology of the shape itself may indeed have to be first simplified, to let the subsequent geometric optimization be free from topological locks. We propose a new bounding shape approximation algorithm which takes as input an arbitrary surface mesh, with potentially complex multi-component structures, and generates automatically a bounding proxy which is tightened on the input and can match even the coarsest levels of approximation. To sustain the nonlinear approximation process that may eventually abstract both geometry and topology, we propose to use an intermediate regularized representation in the form of a shape closing, computed in real time using a new fast morphological framework designed for efficient parallel execution. Once the desired level of approximation is reached in the shape closing, a coarse, tight and bounding polygonization of the proxy geometry is extracted using an adaptive meshing scheme. Our underlying representation is both geometry- and topology-adaptive and can be optionally controlled accurately by a user, through sizing and orientation fields, yielding an intuitive brush metaphor within an interactive proxy design environment. We provide extensive experiments on various kinds of input meshes and illustrate the potential applications of our method in scenarios that benefit greatly from coarse, tight bounding substitutes to the actual high resolution geometry of the original 3D model, including freeform deformation, physical simulation and level of detail generation for rendering.

We introduce a complete morphological analysis framework for 3D point clouds. Starting from an unorganized point set sampling a surface, we propose morphological operators in the form of projections, allowing to sample erosions, dilations, closings and openings of an object without any explicit mesh structure. Our framework supports structuring elements with arbitrary shape, accounts robustly for geometric and morphological sharp features, remains efficient at large scales and comes together with a specific adaptive sampler. Based on this meshless framework, we propose applications which benefit from the non-linear nature of morphological analysis and can be expressed as simple sequences of our operators, including medial axis sampling, hysteresis shape filtering and geometry-preserving topological simplification.