On-line Space Sculpturing for 3D Shape Manipulation Leila De Floriani, Paola Magillo, Enrico Puppo Dip. di Informatica e Scienze dell'Informazione, Universita` di Genova Via Dodecaneso, 35, 16146 Genova, ITALY {deflo,magillo,puppo}@disi.unige.it

Motivations Modern 3D acquisition and reconstruction techniques permit to build high resolution models of complex objects. The shape of an object described as a mesh of triangles of very large size (several thousands to many millions of triangles). Smaller meshes needed for efficient rendering, recognition, classification, collision detection, and manipulation planning.

Paper contributions The Multi-Sculpure: a new data structure to represent 3D shapes at multiple levels of detail. based on a tetrahedral mesh that fills the space between the surface of the object and its convex hull approximations of the objects at different levels of detail and complexity are obtained by pasting more or fewer tetrahedra over the surface nesting property: an approximated mesh contains the actual shape in its enclosed volume higher level of detail = higher degree of concavity

Paper contributions

Paper Contributions An algorithm for on-line extraction of a mesh approximating the given shape and selectively refined in areas of interest. can vary the level of detail through different parts of the object dynamically adapts the extracted mesh to time-varying user parameters by incrementally coarsening or refining a current mesh locally. Applications: recognition, classification, collision detection, planning of manipulation tasks.

Related Work Techniques for the simplification and multiresolution representation of meshes Algorithms for shape reconstruction based on a tetrahedral mesh (Bajaj et al., 1996; Boissonnat, 1984;Veltkamp, 1993; Fageras et al., 1990) Multiresolution representations of convex shapes Our previous work (ICPR’99)

The Multi-Sculpture We consider three meshes: Triangle mesh T = the boundary of a solid object (the finest representation of our shape) Triangle mesh HT = the convex hull of T (the coarsest approximation of our shape) Tetrahedral mesh = filling the portion of space which separates T from HT

The Multi-Sculpture Any other closed mesh formed by triangles of D, T and HT gives a representation of the shape at an intermediate level of detail. T is given in input HT is computed with any algorithm for 3D convex hull D is obtained with any algorithm for constrained 3D triangulation [e.g., Borouchaki et al., 1996; Shewchuck, 1997].

Data Structure Mesh D : a graph in which nodes = tetrahedra arcs = adjacency relations between pairs of tetrahedra sharing a face. Mesh T : the external faces of D. Mesh HT : the faces of D which belong to one tetrahedron.

Approximated Meshes Any mesh M at an intermediate level of detail between T and HT is a closed triangular mesh formed of triangles that are faces of D partitions the set of the tetrahedra of D into: inner set = tetrahedra lying between T and M outer set = tetrahedra lying between M and HT M is formed by faces that separate the inner set from the outer set, plus possibly some faces of T and/or HT.

Approximated Meshes The level of detail of M can be modified locally by Sculpturing a tetrahedron t of the inner set to increase detail: move t to the outer set modify M Pasting a tetrahedron t of the outer set to reduce detail: move t to the inner set

Selective Shape Refinement Extract a mesh which fulfills user requirements. Input: A detail threshold = the desired level of detail = the maximum approximation error admitted (it can be variable through space) Error of a tetrahedron t = distance of t from the true surface A focus set = the portion of space that is of interest = an entity in 3D space (a box, a sphere, a trajectory, etc.) Output: A mesh M whose detail inside the given focus set lies within the given detail threshold. Effective heuristics maintain the resulting mesh as small as possible Dynamic approach: modify the mesh extracted with a previous query

The Algorithm Current state consisting of two queues of tetrahedra: in-out candidates = inner tetrahedra which can be to be sculptured out-in candidates = outer tetrahedra, which can be pasted Two stages: Coarsening stage Refining stage:

The Algorithm Coarsening stage: Refining stage: iteratively extract a tetrahedron t from the out-in queue; if t either does not intersect the focus set, or it is compatible with the detail threshold, then past t Refining stage: iteratively extract a tetrahedron t from the in-out queue; if t intersects the focus set and does not satisfy the detail threshold, then sculpture t

Examples In collision detection and manipulation focus set = an object moving through space (e.g., a robot arm) detail required is maximum inside the focus set, and arbitrarily low elsewhere. In shape recognition and pose estimation use first a detail threshold that is uniformly low over the whole object (focus set = the whole space), in order to detect a raw shape. later, increase detail over certain regions of interest (= new focus sets), in order to refine the result.

The Chair Example (data set courtesy of OCNUS) The input mesh T (1906 triangles) Its convex hull HT (152 triangles)

The Chair Example An intermediate mesh at uniform An intermediate mesh with high detail resolution (996 triangles) inside a box (310 triangles, 56 in the box)

Conclusions A new multiresolution model for 3D shapes which satisfies the nesting property A dynamic algorithms for extracting selectively refined meshes Advantages: trade-off on-line between the detail and the complexity of representation (detail threshold) focus detail over regions of interest (focus set) change representation dynamically through time (dynamic nature of the algorithm)