1. Non-Manifold Multi-Tesselations From Meshes to Iconic Representations of Objects L. De Floriani, P. Magillo, E. Puppo, F. Morando DISI - University of Genova, Italy

2. Motivations Increasing diffusion of 3D models: –3D model databases (industry, biology, art, enterteinment…)

3. Motivations Common 3D file formats (VRML, 3D studio...) : –geometry (polygonal or triangle meshes) –appearance (colors…) Good for visualization Not good for reasoning –verbose –no information about structure and shape features

4. Iconic representations –concise –part-based –scalable to different levels of abstractions Good for interpretation, analysis, search of 3D models

5. Our Proposal 1- An iconic model of 3D shapes consisting of a structured aggregation of parts of different dimensions 2- A method to obtain such model from triangle meshes 3- Multiresolution extension of the iconic model to incorporate several levels of abstraction which parts? which dimension? connecting structure

6. An iconic model 3D object as an aggregation of parts non-manifold, mixed-dimensional mesh

7. An iconic model structured representation –collection of uniformly dimensional, manifold parts –different parts join at non- manifold points and lines –a hypergraph describes the assembly structure

8. Construction of an iconic model Input: –a triangle mesh describing a 3D object –no structure, the whole object is one part –the iconic model is a hypergraph with one node Output: –a mesh of mixed dimensions –object parts have been identified and simplified –the iconic model describes the part-based structure

9. Non-manifold simplification Iterative process that simplifies geometry through edge collapses May create non-manifold conditions Reveals the part-based structure of the object FIGURA DA FRANCO?

10. Edge collapse One step of the simplification process Local modification of the mesh May produce non- manifold situations and thus changes in the iconic model

11. Criteria to drive simplification Must identify meaningful parts of the object –a cost function gives a cost to each edge –at each step collapse the edge of minimum cost –the algorithm is parametric on the cost function edge length (implemented) variation in surface area smoothness sharp features topological changes more …

12. Multiresolution iconic model A single model that spans several levels of abstraction Possibility of extracting iconic models at a desired level of abstraction the level of abstraction may vary in space and time

13. From a simplification sequence to a multiresolution iconic model Relax the total order to a partial order –dependency relation –a collapse C1 precedes a collapse C2 iff C1 creates some of the cells deleted by C2 –transitive closure FIGURA LA METTIAMO?

14. The partially ordered set of edge collapses can be visualized as a DAG

15. Different levels of abstraction of the mesh and of the iconic model are provided by subsets of collapses consistent with the partial order –a collapse can be applied only if all collapses preceding it have been applied –consistent subsets can be visualized as cuts in the DAG

16. This model captures levels of abstraction with the granularity of a single edge collapse –too fine-grained Group together a set of mutually related collapses that produce a meaningful modification of the iconic model –such collapses are seen as one, atomic modification –modify the DAG by merging the corresponding nodes Non-manifold Multi-Tesselation (NMT)

17. Ongoing and future research Short term –experimentation of different cost functions for non- manifold simplification –clustering strategies for the NMT Long term –use of the NMT for organizing a 3D model database support for search and recognition shapes with a similar structure have a congruent iconic model at least at a coarse level of abstraction two similar shapes in a database may share the first levels of their NMT’s