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Overview (Part 1)

Overview (Part 1). Background notions A reference framework for multiresolution meshes Classification of multiresolution meshes An introduction to LOD queries. A Reference Framework for Multiresolution Meshes. Multiresolution mesh consists of:

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Overview (Part 1)

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  1. Overview (Part 1) • Background notions • A reference framework for multiresolution meshes • Classification of multiresolution meshes • An introduction to LOD queries

  2. A Reference Framework for Multiresolution Meshes • Multiresolution mesh consists of: • a collection of mesh modifications describing small portions of a spatial object at different LODs • suitable dependency relation that allows selecting subsets of modifications (according to application-dependent requirements) • Reference framework: • independent of the properties of the modifications • based on a “natural” notion of dependency • dimension-independent

  3. A Reference Framework for Multiresolution Meshes • Multiresolution Meshes • Conforming Multiresolution Meshes • Properties of Multiresolution Meshes

  4. Multiresolution Meshes • G0: an n-dimensional mesh • {M1, M2,…,Mh}: a set of n-dimensional refinement modifications such that for any cell g in Mi-, g belongs either to G0 or to Mj+ for exactly one j<>i

  5. Multiresolution Meshes: Dependency Relation • A modification Mjdirectly depends on a modification Mi, with i<>j, if Mj removes some n-cell introduced by Mi Example: • M1 and M2 directly depend on G0 • M5 directly depends on both M1 and M2

  6. Multiresolution Mesh: Definition • If the transitive closure <of the dependency relation is a partial order, then M=(G0, {M1, M2,…,Mh} ,<) is a multiresolution mesh. • G0 is called the base mesh

  7. Multiresolution Mesh as a DAG • Described as a Directed Acyclic Graph (DAG) in which • nodes: modifications • arcs: direct dependency links

  8. Closed Subsets of Modifications • A subset S of modifications in a multiresolution mesh is closed (wrt the partial order) if, for each modification Mj in S, all modifications on which Mj depends are in S • A closed subset S plus the base mesh G0 define a collection of cells GS: result from applying the modifications in S to G0 • If GS is a mesh, then GS is called the extracted mesh associated with S

  9. A Closed Subset and the Corresponding Extracted Mesh

  10. A Closed Subset and the Corresponding Extracted Mesh

  11. Reference Mesh Any sequence of modifications in M corresponding to a total order extending < produces the reference mesh

  12. Conforming Multiresolution Meshes • Objective: characterize the set of conforming meshes that can be extracted from a given multiresolution mesh • Transform a generic (non-conforming) multiresolution mesh M into a conforming one • Through its closed subsets, the conforming representation encodes all conforming meshes that can be extracted from M • Process: aggregate single (non-conforming) modifications, such that their union is conforming, into minimal-size clusters of conforming modifications

  13. Matching Modifications • Two modifications Mi and Mj are matching if they are both non-conforming at some facet of their boundary, while their union M consists of two conforming meshes (M may not be a conforming modification)

  14. Clusters of Modifications • Cluster: defined by a minimal set C of modifications in a multiresolution mesh such that • for every modification Mi in C, there exists a modification Mj, belonging to C, that is matching with Mi • the union of all modifications in C is a conforming modification

  15. Conforming Representation of a Multiresolution Mesh • M: a non-conforming multiresolution mesh • IfM admits a collection C of clusters of modifications such that • for each modification M in M, there exists exactly one cluster in C which contains M • the transitive closure of relation < extended to C is a partial order • Then, MC =(G0, C, <) is the conforming multiresolution mesh associated with M

  16. Conforming Representation of a Multiresolution Mesh

  17. Properties of Multiresolution Meshes • Growth ratio between the total number of n-cells in a multiresolution mesh and the total number of n-cells in its reference mesh

  18. Properties of Multiresolution Meshes • Width maximum number of n-cells in M+ over all modifications M in a multiresolution mesh M • Height maximum length of a path in the DAG describing a multiresolution mesh M • Expressive power ratio between the number of all closed sets in a multiresolution mesh M and the total number of n-cells in M

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