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

2. 2 Classification of Multiresolution Meshes Multiresolution meshes proposed in the literature differ in the type of modifications involved the properties of the dependency relation Multiresolution meshes can be classified into: Nested models: modifications expand just a single n-cell into a mesh Non-nested models: modifications affect several n-cells and refine them collectively into a mesh

3. 3 Nested Multiresolution Models They are based on the recursive subdivision of an n- dimensional cell into scaled copies of it The dependency relation corresponds to a containment relation The DAG reduces to a tree They can be classified into regular and irregular nested meshes depending on the distribution of the vertices

4. 4 Nested Multiresolution Models Typical subdivision rules for regular nested models in the two-dimensional case are: Quadtree rule Quaternary triangle subdivision rule Right triangle bisection rule Such subdivision rules have been extended to three and higher dimensions

5. 5 Quadtree Subdivision rule: –Replace a square cell  with four squares of 1/4 of the size of  obtained by splitting  in its center Vertices must be on a regular square grid Extension to n-dimensions: subdividing a hyper-cube into 2 n hyper-cubes of size 1/2 n

6. 6 Nested Models as Trees of Meshes In nested models we identify a modification M with M+ (since M- consists just of one cell) We can draw the model as a tree of meshes

7. 7 Quaternary Triangulation Subdivision rule: –Replace a triangle  with four triangles obtained by connecting the midpoints of the edges of  Vertices must be on a regular grid Extensions to higher dimensions: not an immediate extension of the subdivision rule

8. 8 Right Triangle Bisection Subdivision rule: –Replace a right isosceles triangle  with two right triangles obtained by splitting  through the midpoint of its longest edge, resulting into a hierarchy of right triangles Vertices must be on a regular square grid Extensions to higher dimensions: recursive bisection of an n-simplex along its longest 1-simplicial facet

9. 9 Conforming Representation for Quadtrees and Quaternary Triangulations Each level of the tree forms a cluster Closed subsets and associated conforming meshes correspond to the different levels in the tree An extracted non-conforming mesh can be made conforming through a post-processing adjustment step

10. 10 Conforming Representation for a Quadtree

11. 11 Making a Quadtree Mesh Conforming S = the closed subset of nodes forming an extracted (non- conforming) mesh  S Add nodes to S in such a way that any two adjacent cells in  S differ at most for one level in the tree Triangulate the cells according to predefined patterns

12. 12 Making a Quadtree Mesh Conforming Patterns for cell triangulation

13. 13 Making a Quaternary Triangulation Conforming S = the closed subset of nodes forming an extracted (non- conforming) mesh Add nodes to S in such a way that any two adjacent extracted triangles differ at most for one level in the tree Further subdivide the triangles according to predefined patterns

14. 14 Conforming Representation for a Hierarchy of Right Triangles Clusters formed by pair of modifications: in a cluster a pair of triangles is replaced by four triangles which share the splitting vertex

15. 15 Conforming Representation for a Hierarchy of Right Triangles Described by a DAG in which each node (with the exception of boundary nodes) has two parents and four children

16. 16 Conforming Representation for a Hierarchy of Right Triangles Conforming meshes at variable resolution (i.e., formed by cells of different sizes) can be extracted

17. 17 Properties of Regular Nested Models Bounded width: each modification M+ contains a predefined number of cells Linear growth Logarithmic height High expressive power For their conforming representations: high expressive power for hierarchies of right triangles low expressive power for quadtrees and quaternary triangulations Very effective as spatial indexes

18. 18 Irregular Nested Models Hierarchical triangulation (Scarlatos and Pavlidis, 1992): a triangle  is refined by inserting up to four vertices (one on the interior and one on each of the three edges) according to predefined patterns Hierarchical Triangulated Irregular Network (HTIN) (DeFloriani and Puppo, 1995); a triangle  is refined by inserting an arbitrary number of points inside it, or on its edges, and then triangulating them according to the Delaunay criterion In general, they are non-conforming models, since points may be inserted on the edges of a triangle 

19. 19 Non-Nested Multiresolution Models Deal with irregularly distributed data Based on irregular simplicial meshes Built through mesh simplification algorithms: Refinement algorithms: start from the base mesh  0 and progressively refine it into a mesh using all available data points (the reference mesh  mm ) Decimation algorithms: progressively coarsen the reference mesh  mm until the base mesh  0 is obtained

20. 20 Simplification Algorithms Each simplification algorithm uses a specific type of modification Choice where to apply next modification can be driven by several criteria, e.g.: approximation error of the current representation shape of the simplexes number of simplexes involved in a modification possibility of applying independent sets of modifications

21. 21 Modifications in Non-Nested Models Insertion of a vertex P: remove a sub-mesh defined by the specific triangulation criterion (e.g., the Delaunay criterion) from mesh  re-triangulate the hole left in  by the sub-mesh with a set of simplexes incident at P

22. 22 Modifications in Non-Nested Models Deletion of a vertex P: remove the set of simplexes incident at P re-triangulate the polyhedron bounding the resulting hole, usually according to some criterion (e.g, the Delaunay criterion)

23. 23 Modifications in Non-Nested Models The reverse modification (never applied by a refinement algorithm) is called a vertex split Edge collapse (or contraction): shrinks an edge to a vertex replace an edge e with a vertex: either one of the endpoints of e, or a new vertex (e.g., the middle point of e) the n-simplexes incident at e are modified accordingly (some of them become (n-1)-dimensional simplexes)

24. 24 Modifications in Non-Nested Models Triangle (or tetrahedron) collapse (Hamann et a., 1994): contracts a triangle (tetrahedron) to a vertex can be expressed as two consecutive edge collapses Hierarchical 4-k meshes (Velho et al., 1999): multiresolution meshes based on two basic modifications: edge-swap removal of a vertex of degree 4

25. 25 Hierarchical 4-k Meshes A vertex removal can be expressed as a sequence of edge swaps followed by a degree-four vertex removal Described by a DAG in which a node has two parents and two or four children Similar properties to hierarchies of right triangles in terms of expressive power, growth, width and height

26. 26 Properties of Non-Nested Models A bounded width (and a linear growth) guaranteed by imposing restrictions on the modifications: –removing vertices with a bounded degree –collapsing edges with a bounded number of incident simplexes –inserting only vertices which affect a bounded number of cells A logarithmic height guaranteed by decimation algorithms which apply a maximal set of independent modifications simultaneously

27. 27 Expressive Power of Non-Nested Models Non-nested models have usually a lower expressive power than hierarchies of triangles: –modifications in a non-nested model involve a larger number of cells Edge-based multiresolution meshes have a lower expressive power than vertex-based ones: –modifications based on vertex split/edge removal tend to affect more cells than those based on vertex insertion/ deletion right triangles: #M + = 2 (non-conforming) or 4 (conforming) vertex-based: #M + = 5 or 6 on average edge-based: #M + = 8 or 10 on average