1. 1 L O D M O D E L S (2)...Construction through simplification... o Key issues: mselection of vertices to remove mtriangulation method mdegree of vertices merror estimation mheight of the resulting hierarchy

2. 2 L O D M O D E L S (2) Decimation of independent vertices

3. 3 L O D M O D E L S (2) Edge collapse Edge collapse on midpoint Edge collapse on endpoint

4. 4 L O D M O D E L S (2)...Construction through simplification... Illegal edge collapse: one or more triangles flip over because of collapse operation.

5. 5 L O D M O D E L S (2)...Construction through simplification... o Edge collapsing rules: mGreedy: collapse an edge at each step: Gthe shortest edge Gthe edge causing the least error increase Gan edge surrounded by almost coplanar faces mIndependent set: Gtwo edges are independent if they have disjoint influence regions Gselect a maximal set of independent edges and collapse them all together u Results similar to decimation: sremoving vertices of bounded degree guarantees bounded width and linear growth sremoving an independent set guarantees logarithmic height tin greedy collapse fragments may pile-up in a high hierarchy

6. 6 L O D M O D E L S (2)...Construction through simplification... Extension to 3D [Cignoni et al., 1997] o Edge collapse in a tetrahedrization: collapse an edge and the star of tetrahedra surrounding it o Edge selection as in 2D o Applicable to all kinds of volume data sets

7. 7 L O D M O D E L S (2)...Construction through simplification... Decimation s smaller influence regions s more regular triangles s always possible for any vertex t more complex update rules t geo-morphing difficult Collapse s simple update rules s supports geo-morphing t larger influence regions t slivers t collapse not always legal

8. 8 L O D M O D E L S (2) Data structures Relevant information on evolutionary models o Geometry: coordinates of vertices o Connectivity: triplets of vertices forming triangles o Topology: adjacency and incidence relations o Partial order relation o Additional information: accuracy, material, surface normal, etc.

9. 9 L O D M O D E L S (2) Data structures o Linear sequences: encode sequences of local modifications that produce a refined mesh from a coarse one o Explicit structures: encode the geometry, the topology of the components plus the partial order explicitly o Compressed structures: encode the partial order and the local modification which defines each component

10. 10 L O D M O D E L S (2)...Data structures... o Different data structures characterized by the amount of information stored o Trade-off between spatial complexity and efficiency mcompact data structures more suitable to storage and transmission mextended data structures more suitable to complex operations: Gselective refinement Gspatial queries o Compactness can be achieved by exploiting properties of special models

11. 11 L O D M O D E L S (2)...Data structures... Linear sequences: store sequences of local modifications that produce a refined mesh starting at a coarse mesh o List of vertices [Klein & Strasser, 1996] : mstore initial mesh plus sequence of vertices in suitable order meach vertex in the sequence is inserted iteratively to refine mesh mmesh is updated with Delaunay (implicit) rule at each insertion svery compact tsuitable only to explicit and parametric surfaces tlocal update requires numerical computation tselective refinement is computationally expensive tno connectivity, topological, and interference information maintained

12. 12 L O D M O D E L S (2)...Data structures... o List of triangles [Cignoni et al., 1995] : mstore all triangles appearing during refinement/simplification meach triangle is tagged with a life: range of accuracies through which it “survives” during refinement/simplification mlife is used to extract meshes at a given (uniform) LOD mextended to 3D for volume data [Cignoni et al., 1997] sapplicable to all kinds of surface sextraction of a uniform LOD very efficient smoderately compact tselective refinement not possible tno topology and interference maintained

13. 13 L O D M O D E L S (2)...Data structures... An example of a linear sequence o Progressive Meshes [Hoppe, 1996/98] : mstore initial coarse mesh plus a sequence of vertex splits (inverse operation of edge collapse) in suitable order meach vertex split is maintained in compressed format ma uniform LOD is extracted by expanding the sequence up to the desired level scompact sextraction of a uniform LOD efficient without numerical computation ssuitable additional structures to maintain attributes tselective refinement needs additional information tno connectivity, topology and interference maintained

14. 14 L O D M O D E L S (2)...Data structures... o Sequence of vertex insertions [De Floriani et al., 1998]: mstore initial mesh plus a set to vertex insertions in suitable order manalogous to PM but based on vertex insertion rather than split msupports arbitrary triangulations including: GDelaunay triangulation GConstrained Delaunay triangulation GData dependent triangulations smore flexible than PM tslightly more expensive than PM

15. 15 L O D M O D E L S (2)...Data structures... An example of an explicit structure Explicit MT representation [De Floriani et al., 1996/1998] o Geometry: vertex coordinates o Connectivity: for each triangle, references to its three vertices o Partial order: DAG structure with arcs labeled with the attributes of the parent-child relation o Additional information: for each triangle, its approximation error

16. 16 L O D M O D E L S (2)...Data structures... Compressed structures: o Key ideas: meach component is a local modification that can be encoded in compressed form mthe partial order is somehow encoded explicitly o Different structures for models based on edge collapse (PM): mVertex tree [Xia et al., 1996/97] mPM hierarchy [Hoppe, 1997] mCollapse DAG [Gueziec et al., 1998] o One structure for models based on vertex decimation: mImplicit MT [De Floriani et al., 1997/98]

17. 17 L O D M O D E L S (2)...Data structures... Compressed hierarchies for PMs Vertex tree (forest) [Xia et al., 1996/97] mBinary forest of vertices Gtopmost level: vertices of the base mesh Gchildren of a vertex: vertices resulting from split mVertex split can be encoded in compressed form  Rule for selective refinement: a vertex can split if and only if all boundary vertices of the corresponding fragment belong to the current mesh  need for additional interference links mFor each vertex: Gparent-child relation in forest Gadditional links to vertices that must exist in order to allow split sMore compact than explicit MT tLess general than explicit MT tNo control on accuracy of triangles

18. 18 L O D M O D E L S (2)...Data structures... Extended PM structure [Hoppe, 1997] mSame hierarchy as in [Xia et al.], but vertices are renamed at each split mArray of vertices - for each vertex: Gcompressed split information Gparent-child information Gconstant number of dependencies for split/collapse with triangles and vertices of influence region mArray of faces - for each face: Gindexes of vertices Gindexes of (local) neighbors sNo variable-length fields sLocal topology tHigh storage cost tIllegal split may occur

19. 19 L O D M O D E L S (2)...Data structures... Example of illegal split

20. 20 L O D M O D E L S (2)...Data structures... Collapse DAG [Gueziec et al., 1998] mDAG of edge collapse operations: one node per collapse  Edge collapse always on endpoint  one vertex survives mAlmost the same hierarchy as MT mDependencies maintained in hash tables uCost and performances similar to [Xia et al.]

21. 21 L O D M O D E L S (2)...Data structures... Compressed hierarchies for MT based on decimation Implicit MT [De Floriani et al., 1997/98] mEach node corresponds to vertex insertion/decimation mPartial order of nodes is maintained mArray of vertices, each entry storing vertex coordinates mArray of arcs - for every arc a: Gindexes of source and destination nodes Gindex of next arc with same destination node mArray of nodes - for every node n: Gindex of first outgoing and incoming arcs Gnumber of outgoing arcs Gmaximum error of its triangles Gcompressed information to perform vertex insertion/decimation

22. 22 L O D M O D E L S (2)...Data structures... …Implicit MT... o Storage cost: s3.5 times cheaper than Explicit MT uComparable with vertex tree, and collapse DAG o Selective refinement: tSlower than Explicit MT  Accuracy not stored for each triangle  resulting mesh not minimal (about twice the size of mesh obtained from Explicit MT) uComparable with vertex tree, and collapse DAG

23. 23 L O D M O D E L S (2)...Data structures... Hypertriangulation [Cignoni et al., 1995/97] o Interpretation of an MT in a higher dimensional space: triangles of a new node are lifted along a “resolution axis” and welded on floor at the node boundary

24. 24 L O D M O D E L S (2)...Data structures......Hypertriangulation... o Data structure based on topological information (global adjacency) stopology is maintained explicitly sefficient support of queries requiring navigation of domain sinteractive mesh editing thigh storage cost tno interference information tselective refinement super-linear and possible only for some LOD functions

25. 25 L O D M O D E L S (2) Selective refinement o Top-down traversal of hierarchy: mon generic MT [Puppo, 1996, De Floriani et al., 1997/98] mon PM [Hoppe, 1997, Xia et al., 1997] mon restricted quadtrees [Von Herzen and Barr, 1987, Gross et al., 1996] mon hierarchy of right triangles [Evans et al., 1997, Duchaineau et al., 1997] mon hierarchy of irregular triangles [De Floriani & Puppo, 1995] o Bottom-up traversal of hierarchy: mon PM [Xia et al., 1996] mon hierarchy of right triangles [Lindstrom et al., 1996] o Breadth-first traversal of surface: mon hypertriangulation [Cignoni et al., 1995/97] mon hierarchical triangulation [De Floriani & Puppo, 1995]

26. 26 L O D M O D E L S (2) …Selective refinement... Top-down on MT o Visit DAG starting at cut just below its root o Recursively move cut below a node n when a triangle labeling an arc entering n has accuracy worse than LOD Algorithm on explicit MT: sOptimally efficient (on MT with linear growth) sApplicable to all models tNeeds expensive data structure Algorithm on implicit MT: sLighter data structure tSlower tOutput mesh larger tApplicable only to special models

27. 27 L O D M O D E L S (2) …Selective refinement... Top-down on PM (vertex forest or DAG) o Visit forest/DAG starting at topmost level o Recursively expand a vertex when accuracy worse than LOD o Find all vertices that constrain selected vertices o Perform all splits corresponding to selected vertices in the default order s Fast t Needs expensive data structure

28. 28 L O D M O D E L S (2) …Selective refinement... Top-down on restricted quadtrees o Visit tree starting at its root o Recursively expand a quadrant when accuracy worse than LOD o Balance quadtree o Triangulate each quadrant according to its neighbors s Fast t No control on error of triangles generated a posteriori t Needs suitable procedures for neighbor finding

29. 29 L O D M O D E L S (2) …Selective refinement... Top-down on tree of right triangles o Visit tree starting at its root o Recursively refine a triangle when accuracy worse than LOD o Propagate vertex dependencies to obtain a conforming mesh s Easy and fast if all vertex dependencies are available Top-down on trees of irregular triangles o Visit tree starting at its root o Recursively expand a triangle when accuracy worse than LOD o Triangulate mesh a posteriori to make it conforming s Easy and fast t No control on error of triangles generated a posteriori

30. 30 L O D M O D E L S (2) …Selective refinement... Bottom-up on PM (vertex forest): o Visit forest starting at leaves o Recursively discard leaves that can be collapsed o Perform splits corresponding to selected vertices in default order Bottom-up on hierarchy of right triangles: o Visit tree starting at leaves o Recursively merge sibling leaves when possible t All vertices must be analyzed even to extract a coarse LOD

31. 31 L O D M O D E L S (2) …Selective refinement... Breadth-first traversal of domain on hypertriangulations and on hierarchical triangulations o Start with a triangle where highest LOD is required o Incrementally add triangles adjacent to the boundary of current triangulation through global adjacencies o Each time a boundary edge e is crossed, select a triangle which is as coarse as possible, satisfied the LOD, and is compatible with current mesh s Supports dynamic local refinement/abstraction of detail (resolution editing) s Ideal for propagating LOD through the surface rather than through space t Applicable only for LOD monotonically decreasing with distance from a given point t Computational complexity is super-linear t Needs expensive data structure

32. 32 L O D M O D E L S (2) Discussion Taxonomy o Modeling issues: madaptivity and data distribution mshape of triangles mstorage o Processing issues: mselective refinement mnavigation mgeometric queries mconstruction

33. 33 L O D M O D E L S (2)...Discussion... Types of surfaces supported o All types: mMT, PM, HyT, quaternary triangulations o Explicit and parametric surfaces only: mquadtrees, restricted quadtrees, hierarchies of right triangles (problems with trimming curves) mAdaptive hierarchical triangulations

34. 34 L O D M O D E L S (2)...Discussion... Adaptivity and data distribution o Adaptivity: possibility to increase LOD only where necessary o Data distribution: irregular distribution vs regular grid More adaptive, more general Less adaptive, less general MT PM Adaptive hierarchical triangulations Triangle quadtrees Quadtree, hierarchy of right triangles

35. 35 L O D M O D E L S (2)...Discussion... Expressive power o Number of different meshes that can be extracted o Possibility to adapt a mesh to arbitrary LOD o Ratio accuracy/size More expressive, higher ratio Less expressive, lower ratio Explicit MT PM, Implicit MT Triangle quadtree Hierarchy of right triangles, quadtree,restricted quadtree, Adaptive hierarchical triangulations

36. 36 L O D M O D E L S (2)...Discussion... Shape o Regions with regular shape are desirable o Slivers cause numerical errors, and unpleasant visual effects More regular Less regular Quadtree, restricted quadtree, hierarchy of right triangles Triangle quadtree MT PM Adaptive hierarchical triangulations

37. 37 L O D M O D E L S (2)...Discussion... Storage More expensive Less expensive Adaptive hierarchical triangulations Explicit MT, vertex hierarchies Implicit MT Linear sequences (linear PM) Quadtree, triangle quadtree, hierarchy of right triangles

38. 38 L O D M O D E L S (2)...Discussion... Selective refinement o High speed is desirable More efficient Less efficient Explicit MT PM vertex hierarchies Implicit MT Hierarchy of right triangles Triangle quadtree, restricted quadtree Linear sequences (linear PM)

39. 39 L O D M O D E L S (2)...Discussion... Progressive transmission o Compact coding and fast decompression at client are relevant More efficient Less efficient Quadtree, hierarchy of right triangles Linear PM Implicit MT, PM vertex hierarchies Other linear sequences Explicit MT Restricted quadtree, quaternary triangulation

40. 40 L O D M O D E L S (2)...Discussion... Navigation o Ability to move across surface and through LOD More efficient Less efficient Quadtree, hierarchy of right triangles Restricted quadtree, triangle quadtree Adaptive hierarchical triangulation Explicit MT (with adjacencies) Implicit MT, PM vertex hierarchies Linear sequences

41. 41 L O D M O D E L S (2)...Discussion... Geometric queries o Point location, windowing, segment intersection, ray shooting, clip, etc… More efficient Less efficient Quadtree, hierarchy of right triangles Restricted quadtree, triangle quadtree Adaptive hierarchical triangulation Explicit MT Implicit MT, PM vertex hierarchies Linear sequences

42. 42 L O D M O D E L S (2)...Discussion... Construction o Lower time complexity of construction algorithm is desirable o Easy-to-code algorithms are desirable More complex, more difficult Less complex, less difficult Adaptive hierarchical triangulations MT, PM hierarchy, vertex hierarchies Linear sequences (linear PM) Triangle quadtree Quadtree, hierarchy of right triangles