SIMPLIFICATION AND LEVEL-OF-DETAIL HIERARCHIES

• A basic technique for visualization of large-scale models is to provide a level-of-detail (LOD) hierarchy of each model. Thus, depending on the perceptual importance the model in the scene, we can choose a suitable level-of-detail for display.
• There two fundamentally different approaches here: we call them clustering and decimation approaches, respectively.
• Assuming a "vertex based" model (like in OpenGL), we can illustrate them as follows: In clustering, we collapse two vertices when they are close. In decimation approches, we pick a subset of the vertices for simplification.
• In decimation, we are usually constrained by adjacency requirements. In clustering, we have no such constraints. Thus it is more general, and can be applied to regular models such as images as well. For now, we focus on irregular or geometric models.
• Geometric modeling in CAD. Models are of two types: surface models or volumetric model.
• Can view them as simplicial complexes (to be defined).
• The task before us is to (1) represent them, (2) transmit them, (3) render them.
• In thinwire visualization, we want to combine (2) and (3). Here, we want to see the model progressively as the transmission is continuing.

• Fundamental technique here: Level-Of-Detail (LOD) representation. This was first introduced by Clark (1978).
• Here is an illustration using the bunny model:

  

• We will discuss the results in several papers: Hoppe (Progressive Mesh, Siggraph'96), Popovich and Hoppe (Progressive Simplicial Complexes, Siggraph'97), View-Dependent Simplification (Varshney et al, Hoppe), Multiresolution Adaptive Parameterization of Surfaces (Lee et al, Siggraph'00).
• Issues: given a model M₀, how do we construct various level of detailed versions? And how do we use them?
• Simplification Step: M₀ ® M₁ where M₁ is a simplified version of M₀.
• LOD Hierarchy
  

  M0 ® M1 ® ¼® Mh

• Let the input be M₀. We assume M₀ is a oriented 2-manifold.
• We define a pair of operators: ( Split)
  
   
  (u,v,w) and ( Collapse)
  
   
  (u,v,w) In split, we say ßplit vertex u to v,w", and in collapse, we say "collapse v,w to u". These operations are dual.
• This operator is defined for M₀ provided the result has the same genus. This is a restriction.

• Let V Í R^d.
• K is an abstract complex viewed as a set of subsets of V such that if s Î K,
  (i) any subset of s is in K
  (ii) |s| £ d+1.
• Each s Î K gives rise to an open |s|-1 dimensional cell, denoted ásñ.
• K is simplicial complex if all the cells are pairwise disjoint.

• Simplest application of clustering from Rossignac-Borrel [2].
• STEP 1: Grading of Vertices

  • Based on ``visual importance''
  • (A) Favor vertices with high probability of being in silhouette
  • (B) Favor vertices that bound large faces
  • (a) Estimated by inverse of maximum angle q of incident faces.
  • (b) Estimated by maximum length of incident edges.
  • Low [1] argues that cos(q/2) is a better estimate than 1/q.
• STEP 2: Triangulation of Faces
  - standard routine
• STEP 3: Clustering of Vertices
  - SIMPLEST: truncate low order bits of coordinates
• STEP 4: Synthesis of Representative Vertices
  - E.g. Center of mass of cluster
  - Choose vertex of maximum weight.
  - SIMPLEST: use truncated coordinates
• STEP 4: Elimination of Vertices, Edges and Faces
  - Eliminate duplicated trianges/edges/vertices
  - Edges or Triangles collapse to points
  - Eliminate point only if non isolated
  - Triangles collapse to edges
  - Eliminate only if the edge bounds a face in simplified model
• STEP 5: CLEANUP
  - result may not be ``valid 3D models''
• Evaluation
  - method is fast
  - no need to know surface topology
  - topology not preserved
  - grading of vertices is rather subjective
• REFINEMENTS
  - Local clustering: use vertices of large weight as centers of clustering coordinates

References

[1]

K. L. Low. Object representation in computer graphics. Honors year project report, National University of Singapore, Department of Information Systems and Computer Science, 1996.

[2]

J. Rossignac and P. Borrel. Multi-resolution 3D approximations for rendering. In Modeling in Computer Graphics, pages 455-465. Springer-Verlag, June-July 1993.