1. Level of Detail: Generating LODs David Luebke University of Virginia

2. Generating LODs Simplification operator: –Cell collapse –Vertex removal –Edge collapse Full edge collapse Half edge collapse Vertex-pair merge a.k.a. “virtual edge collapse”

3. Generating LODs Simplification operator: –Cell collapse –Vertex removal –Edge collapse Full edge collapse Better fidelity (show why) Half edge collapse Less memory Sort vertices, tris into VAR array for fast rendering Vertex-pair merge a.k.a. “virtual edge collapse” Merge separate objects

4. Generating LODs Simplification algorithm –Outer optimization: where to simplify No optimization Ex: uniform grid cells Greedy optimization Ex: floating cell clustering Ex: sort edge collapses by error, resort on collapse Lazy optimization Ex: sort edge collapses by error, dirty bit on collapse –Inner optimization: how to simplify Ex: placement of new vertex, retriangulation –Coming up: a practical example algorithm

5. Generating LODs Measuring error –Hausdorff distance –METRO –Quadrics (coming up) –Image-based ideas

6. Quadric Error Metric Minimize distance to all planes at a vertex Plane equation for each face: 0 :p  DCzByAx v           1 z y x DCBA T vp Distance to vertex v :

7. Squared Distance At a Vertex    )( ))(( vplanesp TT vppv    )( )( v p TT vppv v v vplanesp TT            )(    )( 2 )()( v p T vpv

8. Quadric Derivation (cont’d) pp T is simply the plane equation squared: The pp T sum at a vertex v is a matrix, Q:  vQvv T  )(              2 2 2 2 DCDBDAD CDCBCAC BDBCBAB ADACABA pp T

9. Construct a quadric Q for every vertex Q1Q1 Q2Q2 v2v2 v1v1 Q 1 T 1 Qvvcost edge  Sort edges based on edge cost –Suppose we contract to v 1 : –v 1 ’s new quadric is simply: Q 21 QQQ  The edge quadric: Using Quadrics

10. Optimal Vertex Placement Each vertex has a quadric error metric Q associated with it –Error is zero for original vertices –Error nonzero for vertices created by merge operation(s) Minimize Q to calculate optimal coordinates for placing new vertex –Details in paper –Authors claim 40-50% less error

11. Boundary Preservation To preserve important boundaries, label edges as normal or discontinuity For each face with a discontinuity, a plane perpendicular intersecting the discontinuous edge is formed. These planes are then converted into quadrics, and can be weighted more heavily with respect to error value.

12. Preventing Mesh Inversion Preventing foldovers: Calculate the adjacent face normals, then test if they would flip after simplification If foldver, that simplification can be weighted heavier or disallowed. 10 54 6 1 3 2 9 A 7 8 54 6 3 2 8 9 A merge

13. Quadric Error Metric Pros: –Fast! (70K poly bunny to 100 polygons in seconds) –Good fidelity even for drastic reduction –Robust -- handles non-manifold surfaces –Aggregation -- can merge objects

14. Quadric Error Metric Cons: –Introduces non-manifold surfaces (bug or feature?) –Tweak factor t is ugly Too large: O(n 2 ) running time Correct value varies with model density –Needs further extension to handle color (7x7 matrices)

15. Measuring Error Most LOD algorithms measure error geometrically –What is the distance between the original and simplified surface? –What is the volume between the surfaces? –Etc Really this is just an approximation to the actual visual error, which includes: –Color, normal, & texture distortion –Importance of silhouettes, background illumination, semantic importance, etc etc etc

16. Measuring Geometric Error Hausdorff distance Average distance Surface-surface vs vertex-surface vs vertex-plane vs vertex-vertex Quadric error metrics: vertex-plane measure that works well in practice