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New Quadric Error Metric for Mesh Simplification with Appearance Attributes

A detailed overview of a novel methodology for simplifying complex meshes with appearance attributes, employing a new quadric error metric for enhanced accuracy and efficiency. The approach involves utilizing quadric matrices to minimize distances and preserve attributes during simplification. The new scheme outperforms existing techniques, resulting in more accurate simplifications with reduced storage requirements and quicker evaluation.

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New Quadric Error Metric for Mesh Simplification with Appearance Attributes

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  1. New quadric metric for simplifying meshes with appearance attributes Hugues Hoppe Microsoft Research IEEE Visualization 1999

  2. Triangle meshes Mesh V F Vertex 1 x1 y1 z1 Vertex 2 x2 y2 z2 … Face 1 2 3 Face 3 2 4 Face 4 2 7 … p R3 - geometry - attributesnormals, colors, texture coords, ... s Rm

  3. Mesh simplification 43,000 faces 2,000 1,000 43 faces complex mesh, expensive

  4. Selection? Edge collapse v1 v v2

  5. Previous selection techniques • Heuristics (edge lengths, …) • Residuals at sample points[Hoppe et al 1993], [Kobbelt et al 1998] • Tolerance tracking[Gueziec 1995], [Bajaj & Schikore 1996],[Cohen et al 1997] • Quadric error metric (QEM)[Garland & Heckbert 1997,1998]very fast, reasonably accurate

  6. Review of QEM [Garland & Heckbert 1997] • Minimize sum of squared distances to planes (illustration in 2D)

  7. n (nTv + d) Qf(v) = (nTv + d)2 = vT(nnT)v + 2dnTv + d2 = vT(A)v + bT v + c = ( A , b , c ) 6 + 3 + 1  10 coefficients Squared distance to plane is quadric v • Given f=(v1,v2,v3): v3 v1 v2

  8. Qf Qf Qf v Qf Qf Qf Initialization of quadrics • For each vertex v in the original mesh: [Garland & Heckbert 1997]

  9. Simplification using quadrics v1 v v2 Qv(v) = Qv1(v) + Qv2(v) = (A,b,c) vmin = minv Qv(v) = -A-1b Prioritize edge collapses by Qv(vmin)

  10. Projection inR3+m v’=(p’,s’) not geometrically closest QEM for attributes [Garland & Heckbert 1998] position p in R3 s in Rm attributes v=(p,s) (p3,s3) (p1,s1) (p2,s2) Q(v) = | v – v’ |2

  11. Resulting quadric dense (3+m) x (3+m) matrix  quadratic space

  12. Time&space complexity

  13. v’=(p’,s’) Q = geometric error + attribute error = | p - p’ |2 +  | s - s’ |2 Contribution: new quadric metric v=(p,s) Projection inR3 ! (p3,s3) (p1,s1) (p2,s2)

  14. Geometric error term Zero-extended version of [Garland & Heckbert 1997]: p s

  15. New quadric metric (cont’d) v=(p,s) (p3,s3) (p1,s1) (p2,s2) (p’,s’) Q = geometric error + attribute error = | p - p’ |2 +  | s - s’ |2 s’(p) is linear  still quadratic

  16. Predicted attribute value positionson face attributeson face face normal attribute gradient

  17. Attribute error term p sj

  18. New quadric m x m matrix is identity  linear space

  19. Time&space complexity

  20. Advantages of new quadric • Defined more intuitively • Requires less storage (linear) • Evaluates more quickly (sparse) • Results in more accurate simplification

  21. simplified (1,000 faces) [G&H98] New quadric Results: image mesh original (79,202 faces)

  22. Other improvements Inspired by [Lindstrom & Turk 1998] (details in paper) • Memoryless simplificationQv = Qv1 + Qv2 re-define Q • Volume preservation linear constraint (Lagrange multiplier)

  23. Results: mesh with color original (135,000 faces) simplified (1,500 faces) [G&H98] New scheme

  24. fuzzy sharp Q is just geometry Q includes normals Results: mesh with normals original(900,000 faces) simplified (10,000 faces)

  25. Wedge attributes >1 attribute vectorper vertex vertex wedge Qv(p, s1 , … , sk)

  26. Results: wedge attributes original (43,000 faces) simplified (5,000 faces)

  27. Results: radiosity solution original(300,000 faces) simplified(5,000 faces)

  28. Summary • New quadric error metric • more intuitive, efficient, and accurate • Other improvements: • memoryless simplification • volume preservation • Wedge-based quadrics

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