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New quadric metric for simplifying meshes with appearance attributes. Hugues Hoppe Microsoft Research IEEE Visualization 1999. Triangle meshes. Mesh. V. F. Vertex 1 x 1 y 1 z 1 Vertex 2 x 2 y 2 z 2 …. Face 1 2 3 Face 3 2 4 Face 4 2 7 …. p R 3. - geometry
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New quadric metric for simplifying meshes with appearance attributes Hugues Hoppe Microsoft Research IEEE Visualization 1999
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
Mesh simplification 43,000 faces 2,000 1,000 43 faces complex mesh, expensive
Selection? Edge collapse v1 v v2
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
Review of QEM [Garland & Heckbert 1997] • Minimize sum of squared distances to planes (illustration in 2D)
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
Qf Qf Qf v Qf Qf Qf Initialization of quadrics • For each vertex v in the original mesh: [Garland & Heckbert 1997]
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)
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
Resulting quadric dense (3+m) x (3+m) matrix quadratic space
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)
Geometric error term Zero-extended version of [Garland & Heckbert 1997]: p s
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
Predicted attribute value positionson face attributeson face face normal attribute gradient
Attribute error term p sj
New quadric m x m matrix is identity linear space
Advantages of new quadric • Defined more intuitively • Requires less storage (linear) • Evaluates more quickly (sparse) • Results in more accurate simplification
simplified (1,000 faces) [G&H98] New quadric Results: image mesh original (79,202 faces)
Other improvements Inspired by [Lindstrom & Turk 1998] (details in paper) • Memoryless simplificationQv = Qv1 + Qv2 re-define Q • Volume preservation linear constraint (Lagrange multiplier)
Results: mesh with color original (135,000 faces) simplified (1,500 faces) [G&H98] New scheme
fuzzy sharp Q is just geometry Q includes normals Results: mesh with normals original(900,000 faces) simplified (10,000 faces)
Wedge attributes >1 attribute vectorper vertex vertex wedge Qv(p, s1 , … , sk)
Results: wedge attributes original (43,000 faces) simplified (5,000 faces)
Results: radiosity solution original(300,000 faces) simplified(5,000 faces)
Summary • New quadric error metric • more intuitive, efficient, and accurate • Other improvements: • memoryless simplification • volume preservation • Wedge-based quadrics