New quadric metric for simplifying meshes with appearance attributes

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New quadric metric for simplifying meshes with
appearance attributes

• Hugues Hoppe
• Microsoft Research
• IEEE Visualization 1999

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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
- geometry - attributes normals, colors,
texture coords, ...
p ? R3
s ? Rm
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Mesh simplification
43,000 faces
2,000
1,000
43 faces
complex mesh, expensive
4
Edge collapse
v1
v
v2
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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

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Review of QEM
Garland Heckbert 1997

• Minimize sum of squared distances to planes

(illustration in 2D)
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Squared distance to plane is quadric
v

• Given f(v1,v2,v3)

v3
v1
v2
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Initialization of quadrics

• For each vertex v in the original mesh

Garland Heckbert 1997
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Simplification using quadrics
v1
v
v2
Qv(v) Qv1(v) Qv2(v) (A,b,c)
vmin minv Qv(v) -A-1 b
Prioritize edge collapses by Qv(vmin)
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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
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Resulting quadric
dense (3m) x (3m) matrix
? quadratic space
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Timespace complexity
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Contribution new quadric metric
v(p,s)
Projection in R3 !
(p3,s3)
(p1,s1)
(p2,s2)
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Geometric error term
Zero-extended version of Garland Heckbert
1997
p
s
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New quadric metric (contd)
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
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Predicted attribute value
positionson face
attributeson face
face normal
attribute gradient
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Attribute error term
p
sj
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New quadric
m x m matrix is identity
? linear space
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Timespace complexity
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Advantages of new quadric

• Defined more intuitively
• Requires less storage (linear)
• Evaluates more quickly (sparse)
• Results in more accurate simplification

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Results image mesh
original (79,202 faces)
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Other improvements

• Inspired by Lindstrom Turk 1998
• (details in paper)
• Memoryless simplification Qv Qv1 Qv2
  re-define Q
• Volume preservation linear constraint (Lagrange
  multiplier)

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Results mesh with color
original (135,000 faces)
simplified (1,500 faces)
GH98
New scheme
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Results mesh with normals
original(900,000 faces)
simplified (10,000 faces)
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Wedge attributes
gt1 attribute vectorper vertex
vertex
wedge
Qv(p, s1 , , sk)
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Results wedge attributes
original (43,000 faces)
simplified (5,000 faces)
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Results radiosity solution
original(300,000 faces)
simplified(5,000 faces)
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Summary

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

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Future work

• Measuring parametric error for simplification of
  texture coordinates.