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

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
…
- geometry p  R3
- attributes s  Rm
normals, colors, texture coords, ...
Mesh simplification
43,000 faces
2,000
1,000
complex mesh, expensive
43 faces
Edge collapse
v1
v
v2
Selection?
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)
Squared distance to plane is quadric

Given f=(v1,v2,v3):
v
v3
n
v1
v2
Qf(v) = (nTv + d)2 = vT(nnT)v + 2dnTv + d2
=
vT(A)v + bT v + c
=
(A,b,c)
6 + 3 + 1  10 coefficients
Initialization of quadrics

For each vertex v in the original mesh:
Qf
Qf
Qf
Qf
v
Qf
Qf
Q (v )   Q (v )
v
f
f v
[Garland & Heckbert 1997]
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)
QEM for attributes
[Garland & Heckbert 1998]
p in R3 position
Projection in R3+m
v=(p,s)
s in Rm attributes
(p3,s3)
(p1,s1)
v’=(p’,s’)
Q(v) = | v – v’ |2
(p2,s2)
not geometrically closest
Resulting quadric
 p
v     R 3 m
s
T
 p
Q(v )   
s





T
A

 
  p  
     b 
 s  

 
 p
   c
s
dense (3+m) x (3+m) matrix
 quadratic space
Time&space complexity
Example
m
[G&H98]
geometry
0
10
+ color
3
28
+ normals
6
55
+ texture coord.
8
78
In general
m > 0 (4+m)(5+m)/2
Contribution: new quadric metric
Projection in
R3
!
v=(p,s)
(p3,s3)
(p1,s1)
v’=(p’,s’)
(p2,s2)
Q = geometric error + attribute error
=
| p - p’ |2
+  | s - s’ |2
Geometric error term
Zero-extended version of [Garland & Heckbert 1997]:
p
Qp
 nnT

0


 

 0
s
0  0

0  0
,
  

0  0
 dn 
 0 

 , d2
  


 0 
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
s' j p   gTj p  d j , p  R 3



positions 
on face 


T
1
T
2
T
3
T
p
p
p
n
1     s1, j 
  

1   g j   s2, j 
 

1
s
   3, j 
0   d j   0 
face normal
attribute gradient
attributes
on face
Attribute error term

Qs j v   s' j p   s j   g p  d j  s j
2
p
Qs j
 g j gTj

 0
 
T
 gj

 0
T
j

2
sj
0  gj
0
0
0
1
0
0
0  d j g j 
 

0  0 
2
, 
,
d
j



d
0
j
 

0  0 
New quadric
m
Q (v )  Qp v   Qs j v  
j 1
 nnT   g j g Tj  g1   gm   dn 

d
g

j
j


j


j

 
T
 d1
 2
2

g
1
, d  d j

,


j

 

I



 
T

d
m

 gm

 
m x m matrix is identity
 linear space
Time&space complexity
Example
m
[G&H98]
New Q
geometry
0
10
10
+ color
3
28
23
+ normals
6
55
35
+ texture coord.
8
78
43
In general
m > 0 (4+m)(5+m)/2
11+4m
Advantages of new quadric

Defined more intuitively

Requires less storage (linear)

Evaluates more quickly (sparse)

Results in more accurate simplification
Results: image mesh
original (79,202 faces)
simplified (1,000 faces)
[G&H98]
New quadric
Other improvements
Inspired by [Lindstrom & Turk 1998]
(details in paper)

Memoryless simplification
Qv = 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
Results: mesh with normals
original
(900,000 faces)
simplified (10,000 faces)
fuzzy
sharp
Q is just geometry Q includes normals
Wedge attributes
>1 attribute vector
per 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


Other improvements:



more intuitive, efficient, and accurate
memoryless simplification
volume preservation
Wedge-based quadrics