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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

New quadric metric for simplifying meshes with appearance attributes

Hugues Hoppe

Microsoft Research

IEEE Visualization 1999

triangle meshes
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
Mesh simplification

43,000 faces

2,000

1,000

43 faces

complex mesh, expensive

previous selection techniques
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
Review of QEM

[Garland & Heckbert 1997]

  • Minimize sum of squared distances to planes

(illustration in 2D)

squared distance to plane is quadric

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

initialization of quadrics

Qf

Qf

Qf

v

Qf

Qf

Qf

Initialization of quadrics
  • For each vertex v in the original mesh:

[Garland & Heckbert 1997]

simplification using quadrics
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)

qem for attributes

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
Resulting quadric

dense (3+m) x (3+m) matrix

 quadratic space

contribution new quadric metric

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
Geometric error term

Zero-extended version of [Garland & Heckbert 1997]:

p

s

new quadric metric cont d
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
Predicted attribute value

positionson face

attributeson face

face normal

attribute gradient

new quadric
New quadric

m x m matrix is identity

 linear space

advantages of new quadric
Advantages of new quadric
  • Defined more intuitively
  • Requires less storage (linear)
  • Evaluates more quickly (sparse)
  • Results in more accurate simplification
results image mesh

simplified (1,000 faces)

[G&H98]

New quadric

Results: image mesh

original (79,202 faces)

other improvements
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
Results: mesh with color

original (135,000 faces)

simplified (1,500 faces)

[G&H98]

New scheme

results mesh with normals

fuzzy

sharp

Q is just geometry

Q includes normals

Results: mesh with normals

original(900,000 faces)

simplified (10,000 faces)

wedge attributes
Wedge attributes

>1 attribute vectorper vertex

vertex

wedge

Qv(p, s1 , … , sk)

results wedge attributes
Results: wedge attributes

original (43,000 faces)

simplified (5,000 faces)

results radiosity solution
Results: radiosity solution

original(300,000 faces)

simplified(5,000 faces)

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