Surface Simplification Using Quadric Error Metrics

Michael Garland Paul S. Heckbert

Why Simplification? Full complexity of models is not

always required Reduce the computational cost Get fast processing speed

Objective Find a surface simplification

algorithm that can be used in rendering systems for multiresolution model Efficient Good quality Generality

Join unconnected regions of the model together ---- aggregation

Categories of surface simplification algorithms Vertex Decimation Vertex Clustering Iterative Edge Contraction

Vertex Decimation Basic idea

Iteratively selects a vertex for removal, removes all adjacent faces, and retriangulates the resulting hole.

Pros: provide reasonable efficiency and quality

Cons: only limited to manifold surfaces

Vertex Clustering Basic idea

A bounding box is placed around the original model and divided into grids

Vertices within each cell are clustered together into a single vertex and model faces are updated accordingly

Pros: Fast Can make drastic topological alterations

Cons Output quality is very low

Iterative Edge Contraction

Cons: do not support aggregation

Assumptions Model only consists of triangles The topology of the model need

not to be maintained In a good approximation, points do

not move far from their original positions

Algorithm overview Iteratively contract vertex pairs ( a

generalization of edge contraction) As the algorithm proceeds, a

geometric error approximation is maintained at each vertex of the model which is represented using quadric matrices

The algorithm proceeds until the simplification goals are satisfied.

Pair Contraction

Advantages of pair contraction Achieve aggregation, which can

join previously unconnected regions of the model together

Algorithm is less sensitive to the mesh connectivity of the original model

Edge contraction vs. pair contraction

Regular grid of 100 closely spaced cubes

Approximation using edge contraction

Approximation using pair contraction

Pair Selection Select the set of valid pairs at

initialization time A pair (V1, V2) is a valid pair for

contraction if either: (V1, V2) is an edge, or ||V1 – V2|| < t, where t is a threshold

parameter

Pair contraction Initially, each vertex is associated with

the set of pairs of which it is a member After performing contraction (V1, V2) → V1

V1 acquires all the edges that were linked to V2

Merge the set of pairs from V2 into its own set and remove duplicate pairs

Error approximation Associate a symmetric 44 matrix Q

with each vertex The error at vertex v is defined as

the quadratic form: ∆(v) = vTQv. For a given contraction (v1, v2) → v’,

a new matrix Q’ needs to be derived to approximate the error at v’. Q’ = Q1 + Q2

Find the position for v’

Simply select either v1, v2, or (v1+v2)/2 which produces the lowest value of error at v’ (∆(v’) = v’TQ’v’).

Find a position for v’ which minimizes ∆(v’)

1

0

0

0

1000

1

34332313

24232212

14131211

qqqqqqqqqqqq

V

Compute the initial Q matrices Associate a set of planes with each

vertex Define the error of the vertex as

the sum of squared distances to the planes 2

)(

1

vplanesp

TT

zyxVV pvvv

Here p=[a b c d]T represents the plane defined by equation ax+by+cz+d=0, and a2+b2+c2=1

Quadratic form

VKVVVplanespp

T

)(

)(

2

2

2

2

dcdbdad

cdcbcac

bdbcbab

adacaba

ppTpK

Kp is defined as “fundamental error quadric”.

Algorithm summary Compute the Q matrices for all the initial vertices Select all valid pairs and compute the optimal

contraction target v’ for each valid pair (v1, v2). The cost of contracting the pair is computed as:

v’T(Q1+Q2)V’ Place all the pairs in a heap keyed on cost with

the minimum cost pair at the top Iteratively remove the pair (v1, v2) of the least

cost from the heap, contract this pair and update the costs of all valid pairs involving v1, v2.

Approximation evaluation The approximation error Ei of the

simplified model Mi is defined as:

in Xvn

Xvi

ini MvdMvd

XXE ,,

1 22

Mn ---- the initial model. Xn ---- sets of points sampled on model Mn

Xi ---- sets of points sampled on model Mi

pvMvd Mp min),(

Result(1) ---- an example sequence of approximations

An example sequence of approximations generated by the algorithm. The entire sequence was constructed in about one second.

5,804 faces

994 faces

532 faces

248 faces

64 faces

Result(2) – sample running times

Time needed to make a 10 face approximation of the given model.

Result(3) – effect of optimal vertex placement

Choosing an optimal position can significantly reduce approximation error.

Result(4) ---- bunny model

69,451 triangles

1,000 triangles

100 triangles1.4% of original

size0.14% of original size

Result(5) ---- Terrain model

199,114 faces

999 faces (46 secs)

Result(6) ---- comparison between different methods

Original model

Uniform Vertex Clustering

Edge Contractions

Pair Contractions

(4,204 faces)

(262 faces)

(250 faces)

(250 faces)

Result(7) ---- level surfaces of error quadrics

1000 face approximation

250 face approximation

Result(8) ---- initially valid pairs

Result(9) -- effect of pair thresholds

Conclusion

This paper presents a surface simplification algorithm using iterative pair contractions and quadric error metrics.

The algorithm can provide high efficiency, high quality and high generality.

Future works Use a more sophisticated adaptive

scheme to select valid pairs Take the color of surface into

account