1 Note on QEM Implementation. 2 Algorithm Summary.
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Transcript of 1 Note on QEM Implementation. 2 Algorithm Summary.
1
Note on QEM Implementation
2
Algorithm Summary
3
2D Version
Think clearly on how 2D is done and
extrapolate how this will be done in 3D
4
Required on Data Structure
VertexPosList edgesList pairs …?!
EdgeVertex v1, v2
Preparation:Modify your sketch with the proposed data structureAble to remove points on contour
5
Visualizing Ellipsoids
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Visualizing QCenter of ellipse:
Reference
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Q for Initial Vertices
l 1: a 1
x+b 1y+c 1
=0 l2 : a
2 x+b2 y+c
2 =0
v
d2 (v,l1) = (a1x+b1y+c1)2 = (p1T v)2 = vT(p1p1
T) v
d2 (v,l2) = … = vT(p2p2T) v
Q (v) = vT(p1p1T + p2p2
T) v vpy
x
cbacybxa T1111111
1
1 :ionnormalizat 22 ii ba
8
Error Metric at Vertices
23323
2221312
211
332313
232212
131211
22
222
is
)1()(
qyqyqxqxyqxqQvve
symmetric
qqq
qqq
qqq
Q
Qvvvppv
vpvcbayxv
T
TTT
TT
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Optimal Position of V
used)not is 3(equation
0
0
1
0222
0222
: optimalFor
332313
232212
131211
232212
131211
y
x
qqq
qqq
qqq
qyqxqy
e
qyqxqx
e
v
23
13
2212
1211
q
q
y
x
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Valid Pairs (1:edge)
v1
v2
Q1=Q(v1) Q2
For each edge (v1v2): Compute v-bar from Q1 and Q2
Evaluate the contracting cost: vT(Q1+Q2)v
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Valid Pair (2: aggregation)
C(n,2) to select the pair whose distance is less than
v1
v2
Q1
Q2
Computation of v-bar and cost: same as case 1
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Info Stored in a Pair
(v1,v2, Q1, Q2, v-bar, cost)Sorted by cost value (start contracting from minimum cost)More than one pair may be associated with the same vertexWhen a vertex is changed, updates need to be done …
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Example: update
v1,Q1
v2,Q2
v3,Q3
Pair contraction
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Example: update
v1,Q1
v2,Q2
v3,Q3
v1, Q1+Q2
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Example: update
v3,Q3
v1, Q1+Q2
Update all pairs involving v1
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Iteration
Least cost pair: v1 to stay, v2 to die Update v1 position New Q for v1 = Q1+Q2 Replace occurrence of
v2 in all edges of v2 as v1
[doubly link] make v1 aware of these new edges
remove degenerate edge (for case1)
Update the pairs involving v1
v1
v2
Q1 Q2
v3
Q3
v
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Numerical ExampleL1: y-5=0L2: x+y-8 = 0L3: x-5=0
(3,5)
(5,3)
l1l2
l3
v1
v2
5749
4
9
2505
000
501
3244
4
4
5794
9
4
3244
4
4
2550
510
000
5
0
1
,,
5
1
0
21
21
21
23
21
21
21
21
33222
23
21
21
21
21
21
21
21
22111
3
28
212
1
21
TT
TT
ppppQ
ppppQ
ppp
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Example (cont)
11141313
1321
1312
1 :cost gContractin
3
13
13
13
21
12: Optimal
1141313
1321
1312
:gcontractinAfter
313
313
313
313
21
vQv
yxy
xv
QQQ
T
(3,5)
(5,3)
l1l2
l3
v1
v2
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Line Equation Thru Two Pts
Reference