The Visibility Problem and Binary Space Partition - PowerPoint PPT Presentation

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The Visibility Problem and Binary Space Partition

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The Visibility Problem and Binary Space Partition The Visibility Problem Z-buffering Draw objects in arbitrary order For each pixel, maintain depth ( z ) Only ... – PowerPoint PPT presentation

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Title: The Visibility Problem and Binary Space Partition


1
The Visibility ProblemandBinary Space Partition
2
The Visibility Problem
b
a
c
e
d
3
Z-buffering
  • Draw objects in arbitrary order
  • For each pixel, maintain depth (z)
  • Only draw pixel if new z is closer
  • Instead
  • draw objects in order, from back to front
  • (painters algorithm)

4
Binary Planar Partitions
0
b
0
1
2
ab
cde
a
c
3
de
a
b
c
2
e
d
e
d
5
Painters Algorithm
0
b
0
1
2
a
c
3
a
b
c
2
e
d
e
d
6
Binary Planar Partitions
b1
b
b2
a
c
7
Auto-partitions
a
b1
b1
c
b2,c
b
b2
b2
a
c
8
Auto-partitions
a
b1
b1
b2
b2,c
b2
c1
c2
a
c1
c
c2
9
Auto-partitions
123n n(n1)/2 O(n2)
10
Binary Planar Partitions
  • Goal
  • Find binary planer partition,
  • with small number of fragmentations

11
Random Auto-Partitions
  • Choose random permutation of segments
  • (s1, s2, s3,, sn)
  • While there is a region containing more than one
    segment,
  • separate it using first si in the region

12
Random Auto-Partitions
u
v1
v4
v3
v2
  • u can cut v4 only if u appears before v1,v2,v3,v4
    in random permutation
  • P(u cuts v4) 1/5

13
Random Auto-Partitions
u
v1
v4
v3
v2
  • Enumber of cuts u makes
  • Enum cuts on right Enum cuts on left
  • ECv1Cv2 ECt1Ct2
  • ECv1 ECv2 ECt1 ECt2
  • 1/21/31/41/n 1/21/31/41/n
  • O(log n)
  • Etotal number of fragments n Etotal number
    of cuts
  • n ?uEnum cuts u makes nnO(log n) O(n
    log n)

Cv11 if u cuts v1, 0 otherwise
14
Random Auto-Partitions
  • Choose random permutation of segments
  • (s1, s2, s3,, sn)
  • While there is a region containing more than one
    segment,
  • separate it using first si in the region
  • O(n log n) fragments in expectation

15
Free cuts
Use internal fragments immediately as free cuts
16
Binary Space Partitions
  • Without free cuts O(n3)
  • With free cuts O(n2)
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