Dr. Scott Schaefer

1
Surface Simplification

• Dr. Scott Schaefer

2
Surface Simplification

• Given a closed polygon model, reduce the number
  of polygons and maintain appearance of the shape

5804 tris
2500 tris
1000 tris
500 tris
3
Reducing Polygons

• Perform local, topological operations to reduce
  number of polygons
• Vertex removal

4
Reducing Polygons

• Perform local, topological operations to reduce
  number of polygons
• Vertex removal
• Edge Collapse

5
Reducing Polygons

• Perform local, topological operations to reduce
  number of polygons
• Vertex removal
• Edge Collapse
• Face Collapse,

6
Reducing Polygons

• Perform local, topological operations to reduce
  number of polygons
• Vertex removal
• Edge Collapse
• Face Collapse,

7
Surface Simplification

• How do we determine the order of edge collapse
  operations?
• Where do we place new vertex after collapse?

8
Error Metrics For Simplification

• QEF Quadratic Error Function
• Measures distance to infinite planes

9
Error Metrics For Simplification

• QEF Quadratic Error Function
• Measures distance to infinite planes

10
Error Metrics For Simplification

• QEF Quadratic Error Function
• Measures distance to infinite planes

symmetric 3x3
3x1
1x1
11
Error Metrics For Simplification

• QEF Quadratic Error Function
• Measures distance to infinite planes

symmetric 3x3
3x1
1x1
Requires 10 floats independent of number of
polygons!!!
12
Combining QEFS
13
Combining QEFS
Add 10 numbers to combine QEFs!!!
14
Placement of Vertices Using QEFs

• Place new vertex at minimum of error function

15
Placement of Vertices Using QEFs

• Place new vertex at minimum of error function

16
Placement of Vertices Using QEFs

• Place new vertex at minimum of error function

17
Placement of Vertices Using QEFs

• Place new vertex at minimum of error function

Not invertible in flat areas or straight edges!!!
18
Placement of Vertices Using QEFs

• Place new vertex at minimum of error function

Pseudoinverse minimizes v
19
Placement of Vertices Using QEFs

• Let where c is a point we want to
  minimize the distance to

20
Placement of Vertices Using QEFs

• Let where c is a point we want to
  minimize the distance to

21
Placement of Vertices Using QEFs

• Let where c is a point we want to
  minimize the distance to

22
Placement of Vertices Using QEFs

• Let where c is a point we want to
  minimize the distance to

23
Plane-Based Quadratic Error Function

• Compact representation (10 numbers)
• Fast to combine multiple functions (addition)
• Relatively easy to minimize (pseudoinverse)
• Suffers from numerical instabilities

24
Surface Simplification Algorithm

• Build QEFs for each vertex
• For each edge
• Compute combined QEF and error
• Insert edge into priority queue sorted by error
• While poly gt target
• Collapse edge

25
Surface Simplification Edge Collapse

• Place new vertex at minimizer of QEF

26
Surface Simplification Edge Collapse

• Place new vertex at minimizer of QEF
• QEF of new vertex is combined QEF

27
Surface Simplification Edge Collapse

• Place new vertex at minimizer of QEF
• QEF of new vertex is combined QEF
• Remove all edges touching collapsed edge from
  priority queue

28
Surface Simplification Edge Collapse

• Place new vertex at minimizer of QEF
• QEF of new vertex is combined QEF
• Remove all edges touching collapsed edge from
  priority queue
• Recompute QEF/error for all edges touching new
  vertex and insert into priority queue

29
Surface Simplification Edge Collapse

• Edge collapse may alter topology of surface
• Test for topology change and exclude unsafe edge
  collapses
• Unsafe edge may become safe after another
  collapse
• Alternatively, perform two edge collapses

30
Surface Simplification Edge Collapse

• Edge collapse may alter topology of surface
• Test for topology change and exclude unsafe edge
  collapses
• Unsafe edge may become safe after another
  collapse
• Alternatively, perform two edge collapses

unsafe edge collapses
31
Surface Simplification Edge Collapse

• Edge collapse may alter topology of surface
• Test for topology change and exclude unsafe edge
  collapses
• Unsafe edge may become safe after another
  collapse
• Alternatively, perform two edge collapses

32
Surface Simplification

• Extremely fast
• Somewhat memory intensive
• Limits maximum surface size
• Greedy algorithm
• Does not guarantee optimal sequence of edge
  collapses!!!