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Surface Simplification. Dr. Scott Schaefer. 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. Reducing Polygons.

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surface simplification

Surface Simplification

Dr. Scott Schaefer

surface simplification1
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

reducing polygons
Reducing Polygons
  • Perform local, topological operations to reduce number of polygons
  • Vertex removal
reducing polygons1
Reducing Polygons
  • Perform local, topological operations to reduce number of polygons
  • Vertex removal
  • Edge Collapse
reducing polygons2
Reducing Polygons
  • Perform local, topological operations to reduce number of polygons
  • Vertex removal
  • Edge Collapse
  • Face Collapse, …
reducing polygons3
Reducing Polygons
  • Perform local, topological operations to reduce number of polygons
  • Vertex removal
  • Edge Collapse
  • Face Collapse, …
surface simplification2
Surface Simplification
  • How do we determine the order of edge collapse operations?
  • Where do we place new vertex after collapse?
error metrics for simplification
Error Metrics For Simplification
  • QEF: Quadratic Error Function
  • Measures distance to infinite planes
error metrics for simplification1
Error Metrics For Simplification
  • QEF: Quadratic Error Function
  • Measures distance to infinite planes
error metrics for simplification2
Error Metrics For Simplification
  • QEF: Quadratic Error Function
  • Measures distance to infinite planes

symmetric 3x3

3x1

1x1

error metrics for simplification3
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!!!

combining qefs1
Combining QEFS

Add 10 numbers to combine QEFs!!!

placement of vertices using qefs
Placement of Vertices Using QEFs
  • Place new vertex at minimum of error function
placement of vertices using qefs1
Placement of Vertices Using QEFs
  • Place new vertex at minimum of error function
placement of vertices using qefs2
Placement of Vertices Using QEFs
  • Place new vertex at minimum of error function
placement of vertices using qefs3
Placement of Vertices Using QEFs
  • Place new vertex at minimum of error function

Not invertible in flat areas or straight edges!!!

placement of vertices using qefs4
Placement of Vertices Using QEFs
  • Place new vertex at minimum of error function

Pseudoinverse minimizes |v|

placement of vertices using qefs5
Placement of Vertices Using QEFs
  • Let where c is a point we want to minimize the distance to
placement of vertices using qefs6
Placement of Vertices Using QEFs
  • Let where c is a point we want to minimize the distance to
placement of vertices using qefs7
Placement of Vertices Using QEFs
  • Let where c is a point we want to minimize the distance to
placement of vertices using qefs8
Placement of Vertices Using QEFs
  • Let where c is a point we want to minimize the distance to
plane based quadratic error function
Plane-Based Quadratic Error Function
  • Compact representation (10 numbers)
  • Fast to combine multiple functions (addition)
  • Relatively easy to minimize (pseudoinverse)
  • Suffers from numerical instabilities
surface simplification algorithm
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 # > target #
    • Collapse edge
surface simplification edge collapse
Surface Simplification: Edge Collapse
  • Place new vertex at minimizer of QEF
surface simplification edge collapse1
Surface Simplification: Edge Collapse
  • Place new vertex at minimizer of QEF
  • QEF of new vertex is combined QEF
surface simplification edge collapse2
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
surface simplification edge collapse3
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
surface simplification edge collapse4
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
surface simplification edge collapse5
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

surface simplification edge collapse6
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
surface simplification3
Surface Simplification
  • Extremely fast
  • Somewhat memory intensive
    • Limits maximum surface size
  • Greedy algorithm
    • Does not guarantee optimal sequence of edge collapses!!!
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