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## PowerPoint Slideshow about ' Surface Simplification' - jenski

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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

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

Reducing Polygons

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

Reducing Polygons

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

Reducing Polygons

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

Surface Simplification

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

Error Metrics For Simplification

- QEF: Quadratic Error Function
- Measures distance to infinite planes

Error Metrics For Simplification

- QEF: Quadratic Error Function
- Measures distance to infinite planes

Error Metrics For Simplification

- QEF: Quadratic Error Function
- Measures distance to infinite planes

symmetric 3x3

3x1

1x1

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 QEFS

Add 10 numbers to combine QEFs!!!

Placement of Vertices Using QEFs

- Place new vertex at minimum of error function

Placement of Vertices Using QEFs

- Place new vertex at minimum of error function

Placement of Vertices Using QEFs

- Place new vertex at minimum of error function

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 QEFs

- Place new vertex at minimum of error function

Pseudoinverse minimizes |v|

Placement of Vertices Using QEFs

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

Placement of Vertices Using QEFs

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

Placement of Vertices Using QEFs

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

Placement of Vertices Using QEFs

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

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

- 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

- Place new vertex at minimizer of QEF

Surface Simplification: Edge Collapse

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

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 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 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 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 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

- Extremely fast
- Somewhat memory intensive
- Limits maximum surface size

- Greedy algorithm
- Does not guarantee optimal sequence of edge collapses!!!

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