1. 3D Model Simplification (GATE-540)
Dr.Çağatay ÜNDEĞER
Instructor
Middle East Technical University, GameTechnologies
&
General Manager
SimBT Inc.
References:
David P. Luebke, A Developer’s Survey of Polygonal Simplification Algorithms
Mark Duchaineau et al. ROAMing Terrain: Real-Time Optimally Adapting Meshes
Game Technologies Program – Middle East Technical University – Spring 2010

2. Outline Introduction to Simplification
Simplification Algorithm Selection Criteria
Kinds of Simplification Algorithms
Brief Description of Simplification Algorithms

3. Polygonal Models
Currently dominate interactive computer graphics because of
Their mathematical simplicity and
Their simple, regular rendering algorithms that embed well in hardware.

4. Difficulties of Polygonal Models
Complexity of polygonal models are measured by the number of polygons
Complexity of required models seems to grow faster than the ability of our graphics hardware to render them interactively.

5. Polygonal Simplification
Offers one solution for developers to deal with these complex models.

6. Polygonal Simplification
Flight simulators have long used hand-crafted multi-resolution airplane models to guarantee a constant frame rate.
In this course, we will deal with automated simplification of these models.

7. Determining Appropriate Algorithm
Need to answer several questions:
Why do I need to simplify models?
What do my models seem like?
What is important to me the most?

8. Why do I Need to Simplify?
Trying to eliminate redundant geometry?
e.g. Volumetric iso-surfaces generated by the marching cubes algorithm tile the model’s flat regions with many small, coplanar triangles?
After subdivision of a model for finite-element analysis, a simplification algorithm is required to remove unnecessary geometry?

9. Why do I Need to Simplify?
Trying to reduce model size?
Geometric compression?
Creating a downloadable models for a Web site?
Minimizing storage requirements?

10. Why do I Need to Simplify?
Trying to improve runtime performance?
Generating levels of detail (LODs) of the objects in a scene?
Simplifying the polygonal scene being rendered,
By representing distant objects with a lower LOD and nearby objects with a higher LOD?

11. What do My Models Seem Like?
No algorithm today exists that simplifying all models sucessully.
Organic forms with smooth curves.
Mechanical objects with sharp corners, flat faces, and regular curves.
Precomputed colors, lighting or texture that must be considered.

12. What do My Models Seem Like?
A few large, high-complexity, individual objects (e.g. terrain data, medical data).
Multiple objects of moderate complexity.
Large assemblies of many small objects.

13. What is Important to Me? Preserve accuracy in the simplified models?
Control the Hausdorff distance of the simplified vertices / surface to the original
Bound the volumetric deviation of the simplified mesh from the original...
Just want high visual fidelity?
Perception is more difficult to quantify than geometry
Preserve model’s topological genus?

14. What is Important to Me? If run-time performance is crucial,
You may need an algorithm capable of drastic simplification.
A drastic simplification may require:
View-depended simplification
Topology-reduction simplification

15. Kinds of Algorithms
Many approaches, each with strengths and weaknesses.
Treatment of mesh topology
Simplification mechanism
Static, dynamic, view-depended

16. Treatment of Mesh Topology
Provides an important distinction
Topology: connected polygonal mesh structure
Genus: number of holes in the mesh surface
Genus of zero
Genus of one
Genus of three

17. Treatment of Mesh Topology
The mesh forms a 2D manifold
if the neighborhood of every feature consists of a connected ring of polygons forming a single surface.
In a triangulated manifold mesh:
Exactly two triangles share every edge, and
Every triangle shares an edge with exactly three neighboring triangles.
(accept borders)

18. Treatment of Mesh Topology
Manifold meshes result in well-behaved models.
Any simplification algorithm can successfully operate on any manifold object.
Some algorithms (e.g. marching-cubes) guarantee manifold output.
Some models have cracks, T-junctions, and nonmanifold points or edges.

19. Treatment of Mesh Topology
A topology-preserving simplification algorithm:
Preserves manifold connectivity at every step.
Fidelity tends to be relatively good.
Can’t be simplified drastically.
Requires a mesh with manifold topology.
A topology tolerant simplification algorithm:
Ignores regions in the mesh with nonmanifold local topology,
Leaves those regions unsimplified.

20. Treatment of Mesh Topology
A topology-modifying algorithm:
Don’t necessarily preserve manifold topology.
May close up holes in the model and aggregate separate objects into assemblies
Permitting drastic simplification beyond the scope of topology-preserving schemes.
Often provides poor visual fidelity.
Good at real-time visualization of complex scenes.

21. Simplification Mechanism
Four basic polygon removal mechanisms:
Sampling,
Adaptive subdivision,
Decimation, and
Vertex merging.

22. Sampling Sample the initial model’s geometry:
With points on the model’s surface or
With voxels superimposed on the model in a 3D grid.
More elaborate and difficult to code approaches.
Trouble achieving high fidelity.
Work best on smooth organic forms with no sharp corners.

23. Adaptive Subdivision
Finds a simple base mesh that can be recursively subdivided to closely approximate the initial model.
Requires creating a base model that captures the original model’s important features, which can be tricky.
Preserve the surface topology, which may limit their capacity for drastic simplification.

24. Decimation
Iteratively remove vertices or faces from the mesh, re-triangulating the resulting hole after each step.
Relatively simple to code and can be very fast.
Use strictly local changes that tend to preserve the genus.
This restricts drastic simplification.

25. Vertex Merge
Collapses two or more vertices of a triangulated model into a single vertex,
Which in turn can be merged with other vertices.

26. Vertex Merge A simple and easy-to-code mechanism.
Range from simple, fast, and crude to complex, slow, and accurate.

27. Static, Dynamic, and View-Dependent Simplification
Traditional static approach:
Polygonal simplification creates several discrete versions of each object in a preprocess, each at a different LOD.
Decoupling simplification and rendering makes this the simplest model to code.
Each LOD can be converted during preprocessing to triangle strips and compiled as a separate display list.

28. Static, Dynamic, and View-Dependent Simplification
Dynamic approach:
Creates a data structure, which encodes a continuous spectrum of detail.
Desired LOD can be extracted from this structure at run-time.
Advantage: uses no more polygons than necessary.

29. Static, Dynamic, and View-Dependent Simplification
Extends dynamic simplification by using view-dependent criteria
Selects the most appropriate LOD for the current view.

30. Brief Description of Some Simplification Algorithms
Triangle Mesh Decimation
Vertex Clustering
Multiresolution Analysis of Arbitrary Meshes
Voxel-Based Object Simplification
Simplification Envelopes
Appearance-Preserving Simplification
Quadric-Error Metrics
Image-Driven Simplification
Progressive Meshes
Real-Time Optimally Adapting Meshes (ROAM)

31. Triangle Mesh Decimation
By Schroeder, Zarge, and Lorenson
Simplify general polygonal models.
Term decimation for iterative removal of vertices.
Designed to operate on the output of the marching cubes algorithm.
Coplanar regions divided into many more polygons than necessary.

32. Triangle Mesh Decimation
Multiple passes over all the vertices:
If the vertex can be removed without violating the local topology, and
If resulting surface would lie within a user-specified distance of the unsimplified geometry,
Deletes vertex and all its associated triangles.
Hole in the mesh is re-triangulated.

33. Triangle Mesh Decimation
Vertices of simplified model are a subset of the original model’s vertices.
Convenient for reusing normals and texture coordinates at the vertices.
Limit the fidelity of the simplifications.
Quite fast and topology tolerant.
Available as part of the Visualization Tool Kit at

34. Vertex Clustering By Rossignac and Borrel
Topology insensitive, neither requiring nor preserving valid topology.

35. Vertex Clustering Assigns an importance to each vertex.
More Important:
Vertices attached to large faces and vertices of high curvature
Less Important:
Vertices attached to small faces and vertices of low curvature.

36. Vertex Clustering Overlays a 3D grid on the model
Collapses all vertices within each cell of the grid to the single most important vertex in cell.

37. Vertex Clustering
Grid’s resolution determines quality of simplification.
During the clustering process,
Triangles whose corners are collapsed together become degenerate and disappear.

38. Vertex Clustering A unique feature in rendering merged corners.
Renders using the graphics hardware line and point primitives.
a triangle with two corners collapsed is a line
a triangle with three corners collapsed is a point

39. Vertex Clustering
A different clustering approach called floating-cell clustering by Low and Tan.
Ranks vertices by importance.
A cell of user-specified size is centered on the most important vertex.
Collapses all vertices falling within the cell to the representative vertex.
The most important remaining vertex becomes the center of the next cell,
And the process repeats.

40. Vertex Clustering Extremely fast. Run in O(n) time for n vertices.
Since algorithm:
Don’t preserve topology, and
Don’t guarantee the amount of error introduced by the simplified surface,
Often less pleasing visually than those of slower algorithms.

41. Multiresolution Analysis of Arbitrary Meshes (MRA)
Adaptive subdivision algorithm by Eck et al.
Uses a compact wavelet representation to guide a recursive subdivision process.
Fast enough to do at runtime
Guarantee that the simplified surface lies within a userspecified distance of the original model.
Wavelet series is a squre-integrable function.

42. Multiresolution Analysis of Arbitrary Meshes (MRA)
Provides a method for finding a simple base mesh
That exhibits subdivision connectivity,
So that recursive subdivision will recover the original mesh.

43. Multiresolution Analysis of Arbitrary Meshes (MRA)
Base mesh by growing Voronoi-like regions across the original surface’s triangles.
When these regions can’t grow anymore,
Voronoi sites form a Delaunay-like triangulation, and
The triangulation forms the base mesh.

44. Multiresolution Analysis of Arbitrary Meshes (MRA)
Manifold topology is required in the input model.
Strict topology-preserving approach.
Original object’s shape and genus limit the potential for drastic simplification.
Fidelity is quite high for smooth, organic forms.
Has difficulty capturing sharp features.

45. Voxel-based Object Simplification
By He et al.
Simplify topology in a gradual and controlled manner using a signal processing approach.
Sample a volumetric representation of the model
By Superimposing a 3D grid of voxels over the polygonal geometry.

46. Voxel-based Object Simplification
Aassigns each voxel a value of 1 or 0,
According to whether the sample point of that voxel lies inside or outside the object.
Resamples the volume with a lower resolution
To elliminate small, high-frequency features.
Applies marching cubes.
Uses topology-preserving algorithm as a postprocess to eliminate redundant geometry.

47. Voxel-based Object Simplification
High-frequency details such as sharp edges and corners are lost.
Greatly restricts its usefulness on mechanical CAD models.
Isn’t topology tolerant,
Since deciding whether sample points lie inside or outside the object requires closed-mesh objects.

48. Simplification Envelopes
By Cohen et al.
Guarantee fidelity bounds
While enforcing global and local topology.
Two offset surfaces, no more than some distance ε from the original surface.
outher ε
original -ε
inner

49. Simplification Envelopes
Outher envelope
Displaces each vertex of the original mesh along its normal by ε,
Inner envelope:
Displaces each vertex by −ε.
Aren’t allowed to self-intersect.
If so, algorithm decreases ε in the local neighborhood.

50. Simplification Envelopes
Envelopes can guide the simplification process.
Keeping simplified surface within envelopes
Iteratively remove triangles or vertices, and
Retriangulate the resulting holes.

51. Simplification Envelopes
Surface never deviates by more than ε from the original model.
Resulting simplifications tend to have excellent fidelity.
Strict preservation of topology
Not good for drastic simplification.
Requires an orientable manifold.
implementation at

52. Appearance-Preserving Simplification
By Cohen, Olano, and Manocha
Takes the error-bounding approach of simplification envelopes a step further.
Provides the best guarantees on fidelity of any simplification algorithm.

53. Appearance-Preserving Simplification
Fidelity is expressed in terms of maximum screenspace deviation.
Rendered image deviate from the original’s appearance by no more than a user-specified number of pixels.

54. Appearance-Preserving Simplification
Attributes affecting appearance:
Surface position:
the coordinates of the vertices.
Surface color:
the color field across the mesh.
Surface curvature:
Field of normal vectors across the mesh.

55. Appearance-Preserving Simplification
The simplification process uses edge collapses.
Edge collapses that cause surface color, normals, or position to shift by more than the maximum user-specified distance ε aren’t allowed.

56. Appearance-Preserving Simplification

57. Quadric Error Metrics
A vertex-merging algorithm by Garland and Heckbert
Perhaps has the best balance yet between speed, fidelity, and robustness.

58. Quadric Error Metrics
Iteratively merging pairs of vertices ( which may not need be connected by an edge).
Candidate vertex pairs include:
All vertex pairs connected by an edge,
Plus all vertex pairs separated by less than a user-specified distance threshold t.

59. Quadric Error Metrics
All candidate vertex pairs are sorted into a priority queue according to the quadric error metric calculated for merging them.
Removes vertex pair with the lowest merge error from the top of the queue and merges it.
Updates the errors of all vertex pairs involving the merged vertices and
Repeats the process.

60. Quadric Error Metrics
A simple way to guide the simplification process.
Extremely fast.
Relatively minor storage costs.
Visual fidelity tends to be quite high.
Doesn’t require manifold topology.
Lets holes close and objects merge.
QSlim, available at

61. Image-Driven Simplification
By Lindstrom and Turk
First to address simplification directly in terms of how the simplified model will look when rendered.
Occurs through a sequence of edge collapse operations using a priority queue.
Uses a purely image-based metric.

62. Image-Driven Simplification
Determines the cost of an edge collapse operation by;
Performing the collapse and
Rendering the model from multiple viewpoints.
Comparing rendered images to images of the original model and
Sums up the mean-square error in luminance across all pixels of all images.

63. Image-Driven Simplification
Lindstrom and Turk used 20 viewpoints in their implementation.
Rendering the entire model for every edge and from many viewpoints is clearly an expensive step.
Optimization:
Edge collapse affects only a small region of the screen.
Only a small portion of the triangles need be rendered per image.

64. Progressive Meshes
Represents polygonal models as a sequence of edge collapses.
First dynamic simplification algorithm for general polygonal manifolds.
Later extended them to support view-dependent simplification.

65. Progressive Meshes Consists of a simple base mesh created by:
A sequence of edge collapse operations
A series of vertex split operations.
A vertex split is the dual of an edge collapse.

66. Progressive Meshes Each vertex split:
Replaces a vertex by two edge-connected vertices,
Creating one additional vertex and two additional triangles.
Correspond to the dual of edge-collapse operations used to create the base mesh.
This will recapture the original model exactly.
Fast enough to perform in real-time.

67. Progressive Meshes
Models mesh complexity and fidelity as an energy function to be minimized.
Sorts edges into a priority queue.
Energy function is minimized in a greedy fashion by performing edge collapse operations at the head of the queue.
Sequence of edge collapse operations performed becomes the hierarchy of vertex split operations.

68. Progressive Meshes 3 view-dependent simplification criteria:
A view frustum test
A backfacing test
A screenspace error threshold

69. Progressive Meshes Manifold topology is required.
Prevents holes from closing and objects from aggregating.
Hoppe’s energy-minimization approach
Produces high-fidelity simplifications
But is relatively slow.
Instead, any algorithm based on edge collapses can be used to generate a progressive mesh.

70. Real-Time Optimaly Adapting Meshes (ROAM)
By Mark Duchaineau et al.
A view-dependent simplification algorithm.
Dynamic terrain and simple vertex morphing.

71. Real-Time Optimaly Adapting Meshes (ROAM)
Uses:
A bintree.
2 priority queues to drive triangle split and merge.

72. Real-Time Optimaly Adapting Meshes (ROAM)
Originally developed for flight simulators.
Provides high quality triangulations at high frame rates.