Simplification

1
Simplification
low LOD
70000 triangles

• Jarek Rossignac
• GVU Center
• Georgia Institute of Technology

2
Loss-less or lossy compression?

• Loss-less compression
• Quantize parameters (coordinates) based on
  application needs
• Finite precision measurements, design,
  computation
• Limited needs for accuracy in some applications
• Encode quantized location and exact incidence
• Lossy compression
• Encode approximations of the surface using a
  different representation
• How to measure the error to ensure that tolerance
  is not exceeded?

Lossy
Loss-less
3
Triangle count reduction techniques (LOD)

• Quantize cluster vertex data (RossignacBorrel9
  2)
• remove degenerate triangles (that have coincident
  vertices)
• Adapted by P. Lindstrom for out-of-core
  simplification
• Repeatedly collapse best edge (RonfardRossignac96
  )
• while minimizing maximum error bound
• Adapted by M. Garland for least square error

4
Vertex clustering (Rossignac-Borrel)

• Subdivide box around object into grid of cells
• Coalesce vertices in each cell into one
  attractor
• Remove degenerate triangles
• More than one vertex in a cell
• Not needed for dangling edge or vertex

5
RossignacBorrel 93
6
RossignacBorrel 93
7
Improving on Vertex Clustering

• Advantages
• Trivial to implement
• Fast
• Works on any mesh or triangle soup
• Guaranteed Hausdroff error to diagonal of cell
• Reduces topology
• Removes holes. Never creates one
• Merges connected shells components. Never splits
  them.
• Drawbacks
• Produces sub-optimal results
• Too much error for a given triangle count
  reduction
• Prevents the merging of distant vertices on flat
  portions of the surface
• Fix limit vertex moves by the resulting error
• Not a fixed grid

8
Simplification through edge collapse
9
How to decide which edges to collapse?

• Minimize the error between original and resulting
  LOD
• How to compute/estimate error
• Peformance
• Geometric proximity clustering of vertices
  (pessimistic)
• RossignacBorrel quantizing vertices identifies
  candidate edges
• Error is bounded by the quantization error
• Fast, easy, robust, but sub-optimal results
• Collapse edges
• Longer edges in almost planar regions
• Estimate error as max distance to supporting
  planes (RonfardRossignac)
• Must keep list of all planes supporting triangles
  incident on contracted edges
• Use sum of squares instead of max
  (HeckbertGarland) faster, no bound
• L2 norm, needs only add 4x4 matrices when
  clusters are merged

10
Distance and quadratic error

• Point-plane distance
• Point P(x,y,z)
• Plane containing point Qm and having unit normal
  Nm
• Distance PQm?Nm
• Can compute max (conservative, RonfardRossignac)
  or sum (cheap, HeckbertGarland) of (PQm?Nm)2 for
  the planes of all the triangles Tm incident upon
  vertices merged at P
• Distance squared (PQm?Nm)2 amx2bmy2cmz2dmxy
  emyzfmzxgmxhmyimzjm
• Sum of distances squared (PQm?Nm)2 (PQn?Nn)2
  
• (aman)x2 (bmbn)y2 (cmcn)z2 (dmdn)x
  (emen)y (fmfn)z gm gn
• As vertices are merged recursively
• With max, you need to remember all the planes
• With sum, you just add the coefficients

11
RonfardRossignac EG96
12
Shape complexity

• Optimal bit allocation in 3D compression
• KingRossignac, Computational Geometry, Theory
  Applications99
• Approximate ET by K/T
• Assumes uniform error distribution (all edge
  collapses increase ET)
• Assumes smooth shapes with no features smaller
  than tesselation
• Use integral of curvature to estimate K
• K estimate computed efficiently using sphere-fit
  for each edge
• Formula derived for objects made of relatively
  large spherical caps
• Yields crude estimate for doubly curved surfaces
  (saddle points...)

K/T