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```					           CS 551/651:

LOD:
A Sampling of Influential Algorithms

David Luebke                    1                     7/19/2011

 What is a quadric in this algorithm?
 How is it calculated?
 How is it used?

David Luebke           2                 7/19/2011

   Minimize distance to all planes at a vertex
   Plane equation for each face:
v
p:        Ax + By + Cz + D = 0

   Distance to vertex v :
x 
p  v = [A
T
B C   D ]  
y

1 
z
 
David Luebke                             3                           7/19/2011
D( v ) =         ( p        T
v) 2
p planes ( v )

=          (v         T
p )( pT v )
p planes ( v )

=          v     T
( ppT )v
p planes ( v )

                             
= v  ppT

T
v

pplanes ( v )               

David Luebke                           4                          7/19/2011
(cont’d)
    ppT is simply the plane equation squared:

AC
                   
 AB       B2    BD 
BC
pp =
T

 AC       BC C2 CD 
                 2 

    The ppT sum at a vertex v is a matrix, Q:
D( v ) = vT (Q )v

David Luebke                        5                      7/19/2011
    Construct a quadric Q for every vertex

v1                        v2
The edge quadric:           Q1            Q          Q2
Q = Q1 + Q2

   Sort edges based on edge cost
   Suppose we contract to v1:       edge cost = v1T Qv1
   v1’s new quadric is simply:      Q

David Luebke                           6                              7/19/2011
Recap:

    Minimize Q to calculate optimal coordinates
for placing new vertex
       Details in paper; involves inverting Q
       Authors claim 40-50% less error
perpendicular to boundary edges
 Prevent foldovers: check for normal flipping
 Create virtual edges between vertices closer
than some threshold t
David Luebke                         7                  7/19/2011
Boundary Preservation

 To preserve important boundaries, label
edges as normal or discontinuity
 For each face with a discontinuity, a
plane perpendicular intersecting the
discontinuous edge is formed.
 These planes are then converted into
quadrics, and can be weighted more
heavily with respect to error value.

David Luebke                8                       7/19/2011
Recap:
Simplification Envelopes
 What is the basic approach of SE?
 What is the underlying mechanism?
 What do they guarantee?

David Luebke               9              7/19/2011
Simplification Envelopes

 Idea: keep simplification between inner and
outer offset surfaces (draw it)
 Algorithm:
       Set maximum offset 
       Generate offset surfaces (avoid self-intersections!)
       Do all simplification operations that don’t cause
intersection (using decimation, edge collapse, etc)
    Guarantee: silhouette will not deviate by more
than /z pixels
David Luebke                          10                         7/19/2011
Simplification Envelopes

    Pros:
       Guaranteed silhouette fidelity bounds
       Guaranteed to preserve global topology:
 Same  genus
 No self-intersections

 Why might these be important?

       Very high fidelity in practice

David Luebke                           11                7/19/2011
Simplification Envelopes

    Cons
       Slow, very slow
       Complex and finicky to code
       Requires valid manifold topology to begin with
       Even if silhouette distortion is < 1 pixel, can still
introduce visible artifacts
       Can you think of a surface that could be simplified
to within a very small distance and look completely
different?

David Luebke                          12                         7/19/2011
Appearance-Preserving
Simplification
 Cohen, Olano, and Manocha, SIGGRAPH 98
 Track texture distortion as underlying mesh is
simplified, as well as geometric distortion
       Bound this distortion with world-space , can
translate to screen-space pixel bound
       Problem: color comes from lighting calculations as
well as intrinsic color captured in texture map
       Solution: store lighting parameters in normal map
       If  < 1 pixel, can truly guarantee visual fidelity

David Luebke                         13                        7/19/2011
Appearance-Preserving
Simplification
    A bumpy object and its normal map:

David Luebke              14              7/19/2011
Appearance-Preserving
Simplification
    LODs with and without normal mapping:

David Luebke              15                 7/19/2011
Image-Driven Simplification
    Lindstrom & Turk,
ACM TOG 2000
    Compare simplifications
to original via images
       Do edge collapse, render
from 12-20 views
       Evaluate difference
(how?)
       “Unrender” (how?) and
try again
       Pick cheapest and apply
David Luebke                              16   7/19/2011
Image-Driven Simplification

    Pros:
       Does a great job preserving appearance
       Solves the question of how to trade off geometric
distance versus color/normal/texture attributes
       Pays attention to shading artifacts
       Sensitive to texture content
       Drastically simplifies invisible regions!

David Luebke                         17                        7/19/2011
Image-Driven Simplification

    Cons:
       Slow, very slow
       Can always contrive an example that breaks it:
 Hard to know how many images is enough
 Hard to know how to evaluate images (RMS vs
Bolin-Meyer)

David Luebke                          18                        7/19/2011
Progressive Meshes

 Hoppe, SIGGRAPH 96
 Idea: apply a stream of edge collapse (ecol)
operations ordered by the error they introduce
       Measuring error: energy-minimization problem
       Reorder local neighborhood edges after collapse
    Extension: organize ecol operations in

David Luebke                        19                           7/19/2011
Progressive Meshes

    Pros:
       Elaborate energy-minimization metric leads to
very high-quality simplifications
       Fine-grained control over degree of simplification
       Ecol operation reversible (vsplit) and fast enough
to support dynamic/view dependent simplification
       Supports some neat features:
 Can  geomorph between LODs
 Stream vsplits over network for progressive refinement

 Can dynamically generate triangle strips to render faster

David Luebke                             20                             7/19/2011
Progressive Meshes

    Cons
       Slow preprocess
 Result   of elaborate energy metric, not inherent to PM
       Not clear that dynamic/view-dependent LOD wins
       Extremely fine-grained simplification might be
less optimal than coarser clustering at times

David Luebke                              21                                7/19/2011
The End

David Luebke       22    7/19/2011

```
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