1. Progressive coding Motivation and goodness measures
Siggraph'98 Course21: 3D Geometry Compression
Progressive coding Motivation and goodness measures
Jarek Rossignac
(with Renato Pajarola)
GVU Center and College of Computing
Georgia Tech, Atlanta
Jarek Rossignac 1

2. Progressive refinements
Siggraph'98 Course21: 3D Geometry Compression
Progressive refinements
Previously covered connectivity compression is loss-less
It is complemented by the compression of vertex data
3D coordinates, normals, colors, texture coordinates
Discussed later in the “Geometry” section
Exploiting a lossy quantization
When these two are insufficient, we can simplify the model
Reduce triangle and vertex count through a sequence of edge-collapses
Select sequence that minimizes the resulting (geometric or visual) error
We can use progressive transmission that increases accuracy
Download compressed crude model first
When more accuracy is needed, download upgrades and refine model
You may start navigation right away using crude model
Often you may not need to download the complete model at all
Jarek Rossignac 2

3. ? Mesh refinement Siggraph'98 Course21: 3D Geometry Compression
Jarek Rossignac 3

4. Just-in-time upgrades of features
Download geometry only when images won’t do
Subdivide surface into features (hierarchical)
Use approximations of feature when acceptable
Replace by more detailed sub-features as necessary
To upgrade a feature, given its boundary, we must construct:
A detailed representation of its (approximate) shape
A partition of the feature’s surface into sub-features B C A D E
So what is an upgrade? Any information that would describe how to refine the geometric representation of an object or feature.
For example, only the highest significant bits of the coordinates of the vertices of a model may have been initially downloaded. An upgrade may provide subsequent less-significant bits for each coordinate.
As discussed earlier, an upgrade may be the description of an entirely new representation of an object or feature.
We will make here a specific assumption and later will argue that it does not lead to any loss of generality for our compression objectives. We will assume that an upgrade is the geometric representation of the interior of a surface patch of which we know the boundary.
Furthermore, we will assume that this refined geometry is annotated so as to define sub-features and their boundary, should some of these need to be further upgraded.

5. Selective download of feature upgrades
Upgrade Geometry Cost Quality Children
0 g a0,r0 p0,e0 1,2,3
1 g a1,r1 p1,e1 4,5,6
2 g a2,r2 p2,e2 7,8,9
User
Server
view&model manipulation
Downloaded upgrades 0,2,3,8,9,10,11,12
Client
The client may save all the lower resolution models in a local--and generally incomplete--version of a multi-resolution hierarchy for situations where some features may be rendered at a resolution that is lower than the best locally available.
In our example, the client has only downloaded the upgrades numbered in green (the center and right part of the tree).
crude model (g0) 1 3 2 12 6 7 9 10 4 8 11 5
detailed features

6. Simultaneous access to remote datasets
3D Servers
Client viewers
Compressed crude models and upgrades

7. Compressed successive upgrades
Siggraph'98 Course21: 3D Geometry Compression
Compressed successive upgrades
upgrade 2
Simplification 1
Compression
upgrade 1
Simplification 2
Compression
Crude model
Compression
Decompression
Decompression
upgrade
Decompression
upgrade
Jarek Rossignac 4

8. What is a good feature for selective upgrade?
Connected subset of the boundary
Surface patch
Entire surface
Defined by hierarchical simplification
Assume its boundary is given
Outer loops of a connected patch
One triangle of a solids boundary
Represent how to stretch the surface between this boundary
Compression: bit-efficient model of the internal shape
The upgrade compression problem starts with a surface patch (which could be an entire object).
The goal is to compress the description of the relative interior of this patch, assuming that the client already has the boundary (loops of edges that bound the outer boundary and the holes).
The case of compressing the surface of a polyhedral solid may be converted to this upgrade-compression problem by sending any one triangle first and by using its boundary as the outer bounding loop for a surface patch that is the rest of the entire object.
So our compression problem will be:
Given the boundary of a surface, how to compute a bit-efficient representation of that surface so that it can be quickly decompressed by a client who already has the boundary. ? ? ? ?

9. Simplification process defines features
As noted earlier, an upgrade that we wish to compress may represent an entire mesh or a small feature. In the context of the progressive transmission of feature upgrades, a feature may be defined as the connected component of the modified parts of the geometry of the object.
Consider for example the most popular simplification step: the edge collapse, which identifies an edge at a time and collapses it, removing two triangles and spreading the adjacent triangles to cover the so created empty space.
Note that the result of a sequence of edge-collapse operations (bottom right) is independent of the order in which the operations were carried out. One can identify (bottom center) all the edges in the original model (bottom left) that should be collapsed during a simplification process that transforms one level-of-detail into a subsequent lower level. The star of these edges (triangles incident upon them) define the part of the geometry that is altered by the simplification process.
The connected components of this area are good candidates for features. A finer decomposition may be defined as the connected components of the difference between the mesh and the triangles and edges that have not been affected by the simplification process.

10. Upgrade = reverse simplification
compress
Upgrade
So one approach is to define the features as explained in the previous slide and to send their simplified representation first. Then, if an upgrade is needed, the client will discard the entire feature and replace it with a more detailed description, derived by decompressing the upgrade information.
Again, at that point, the boundary of the feature is known to the client. The upgrade defines how the hole is to be filled.
The process may be performed recursively by further simplifying the results of previous simplification steps.
Why not allow each edge collapse to be a simplification step? As will be discussed later, this approach offers flexibility, but does not produce the best compression ratios (no economy of scale).
decompress ?

11. Evaluating progressive transmission
Siggraph'98 Course21: 3D Geometry Compression
Evaluating progressive transmission
Time to first picture
Midway accuracy
Error for the received model
Better
Time to full accuracy
Bits transmitted (or time)
Jarek Rossignac 5

12. Non-progressive transmission
Siggraph'98 Course21: 3D Geometry Compression
Non-progressive transmission
approximation error
bits (time)
nothing
complete model
Jarek Rossignac 6

13. Two-step approach: Crude-Full
Siggraph'98 Course21: 3D Geometry Compression
Two-step approach: Crude-Full
single resolution model
crude model
complete model
approximation error
bits (time)
nothing
Jarek Rossignac 7

14. Multiple levels of accuracy
Siggraph'98 Course21: 3D Geometry Compression
Multiple levels of accuracy
single resolution model
two-shot
approach
crude
intermediate
approximation error
Longer wait for complete model
complete
bits (time)
nothing
Jarek Rossignac 8

15. Optimal, continuous refinement
Siggraph'98 Course21: 3D Geometry Compression
Optimal, continuous refinement
approximation error
Want smallest area?
bits (time)
crude model
final model
Jarek Rossignac 9

16. Simplification, Compression & Upgrades
Reduce the number of triangles that represent a 3D “feature”
Compression
Reduce the number of bits needed to represent the “feature”
Upgrade
Information used to recover an original feature from its simplification
Progressive representation
Coarse model and a tree of recursive upgrades B C A D E

17. Progressive coding Compressed Progressive Meshes
Siggraph'98 Course21: 3D Geometry Compression
Progressive coding Compressed Progressive Meshes
Jarek Rossignac
(with Renato Pajarola)
GVU Center and College of Computing
Georgia Tech, Atlanta
Jarek Rossignac 1

18. Problem of compressing updates
Location
where does a refinement occur
Incidence
how to update the mesh connectivity
Geometry
new coordinates
update = increases the number of vertices and triangles in the mesh
location
incidence
compressed format

19. Progressive meshes (Hoppe96)
Simple simplification and refinement operators
edge collapse operation (ecol)
vertex split operation (vsplit)
defined by split-vertex and cut-edges
Series of meshes
defined by crude base mesh M0 and sequence of vsplits
cut-edges
ecol
vsplit
split-vertex M0 M1
Mmax-1
Mmax
...
vsplit

20. Vertex splits (Hoppe) The inverse of an edge collapse
Insert two triangles
Cut mesh at 2 adjacent edges
Replicates 2 edges and 1 vertex
Move new vertex to create gap
Fill the crack with 2 new triangles
Specify position of new vertex
Encode relative displacement (arrow)
Incidence cost: 8–13 b/T
Identify a vertex (10<log|V|<20) bits
Identify 2 incident edges (6 bits?)
Hope proposed to invert the sequence of edge collapses and store them as a sequence of upgrades. Each upgrade insets a vertex and two triangles.
To define where the vertex and the triangles are to be inserted, Hoppe needs to specify two adjacent edges in the current stage of the mesh.
These edges, marked in red, are then split, opening a four-sided gap (bleu)that will be filled with a two triangles.
The specification of these two edges may be coded using the ID of their common vertex and the IDs that distinguish any two amongst the many edges incident upon that vertex. The cost per split (per 2 triangles) is:
Index to vertex: log(V) bits
Index to 2 adjacent edges:2 log(max valence) bits
In practice,for small meshes, this amounts to less than 20 bits per split (or 10 bits per triangle).

21. CPM (Pajarola&Rossignac 99)
Siggraph'98 Course21: 3D Geometry Compression
CPM (Pajarola&Rossignac 99)
Sequences of vertex splits (as in Hoppe’s PM)
Grouped in batches
Example
not marked
marked
mesh incidence updates
Jarek Rossignac 2

22. CPM encoding of connectivity
Siggraph'98 Course21: 3D Geometry Compression
CPM encoding of connectivity
Avoid log(V) cost by marking all vertices
1 bit per vertex in batch
30% to 50% vertices are split in each batch
Better encoding of cut-edges
log(6)+log(2)
Amortized total connectivity cost 3.6T bits
1.5 b/T for marking vertices (amortized)
2.1 b/T for identifying cut edges
Pajarola&Rossignac, Compressed Progressive Meshes, IEEE TVCG’99 Mi
Mi+1
{vsplits}
{vsplits} c j d f k h 2 1
Jarek Rossignac 3

23. CPM butterfly prediction
Predict geometry based on previous LOD
approximation using weighted sum of incident vertices and subset of topology 2 neighbors
idea based on Butterfly interpolation of Zorin and Schroeder

24. CPM prediction error encoding
Estimate edge collapse
predict original vertices A’, B’
predicted edge collapse vector v’
prediction error e = v’ - v
CPM collapses edge to its mid-point
Encode prediction error
Laplace distribution L(x)
based on prediction error variance of current refinement batch
create corresponding Huffman encoding for e B’ A B A’ v v’

25. Total amortized cost per triangle
Siggraph'98 Course21: 3D Geometry Compression
Total amortized cost per triangle
Bunny
9666 triangles, 10 LODs, 11.3T bits (3.6 connectivity geometry)
Horse
21622 triangles, 9 LODs, 10.6T bits ( )
Skull
21904 triangles, 7 LODs, 10.9T bits ( )
Fohe
7240 triangles, 7 LODs, 13.7T bits ( )
Fandisk
12950 triangles, 9 LODs, 11.4T bits ( )
Jarek Rossignac 4

26. Comparison with TS and PFS
Siggraph'98 Course21: 3D Geometry Compression
Comparison with TS and PFS
2832 vertices
PFS
254%
Estimated at 18T bits, fewer LODs
approximation error
TS [Taubin, Rossignac 98]
PFS [Taubin et al. 98] CPM [Pajarola, Rossignac 99]
bits
crude initial model TS
100%
CPM
125%
Jarek Rossignac 5

27. Error protection for CPM
G. Al-Regib, Y. Altunbasak and J. Rossignac. “An Unequal Error Protection Method for Progressively Compressed 3-D Meshes”, Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP). May 2002.
Add error correction channel bits to increase the chances of recovering from transmission errors
Adjust level of error protection to the importance of the refinement batch (optimization)

28. Progressive Transmission of Tetrahedral Meshes Implant-Spray
Siggraph'98 Course21: 3D Geometry Compression
Progressive Transmission of Tetrahedral Meshes Implant-Spray
Jarek Rossignac
(with Renato Pajarola and Andrzej Szymczak)
GVU Center and College of Computing
Georgia Tech, Atlanta
Jarek Rossignac 1

29. Progressive Tetrahedral Meshes
Siggraph'98 Course21: 3D Geometry Compression
Progressive Tetrahedral Meshes
ImplantSpray: Pajarola&Rossignac&Szymczak, IEEE VIS’99
Vertex split refinement operator
Extension of vertex split for triangle meshes [Staadt&Gross98]
Defined by split-vertex and set of incident cut-faces
Series of tetrahedral meshes defined by sequence of vertex-splits
Send crude model and batches of refinement updates
Mark split-vertices
Encode cut-faces
split-vertex
ecol
vsplit
cut-faces
Jarek Rossignac 4

30. Identifying cut-faces for the split-vertex
Siggraph'98 Course21: 3D Geometry Compression
Identifying cut-faces for the split-vertex
A vertex has roughly
12 incident edges
30 incident triangular faces
20 incident tetrahedra
A split-vertex has about 6 cut-faces
We must select about 6 triangular faces out of 30
Hull: The boundary of the star of the split-vertex
A manifold surface
Skirt: Cut-faces
connected surface around split-vertex
The skirt boundary
a cycle of k edges in the hull
closed path on planar triangle graph
cut-faces
split-vertex
hull
skirt
Jarek Rossignac 5

31. Results of Progressive Tetrahedral Meshes
Siggraph'98 Course21: 3D Geometry Compression
Results of Progressive Tetrahedral Meshes
Turbine blades
blades and exterior as tetrahedral mesh
tetrahedra
49 LODs
5.02 bits per tetrahedron
connectivity information
no geometry data
Compare to indexed face list:
4x17 bits per tetrahedron
Jarek Rossignac 6