Geometric Compression Through Topological Surgery

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Geometric Compression Through Topological Surgery

• Gabriel Taubin (IBM)
• Jarek Rossignac (Georgia Tech)

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Problem Statement

• Want to send mesh across network
• Compress Losslessly
• Want to save CPU time
• Decompress on load
• Want to get triangle strips
• Save on vertex reads transforms

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Basic Idea

• Use properties of planar graphs to compress
• Extend to graphs 3d meshes using hacks
• Lossily drop bits left right out of vertex
  position since vertex pos dominates size
• Use reasonably good predictor to guess location

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Compressing Topology

• Start with meshes having Euler Characteristic 2
  (Deformable to sphere)

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Compressing Topology

• Want to cut it into a planar mesh with lots of
  long triangle strips

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Compressing Topology

• A MST on vertices indicates some interesting cut
  lines

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Compressing Topology

• Notice how the edge boundary graph has 2 edges
  for each MST cut line

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Compressing Topology

• Splaying our faithful Euler-2 characteristic mesh
  looks as follows

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Compressing Topology

• Note that in the flattened shape a binary graph
  can represent edges

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Compressing Topology

• If we save these two graphs 1 bit per marching
  edge graph as to how it fits to MST, weve got
  topology of Euler-2 meshes

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Not all cuts are equal

• But some will earn you a prize
• from lInstitut Paul Bocuse in Lyons
• Others

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Not all cuts are equal
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Why cuts are not equal

• Vertex edge graph stored as
• Integer runlength
• Bit leaf?
• Bit more runs starting from here?
• Longer triangle strips
• Amortize-out 2 starting vertices
• Faster rendering

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1st cut spanning tree construction

• Use edge length and run MST algorithm
• Disaster with a capital D

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Actual attempt to make MST

• Distance from edge midpoint to tree root

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Modification to improve result

• 2 pass algorithm where only non-branching edges
  are allowed on 1st pass

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Concentric Rings Method

• Choose a root, order by concentric rings

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Connect concentric layers

• with triangle that minimize branches

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Final Mesh
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Concentric Rings applied to bone

• Generating concentric rings makes for much longer
  triangle runs

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Encoding Vertex Position

• Vertex Spanning Tree offers mechanism for vertex
  position prediction
• Build a prediction function based on samples from
  parents in run

A

• Store per-vertex fixed-point delta from given
  predictor

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Vertex Position Predictor

• Assume a linear predictor
• f(v0vK)b0v0bkvk
• Thus estimate b0bk by minimizing the least
  square error over all vertices of depth ngtk for
  given model

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Dealing with real meshes

• Jump Edges
• Ugly hack to deal with non-spheres
• Read paper for juicy details
• Making sure that edges line up after quantization
  prediction

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Results

• They dont all add up
• Feels like they use the red herring called
• ASCII VRML
• To boost their results

(sometimes literally)
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Results (what a) Crock

• About as lossless as a spam stock investment

12 bits per coord
10 bits per coord
8 bits per coord
original
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Results Crocodile
(a) (b) (c) (d)
Bits/tri 5.00 4.23 2.77 2.16
Vertex runs 1292 1528 80 168
Tri runs 2388 1612 1340 526
Bits/v-run 5 4 10 8
Bits/t-run 5 5 6 8
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ResultsArchitecture
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ResultsArchitecture
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Their stated results

• Parsing a file in compressed binary is 20x faster
  than in ascii
• So is transcribing it from stone tablets
• Writing uncompressed binary is 30x faster than
  writing in ascii
• Are they using printf or something?!
• Unless youre writing aligned data, this bitwise
  stuff should kill your perf if you understand
  what you are doing, and it should be no slower
  than 2x-4x (ASCII is about 2-4x bigger)

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Their stated results

• Decompression reconstructs 60-90Ktris/s
• Takes same time to construct scene in VRML as
  compressing it with their algo
• If they would only compare to something
  not-terrible
• Writing scene in compressed form is 10x faster
  than in uncompressed.
• If youre writing to disk or network, sure

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The cool result

• Optimality analysis of their algorithm
• Number of triangulations of simply connected
  polygon of n2 vertices
• If enumerated and an index into triangulation is
  used, encoding requires log2(ceil(Cn)) bits. As
  n-gtinf expr-gt 2

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The cool result
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The cool result

• Optimal fixed-length encoding 2bits per vertex
• On some examples they do better
• They have some evidence of better performance
  than this fixed length scheme
• on highly tesselated models
• Like a bunny subdivided 2 times to have 38,000
  polys (1.5 bits per tri)
• Potentially like a modern game model?
• Around 2.5 for other models

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Triangle Mesh Compression

• Costa Touma, Craig Gotsman
• Technion - Israel Institute of Technology

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Insight

• In polygonal mesh that is orientable
• Vertices incident on any mesh vertex may be
  ordered
• Separation property
• If you cut out a ring of a mesh, it separates it
  into 2 disjoint meshes
• set of vertices inside (may be empty)
• set outside
• Connectivity can be encoded by degree

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Definitions

• Vertex cycle
• Cyclic sequence of vertices along tri edges
• Active list
• Vertex cycle at wavefront of encoding. Mesh
  divided into portion of mesh encoded and portion
  not encoded
• Focus
• Vertex in active list being processed

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Step 0 Make object closed
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Step 1 Pick tri for active list

• Specify node degreesadd 6, add 7, add 4

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Continue working with focus

• Add 4, Add 8

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And now focus is done

• Add 5, Add 5

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Active edge added twice split

• Add 5, Split 5

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Algorithm

• Start with triangle, active list of 3 vertices
• Add n vertices (clockwise order) adjacent to
  current focus to active list with add ltngt
  command
• If active list intersects itself, it is split to
  two active list with a split ltoffsetgt command

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Merge case

• This deals with genus 1 objects (torus,etc)
• If first free edge of active vertex is in active
  list of vertex, 2 possibilities
• It is on current active list
• Split ltoffsetgt
• It is on active list already split
• Merge ltindexgt ltoffsetgt

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Better prediction of vertex position

• Have previous triangle
• Compute plane equation
• Assume next triangle is coplanar
• Encode error term
• Simple, elegant, low error
• Use codebook for most common errors

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Results

• For regular meshes
• Get around .2 bits per vertex for topology
• With RLE
• Increasing quantization from 8-10 bits, size goes
  up by 30-40 (!?)

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Results
Model verts VRML gz IBM conn IBM cord Our conn Our cord
blob 8036 117K 3447 10352 1709 7951
tri 2832 44K 1523 3673 764 2937
eight 766 11K 363 1146 53 683
shape 2562 35K 713 4578 48 2990
beet 2655 36K 1585 4982 781 3576
eng 2164 24K 1041 4703 330 3425
dum 11738 114K 4929 20351 1210 11162
cow 3066 40K 1766 4878 779 3376

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Decompression

• Leaf triangle reconstructed from root id

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Decompression
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Decompression
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Decompression
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Decompression
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Decompression
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Decompression
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Decompression
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