Advances and challenges in the design of compact representations of meshes and complexes

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Advances and challenges in the design of compact
representations of meshes and complexes

• Jarek Rossignac
• School of Interactive Computing
• Georgia Tech

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CATRAT

• We want Constant Amortized Time (CAT) cost
  operators
• for Random Access and Traversal (RAT) of the mesh
• Access vertices and faces using (nearly)
  consecutive IDs
• For (int v 0 vltV v) processVertex(v)
• For (int f 0 fltF f) processFace(f)
• Given vertex, v, obtain an incident face, f(v)
• Given face, f, obtain a bounding vertex, v(f)
• Given tuple (f,v) of a face f and a bounding
  vertex v, obtain
• Next tuple (swing clockwise) around v
• Next tuple (walk cw) around f
• Want the time cost of each operator
• to be constant (on average) and small

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Example of an edge-based scheme

• For each face f, store a reference to one of its
  bounding edges
• For each vertex v, store a reference to one
  incident edge
• For each edge e, store references to
• Its 2 bounding vertices
• Its 2 incident faces
• Its 2 following edges along each face
• Cost F V 6E references (32-bit integers)
• In a planar triangulation
• F 2V 4
• E 3V
• Cost 2V V 18 V 21V 672 bpv (bits per
  vertex)
• Zipper reduces this cost to 12 bpv

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Prior art Mesh connectivity storage cost
r V r 32 bits per vertex
T
T
ESQ (editable SQuad) Castelli, Devillers,
Rossignac12 SIBGRAPI 4V
b bpv b bits per vertex
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Corner-based operators

• Access
• v(f) an arbitrary vertex bounding given face f
  (f 0,1,2F-1)
• f(v) an arbitrary face incident upon given
  vertex (v 0,1,2V-1)
• Traversal
• n(f,v) next vertex after v (walking cw) around
  face f
• s(f,v) next face after f (swinging cw) around
  vertex v
• SIMPLIFIED NOTATION
• Corner c(f,v) as place marker for RAT
• n(f,v) is now written c.n
• s(f,v) is now written c.s
• Bijections between corners and
• half-edges, edge-uses, oriented-edges, darts

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Corners operators for triangle meshes

• Assign consecutive integer IDs to the 3 corners
  of each triangle
• Core operators
• c.v the vertex of corner c
• c.t the triangle of corner c
• v.c an arbitrary corner of vertex v
• t.c an arbitrary corner of triangle t
• c.n next corner around c.t
• c.o opposite (in adjacent triangle)
• Derived operators
• c.s c.n.o.n swing around c.v
• c.p c.n.n (previous in c.t)
• c.l c.n.o (left)
• c.r c.p.o (right)
• c.u c.r.p (unswing around c.v)

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Extended Corner Table (ECT) 13V

• Based on previously proposed Corner Table (CT)
• Edgebreaker on a Corner Table A simple
  technique for representing and compressing
  triangulated surfaces,
• J. Rossignac, A. Safonova, A. Szymczak.
• Book chapter in
• Hierarchical and Geometrical Methods in
  Scientific Visualization,
• Farin, G., Hagen, H. and Hamann, B., eds.
• Springer-Verlag, Heidelberg, Germany. Pages
  41-50, January 2003.
• ECT stores back pointer Cv from vertex to one
  incident corner

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Extended Corner Table (ECT)

• Geometry short int XV, YV, ZV
• 16-bit coordinates relative to AABB
• Yields 1/32 mm accuracy over a 2 m aircraft
  engine
• Connectivity int CV, V3T, O3T
• v.c Cv, c.v Vc, c.o Oc
• stored in look-up tables
• c.t c/3, t.c 3t
• Since corners of t have consecutive IDs
• c.n c.t.c (c1)3
• Since corners of t are sorted (orientation)
• Storage cost (14.5V 464 bpv)
• Geometry 30.5 1.5V
• Connectivity ECT 6TV 13.0V (90 of total
  storage)
• For simplicity, assume low-genus, manifold mesh
  T 2V

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SCT Corner Table for Simplicial Complexes

• SCT Simplicial Corner Table, J. Rossignac X.
  Zhu, in preparation.
• Same data structure as ECT
• Extends topology to support
• Non-manifold edges
• Non-manifold vertices
• Wires (dangling edges)
• 6 corners per triangle
• 3 on each side
• Side-preserving crawl (spider)
• c.s and c.o do not cross surface
• New operator
• To cross surface

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CT extended to tetrahedra

• TetStreamer Compressed Back-to-Front
  Transmission of Delaunay Tetrahedra Meshes, U.
  Bischoff and J. Rossignac, Proceedings of the
  Data Compression Conference, IEEE Computer
  Society, 2005. .
• Same data structure as CT
• V and O
• 4 corners per tet
• Use wedge as place marker for RAT
• Wedge operators (mimic corner ones)
• Swing around edge
• Next wedge in tet from vertex
• Cross wedge
• Use test to compute twist
• Between face-adjacent tets

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SOT (Sorted O Table) for triangle meshes 6V

• SOT Compact Representation for Triangle and
  Tetrahedral Meshes,
• T. Gurung and J. Rossignac,
• Georgia Institute of Technology,
• SIC Technical Report GT-IC-10-01, 2010
• Start with ECT C, V, O tables
• Match each v with unique t
• Sort triangles so that matched t has same ID as
  its matching vertex
• Discard C V tables (no longer needed)!
• To compute c.v, swing (cc.s) until c lt 3V

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Linear cost construction of SOT

• Start with V O tables
• Traversal matches each vertex with unique
  triangle

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Extending SOT to tetrahedron meshes 4T

• SOT Compact representation for Tetrahedral
  Meshes,
• T. Gurung and J. Rossignac.
• ACM Symposium on Solid and Physical Modeling
  (SPM), 79-88. 2009.
• 4 references per tet
• Trivial combination of the extension of CT to tet
  meshes and of the SOT idea

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SOT Construction

• Start with V O tables (for tet meshes)
• Depth-first traversal to match each vertex with
  unique tet
• Renumber tets so that matched tet has same ID as
  its vertex
• Discard the V table
• To compute the ID of vertex v from a corner c, we
  may need to visit all tets incident on v
• Same as visiting all triangles on a sphere
  (boundary of star(v))

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SQuad (sorted quads) 4V

• SQuad Compact Representation for Triangle
  Meshes,
• T. Gurung, D. Laney (LLNL), P. Lindstrom (LLNL),
  and J. Rossignac. Eurographics 2011.
• Published as a journal paper in the
• Computer Graphics Forum Journal (CGF) 30(2)
  355-364.
• Match most v with two adjacent ts (quad),
• sort quads, store only 4 (external) c.s swings
  per quad
• (the internal ones are implicit)

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SQuad construction and representation

• Starts with depth-first traversal (as for SOT)
• But try to pair each matched triangle with an
  unmatched adjacent neighbor that is also incident
  on the matching vertex (only 2 candidates, try
  right first)
• Store only 4 outer swings per quad
• Single triangle groups stored using same
  structure
• Unmatched triangles stored at the end of the
  table
• Do not store V instead swing as in SOT

97 triangles are paired
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SQuad preserves memory coherence

• Squad preserves vertex order
• So it mostly preserves memory coherence
• Except for unmatched triangles (which are stored
  at the end)

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ESQ Editable SQuad

• ESQ Editable SQuad representation for triangle
  meshes,
• L. Castelli Aleardi, O. Devillers, J. Rossignac.
• IEEE sponsored Conference on Graphics, Patterns
  and Images (SIBGRAPI),
• August 22, 2012, Brazil.
• Extend the different types of groups V, VT, VTT,
  T, VVT?
• Use a separate array for each type (fixe size)
• Support constant time local edits
• Valence-3 Vertex deletion
• Edge flip
• Triangle split
• Reform groups locally
• Many configurations
• Update arrays (no holes)

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LR 2V

• LR Compact connectivity representation for
  triangle meshes,
• Topraj Gurung, Peter Lindstrom, Mark Luffel,
  Jarek Rossignac.
• ACM SIGGRAPH 2011.
• Published as a journal paper in the ACM
  Transactions on Graphics (TOG), 30(4) 67, August
  2011.
• Arrange quad diagonals along quasi-Hamiltonian
  cycle (ring), store v.L / v.R, exceptions

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Ring Expander (LR ring construction)
0.005 of vertices are not in the ring
Compute a nearly-Hamiltonian cycle in a mesh
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Triangle types (relative to ring)

• Subscripts number of ring edges
• T0 have no ring edges (splits)
• T1 have one ring edge (most common)
• T2 have two ring edge (dead-ends)
• Superscripts warts exceptions
• Wart adjacent (T0w,T2w) pair
• Non-wart T0s and their T1i neighbors
• are exceptions

Triangles in warts 2.35 Non-wart T0 1.05
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Numbering elements and storing data in LR

• Vertices numbered along ring
• v.P previous ring vertex (implicit)
• v.N next ring vertex (implicit)
• v.L left neighbor (stored as Lv in L array)
• v.R right neighbor (stored in R array)
• Triangles
• Match v with 2 triangles sharing edge (v , v.N)
• Implicitly numbered 2v and 2v1
• Corners Numbered around quad v.i 8vi
• Skip .3 to avoid cost of division by 3

2v1
2v
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Computing c.o
A few cases. Here are two examples
Case 2 test v.L.P.L v.N
Case 1 test v.L v.N.L
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Wart skipping (LR)

• Each T2w is associated with two ring edges (has 2
  slots)
• (v.P,v) and (v,v.N)
• LR stores T0w in the first slot
• u tip vertex of matching T0w
• Lv.Pu and Lvv.P
• Saves 15 refs per wart
• 20 storage cost reduction, resulting in cost
  1.08T

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BELR 52 bpv

• Relative indexing
• Stores delta (v.L v)V instead of v.L
• Can often be represented using 16 bits
• Uses a ring-expander that makes shorter corridors
• Compromise between corridor length and (T0 count)

Depth first
Breadth first
Optimized
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Zipper 12 bpv

• Zipper A compact connectivity data structure
  for triangle meshes,
• T. Gurung, M. Luffel, P. Lindstrom, and J.
  Rossignac.
• ACM Symposium on Solid and Physical Modeling,
  October 2012.
• Journal of Computer-Aided Design. 45(2) 262-269
  (2013).
• Store most v.L and v.R as 2-bit deltas (gaps)
• Use only 6 refs per T0 exception
• An improved ring construction leading to fewer
  exceptions

Stores connectivity using only 6 bits per
triangle 57 better than Surface-Mesh
Sieger11 35 better than Directed-Edge
Campagna98 4.4 better than BELR GLLR
SIGGRAPH11 Linear time cost construction
Local connectivity changes non constant
cost Constant time access and traversal
operators 2-3 faster than BELR, 3
slower than LR 15 slower in app than
Surface-Mesh Sieger11
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2-bit deltas

• Zipper stores a 3-bit code per ring triangle
• 2-bit delta v.?L v.P.Lv.L (when it is 0, 1, 2,
  or 3)
• All deltas are 0, except at T0s
• If v has valence 6, then 0 ?L 3
• Stats v.?L 0 (29), 1 (44), 2 (19), 3 (4)
• 1-bit wart marker
• Code 111 identifies an exception v.L stored
  explicitly
• (The rare warts with delta3 are treated as
  exceptions.)

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Decoding v.L and v.R quickly (Zipper)

• To compute v.L without summing all preceding
  deltas
• Force an exception every 32-vertex block
• Distribute 3-bit (deltawart) codes into three
  32-bit words (hi,mid,lo)
• Use bitwise AND to compute mask identifying
  exceptions (111)
• Use a population count (POPCNT ) instruction to
  count set bits (number of exceptions) and
  identify index of preceding exception
• Compute sum of deltas from it to v using POPCNT
• Using POPCNT instead of loop improves speed 10x

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Computing opposites in T1 and T2 (Zipper)

• From T1 to T1 (trivial when delta is 0 or 1)
• From T0 to a T1 or T2
• With each T0 , we store 3 vertex refs (c.v, u, v)
  
• We use v.L/v.R v.P to infer c.o in constant
  time

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Computing opposites in T0 (Zipper)

• We store T0s in a 4-ary cuckoo hash
  Fotakis03)
• With each T0 we store only 6 references
• 3 IDs of opposite corner (hash values)
• Can discard hash if c.o not required by
  application
• 3 vertex refs (c.v, u, v)
• To get from a corner c in a T1 to c.o in a T0
• Use c as hash key, which returns 4 candidates for
  c.o
• For the correct c.o, c.o.t is incident upon c.p.v
  and c.n.v

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Ring Bender (Zipper)

• Reduce the number of T0 triangles by 2.3x
• Every T0 is connected to a T2 by a corridor of
  T1s
• Rearrange ring to remove T1s in T0T1T1T2
  corridors
• to make (T0,T2 ) adjacent (wart)
• Increases warts from 2.35 to 3.2
• Reduces number of exceptions
• Saves about 1 bpt

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Storage and speed results (Zipper)

• Storage median 5.98 bpt
• Worst case models Buddha (13.37 bpt) and Thai
  (7.97 bpt)
• Average (excluding Buddha Thai) 5.89 bpt
• Fixed storage cost 5 bpt (5 words for 32
  triangles block)
• 3 words for codes, a reference into an exception
  table, and a reference for the forced exception.
• Variable cost increase exception count by 3
  adds 1 bpt
• Performance of c.v and c.o when mesh fits in
  memory
• Zipper ops are 2-3x faster than BELR,
• Zipper ops are 1.8-3.6x slower than LR
• In applications (graph distance)
• 15 slower than Surface Mesh Sieger11

Representation Time
Surface Mesh 1.00
CT 0.85
LR 0.83
Zipper 1.18
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Summary future challenges

• Summary for Zipper
• Stores connectivity using only 6T bits
• 35x better than Directed-Edge Campagna98
• 4.4x better than BELR GLLR SIGGRAPH11
• Linear time cost construction
• No constant cost connectivity changes
• Constant time access and traversal operators
• 2x-3x faster than BELR, 3x slower than LR,
• In app (distance) 15 slower than Surface Mesh
  Sieger11
• Interesting (or impossible?) challenges
• Zipper-64 5 bpt?
• Application to general polygon mesh connectivity
  5 bpv?
• Support non-manifold complexes?
• Support streaming?
• Support connectivity changes?
• Apply to tetrahedron meshes?

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Zipper extended to polygon Y-meshes 6 bpv

• General position polygon mesh
• Assume manifold
• No 4 planes intersect
• Each vertex has valence 3
• Use Zipper to represent its dual graph
• 12 bits per primal face
• Primal mesh has twice as many vertices as faces
• For more general polygon meshes
• Must split vertices
• Dual of triangulating polygonal faces

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Recap

• Extend ECT to support simplicial complexes
• SOT 6V
• Match (VT groups), renumber matched triangles,
  store O
• Extend SOT to tetrahedron meshes
• SQuad 4V
• Match (VTT groups) renumber matched triangles,
  store 2 swings per tri
• ESQ Modify SQuad to support local edit at
  constant cost
• LR 2V (BELR 1.6V)
• Build quasi-Hamiltonian ring, store v.L , v.R,
  exceptions (T0 and neighbors)
• BELR store deltas (v.L - v / v.R v), optimize
  ring deltas fit in 16 bits
• Zipper 0.19 V
• Store gaps (v.L v.P.L / v.R v.P.R), encode
  most of them with 2 bits, improve ring and reduce
  number and cost of exceptions
• Apply Zipper to Y-meshes (valence-3 polygon
  meshes)