Compressing Triangle Meshes

1
Compressing Triangle Meshes

• Leila De Floriani, Paola Magillo
• University of Genova
• Genova (Italy)
• Enrico Puppo
• National Research Council
• Genova (Italy)

2
Why Geometric Compression?

• Availability of large geometric datasets in
  mechanical CAD, virtual reality, medical imaging,
  scientific visualization, geographic information
  systems, etc.
• Need for
• speeding up transmission of geometric models
• reducing the costs of memory and of auxiliary
  storage required by such models
• enhancing rendering performances limitations on
  on-board memory and on data transfer speed

3
...Why Geometric Compression?...

• Compression methods aimed at two complementary
  tasks
• compression of geometry efficient encoding of
  numerical information attached to the vertices
  (position, surface normal, color, texture
  parameters)
• compression of mesh connectivity efficient
  encoding of the mesh topology
• Compression methods developed for triangle meshes

4
Compression of Connectivity

• Two kinds of compression methods
• Direct methods
• Goal minimize the number of bits needed to
  encode connectivity
• Progressive methods
• Goal an interrupted bitstream must provide a
  description of the whole object at a lower level
  of detail

5
Our Proposal

• Direct method
• Sequence of triangles in a shelling order
• Progressive Method
• Sequence of edge swaps
• where destructive operator vertex removal

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Sequence of Triangles in a Shelling Order

• Method based on a shelling order a sequence of
  all the triangles in the mesh with the property
  that the boundary of the set of triangles
  corresponding to any subsequence forms a simple
  polygon
• A triangle mesh is shellable if it admits a
  shelling sequence
• A shellable mesh is extendably shellable if any
  shelling sequence for a submesh can be completed
  to a shelling sequence for the whole mesh
• The method works for every triangulated surface
  homeomorphic to a sphere or a disk
• Encoding four 2-bits codes per edge SKIP,
  VERTEX, LEFT, RIGHT

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...Sequence of Triangles in a Shelling Order...

• Algorithm
• Start from an arbitrary triangle, whose boundary
  forms the initial polygon
• Loop on the edges of the current polygon
• for each edge e
• try to add the triangle t adjacent to e and lying
  outside the polygon
• if successful, update the current polygon
• in any case, send a code
• when necessary, send a vertex
• Each edge is examined at most once
• Each vertex is sent just once

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...Sequence of Triangles in a Shelling Order...

• Algorithm
• if t brings a new vertex gt VERTEX vertex
  coordinates
• if t does not exist or cannot be added gt SKIP

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...Sequence of Triangles in a Shelling Order...

• Algorithm
• if t shares the polygon edge on the left of e gt
  LEFT
• if t shares the polygon edge on the right of e
  gt RIGHT

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...Sequence of Triangles in a Shelling Order...

• Properties of the Shelling Method
• Every vertex is encoded only once
• Compression and decompression algorithms
• work in time linear in the size of the mesh
• no numerical computation necessary
• conceptually simple and easy to implement
• Adjacencies between triangles are reconstructed
  directly from the sequence at no additional cost

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...Sequence of Triangles in a Shelling Order...

• Cost Evaluation
• In theory
• at most two bits of connectivity information for
  each edge
• gt at most 6n bits for a mesh with n vertices
• In practice
• less than 4.5n bits of connectivity

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...Sequence of Triangles in a Shelling Order...

• Experimental Results (on TINs)
• Exp vert tri code bits compress.
  
• bits /vert
  time(tri/s)
• U1 42943 85290 182674 4.2538
  1.644(51879)
• U2 28510 56540 123086 4.3173
  1.077(52483)
• U3 13057 25818 57316 4.3897
  0.479(53899)
• U4 6221 12240 27180 4.3690
  0.215(56930)
• A1 15389 30566 64678 4.2029
  0.565(54099)
• A2 15233 30235 63958 4.1986
  0.561(53894)
• A3 15515 30818 65210 4.2030
  0.572(53877)
• A4 15624 31042 65520 4.1935
  0.577(53798)
• B1 5297 10570 22392 4.2273
  0.182(58076)
• B2 5494 10959 23468 4.2716
  0.188(58292)
• B3 5397 10768 23060 4.2727
  0.186(57892)
• B4 5449 10874 23136 4.2459
  0.187(58149)
• U1--4 uniform resolution (in decreasing order)
• A1--4 one fourth of the area is at high
  resolution, the rest is coarse
• B1--4 one 16th of the area is at high
  resolution, the rest is coarse

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...Sequence of Triangles in a Shelling Order...

• Properties
• The method generalizes to surfaces with arbitrary
  genus
• The algorithm automatically cuts the surface into
  simply connected patches with a small overhead
• No additional control code required
• Vertices belonging to more than one patch are
  repeated
• Cost in practice, less than 5.5n bits of
  connectivity

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...Sequence of Triangles in a Shelling Order...

• Experimental Results (on 3D meshes)
• Mesh vert tri repeated
  code bits
• patches vert
  bits /vert
• eight 766 1536 6 198
  3856 5.0339
• shape 2562 5120 1 0
  10478 4.0897
• cow 3078 5804 25 356
  13984 4.5432
• femur 3897 7798 5 124
  18894 4.8483
• pieta 3475 6976 15 468
  17124 4.9278
• skull 10950 22104 80 3242
  58150 5.3105
• bunny 34834 69451 3 323
  146986 4.2196
• fandisk 6475 12946 1 0
  27298 4.2159
• phone [phone]
  149058 4.4891

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...Sequence of Triangles in a Shelling Order...

• Experimental Results (on 3D meshes)

whole mesh patch 1
patch 2
. other 4 patches with few triangles each
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...Sequence of Triangles in a Shelling Order...

• Experimental Results (on 3D meshes)

whole mesh patch 1
patch 2
. other 78 patches with few
triangles each
17
Sequence of Edge Swaps

• Method based on the iterative removal of a vertex
  of bounded degree (less than a constant b)
  selected according to an error-based criterion
• the vertex which causes the least increase in the
  approximation error is always chosen
• The polygonal hole P left by removing vertex v
  is retriangulated
• The inverse constructive operator inserts vertex
  v and recovers the previous triangulation of P

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...Sequence of Edge Swaps...

• Sequence of Edge Swaps
• The old triangulation T is recovered from the new
  one T' by first splitting the triangle t of T'
  containing vertex v and then applying a sequence
  of edge swaps

T
T
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...Sequence of Edge Swaps...

• Sequence of Edge Swaps
• Encoding
• for each removed vertex v
• a vertex w and an integer number indicating a
  triangle around w (they define the triangle t of
  T' containing v)
• the packed sequence of edge swap which generates
  T from T'

Vertex w Triangle index 0
Sequence of edge swaps
T
T
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...Sequence of Edge Swaps...

• 1) Split triangle t into three triangles

T
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...Sequence of Edge Swaps...

• 2) Swap edge indicated by number 2 around v

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...Sequence of Edge Swaps...

• 3) Swap edge indicated by number 0 around v

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...Sequence of Edge Swaps...

• 4) Swap edge indicated by number 2 around v

T
gt swap sequence 2 0 2
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...Sequence of Edge Swaps...

• Cost Evaluation
• For each removed vertex v
• log n bits for one vertex reference
• log b bits for the index of a triangle
• for edge swap
• log r bits for the index of the edge to swap,
  where r is the current number of triangles
  incident at v
• r is initially 3, and increases by one at each
  edge swap
• at the last swap, r is at most b-1
• gt less than log((b-1)!)-1 bits for the whole
  sequence of swap indexes
• gt n(log n log b log((b-1)!)-1) bits of
  connectivity information
• for instance, for n216 and b23 gt about
  26.5216 bits of connectivity

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...Sequence of Edge Swaps...

• Properties
• Adaptivity to LOD generation is good since
  vertices are removed by taking into account the
  accuracy of the resulting approximation
• Unlike other methods (Hoppe, 1996 Snoeyink and
  van Kreveld, 1997), no specific retriangulation
  criterion is assumed
• The criterion used in the retriangulation is
  encoded in the sequence of swaps
• Coding and decoding algorithms with different
  error-driven selection criteria experimented in
  the context of multiresolution triangulations (De
  Floriani, Magillo, Puppo, IEEE Visualization 1997)