1 Compressing TINs Leila De Floriani, Paola Magillo University of Genova Genova (Italy) Enrico Puppo National Research Council Genova (Italy)
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Compressing TINs
Leila De Floriani, Paola MagilloUniversity of Genova
Genova (Italy)
Enrico PuppoNational Research Council
Genova (Italy)
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Why Terrain Compression?
Availability of large terrain datasets in GIS
Need for
speeding up transmission of terrain models through a communication line
reducing the costs of memory and of auxiliary storage required by terrain models
speeding up loading of a terrain model from disk into memory
enhancing rendering performances: limitations on on-board memory and on data transfer speed
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...Why Terrain Compression?...
Objective:
Design compact structures for encoding a terrain model as a sequential bitstream
Regular Square Grids (RSGs) can be compressed through techniques similar to those used for compressing images
This is not true for Triangulated Irregular Networks (TINs)
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...Why Terrain Compression?...
Compression methods for TINs aimed at two complementary tasks:
compression of geometry: efficient encoding of numerical information attached to the vertices (i.e., position, surface normal, texture parameters)
compression of connectivity: efficient encoding of the topology of the TIN
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Data Structures for TINsList of triangles:
It encodes the list of triangles of the TIN
For each triangle: the geometrical information associated with its three vertices (position in space, surface normal, etc.)
Drawback:
each vertex is repeated for all triangles incident in it
Storage cost:
in a TIN with n vertices, there are ~2n triangles
cost: 18n floats, if geometric information associated with a vertex is just its position in space
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...Data Structures for TINs...Indexed data structure:
List of vertices + list of triangles
For each vertex: its geometrical information
For each triangle: references to its three vertices
Storage cost:
6n log n bits +
3n floats (cost of storing geometrical information)
since a vertex reference for a triangle requires log n bits
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...Data Structures for TINs...Indexed data structure with adjacencies:
List of vertices + list of triangles
For each vertex: its geometrical information
For each triangle: references to its three vertices + references to its three adjacent triangles
Storage cost:
(12n log n + 6n) bits + 3n floats (cost of storing geometrical information), since each triangle reference requires (log n + 1) bits
t
t1
t2t3
P1
P2
P3
t: (P1,P2,P3) (t1, t2, t3)
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...Data Structures for TINs...Comparison of the three data structures
n = number of vertices, let one float = 32 bits
list of triangles: 18n floats
if n = 216 ==> 18*216 floats = 578*216 bits
indexed data structure: 6n log n bits + 3n floats
if n = 216 ==> 96*216 bits+ 3*216 floats = 150*216 bits
indexed data structure with adjacencies: (12n log n + 6n) bits+ 3n floats
if n = 216 ==> 198*216 bits + 3*216 floats = 252*216 bits
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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
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...Compression of Connectivity...
Direct Compression Methods
Triangle strips (and triangle fans) used in graphics API (e.g., OpenGL)
Generalized triangle meshes (Deering 1995; Evans et al., 1996; Chow, 1997)
Topological surgery (Taubin and Rossignac, 1996)
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...Direct Methods...
Triangle Strips
Each strip is a sequence of vertices Each triangle in a strip has its vertices at three consecutive positions A TIN is encoded as a collection of strips
Drawbacks: each vertex is encoded twice on average it is difficult to obtain few long strips [Evans
et al., 1996]
1
2
3
4
5
6
7
. . . .
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Generalized Triangle Meshes (Deering, 1995)
Sequence of vertices with alternate strip-like and fan-like behavior
Behavior at each vertex specified by a code bit A small buffer allows to reuse some past vertices A TIN is encoded as a collection of generalized strips Cost: ~11 bits per vertex for connectivity
...Direct Methods...
1
2
3
4
5
6
7
. . . .
2
6
5
1
4
3
. . . .Strip-like(zig-zag) Fan-like
(turning)
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Topological Surgery (Taubin and Rossignac, 1996)
It cuts a mesh and opens into a connected set of triangles shaped as a tree (triangle spanning tree) The edges along which the mesh is cut form another tree (vertex spanning tree)
The bitstream produced by the method contains the two trees Algorithms rather complicated
...Direct Methods...
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Progressive Compression Efficient encoding of the mesh produced by a simplification algorithm
A sequence of progressive LODs generated by iteratively applying a destructive operator which removes details from a mesh
An inverse constructive operator recovers such details
Encoding:
coarsest mesh produced in the simplification process + sequence of construction operations
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...Progressive Compression...
Progressive Compression Methods
Progressive meshes (Hoppe, 1996)
destructive operator = edge collapse
Sequence of ordered vertex sequences (Snoeyink and van Kreveld, 1997)
destructive operator = removal of a set of vertices
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...Progressive Compression... Progressive Meshes (Hoppe, 1996)
Destructive operator = edge collapse
Inverse constructive operator = vertex split
A TIN is progressively simplified by repeated edge collapses The bitstrean contains the sequence of edge collapses The TIN is reconstructed by applying a vertex split for each edge collapse
v1
v1 v2
e1
e2
e
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...Progressive Compression... Sequence of ordered vertex sequences (Snoeyink and van Kreveld, 1997)
Destructive operator = removal of a set of vertices from a Delaunay triangulation Inverse constructive operator = reinsertion of the vertices
The bitstream contains the sequence of sets of vertices removed
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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
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...Sequence of Triangles in a Shelling Order...
Algorithm
if t brings a new vertex ==> VERTEX + vertex coordinates if t does not exist or cannot be added ==> SKIP
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...Sequence of Triangles in a Shelling Order...
Algorithm
if t shares the polygon edge on the left of e ==> LEFT if t shares the polygon edge on the right of e ==> 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 ==> 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
Exp #vert #tri #code bits compress. decompr. bits /vert time(tri/s) time(tri/s)
U1 42943 85290 182674 4.2538 1.644(51879) 2.414(35331)U2 28510 56540 123086 4.3173 1.077(52483) 1.603(35271)U3 13057 25818 57316 4.3897 0.479(53899) 0.734(35174)U4 6221 12240 27180 4.3690 0.215(56930) 0.348(35172)A1 15389 30566 64678 4.2029 0.565(54099) 0.855(35749)A2 15233 30235 63958 4.1986 0.561(53894) 0.853(35455)A3 15515 30818 65210 4.2030 0.572(53877) 0.867(35545)A4 15624 31042 65520 4.1935 0.577(53798) 0.880(35275)B1 5297 10570 22392 4.2273 0.182(58076) 0.298(35469)B2 5494 10959 23468 4.2716 0.188(58292) 0.308(35581)B3 5397 10768 23060 4.2727 0.186(57892) 0.304(35421)B4 5449 10874 23136 4.2459 0.187(58149) 0.308(35305)
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 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 left by removing vertex v is retriangulated
The inverse constructive operator inserts vertex v and recovers the previous triangulation of
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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
...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: wTriangle 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
==> 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
==> less than log((b-1)!)-1 bits for the whole sequence of swap indexes
==> n(log n +log b+ log((b-1)!)-1 bits of connectivity information
for instance, for n=216 and b=23 ==> about 31.4*216 bits of connectivity
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Conclusions
Direct method:
Very simple, yet efficient, fully implemented
Compression rates better than those of other simple methods (e.g., triangle strips), comparable with those of more complex methods (e.g., topological surgery)
It generalizes to triangulated surfaces with arbitrary genus: It automatically cuts the surface into simply connected patches with a small overhead Cost: experimentally, less than 5.5n bits of connectivity
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…Conclusions...
Experimental Results of the direct method on 3D triangulations
whole mesh patch 1 patch 2 …. + other 4 patches with few triangles each
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…Conclusions...
Progressive method:
More general than other proposed methods (e.g., Hoppe, 1996; Snoeyink and van Kreveld, 1997) since no specific retriangulation criterion is assumed
Adaptivity to LOD generation is good since vertices are removed by taking into account the accuracy of the resulting approximation
Vertex removal algorithms with different error-driven selection criteria experimented in (De Floriani, Magillo, Puppo, IEEE Visualization 1997)