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EG99 Tutorial

Mesh Compression

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EG99 Tutorial

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

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EG99 Tutorial

...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

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EG99 Tutorial

Compression of Geometry(Deering, 1995; Chow, 1997)

Positions, normals and colors quantized to significantly fewer than 32 bits (single-precision IEEE floating point) with little loss in visual quality

Example: quantization for position information:

normalizing object geometry to a unit cube

positions quantized to q bits (q<=16) by truncating the less significant m bits of position components, where m=16-q

Optimizations:

positions are "delta-encoded", i.e., just the difference between a vertex position and that of its predecessor in the bitstream is encoded

quantization assignment can be done by partitioning the triangle mesh into parts of similar detail based on triangle size and curvature

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EG99 Tutorial

Compression of Connectivity

Two kinds of compression techniques:

Direct techniques:

Goal: minimize the number of bits needed to encode connectivity

Progressive techniques:

Goal: an interrupted bitstream must provide a description of the whole object at a lower level of detail

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EG99 Tutorial

Direct Methods

Triangle strips (and triangle fans) used in graphics API (e.g., OpenGL)

Generalized triangle meshes (Deering 1995; Evans et al., 1996; Chow, 1997; Bar Yehuda and Gotsman 1996)

Topological surgery (Taubin and Rossignac, 1996)

Sequence of triangles in a shelling order (De Floriani, Magillo and Puppo, 1998)

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EG99 Tutorial

...Direct Methods... Triangle Strips Defined as a collection of sequences of vertices Each triangle in a strip has its vertices at three

consecutive positions

Encoding: number and lengths of the strips two vertex references per strip one vertex reference per triangle one bit per triangle (swap code)

Drawbacks: each vertex must be encoded twice on average it is difficult to obtain few long strips from

a generic mesh [Evans et al., 1996]

Sequence: 1,2,3,4,5,6,71

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EG99 Tutorial

...Direct Methods... Generalized Triangle Strips (with direct vertex encoding)

1,2,3,4,5,6,7,8 NO!

1,2,3,4,5,6,7,5,8 YES

Another way of encoding with swap codes:

it tells on which open side the new vertex must be attached

1,2,0,3,0,4,0,5,0,6,0,7,1,8

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EG99 Tutorial

...Direct Methods...

Generalized Triangle Meshes (Deering, 1995) Definition of a new syntax for a generalized strip Use of a mesh buffer: a queue of maximum length 16 of previously used vertices

Encoding: one bit per vertex to indicate whether the vertex must be inserted into the mesh buffer two bits per triangle to indicate how to continue the current strip (three possible codes) one bit per triangle to indicate whether a new vertex should be read or a vertex from the mesh buffer must be used 4 bits of address to access a vertex of the mesh buffer

Cost: ~11 bits per vertex for connectivity

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EG99 Tutorial

...Direct Methods...

Generalized Triangle Meshes

Example:

R = Restart RO = Replace Oldest RM = Replace Middle P = Push number = vertex index in mesh -number = vertex index in

buffer (from top)

R 6 P RO 1 RO 7 P RO 2 RO 3 RM 4 RM 8 P RO 5 RO 9 P RO 10 RM 11 RO 17 P RM 16 P RM -3 RO 15 P RM -5 RO 6 RM 14 P RO 13 RM -9 RO 12 RM 18 P RM -3 RO 15 P RO -5 RO 6 RM 14 P RO 13 P RM -9 RO 23 RO -10 RO -7 RM 30 RM 29 RM 28 RM -1 RO -2 RM -3 RM 27 RO 26 RM -4 RO 25 RO -5

QUEUE: 6,7,8,9,17,16,...

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EG99 Tutorial

...Direct Methods...

Topological Surgery (Taubin and Rossignac, 1996)

Vertex spanning tree formed by selecting the minimum number of edges in the input mesh which connect all its vertices

The edges of the vertex spanning tree decompose the mesh into a collection of non-overlapping triangle strips (runs)

Such triangle strips form a spanning tree of the triangles of the mesh

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EG99 Tutorial

...Direct Methods...

Topological Surgery

Example:

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EG99 Tutorial

...Direct Methods...

Topological Surgery

Encoding structure: vertex + triangle spanning trees

Order imposed on the vertices by the spanning tree exploited for computing estimates of the position of a vertex from the position of its 3 or 4 ancestors in the tree

Cost: between 2.5 and 7 bits per vertex to encode connectivity

Drawbacks: decompression stage complicated by large memory requirements (all information must be kept in main memory when reconstructing the mesh)

difficult to implement

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EG99 Tutorial

...Direct Methods...

A Compression Method based on Shelling

(De Floriani, Magillo and Puppo, 1998)

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

The compression algorithm 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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EG99 Tutorial

...Direct Methods...

A Compression Method based on Shelling

Start from an arbitrary triangle, whose boundary forms the current polygon

Loop on the edges of the current polygon:

for each edge e: try to add the triangle t externally adjacent to e if successful, update the current polygon in any case, send a code when necessary, send a vertex

Each vertex is sent just once Each edge is examined at most once

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EG99 Tutorial

...Direct Methods...

A Compression Method based on Shelling

if t brings a new vertex ==> VERTEX + vertex coordinates if t does not exist or cannot be added ==> SKIP

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EG99 Tutorial

...Direct Methods...

A Compression Method based on Shelling

if t shares the polygon edge to the left of e ==> LEFT if t shares the polygon edge to the right of e ==> RIGHT

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EG99 Tutorial

...Direct Methods...

A Compression Method based on Shelling

Properties of the Shelling Method Every vertex is encoded only once Each edge is examined at most once Compression and decompression algorithms:

work in time linear in the size of the mesh no numerical computations conceptually simple and easy to implement

Adjacencies between triangles are reconstructed directly from the sequence at no additional cost Cost:

in theory: at most 6 bits of connectivity per vertex in practice: less than 4.5 bits of connectivity per vertex

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EG99 Tutorial

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

Experimental Results (on TINs)

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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EG99 Tutorial

...Direct Methods...

A Compression Method based on Shelling

The method generalizes to surfaces with arbitrary genus The algorithm automatically cuts the surface into simply

connected meshes with a small overhead

No additional control codes required Vertices belopnging to more than one patch are repeated

Cost: in practice, less than 5.5 bits of connectivity per vertex

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EG99 Tutorial

...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 5.0339shape 2562 5120 1 0 10478 4.0897cow 3078 5804 25 356 13984 4.5432femur 3897 7798 5 124 18894 4.8483pieta 3475 6976 15 468 17124 4.9278skull 10950 22104 80 3242 58150 5.3105bunny 34834 69451 3 323 146986 4.2196fandisk 6475 12946 1 0 27298 4.2159phone 33204 66287 3 12 149058 4.4891

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EG99 Tutorial

...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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EG99 Tutorial

...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

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EG99 Tutorial

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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EG99 Tutorial

...Progressive Compression...

Each LOD can be seen as a form of lossy compression of the original mesh

There is a trade-off between loading/transmission times and loss of detail

Compression rates are usually lower than those achieved by direct techniques

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EG99 Tutorial

...Progressive Compression... Progressive Compression Methods

Progressive meshes (Hoppe, 1996)

destructive operator = edge collapse

Sequence of edge swaps (De Floriani, Magillo and Puppo, 1998)

destructive operator = vertex removal

Sequence of ordered vertex sequences (Snoeyink and van Kreveld, 1997)

destructive operator = removal of a set of vertices

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EG99 Tutorial

...Progressive Compression...

Progressive Meshes (Hoppe, 1996) Edge collapse:

replace an edge e with a vertex v1 and the two triangles sharing e with two edges incident at v1

Vertex split (inverse operation): expand a vertex v1 into an edge e=v1v2 and two edges e1 and e2 among those

incident at v1 into two triangles

v1

v1 v2

e1

e2

e

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EG99 Tutorial

...Progressive Compression... Progressive Meshes (Hoppe, 1996)

Encoding: new vertex v2 reference to v1 code specifying the position of e1 and e2 in the set of edges incident at v1

Cost: n(log n + log(b(b-1))) bits of connectivity, where b = maximum degree of a vertex at any step

for instance, for n=216 and b=23 ==> about 21.8*216 bits of connectivity

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EG99 Tutorial

...Progressive Compression...

Sequence of Edge Swaps

(De Floriani, Magillo, Puppo, 1998)

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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EG99 Tutorial

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

...Progressive Compression...

T

T’

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EG99 Tutorial

...Progressive Compression...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'

T’

T

Vertex: wTriangle index: 0Swap sequence: 2 0 2

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EG99 Tutorial

...Progressive Compression...

Sequence of Edge Swaps

Cost:

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

Properties:

adaptivity to LOD generation is good since the vertices are removed by taking into account the accuracy of the resulting approximation

the criterion used in the retriangulation is encoded in the sequence of swaps: more general than other progressive methods

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EG99 Tutorial

...Progressive Compression...

Snoeyink and van Kreveld's Method

It applies to Triangulated Irregular Networks (TINs) based on Delaunay triangulation

Delaunay triangulation:

a triangulation is called a Delaunay triangulation if the circumcircle of any of its triangles does not contain any vertex in its interior

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EG99 Tutorial

...Progressive Compression... Snoeyink and van Kreveld's Method

LOD generation process:

at each step, a maximal set of independent vertices (i.e., vertices which are not connected by an edge) of bounded degree is removed

The process of removing a set of vertices terminates in a logarithmic number of steps