Mesh Compression Mesh Compression Connectivity: Often, - - PDF document

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Mesh Compression Mesh Compression

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Triangle Meshes

Two main parts: Connectivity:

Often, triangulated graph Sometimes, polygons 3-connected graphs

Geometry:

Positions in space

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Basic Definitions (I)

Vertex: node V vertices Edge: between 2 vertices - E edges Face: face between edges - F faces Euler Formula: F – E + V = 2(c-g) –b

c: # of connected components g: genus b: # of boundary edges

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Basic Definitions (II)

Valence of a vertex: number of emanating edges Degree of a face: number of edges around

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History of Multimedia

Success of Digital Signal Processing (DSP)

Sounds, images, videos Technology followed scientific evolution Compression allows small download times

MP3, JPEG, MPEG…

70 80 90 00

sound images video geometry text CS101b - Meshing

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Compression

(De)Compression of 3D Meshes:

Huge meshes Arbitrary topology Irregular connectivity Non-uniform sampling

I am too fat….

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A Common Format: VRML

Vertices

(geometry)

v1 (x1;y1;z1) v2 (x2;y2;z2) v3 (x3;y3;z3) v4 (x4;y4;z4) v5 (x5;y5;z5) v6 (x6;y6;z6) v7 (x7;y7;z7) Faces

(connectivity)

f1 (v1;v3;v2) f2 (v4;v3;v1) f3 (v4;v1;v5) f4 (v1;v6;v5) f5 (v6;v1;v7) f6 (v2;v7;v1) f7 (…)

V R M L

Valence n: n redundancies → average 6

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Geometry - Naïve Encoding

Vertices

(geometry)

v1 (x1;y1;z1) v2 (x2;y2;z2) v3 (x3;y3;z3) v4 (x4;y4;z4) v5 (x5;y5;z5) v6 (x6;y6;z6) v7 (x7;y7;z7)

Naïve: 3 * float 3 2 = 9 6 b/ v Direct ( quantization) : 3 * 1 0 bits = 3 0 b/ v

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Naïve: 3 * int 3 2 = 9 6 bit/ face ≈1 9 2 b/ v Direct: 3 * log2( V) bit/ face ≈5 0 b/ f for 1 0 0 k vertices ≈1 0 0 b/ v

Faces

(connectivity)

f1 (v1;v3;v2) f2 (v4;v3;v1) f3 (v4;v1;v5) f4 (v1;v6;v5) f5 (v6;v1;v7) f6 (v2;v7;v1) f7 (…)

Connectivity -Naïve Encoding

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And Yet… Far from Optimal

From a census of planar maps [Tutte ’64], #triangulation(V) = For large V (>1000), #triangulation(V) ≈ Theoretical connectivity bound: 3.24b/v ( ) ( ) ( )!

1 ! 2 3 ! 1 4 2 + + + V V V V ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 27 256

V V 24 . 3 27 256 log

2 2

2

=

⎟ ⎠ ⎞ ⎜ ⎝ ⎛

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A Look at Data Compression

Data compression: Data compression:

Storing data in a format that requires Storing data in a format that requires less space than usual = packing data less space than usual = packing data

Improves: Improves:

• Storage
• Transmission
• Retrieval

Big file

Small file

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Lossless Encoding

Well suited for text, files, medical data, …

Goal: Remove all redundancy

Data Data Data Data

encoding encoding decoding decoding

0 1 1 00 1 0 01 0 00 1 00 0 1 0 1 1 00 1 0 01 0 00 1 00 0 1 … …

= =

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Lossy Encoding

Well suited for sound, images, videos, etc…

Goal: Remove unnoticeable features, then all redundancy

Data Data Data Data

encoding encoding decoding decoding

0 1 1 00 1 0 01 0 01 0 1 1 00 1 0 01 0 01 … …

!= !=

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The Notion of Entropy

First, decompose a data set into a sequence of events. Entropy: Smallest possible expected numbers of bits needed to encode a sequence of events. pi = probability of occurrence of event i

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Entropy Encoding

The game:

Find bit-rate closest to entropy

The idea:

Give shortest codewords to most probable events.

Two problems:

• 1. How to turn this notion into an algorithm?
• 2. How to benefit from statistical modeling?

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Run-Length Encoding

Run: consecutive values Example: Sequence: R T A A A A A A S D E E E E E Replaced by: R T *6A S D *5E

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Dictionary methods

The idea: many data types contain repeating patterns → find an earlier occurrence in the input data, and only output a pointer/length. Extensions:

• combined with Huffman, etc.

[Lempel [Lempel-

• Ziv 77]

Ziv 77] lha, arc, arj, zip

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Huffman Encoding

Generate codes with variable length

• Higher probabilities → shorter code words
• Lower probabilities → longer code words

Symbol Code word A B 10 C 110 D 111

Cons:

• Integer code length
• Must transmit probabilities

Extensions:

• Adaptive (non stationary data)
• Extended (groups of symbols)

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Arithmetic Encoding

Pros:

• Non-integer code length
• Optimal for large #symbols -> (entropy + ε) b/symbol

Cons:

• Must transmit probabilities

Extensions:

• adaptive (avoid transmission of probabilities)
• order-n: inter-symbol probability
• stack-run: coupled with detection of runs

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Statistical Modeling

Massaging the data can pay off

• Delta-encoding
• Prediction
• Transform

Prior knowledge on the data helps

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Example of Lossy encoding

Typically: 1) Transform DCT Wavelets Bloc sorting, Distance coding, etc. 2) Quantization Scalar Vector 3) [Detection of runs] 4) Entropy encoding

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Needs in 3D Encoding (I)

Order of transmission:

Single-rate coders: Encode a mesh in one single bitstream. Also called single-resolution coders. Progressive coders: Encode a mesh as successive refinements. Coarse-to-fine style.

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Needs in 3D Encoding (I)

Order of transmission:

Single-rate Progressive

T r a n s m i s s i o n

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Needs in 3D Encoding (II)

Integrity of encoding:

Lossless coders: Preserve connectivity and/or geometry.

For: collaborative design, games, medical data…

Lossy coders: Optimize the ratio rate/distortion

For: geometry over the internet, e-commerce…

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Needs in 3D Encoding (III)

Additional features:

Resilience: robust to packet loss Efficiency: fast, or low memory requirements Guaranteed bounds: ensuring max bit rate Scalability: max speed vs min bit rate Etc…

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Part I

Single-Rate Codecs

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Redundancy and VRML

Vertices

(geometry)

v1 (x1;y1;z1) v2 (x2;y2;z2) v3 (x3;y3;z3) v4 (x4;y4;z4) v5 (x5;y5;z5) v6 (x6;y6;z6) v7 (x7;y7;z7) Faces

(connectivity)

f1 (v1;v3;v2) f2 (v4;v3;v1) f3 (v4;v1;v5) f4 (v1;v6;v5) f5 (v6;v1;v7) f6 (v2;v7;v1) f7 (…)

V R M L

Valence n ~ redundancy

• > average 6

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Naïve Encoding

Connectivity f1 (v1; v3; v2) f2 (v4; v3; v1) f3 (v4; v1; v5) f4 (v1; v6; v5) f5 (v6; v1; v7) f6 (v2; v7; v1) f7 (… )

Geom etry v1 ( x1 ;y1 ;z1 ) v2 ( x2 ;y2 ;z2 ) v3 ( x3 ;y3 ;z3 ) v4 ( x4 ;y4 ;z4 ) v5 ( x5 ;y5 ;z5 ) v6 ( x6 ;y6 ;z6 ) v7 ( x7 ;y7 ;z7 )

• > 3V float
• > 3F log2(V) ~ 6V log2(V)

~ 1 00 b/ v 9 6 b/ v

V = 50k

to give an order of magnitude

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Single-Rate Encoding

• Review:

Review: 3 main approaches based on:

3 main approaches based on: − − Triangle strips Triangle strips − − Edge/gate traversal Edge/gate traversal − − Valence Valence

• Recent developments

Recent developments

− − Improvement Improvement − − Hint of optimality Hint of optimality

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Link to Other Fields

In Graph Theory, some previous work

Graphs can be huge, so encoding is good But mainly designed to compress (nearly)

complete graphs

Far away from our 3-connected graphs Different approaches using Lipton-Tarjan for instance [DL98]

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Triangle Strips

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Deering 95 (I)

Connectivity:

• Generalized triangle strips
• Vertex buffer (size: 16)

Idea: use stack operations to reuse vertices. → reduce random access to vertices Geometry:

• Quantization, Delta coding and Huffman

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Deering 95 (II)

Redundancy ~ 2 One symbol per face → 2V symbols

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Deering 95 (III)

Features:

• Fast
• Local → well suited for hardware
• 8-11 b/v for connectivity
• Geometry: highly variable
• Integrated in JAVA 3D

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Taubin & Rossignac 98 (I)

The idea: Cut a mesh by using 2 interlocked trees:

• a spanning tree of vertices,
• a spanning tree of triangles.

Connectivity:

• Spanning trees
• Marching pattern → zigzag walking

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Taubin & Rossignac 98 (II)

Geometry:

• Prediction from

multiple ancestors

• f the vertex tree

Features:

• average: 4 b/v for connectivity

vertex tree

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Gumhold & Straβer 98 (I)

The idea: use an active boundary with a current gate, and encode the adjacent face insertion using 8 distinct symbols.

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Gumhold & Straβer 98 (II)

Connectivity:

• "Cut-border" machine

One symbol per processed edge, i.e. per conquered face → F symbols ~ 2V Features:

• average ~ 4.36 b/v for connectivity
• Focus on speed (especially decompression)

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Rossignac 99 (I) - EdgeBreaker

The idea:

Use an active boundary with a current gate, and encode the adjacent face insertion using 5 distinct symbols.

→ F symbols ~ 2V Connectivity:

• Spanning tree
• Gate traversal

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Rossignac 99 (II) - EdgeBreaker

Features:

• 4 b/v guaranteed for connectivity

Overall 2V symbols (one symbol per face) V "C" symbols (creates one vertex) → 1 bit (0) V other symbols among [RLSE] → 3 bits (1..) Later improved…

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Improvements

Rossignac & Szym czak 9 9 Wrap&Zip: Linear decompression of triangle meshes King & Rossignac 9 9 Guaranteed 3 .6 7 V Bit Encoding of Planar Triangle Graphs Szym czak, King & Rossignac 00 An Edgebreaker-based Efficient Compression Scheme for Regular Meshes I senburg & Snoeyink 0 0 Spirale Reversi: Reverse Decoding of EdgeBreaker Encoding Gum hold 00 New Bounds on the Encoding of Planar Triangulations. (3 .55 b/ v for EdgeBreaker)

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Valence-based Approaches

Touma & Gotsman 98

One valence code per vertex Successive pivot vertices

Isenburg & Snoeyink 99

Edge collapses One valence code per removed vertex Compression ratios equivalent to

Alliez & Desbrun 01

Improvements upon Theoretical study Improvements

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Valence Distribution (I)

1000 2000 3000 4000 5000 6000 7000 8000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Calls for entropy encoding

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Valence Distribution (II)

10000 20000 30000 40000 50000 60000 70000 80000 90000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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Valence Distribution (III)

2000 4000 6000 8000 10000 12000 14000 16000 18000 1 2 3 4 5 6 7 8 9 10

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The idea:

Deterministic edge conquering from successive pivot vertices along an active edge list.

Connectivity:

• Output one valence symbol per

vertex, + ε

• Huffman encoding of valences

Atomic edge conquest -> valence symbol Full pivot vertex -> no code, hence V codes Split(offset) code (of rare occurrence)

Touma & Gotsman 98 (I)

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Touma & Gotsman 98 (II)

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Geometry:

• Global quantization and

parallelogram prediction

• Entropy encoding

Features:

• average: 2 b/v for connectivity
• benefits from mesh regularity
• bit-rate vanishes to zero when regular
• not seriously challenged since 98

Parallelogram rule

decoded area

Touma & Gotsman 98 (III)

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Isenburg & Snoeyink 99 (I)

The idea:

Sequence of reversible edge contractions, recordable through valence codes for simple digons, and start/end for complex/trivial.

Connectivity:

• Successive edge collapses
• Output one valence code/v, + ε
• Entropy encoding of valences

Valence Valence End Start

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Isenburg & Snoeyink 99 (II)

Features:

• Two faces conquered per valence, hence V codes
• Connectivity: slightly more than TG98 [1-4] b/v
• Also benefits from mesh regularity

→ No split codes, but start/end

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Alliez & Desbrun 01

Connectivity:

• Derived from TG98
• Output one valence code per vertex, + ε
• Arithmetic encoding of valences

Idea: adaptive edge conquering from successive pivot vertices along an active edge list. Features:

• Minimize # split codes
• Theoretical study: Hint of optimality

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The Split Case

Split(offset)

Split(3)

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Intuition behind Split Codes

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Towards Adaptive Conquest

One remaining degree of freedom in TG→ choice in the pivot → valence min / average valence min

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Adaptive Conquest

[ 5 ,5 ,6,6 ,split( 2 ) ] [ 3 ,6 ,6 ]

TG 98 AD 01

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Results

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Discussion

TG98 not really challenged since 98 → valence approach “seems” optimal But how to prove it?

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Valence as Entropy (I)

We want to maximize: with the constraints:

pi = vi/V, so: F – E + V = 1

2E = 3F ⇒ 2E ≈ 6V, so: Solution: Lagrange Multipliers

∑

=

i i i

p p e ) / 1 ( log2 1 =

∑

i i

p 6 =

∑

i i

p i

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Valence as Entropy (II)

Worst-case distribution pi is such as: Necessary condition: We find:

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − +

∑ ∑ ∑

i i i i i i i

p i p p p Max 6 1 ) / 1 ( log2 μ λ

i pi μ λ + − = ) 2 ln( / 1 ) ( log

2

i i

p

−

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 3 4 27 16

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Valence as Entropy (III)

Now: Worst-case entropy: log2(256/27)=3.24… → Matches Tutte’s census! Did we reach optimality? Well, maybe

Splits are a pain (best solution: Kälberer ‘05) Expected behavior is sublinear, though Only asymptotical

) 27 / 256 ( log ) / 1 ( log

2 2

= =∑

i i i

p p e

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Some connectivity results

1 .98 bit/ vertex 1 .02 0 .024 2 .20 2 .71 2 .27 1 .88 CS101b - Meshing 62

Main advantages of Valence

Coding valences guarantees:

• Optimality in the regular case (6…….6)

Typically, less than 0.01 bit per vertex

• (Near) Optimality in the irregular cases

Never more than 3.5bpv

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Deering 95

8-11

Redundancy ~ 2 Connectivity (b/v) Triangle strips Note

Method

Taubin & Rossignac 98 Gumhold & Straβer 98 Touma & Gotsman 98 Rossignac 99 King & Rossignac 99 Gumhold 00 Isenburg & Snoeyink 99 Alliez & Desbrun 01 ~ 4 Spanning trees ~ 4.36 2V codes Cut-border ~ 2 (~0 when regular) V codes Valence

4 guaranteed

3.67 grntd. 3.55 grntd.

2V codes, upper bound

Better upper bound Even better upper bound

Gate traversal ~ TG98 V codes Valence

Tutte 62

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Quad/Polygon/Tetrahedron

Quads

King, Rossignac & Szymczak 99 (~EdgeBreaker)

Polygons

Isenburg & Snoeyink 00 (Face Fixer, edge-based) Kronrod & Gotsman 00 (Face-based) Khodakovsky et al’ 03 (Face and vertex based)

Tetrahedra

Gumhold, Guthe & Straβer 99 -> ~2 b/tet.

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Part II

Progressive Codecs

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Progressive Mesh Encoding

Definition: Definition:

Produces a Produces a “ “multi multi-

• resolution

resolution” ” format format → → Coarse model + refinements Coarse model + refinements Concept: Concept: Creates a bit stream with decreasingly Creates a bit stream with decreasingly important details. important details.

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An Ideal Progressive Encoder

Bit Bit -

• rate

rate Distortion Distortion better better worse worse

Bits must be sent in decreasing order of Bits must be sent in decreasing order of innovation innovation

For each bit For each bit-

• rate, max ratio:

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Review

− − Progressive Meshes 96 Progressive Meshes 96 − − Progressive Forest Split 98 Progressive Forest Split 98 − − Compressed Progressive Meshes 99 Compressed Progressive Meshes 99 − − Cohen Cohen-

• Or et al. 99

Or et al. 99 − − Geometry Geometry-

• based triangulation 00

based triangulation 00 − − Spectral Compression 00 Spectral Compression 00 − − Complete Complete Remeshing Remeshing 00 00 − − Valence Valence-

• centered 01

centered 01 − − Plus recent developments Plus recent developments

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Common idea

Encoding a mesh refinement consists in: Encoding a mesh refinement consists in: 1.

• 1. Localization

Localization → → "where" "where" 2.

• 2. Action

Action → → "how to refine" "how to refine"

localisation action Exem ple:

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Hoppe 96

Encoding (decimation) Decoding (refinement)

• 1. Localization

each vertex to split ~ log2(V) bits to localize

• 2. Action

Connectivity: 2 edges to split log2(C

v)

16 b/ v Geometry: delta encoding ~20 b/ v

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Taubin et al. 98 – Prog. Forest Split

• Localization :

Localization :

~ forest of vertices to split (1 b/edge) ~ forest of vertices to split (1 b/edge) ~ edge collapses grouped in batches ~ edge collapses grouped in batches

• Action :

Action :

• remeshing

remeshing holes holes ~ 3.5 b/v ~ 3.5 b/v

• prediction and pre/post smoothing

prediction and pre/post smoothing for geometry for geometry

Atomic operator: edge collapse Atomic operator: edge collapse

Forest edges Cut Triangulate tree loops Refinement done

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→ → by triangulating tree loops

by triangulating tree loops Encoding using a variation of Encoding using a variation of [TR98] [TR98] (single (single-

• rate)

rate)

Remeshing Remeshing holes holes

Connectivity Connectivity: 10 b/v : 10 b/v Geometry Geometry: 20 b/v : 20 b/v

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Pajarola and Rossignac 99 (CPM)

• Localization :

Localization :

• 2

2-

• coloring of vertices to group

coloring of vertices to group vertex splits into batches vertex splits into batches → → amortized cost: 3 b/v amortized cost: 3 b/v

• Action :

Action :

• Indicates two edges among

Indicates two edges among the valence of each vertex to split the valence of each vertex to split

• "Butterfly" prediction for

"Butterfly" prediction for geometry geometry

Atomic operator: Atomic operator: edge collapse edge collapse Batches of edge Batches of edge collapses collapses

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Metrics in CPM

• Metrics: modified QEM (

Metrics: modified QEM (GarlandHeckbert97 GarlandHeckbert97) ) Idea: Idea: select a subset of the less expensive edges select a subset of the less expensive edges that do not violate independent constraints that do not violate independent constraints → → greedy edge selection by increasing greedy edge selection by increasing approximation error (instead of priority approximation error (instead of priority queue used by queue used by GH97 GH97). ). Connectivity Connectivity: 7.2 b/v : 7.2 b/v Geometry Geometry: 15.4 b/v : 15.4 b/v

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Cohen-Or et al. 99

• Localization :

Localization :

~ 2 or 4 face coloring ~ 2 or 4 face coloring (build an independent set)

(build an independent set)

2 coloring: 2 b/v (many degree 5 and >) 2 coloring: 2 b/v (many degree 5 and >) 4 coloring: 4 b/v (degree 4 or 3) 4 coloring: 4 b/v (degree 4 or 3) LZ encoding of face labels LZ encoding of face labels

• Action :

Action :

• deterministic

deterministic retriangulation retriangulation (inside (inside-

• Z)

Z)

• barycentric prediction for geometry

barycentric prediction for geometry

Connectivity Connectivity: 6 b/v : 6 b/v Geometry Geometry: 15.4 b/v : 15.4 b/v

Atomic operator: vertex removal Atomic operator: vertex removal

4- coloring 2- coloring

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2-coloring

Determ inistic inside- z triangulation / 2- coloring

Coding Decoding

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Discussion

Long thin triangles… 4 coloring 4-coloring 2-coloring

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Alliez & Desbrun 01

( (SIGGRAPH 01

SIGGRAPH 01)

)

• We seek:

We seek:

− − Per Per-

• vertex granularity

vertex granularity − − One valence code per vertex, ideally One valence code per vertex, ideally − − Detection of uniformity/regularity Detection of uniformity/regularity − − Good rate/distortion tradeoff Good rate/distortion tradeoff

• We need:

We need:

− − Mesh simplification Mesh simplification

» » atomic decimation/refinement operator atomic decimation/refinement operator » » error metric error metric

− − No extra No extra “ “how to refine how to refine” ” data data

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Core ideas

• Localization :

Localization :

~ at no cost ~ at no cost →

→ gate

gate-

• based conquest

based conquest

• Action :

Action :

− − valence of inserted vertex, valence of inserted vertex, − − adaptive adaptive remeshing remeshing table, at no cost. table, at no cost.

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Valence-centered Approach?

We showed (Eurographics ’01) that valence leads to optimal connectivity encoding

1000 2000 3000 4000 5000 6000 7000 8000

1 2 3 4 5 6 7 8 9 10 11 12 13 14

V alence Occurrences

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Main Ideas

Connectivity: Only one valence per vertex

• S

ufficient for both localization and action

• Close to optimal compression [AD 01]

Geometry: Normal/tangential separation

• S

eparate geometry/ parameterization [KSS 00]

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Our Method at a Glance

Decimation Strategy

Passes of vertex removals Automatic re-triangulation

Entropy Encoding

Compression of the list of symbols

(essentially valences)

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Basic Primitives

Ordinary patch

1 input gate N-1 output gates 1 vertex removal

Null patch Gate

(oriented edge)

1 input gate 2 output gates 0 vertex removal

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Decimation Strategy

Gate-based deterministic conquest

Vertex removal Fifo of gates Eliminate localization cost

Targeting special vertices

Low valences to respect balance (Denny - Sohler 97) Cosmetic decisions

Automatic re-triangulation

• Favor regular remeshing

Look-up table

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Example of Conquest

Codes

5

Fifo

g1 g2 g3 g4 g5

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Conquest - Step 2

6 5 g2 g3 g4 g5 g6 …

Codes Fifo

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Conquest - Step 3

6 6 5 g3 g4 g5 g6 g7 …

Codes Fifo

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Conquest - Step 4

6 6 6 5 g4 g5 g6 g7 g8 …

Codes Fifo

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Conquest - Step 5

6 6 6 6 5 g5 g6 g7 g8 g9 …

Codes Fifo

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Conquest - End

6 5 6 4 5 …

Codes Fifo

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Decimation Strategy

Gate-based deterministic conquest

Vertex removal Fifo of gates Eliminate localization cost

Targeting special vertices

Low valences to respect balance [Denny - Sohler 97] Cosmetic decisions

Automatic re-triangulation

• Favor regular remeshing

Look-up table

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Valence Dispersion

→ remove only valences [3-6] → [3-4] on boundaries Before : V = ∑(valences, central vertex s excluded) After : V' = ∑(valences)

V' = V + (valence(s)-6)

before after

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Vertex Selection

Valence > 6 Already visited

Null patch if:

• Valence > 6
• Already visited
• Metric-related decision

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Decimation Strategy

Gate-based deterministic conquest

Vertex removal Fifo of gates Eliminate localization cost

Targeting special vertices

Low valences to respect balance (Denny - Sohler 97) Cosmetic decisions

Automatic re-triangulation

• Favor regular remeshing

Look-up table

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Remeshing strategy

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Remeshing strategy

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Remeshing strategy

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Remeshing strategy

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Remeshing strategy

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Remeshing strategy

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Retriangulation Look-up Table

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Regular Decimation

[Kobbelt 00]

Targeting valences [3-6]

√3

Targeting valences [3] Substitution: 3->6

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Irregular Decimation

Targeting valences [3-6] Targeting valences [3] Substitution: 3->6

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Entropy Encoding

[3-6;N] [6;N] [3-6;N] [6;N] Adaptive arithmetic encoding [Schindler 99]

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Geometry encoding

Barycentric prediction

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Normal / tangential separation

~ [Khodakovsky et al. 00]

Local Frenet frame CS101b - Meshing

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Results

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10 Decimation 19851 -> 4 vertices Quantization 12 bits Connectivity 4.61 b/v Geometry 16.24 b/v Decimation 36450 -> 24 vertices Quantization 10 bits Connectivity 0.39 b/v Geometry 3.58 b/v

Results 20.87 b/v 3.97 b/v

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Rate / Distortion

1 2 3 4 5 6

20000 40000 60000 80000 100000 120000 140000 160000 180000

rate (# bits) Distortion (% Bounding Box Mean square error)

20 11 217 vertices

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Discussion

• Optimality ?

Optimality ?

• One unique valence code per vertex

One unique valence code per vertex

• Does decimation hide entropy ?

Does decimation hide entropy ? → → non non-

• optimal tiling, valence > 6 or
• ptimal tiling, valence > 6 or

metric decision lead to null patch codes metric decision lead to null patch codes (unused by refinement). (unused by refinement).

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Perspectives

• Entropy

Entropy-

• driven

driven remeshing remeshing process, process, minimizing minimizing occurency

• ccurency of null
• f null

patches (seek optimal tiling). patches (seek optimal tiling).

• Prediction on metric decision

Prediction on metric decision (regular sharp edges, corners) (regular sharp edges, corners)

• Inverse Loop in the regular case ?

Inverse Loop in the regular case ?

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Devillers & Gandoin 00 (I)

• Localization :

Localization :

~ implicitly transmitted by the order on ~ implicitly transmitted by the order on the data, via a space subdivision. the data, via a space subdivision. ~ progressive quantization ~ progressive quantization

• Action :

Action :

• transmit only

transmit only occurences

• ccurences
• use geometric triangulation

use geometric triangulation i.e. Delaunay/Crust/Natural i.e. Delaunay/Crust/Natural Neighbors Neighbors… …

• > well suited to terrains

> well suited to terrains

Drop the order upon the vertices CS101b - Meshing 11

Devillers & Gandoin 00 (II)

Transmission Subdivision

Theoretical analysis: Theoretical analysis: Vertex precision: Q bits Vertex precision: Q bits Direct encoding: VQ Direct encoding: VQ Occurrence Occurrence-

• based approach: V(Q

based approach: V(Q-

• log

log2

2V+2.402)

V+2.402) -

• >

> worst case, i.e. uniformity

worst case, i.e. uniformity CS101b - Meshing

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Karni & Gotsman 00 - Spectral Compression

The idea: The idea: Apply ~ Signal Processing theory Apply ~ Signal Processing theory Similar in spirit to Similar in spirit to lossy lossy JPEG JPEG → → requires mesh requires mesh partitionning partitionning ( (MeTiS MeTiS) ) Then each Then each submesh submesh is turned into a compact is turned into a compact linear combination of orthogonal basis linear combination of orthogonal basis functions functions Basis functions Basis functions: : eigenvectors of the eigenvectors of the topological topological Laplacian Laplacian of a

• f a submesh

submesh

• > Recently improved by fixed spectral bases

> Recently improved by fixed spectral bases

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Khodakovsky, Schröder & Sweldens 00 Progressive Geometry Compression

We are not interested in the connectivity Let’s remesh the object! Coarse mesh Then subdivision connectivity (regular) Bottom-up: loop-based wavelet decomposition Top-to-bottom: Zero tree to locate efficiently

the details

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Wavelet Transform

Effect of wavelet transform

Changes distribution of coefficients Almost all coefficients close to zero

0.27 0.34 0.41 0.49 0.56 0.0 0.8 1.7 2.5 3.3

Vertex coordinates Wavelet coefficients CS101b - Meshing

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Zero-Tree

Need tree structure for coefficients

wavelets live on edges

Test whole tree for significance

split tree isolating significant coefs

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Results (I)

21046B 8517B 3411B 1369B 1358B 2784B 701B 6063B

21K 6K

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Results (II)

956B 2004B 4806B 26191B 1253B 2804B 6482B 14844B

4 b/v 2.5 b/v

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Next Challenges?

• Progressive polygon meshes
• Progressive topology encoding
• Progressive animated geometry
• Multi-connected components
• What is entropy of geometry ?
• Resilient encoding
• …