Multiresolution Point-set Surfaces Franois Duranleau Philippe - - PowerPoint PPT Presentation

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Multiresolution Point-set Surfaces Franois Duranleau Philippe - - PowerPoint PPT Presentation

Multiresolution Point-set Surfaces Franois Duranleau Philippe Beaudoin Pierre Poulin Dp. dinformatique et de recherche oprationnelle GI 2008 Introduction def Outline 1 Introduction 2 Analysis 3 Synthesis 4 Results 5 Conclusion &


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Multiresolution Point-set Surfaces

François Duranleau Philippe Beaudoin Pierre Poulin

Dép. d’informatique et de recherche opérationnelle

GI 2008

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Introduction

Outline

1 Introduction 2 Analysis 3 Synthesis 4 Results 5 Conclusion & Future Work

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Introduction

Point-set Surfaces and Surface Editing

  • Point-set surfaces are becoming popular for

shape modeling

  • Surface editing in the presence of fine geometric

details can be problematic

  • Multiresolution representations for meshes are

well known

  • Interest for multiresolution representation for

point-set surfaces

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Introduction

Decomposition

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Introduction

Surface Editing

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Introduction

Overview

⊖ ⊕ ⊖ ⊕

smooth, downsample downsample smooth,

Level 1 Level 1 Level L Level L Level L − 1 Level L − 1

Synthesis

Level 0 . . . Details 1

Analysis

Details L

upsample upsample

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Introduction

Previous Work

  • Multiresolution meshes

[Eck+ 95] [Lounsbery+ 97] [Zorin+ 97] [Kobbelt+ 98] [Guskov+ 99] [Lee+ 00] [Guskov+ 00] [Hubeli-Gross 01] ...

  • “Multiresolution” for points: mostly hierarchical

structures geared for rendering

[Pfister+ 00] [Rusinkiewicz+ 00] [Botsch+ 02] [Pajarola 03] [Park+ 04] [Pajarola+ 05] [Wu+ 05] ...

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Introduction

Previous Work

  • Progressive point-set surfaces

[Fleishman+ 03] [Singh-Narayanna 06]

  • Triangle fans

[Linsen-Prautzsch 03]

  • Multiscale point-set surfaces

[Pauly+ 06] [Zhang+ 05]

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Introduction

Previous Work

  • Progressive point-set surfaces

[Fleishman+ 03] [Singh-Narayanna 06]

  • Triangle fans

[Linsen-Prautzsch 03]

  • Multiscale point-set surfaces

[Pauly+ 06] [Zhang+ 05] + [Boubekeur+ 07]

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Analysis

Outline

1 Introduction 2 Analysis 3 Synthesis 4 Results 5 Conclusion & Future Work

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Analysis

Analysis

⊖ ⊕ ⊖ ⊕

smooth, downsample downsample smooth,

Level 1 Level 1 Level L Level L Level L − 1 Level L − 1

Synthesis

Level 0 . . . Details 1

Analysis

Details L

upsample upsample

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Analysis

Analysis

⊖ ⊖

smooth, downsample downsample smooth,

Level 1 Level 1 Level L Level L Level L − 1 Level L − 1 Level 0 . . . Details 1

Analysis

Details L

upsample upsample

Synthesis

⊕ ⊕

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Analysis

Analysis

smooth, downsample

Level 1 Level 1 Level L Level L − 1 Level L-1 Level 0 . . . Details 1

Analysis

Details L

upsample upsample

Synthesis

⊕ ⊕ Level L

downsample smooth,

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Analysis

Coarser Level Generation

Level L-1 ⊖

smooth, downsample

Level L Details L

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Analysis

Coarser Level Generation

smooth, downsample

Level L Details L Level L-1

  • MLS surfaces ⇒ smoothing by MLS projection

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Analysis

Coarser Level Generation

smooth, downsample

Level L Details L Level L-1

  • MLS surfaces ⇒ smoothing by MLS projection
  • Downsample point set before projection

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Analysis

Coarser Level Generation

smooth, downsample

Level L Details L Level L-1

  • MLS surfaces ⇒ smoothing by MLS projection
  • Downsample point set before projection
  • Similar to [Pauly+ 06], but constant

downsampling factor

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Analysis

Coarser Level Generation

smooth, downsample

Level L Details L Level L-1

  • MLS surfaces ⇒ smoothing by MLS projection
  • Downsample point set before projection
  • Similar to [Pauly+ 06], but constant

downsampling factor

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Analysis

Coarser Level Generation

smooth, downsample

Level L Details L Level L-1

  • Downsampling: same as smoothing

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Analysis

Coarser Level Generation

smooth, downsample

Level L Details L Level L-1

  • Downsampling: same as smoothing
  • However:

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Analysis

Coarser Level Generation

smooth, downsample

Level L Details L Level L-1

  • Downsampling: same as smoothing
  • However:
  • Add extra refinement step using heuristics

based on k-neighborhood analysis

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Analysis

Detail Extraction

smooth, downsample

Details L Level L Level L − 1

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Analysis

Detail Extraction

smooth, downsample

Details L Level L Level L − 1 Level L Level L − 1 ⊕

upsample

  • Main difficulty: represent detail information

coherently with upsampling procedure

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Analysis

Detail Extraction

smooth, downsample

Details L Level L Level L − 1

  • Main difficulty: represent detail information

coherently with upsampling procedure

  • Meshes profit from explicit connectivity

information

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Analysis

Detail Extraction

smooth, downsample

Details L Level L Level L − 1

  • Main difficulty: represent detail information

coherently with upsampling procedure

  • Meshes profit from explicit connectivity

information

  • [Linsen-Prautzsch 03]: store full k-neighborhood

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Analysis

Detail Extraction

smooth, downsample

Details L Level L Level L − 1

  • Main difficulty: represent detail information

coherently with upsampling procedure

  • Meshes profit from explicit connectivity

information

  • [Linsen-Prautzsch 03]: store full k-neighborhood
  • Intrinsic reformulation [Boubekeur+ 07]

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Analysis

Extraction Procedure

Level L − 1

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Analysis

Extraction Procedure

Point from level L

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Analysis

Extraction Procedure

δ

1 Project on level L − 1

δ = geometric detail information

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Analysis

Extraction Procedure

δ

1 Project on level L − 1 2 Find a surrounding triangle

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Analysis

Extraction Procedure

δ

1 Project on level L − 1 2 Find a surrounding triangle 3 Reformulate projection relative to the triangle

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Analysis

Extraction Procedure

δ

(t0, t1, t2, β1, β2, δ)

1 Project on level L − 1 2 Find a surrounding triangle 3 Reformulate projection relative to the triangle

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Analysis

Extraction Procedure

δ

(t0, t1, t2 , β1, β2, δ)

1 Project on level L − 1 2 Find a surrounding triangle 3 Reformulate projection relative to the triangle

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Analysis

Extraction Procedure

δ

(t0, t1, t2, β1, β2 , δ)

1 Project on level L − 1 2 Find a surrounding triangle 3 Reformulate projection relative to the triangle

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Analysis

Extraction Procedure

δ

(t0, t1, t2, β1, β2, δ )

1 Project on level L − 1 2 Find a surrounding triangle 3 Reformulate projection relative to the triangle

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Analysis

Extraction Procedure

δ

(t0, t1, t2, β1, β2, δ, ˜ δ )

˜ δ

1 Project on level L − 1 2 Find a surrounding triangle 3 Reformulate projection relative to the triangle

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Analysis

Triangle selection

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Analysis

Triangle selection

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Analysis

Triangle selection

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Analysis

Triangle selection

t1

2π−θ 2

t2 t0 θ

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Analysis

Triangle selection

> π

t0 t2 t1

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Analysis

Reformulation

r q ˜ δ

  • Find a point r on the triangle such that

q = r + ˜ δn(r) for some ˜ δ (n(r) = normal estimation at r)

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Analysis

Reformulation

r q ˜ δ

  • Find a point r on the triangle such that

q = r + ˜ δn(r) for some ˜ δ

  • Iterative procedure (gory details in the paper)

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Analysis

Reformulation

r q ˜ δ

  • Find a point r on the triangle such that

q = r + ˜ δn(r) for some ˜ δ

  • Iterative procedure
  • β1, β2 computed from r

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Synthesis

Outline

1 Introduction 2 Analysis 3 Synthesis 4 Results 5 Conclusion & Future Work

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Synthesis

Synthesis

⊖ ⊕ ⊖ ⊕

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Level 1 Level 1 Level L Level L Level L − 1 Level L − 1

Synthesis

Level 0 . . . Details 1

Analysis

Details L

upsample upsample

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Synthesis

Synthesis

⊕ ⊕ Level 1 Level 1 Level L Level L Level L − 1 Level L − 1

Synthesis

Level 0 . . . Details 1

Analysis

Details L

upsample upsample downsample smooth,

smooth, downsample

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Synthesis

Synthesis

⊕ Level 1 Level 1 Level L Level L Level L-1 Level L − 1

Synthesis

Level 0 . . . Details 1

Analysis

Details L

upsample upsample downsample smooth,

smooth, downsample

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Synthesis

Synthesis Procedure

(t0, t1, t2, β1, β2, δ)

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Synthesis

Synthesis Procedure

(t0, t1, t2, β1, β2

  • , δ)

1 Compute base position

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Synthesis

Synthesis Procedure

(t0, t1, t2, β1, β2, δ)

1 Compute base position 2 Estimate normal direction at base position

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Synthesis

Synthesis Procedure

(t0, t1, t2, β1, β2, δ)

1 Compute base position 2 Estimate normal direction at base position 3 Intersect ray with surface

(simplification of [Adamson-Alexa 04])

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Synthesis

Synthesis Procedure

(t0, t1, t2, β1, β2, δ, ˜ δ )

˜ δ

1 Compute base position 2 Estimate normal direction at base position 3 Intersect ray with surface

(fast estimation with ˜ δ)

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Synthesis

Synthesis Procedure

(t0, t1, t2, β1, β2, δ)

1 Compute base position 2 Estimate normal direction at base position 3 Intersect ray with surface 4 Estimate normal direction at intersection

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Synthesis

Synthesis Procedure

δ

(t0, t1, t2, β1, β2, δ )

2 Estimate normal direction at base position 3 Intersect ray with surface 4 Estimate normal direction at intersection 5 Displace by δ

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Synthesis

Synthesis Procedure

(t0, t1, t2, β1, β2, δ ∆

  • )

δ′

2 Estimate normal direction at base position 3 Intersect ray with surface 4 Estimate normal direction at intersection 5 Displace by δ′ = δ

∆∆′ [Boubekeur+ 07]

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Results

Outline

1 Introduction 2 Analysis 3 Synthesis 4 Results 5 Conclusion & Future Work

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Results

Igea

134345 43636 14503 4551 1512 500

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Results

Armadillo

172974 48312 15218 6510

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Results

Bumps

97284 32028 10395 3583 1140 374

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Results

Isis

196256 53883 14953 4368 1600

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Results

Statistics

Analysis Synthesis RMS error Igea Armadillo Bumps Isis 27.4 / 60.1 137.4 / — 17.9 / 36.7 39.9 / 67.3 7.6 / 12.2 9.5 / — 5.0 / 6.1 10.5 / 14.4 2.64×10−4 9.5×10−5 4.90×10−4 1.75×10−4

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Results

Deformation

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Results

Detail Emphasis

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Conclusion & Future Work

Outline

1 Introduction 2 Analysis 3 Synthesis 4 Results 5 Conclusion & Future Work

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Conclusion & Future Work

Conclusion

  • Point-set surfaces are flexible with simple data

structures

  • No connectivity information can be a pain
  • Multiresolution point-set surfaces
  • Verify special conditions when downsampling
  • Detail information with partial topology information

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Conclusion & Future Work

Conclusion

  • Point-set surfaces are flexible with simple data

structures

  • No connectivity information can be a pain
  • Multiresolution point-set surfaces
  • Verify special conditions when downsampling
  • Detail information with partial topology information
  • Faster processing for editing (coarser levels),

but synthesis time prevents full interactivity.

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Conclusion & Future Work

Conclusion

  • Point-set surfaces are flexible with simple data

structures

  • No connectivity information can be a pain
  • Multiresolution point-set surfaces
  • Verify special conditions when downsampling
  • Detail information with partial topology information
  • Faster processing for editing (coarser levels),

but synthesis time prevents full interactivity. But...

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Conclusion & Future Work

Future Work

  • ... there is hope:
  • Adaptive multiresolution [Zorin+ 97]
  • Highly parallelizable operations (multi-core CPUs,

GPU)

  • Room for improvement of heuristics’ robustness
  • Compression

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Conclusion & Future Work

Future Work

  • ... there is hope:
  • Adaptive multiresolution [Zorin+ 97]
  • Highly parallelizable operations (multi-core CPUs,

GPU)

  • Room for improvement of heuristics’ robustness
  • Compression
  • Wavelets?

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Conclusion & Future Work

Acknowledgements

  • Di Jiang
  • This work was supported in parts by grants from

NSERC, FQRNT, and MITACS.

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