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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Shape Modeling with Point-Sampled Geometry
Mark Pauly Richard Keiser Leif Kobbelt Markus Gross ETH Zürich ETH Zürich ETH Zürich RWTH Aachen
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Motivation
• Surface representations – Explicit surfaces (B-reps)
• Polygonal meshes • Subdivision surfaces • NURBS
– Implicit surfaces • Level sets • Radial basis functions • Algebraic surfaces
- Efficient rendering - Sharp features - Intuitive editing
- Boolean operations - Changes of topology - Extreme deformations
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Motivation
• Surface representations – Explicit surfaces (B-reps)
• Polygonal meshes • Subdivision surfaces • NURBS
– Implicit surfaces • Level sets • Radial basis functions • Algebraic surfaces
- Boolean operations - Changes of topology - Extreme deformations
- Efficient rendering - Sharp features - Intuitive editing
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Motivation
• Surface representations – Explicit surfaces (B-reps)
• Polygonal meshes • Subdivision surfaces • NURBS
– Implicit surfaces • Level sets • Radial basis functions • Algebraic surfaces
– Hybrid Representation • Explicit cloud of point
samples • Implicit dynamic surface
model
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Outline
• Implicit surface model – Moving least squares approximation
• Interactive shape modeling – Boolean operations
– Free-form deformation
• Demo
• Results & Conclusions
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Surface Model
• Goal: Define continuous surface from a set of discrete point samples
discrete set of point samples
P = { pi, ci, mi, ... }
continuous surface S interpolating or
approximating P
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Surface Model
• Moving least squares (MLS) approximation (Levin, Alexa et al.) – Surface defined as stationary set of projection
operator ΨP implicit surface model
– Weighted least squares optimization • Gaussian kernel function
– local, smooth – mesh-less, adaptive
{ }xxxS PP =Ψ∈= )(3R
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Adaptive MLS
Adapt kernel size in the moving least squares(MLS) approximation to local sample density
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Boolean Operations
+ - -
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Boolean Operations
• Classification – Inside-outside test using signed distance function
induced by MLS projection
• Sampling – Compute exact intersection of two MLS surfaces
to sample the intersection curve
• Rendering – Accurate depiction of sharp corners and creases
using point-based rendering
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Boolean Operations
S
p
• Classification: – given a smooth, closed
surface S and point p. Is p inside or outside of the volume V bounded by S?
V
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Boolean Operations
S
p
q
• Classification: – given a smooth, closed
surface S and point p. Is p inside or outside of the volume V bounded by S?
1. find closest point q on S V
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Classification: – given a smooth, closed
surface S and point p. Is p inside or outside of the volume V bounded by S?
1. find closest point q on S
2. classify p as – inside V, if (p-q)·n < 0 – outside V, if (p-q)·n > 0
Boolean Operations
p
qn
S
V
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Boolean Operations
• Classification: – represent smooth surface S
by point cloud P m
PS
V
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Boolean Operations
p
• Classification: – represent smooth surface S
by point cloud P m
1. find closest point q in P
2. classify p as – inside V, if (p-q)·n < 0 – outside V, if (p-q)·n > 0
P
qn
S
V
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Boolean Operations
• Classification: for points close to surface, use not p, but MLS projection y for correct classification
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Sampling the intersection curve
Boolean Operations
A
classification
sampling the intersection curve
B
A ∪ B
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Sampling the intersection curve 1. identify pairs of closest points
Boolean Operations
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Sampling the intersection curve 1. identify pairs of closest points
Boolean Operations
1q 2q
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Sampling the intersection curve 1. identify pairs of closest points
2. find closest point on intersection of tangent spaces
Boolean Operations
r
1q 2q
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Sampling the intersection curve 1. identify pairs of closest points
2. find closest point on intersection of tangent spaces
3. re-project point on both surfaces
Boolean Operations
1qʹ′ 2qʹ′
r
1q 2q
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Sampling the intersection curve 1. identify pairs of closest points
2. find closest point on intersection of tangent spaces
3. re-project point on both surfaces
4. iterate
Boolean Operations
r ʹ′1qʹ′ 2qʹ′
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Rendering intersection curve
(a) mutual clipping of two surfels on intersection curve, (b) boolean differences on bunny model, (c) zoom of the intersection curves, (d) sampling distribution (e) an example of a corner.
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Free-form Deformation
• Smooth deformation field F:R3→R3 that warps 3D space
• Can be applied directly to point samples
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Free-form Deformation
Rotational deformations of a cylinder. (a) original, (b) color-coded scale parameter, (c) rotation around axis a1, (d) rotation around axis a2
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Free-form Deformation
• Intuitive editing using painting metaphor – Define rigid surface part and handle using
interactive painting tool
– Displace handle using translation and/or rotation
– Create smooth blend towards rigid part
control handle
rigid part
deformable part
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Dynamic Sampling
• Large deformations lead to strong distortions
Deformation of a plane. (a) local stretching: blue =zero stretch, red=maximum stretch, (b) after re-sampling, (c) sampling distribution. Bottom row: illustration of point insertion.
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Results
• Combination of free-form deformation with collision detection, boolean operations, particle-based blending, embossing and texturing
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Results
• Interactive modeling with scanned data: noise removal, free-form deformation, cut-and-paste editing, interactive texture mapping
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Results
• The Octopus: Free-form deformation with dynamic sampling
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Conclusions
• Point cloud: Explicit representation – Minimal consistency constraints allow efficient
dynamic re-sampling
– Modeling of sharp features
– Fast rendering
• MLS approximation: Implicit surface model – Fast inside/outside tests for boolean classification
and collision detection
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
Future Work
• Physics-based modeling
• Haptic interfaces
• Robust handling of singularities for boolean operations
• More complex surfaces, e.g. hairy or furry models
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Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry SIGGRAPH 2003
• Tim Weyrich, Matthias Zwicker • European Graduate Program on
Combinatorics, Geometry, and Computation
• Check out:
www.pointshop3d.com
Acknowledgements