given a smooth, closed surface S and point p. Is p ... represent smooth surface S by point cloud P m. P. S .... Tim Weyrich, Matthias Zwicker. ⢠European Graduate ...
Shape Modeling with Point-Sampled Geometry Mark Pauly
Richard Keiser
Leif Kobbelt
Markus Gross
ETH Zürich
ETH Zürich
RWTH Aachen
ETH Zürich
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Motivation • Surface representations – Explicit surfaces (B-reps) • Polygonal meshes • Subdivision surfaces • NURBS
- Efficient rendering - Sharp features - Intuitive editing
– Implicit surfaces • Level sets • Radial basis functions • Algebraic surfaces
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
- Boolean operations - Changes of topology - Extreme deformations
SIGGRAPH 2003
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Motivation • Surface representations – Explicit surfaces (B-reps) • Polygonal meshes • Subdivision surfaces • NURBS
- Boolean operations - Changes of topology - Extreme deformations
– Implicit surfaces • Level sets • Radial basis functions • Algebraic surfaces
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
- Efficient rendering - Sharp features - Intuitive editing
SIGGRAPH 2003
Motivation • Surface representations – Explicit surfaces (B-reps) • Polygonal meshes • Subdivision surfaces • NURBS
– Implicit surfaces • Level sets • Radial basis functions • Algebraic surfaces
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
– Hybrid Representation • Explicit cloud of point samples • Implicit dynamic surface model
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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, ... } Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
continuous surface S interpolating or approximating P SIGGRAPH 2003
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Surface Model • Moving least squares (MLS) approximation (Levin, Alexa et al.) – Surface defined as stationary set of projection operator ΨP implicit surface model
{
SP = x " R 3 !P ( x ) = x
}
– Weighted least squares optimization • Gaussian kernel function – local, smooth – mesh-less, adaptive
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations
+
-
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
-
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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 Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • Classification: – given a smooth, closed surface S and point p. Is p inside or outside of the volume V bounded by S?
S
p
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
V
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Boolean Operations • 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
S
V
p
q
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • 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
S
p
V
n
q
– inside V, if (p-q)·n < 0 – outside V, if (p-q)·n > 0
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
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Boolean Operations • Classification: – represent smooth surface S by point cloud P m
S
P
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 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
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
S
P
p
V
n
q
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Boolean Operations • Classification: – piecewise constant surface approximation leads to false classification close to the surface
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • Classification: – piecewise constant surface approximation leads to false classification close to the surface
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
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Boolean Operations • Classification: – piecewise constant surface approximation leads to false classification close to the surface
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • Classification: – piecewise constant surface approximation leads to false classification close to the surface
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
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Boolean Operations • Classification: – piecewise constant surface approximation leads to false classification close to the surface
q
n p
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • Classification: – use MLS projection of p for correct classification
n
p p
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
!P (p )
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Boolean Operations • Sampling the intersection curve
classification A
B
sampling the intersection curve
A∪B Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • Sampling the intersection curve 1. identify pairs of closest points
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
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Boolean Operations • Sampling the intersection curve 1. identify pairs of closest points
q1
q2
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • Sampling the intersection curve 1. identify pairs of closest points 2. find closest point on intersection of tangent spaces
r
q1 Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
q2 SIGGRAPH 2003
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Boolean Operations • 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
q1!
q2! r
q1
q2
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Boolean Operations • 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 q1!
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
r!
q2!
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Free-form Deformation • Smooth deformation field F:R3→R3 that warps 3D space • Can be applied directly to point samples
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 rigid part deformable part control handle
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
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Dynamic Sampling • Large Robustdeformations free-form deformation lead to strong requires distortions dynamic adaptation of the sampling density
10,000 points
271,743 points
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Dynamic Sampling • Dynamic insertion of point samples: – measure local surface stretch – split samples that exceed stretch threshold – regularize distribution by relaxation – interpolate scalar attributes
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
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Results • Combination of free-form deformation with collision detection, boolean operations, particle-based blending, embossing and texturing
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
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Results • The Octopus: Free-form deformation with dynamic sampling
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
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Future Work • Physics-based modeling • Haptic interfaces • Robust handling of singularities for boolean operations • More complex surfaces, e.g. hairy or furry models
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
Acknowledgements • Tim Weyrich, Matthias Zwicker • European Graduate Program on Combinatorics, Geometry, and Computation • Check out:
www.pointshop3d.com
Pauly, Keiser, Kobbelt, Gross: Shape Modeling with Point-Sampled Geometry
SIGGRAPH 2003
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