Shape Modeling with Point-Sampled Geometry

Transcription

1 Shape Modeling with Point-Sampled Geometry Mark Pauly Richard Keiser Leif Kobbelt Markus Gross ETH Zürich ETH Zürich RWTH Aachen ETH Zürich 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 1

2 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 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 2

3 Outline Implicit surface model Moving least squares approximation Interactive shape modeling Boolean operations Free-form deformation Demo Results & Conclusions Surface Model Goal: Define continuous surface from a set of discrete point samples discrete set of point samples P = { p i, c i, m i,... } continuous surface S interpolating or approximating P 3

4 Surface Model Moving least squares (MLS) approximation (Levin, Alexa et al.) Surface defined as stationary set of projection operator Ψ P implicit surface model S 3 { x R Ψ ( x x} P = P ) = Weighted least squares optimization Gaussian kernel function local, smooth mesh-less, adaptive Adaptive MLS Adapt kernel size in the moving least squares (MLS) approximation to local sample density 4

5 Boolean Operations 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 5

6 Boolean Operations Classification: given a smooth, closed surface S and point p. Is p inside or outside of the volume V bounded by S? p S V 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 p q S V 6

7 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 p n S V 2. classify p as q inside V, if (p-q) n < 0 outside V, if (p-q) n > 0 Boolean Operations Classification: represent smooth surface S by point cloud P m P S V 7

8 Boolean Operations Classification: represent smooth surface S by point cloud P m P S 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 n q V Boolean Operations Classification: for points close to surface, use not p, but MLS projection y for correct classification 8

9 Boolean Operations Sampling the intersection curve classification A B sampling the intersection curve A B Boolean Operations Sampling the intersection curve 1. identify pairs of closest points 9

10 Boolean Operations Sampling the intersection curve 1. identify pairs of closest points q 1 q 2 Boolean Operations Sampling the intersection curve 1. identify pairs of closest points 2. find closest point on intersection of tangent spaces r q 1 q 2 10

11 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 qʹ 1 r qʹ 2 q 1 q 2 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 qʹ 1 r ʹ qʹ 2 11

12 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. Free-form Deformation Smooth deformation field F:R 3 R 3 that warps 3D space Can be applied directly to point samples 12

13 Free-form Deformation Rotational deformations of a cylinder. (a) original, (b) color-coded scale parameter, (c) rotation around axis a 1, (d) rotation around axis a 2 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 13

14 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. Results Combination of free-form deformation with collision detection, boolean operations, particle-based blending, embossing and texturing 14

15 Results Interactive modeling with scanned data: noise removal, free-form deformation, cut-and-paste editing, interactive texture mapping Results The Octopus: Free-form deformation with dynamic sampling 15

16 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 Future Work Physics-based modeling Haptic interfaces Robust handling of singularities for boolean operations More complex surfaces, e.g. hairy or furry models 16

17 Acknowledgements Tim Weyrich, Matthias Zwicker European Graduate Program on Combinatorics, Geometry, and Computation Check out: 17