Interactive surface reconstruction on triangle meshes with subdivision surfaces

1. Interactive surface reconstruction on triangle meshes with subdivision surfaces Matthias Bein Fraunhofer-Institut für Graphische Datenverarbeitung IGDFraunhoferstraße 564283 Darmstadt Tel.: +49 6151 155 – 465 Email: mbein@igd.fraunhofer.dehttp://www.igd.fraunhofer.de

2. Exposure of the problem Input: triangle mesh (scanned data) Aim: connected control mesh for subdivision surfaces Constraints: time, complexity, control, accuracy and more

3. Exposure of the problem • Difficulties • Holes • Varying point density • Errors • Bad triangulation

4. Related Work • Academic • Prof. Klein (primitive fitting, parametrization) • Prof. Kobbelt (quad dominant remeshing)

5. Related Work • Commercial • Geo Magic

6. Motivation • Technical models can be reconstructed pretty good • Freeform models too? • Fully automatic reconstruction is hardly possible (and not even wanted) • Human‘s shape recognition unreached • Designer‘s intention can not be predicted • Control over the reconstructed model has to be assured • Challenge: What is the minimum user interaction for surface reconstruction?

7. Idea Pick2 Pick1 Pick3 Pick4 User picks vertices Patch borders are extracted automatically Points in borders are approximated automatically Hole filling is implicit

8. Components • Principle curvature analysis • Feature recognition • Patch border alignment • Visualisation and tools • Support the user to understand the shape • Patch approximation • Patch borders need to track curvature lines for alignment • Approximate the surface inside the patch • Holes • Bad triangulation • Regular grid wanted

9. Principle Curvature Analysis • Principle Curvature Analysis • Discrete • Taubin, modifications and others • Inadequate for noisy scanned meshes • Analytic • Primitive fitting • Polynom fitting • Moving Least Squares • ...

10. Our Principle Curvature Analysis Local coordinate system with radius Polynom surface and normal Vertices Main curvature 1 Main curvature 2 • Polynom fitting in a local coordinate system • Bivariate polynom of total grade 2 or 3 • Radius neighbourhood search • triangulation used for neighbourhood information only • Least squares approximation • Analytic derivation of main curvature direction and values • 25 seconds for 100.000 vertices (2GHz, 2GB ram)

11. Visualisation and tools • Visualisation • Main curvature direction • Scaled normals

12. Visualisation and tools • Tools • Region growing • Pick a vertex • BFS growing with constraints • Main curvature lines • Pick a vertex • Track main curvature lines • Modified shortest path • Pick two points • Search path following main curvature lines

13. Modified shortest path • Experiments with Dijkstra algorithm • Robust • Symmetric • Works with the intended user interaction • Weight every edge length • reduce its length if the edge is „good“ • small angle to a main curvature direction • Small angle to the current path • Surface around the edge is orientable • Prefere the main curvature direction with the lower curvature value (along an edge, not across it) • 1-neighbourhood (triangulation) is not sufficient • Radius neighbourhood search • Path calculation within a second

14. Patch Approximation • Sequential in u and v direction • Input: four borders and number of segments in u and v • Analyse the patch (aspect ratio) • Calculate cutting planes and cutting curves inside the patch in u or v • Robust against holes and bad triangulations • Approximate the border curves and cutting curves in one direction => first set of controle points • Interpolate first set of control points in the other direction => final set of controle points • Reduction of the patch approximation to several curve approximations

15. B-Spline Curve Approximation • Input: set of points dk with parameters tk • B-Spline definition: c(t) = Σ Ni(t)pi • Linear equation system: dk = Σ Ni(tk)pi D = N * P • Overestimated (# points > # control points) • Multiply with transposed N • NT * D = NT * N * P • Q = M * P • Solve this linear equation system to gain control points P • M is symmetric and positive definite => LU decomposition • Least squares approximation. Error = Σ || dk - c(tk) ||2 • Catmull&Clark subdivision derived from uniform B-Splines • => Subdivision control net with this approximation

16. Results • Whole seat • 206k triangles • 105k vertices • Backrest • 108k triangles • 57k vertices • Reconstruction • 19 patches • ~300 quads • 7 minutes

17. Results • Vase • 50k triangles • 25k vertices • Reconstruction • 11 patches • ~150 quads

18. Results • Vase • 50k triangles • 25k vertices • Reconstruction • 11 patches • ~150 quads

19. Results • Vase • 50k triangles • 25k vertices • Reconstruction • 11 patches • ~150 quads

20. Future Work • Connecting patches and iterative reconstruction (in progress) • Error visualisation (in progress) • Refining patch approximation • Parametrize all points inside a patch • Approximate patch by solving one linear equation system • Attach semantics to features • Extrapolate parametric GML model

21. Acknowledgement European Project Focus K3D European Project 3D-COFORM Volkswagen AG AIM@SHAPE Digital Shape Workbench

22. Questions and Discussion Thank you for listening Feel free to ask any questions. Suggestions for improvements welcome...