Interactive surface reconstruction on triangle meshes with subdivision surfaces Matthias Bein Fraunhofer-Institut für Graphische Datenverarbeitung IGD Fraunhoferstraße Darmstadt Tel.: –

07_10_26_Presentation.ppt 2 Exposure of the problem Input: triangle mesh (scanned data) Aim: connected control mesh for subdivision surfaces Constraints: time, complexity, control, accuracy and more

07_10_26_Presentation.ppt 3 Exposure of the problem Difficulties Holes Varying point density Errors Bad triangulation

07_10_26_Presentation.ppt 4 Related Work Academic Prof. Klein (primitive fitting, parametrization) Prof. Kobbelt (quad dominant remeshing)

07_10_26_Presentation.ppt 5 Related Work Commercial Geo Magic

07_10_26_Presentation.ppt 6 Related Work Commercial Geo Magic

07_10_26_Presentation.ppt 7 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?

07_10_26_Presentation.ppt 8 Idea User picks vertices Patch borders are extracted automatically Points in borders are approximated automatically Hole filling is implicit Pick1 Pick2 Pick3 Pick4

07_10_26_Presentation.ppt 9 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

07_10_26_Presentation.ppt 10 Principle Curvature Analysis Discrete Taubin, modifications and others Inadequate for noisy scanned meshes Analytic Primitive fitting Polynom fitting Moving Least Squares...

07_10_26_Presentation.ppt 11 Our Principle Curvature Analysis 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 vertices (2GHz, 2GB ram) Local coordinate system with radius Polynom surface and normal Vertices Main curvature 1 Main curvature 2

07_10_26_Presentation.ppt 12 Visualisation and tools Visualisation Main curvature direction Scaled normals

07_10_26_Presentation.ppt 13 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

07_10_26_Presentation.ppt 14 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

07_10_26_Presentation.ppt 15 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

07_10_26_Presentation.ppt 16 B-Spline Curve Approximation Input: set of points d k with parameters t k B-Spline definition: c(t) = Σ N i (t)p i Linear equation system: d k = Σ N i (t k )p i D = N * P Overestimated (# points > # control points) Multiply with transposed N N T * D = N T * 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 || d k - c(t k ) || 2 Catmull&Clark subdivision derived from uniform B-Splines => Subdivision control net with this approximation

07_10_26_Presentation.ppt 17 Results Whole seat 206k triangles 105k vertices Backrest 108k triangles 57k vertices Reconstruction 19 patches ~300 quads 7 minutes

07_10_26_Presentation.ppt 18 Results Vase 50k triangles 25k vertices Reconstruction 11 patches ~150 quads

07_10_26_Presentation.ppt 19 Results Vase 50k triangles 25k vertices Reconstruction 11 patches ~150 quads

07_10_26_Presentation.ppt 20 Results Vase 50k triangles 25k vertices Reconstruction 11 patches ~150 quads

07_10_26_Presentation.ppt 21 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

07_10_26_Presentation.ppt 22 Acknowledgement European Project Focus K3D European Project 3D-COFORM Volkswagen AG Digital Shape Workbench

07_10_26_Presentation.ppt 23 Questions and Discussion Thank you for listening Feel free to ask any questions. Suggestions for improvements welcome...