1 Complex Topology The talk includes Large Acquisition Holes - PDF document

Large and Accurate reconstructions Towards Ideal Surface Reconstruction Daniel Cohen-Or Structured Light Scanners � Acquired in few seconds (<30s) � Few structured light shots (<12) The 3D scanning Imperfect Acquisition � 3D Scanning, many � The problem of coverage: challenges : � Large missing parts � Acquisition of multiple views � Non uniform sampling � Non-uniform sampling � Registration of range maps f � Outliers � Consolidation � Noisy data � Surface tessellation � Orientation Involve extensive user intervention 1

Complex Topology The talk includes � Large Acquisition Holes � Completion of Large missing parts � Consolidation of point clouds � Consolidation of point clouds play p y � Registration of noisy data Topology-aware Reconstruction Competing Fronts for Coarse Competing Fronts for Coarse– –to to– –Fine Fine Surface Reconstruction Surface Reconstruction (Sharf et al. Eurographics (Sharf et al. Eurographics 2006 2006) ) play Deformable model Deformable Model Reconstruction � Implicit coarse guidance � Watertight guarantee field or attraction field � Topology control � Explicit deformable model (a mesh) play 2

Competing Fronts Overview Mesh Fronts � Incremental steps � front : set of connected vertices not ε -close to zero level-set � Multiple fronts � Front control � Front control � Coarse-to-fine play play Attraction Field Initialization � Vertex Attraction ( E a ) � Guidance field: � Outward normal direction � RBF distance-field � Unsigned field speed � Unsigned field speed � Uphill evolution � Deformable model D : � sphere mesh placed in interior play Competing fronts Starting position invariance � Coarse-to-Fine � Flexible evolution � Initial coarse reconstruction � Multiple fronts � Incremental fine detail recovery � Incremental fine detail recovery � Adaptation control � Tension factor play � Triangle density play 3

Coarse-to-fine Competition Topology ambiguity Topology ambiguity solution: Coarse-to-fine Competition : � � First: hole – complete hole - complete � � Next: narrow tunnel - intrude Next: narrow tunnel intrude narrow tunnel intrude narrow tunnel - intrude � � � � Coarse-to-fine Competition Coarse-to-fine Competition � 3D scenario play play Power-crust Competing-fronts Final Projection Evolution Parameters E a � ε -close vertices: normal project � Attraction ( E a ) Outward normal direction � far-vertices: interpolate � Unsigned distance field speed � � Tension ( w·E t i ) E t Smoothness factor � ⎧ ⎫ � Laplace system: ⎪ ⎪ ∑ ⋅ + 2 ⎨ i i ⎬ arg min ( w E E ) t a ⎪ ⎪ ⎩ ⎭ v ∈ v front i i � Local remeshing and subdivision play 4

ε -close Stop Criteria Topology Aware � High genus cases: � satisfied points: points ε -close to model � Collision detection � Merge fronts � Merge fronts � “wake up” procedure for wake up procedure for � unsatisfied points: � Fronts revived and subdivided � Tension released play Tunnel Results Limitations play play User Interaction! Topology-aware � An ill posed problem: infinite So far: surfaces pass through or near the data points � Watertight guarantee � Watertight guarantee � Heuristics (Coarse–to–fine) � Reconstructed object is not necessarily the expected one! � Topology aware – but… � UI for correct interpretation! 5

Interactive Topology-aware Interactive Topology-aware Surface Reconstruction (Sharf et al.) � Automatic detection of ill conditioned cases � Ask the user for inside/outside constraints � Resolve locally and achieve expected y p shape. FEM Fields Implicit Formulation � Underlying structure: � Goal: construct surface: smooth � Dynamically adaptive � close to the input points octree � separates the in/out scribbles � � Dual hierarchical mesh � Implicit representation u ( p ) s.t.: graph Z = u -1 ( { 0}) � � Penalty functions: u ( p ) ≈ 0 for p ∈ P s � u ( p ) > 0 for p ∈ P in � ( ) Ψ = u ( p ) < 0 for p ∈ P out T u u Ku � smoothness M M Method Overview FEM Fields cont’d Ψ point constraints + Ψ smoothness o ptimization problem: � Automatically generated, � loose constraints We solve using a fast sparse Cholesky factorization � (CHOLMOD) � Compute a smooth coarse Update factorization when user adds/removes constraints Update factorization when user adds/removes constraints. � approximation � Analyze the implicit function and identify weak regions � Allow the user to draw scribbles to specify local sign. 6

Topological Critical Points Topological Critical Points � Topological non-stability detection: u ( p ) - ε � Where should interaction occur? and u ( p )+ ε have different topologies User Interface Weak Regions � Provide 2D tablets at weak regions near zero � Discrete critical points level-set (Morse): � 2D tablets are located at critical points, � Partition link graph into � Partition link graph into perpendicularly to critical lines perpendicularly to critical lines positive and negative groups � The user draws scribbles to correct or reinforce the � Detect critical points by topology number of connected groups Video 7

Surface Reconstruction using Surface Reconstruction using Local Shape Priors Local Shape Priors (SGP07) Surface Reconstruction using Surface Reconstruction using Local Shape Priors Local Shape Priors Parametrization-free Sharp Features, Faithful Reconstruction (Lipman et al. SGP07) projection (Lipman et al.) Initial guess Original data (projected set) MLS projection LOP projection Lop 07 8

4 -Points Congruent Sets for Robust Pairwise Surface Skeleton extraction from Registration incomplete point cloud (Aiger et al.) Andrea Tagliasacchi, Hao Zhang, Daniel Cohen-Or 4 points SIGGRAPH 2009 50 Skeleton aided reconstruction Direct reconstruction through RBF 52 51 Thank you 9