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Sandia Sandia Presentation Transcript

  • A Topological Approach to Shape Analysis and Alignment David O’Gwynn University of Alabama at Birmingham Sandia National Laboratories – p. 1/5
  • The Problem: Shape Alignment Importance of Part Structure Part Structure from TopologyTopology → Parts → Alignment Results Sandia National Laboratories – p. 2/5
  • The Problem: Shape Alignment Importance of Part Structure Part Structure from TopologyTopology → Parts → Alignment Results Sandia National Laboratories – p. 2/5
  • The Problem: Shape Alignment Importance of Part Structure Part Structure from TopologyTopology → Parts → Alignment Results Sandia National Laboratories – p. 2/5
  • The Problem: Shape Alignment Importance of Part Structure Part Structure from TopologyTopology → Parts → Alignment Results Sandia National Laboratories – p. 2/5
  • The Problem: Shape Alignment Importance of Part Structure Part Structure from TopologyTopology → Parts → Alignment Results Sandia National Laboratories – p. 2/5
  • Problem Definition Data Problem Existing solutions Sandia National Laboratories – p. 3/5
  • Surface data from very different sourcesMRI Stored in many different databases Range scansCAT [Levoy 2000] Sandia National Laboratories – p. 4/5
  • Predominantly stored as a meshMRI Range scansCAT [MeshLab] [Levoy 2000] Sandia National Laboratories – p. 4/5
  • Information stored in a mesh... Sandia National Laboratories – p. 5/5
  • Information stored in a mesh... Surface sample points Sandia National Laboratories – p. 5/5
  • Information stored in a mesh... Surface Local connectivity of sample pointssample points Sandia National Laboratories – p. 5/5
  • The ProblemOrientation of shapes rarely consistent across databases ... Or even in the same database Sandia National Laboratories – p. 6/5
  • The ProblemHow do we align two similar mesh shapes? Sandia National Laboratories – p. 7/5
  • Further ComplicationsVariations in sampling density Variations in pose, physiology Sandia National Laboratories – p. 8/5
  • Given: Input shapes P and Q 2-manifold triangle meshes Same underlying shape class (hands, humans) Transform class A Isometry: Rigid or distance-preserving transformsFind:Transformation α ∈ A that minimizes E(P, Q) = min d2 (α(P), Q) α Sandia National Laboratories – p. 9/5
  • Existing SolutionsSurface Registration Local GlobalOrientation Normalization Principle component analysis (PCA) Function-space search Sandia National Laboratories – p. 10/5
  • Surface Registration → → [Gelfand 2006] [Levoy 2000] [Levoy 2000] Sandia National Laboratories – p. 11/5
  • Local RegistrationIterative Corresponding Point (ICP)[Besl and McKay 1992, Chen and Medioni 1991][Rusinkiewicz and Levoy 2001]Given: initial guess α0correspond α0 (P),Q → solve for α1 [Gelfand 2006]correspond α1 (P),Q → solve for α2 Correspondence is based on a... modified nearest neighboruntil convergence approach.Process iterates to local minimum of E(P, Q). Sandia National Laboratories – p. 12/5
  • Global RegistrationRobust Global Registration[Gelfand et al. 2005] Select unique points on P and Q based on volumetric feature space Search correspondence space for optimal corre- spondenceFour Points Congruent Sets (4PCS)[Aiger et al. 2008] Search 4-point set correspondence space [Gelfand 2005]The goal is to find the global minimum of E(P, Q). Sandia National Laboratories – p. 13/5
  • Orientation Normalization [Chaouch and Verroust-Blondet 2009] Sandia National Laboratories – p. 14/5
  • Principal Component Analysis (PCA)Each shape has a “correct” orientation characterized by itsprincipal components.Continuous PCA (CPCA)[Vranic et al. 2001]Plane Reflection Symmetry PCA[Chaouch and Verroust-Blondet 2009]Find the principal axes of P and Q(best fitting ellipsoid) and align those axes Sandia National Laboratories – p. 15/5
  • Function-space AlignmentMap P and Q to some functional space and solve for therotation in that space that best aligns them. Then map thatrotation back to R3 .Axially Symmetric Alignment viaSpherical Harmonics[Kazhdan 2007]GPU-based Rotational Alignment[Martinek and Grosso 2009] [Kazhdan 2007] Sandia National Laboratories – p. 16/5
  • Issues...Surface Registration Predicated on “ground truth” model to which input shapes conform Exceptions: Anguelov et al. 2005 Chang and Zwicker 2008Orientation Normalization Reliance on well-defined axes/planes of symmetry Poor performance on non-symmetric objectsTreat the shape as a geometric monolith. Sandia National Laboratories – p. 17/5
  • Our ProblemHow do we align these horses? surface characteristics? principle axes? Sandia National Laboratories – p. 18/5
  • Solution... Helps to remember they’re horses... and not a bag of surface samples, or planes of symmetry Sandia National Laboratories – p. 19/5
  • Solution... They have a part structure.Aligning horses requires aligning their parts. Sandia National Laboratories – p. 19/5
  • So we need part structure. What do we mean by parts?How are these parts related?How do we get this part structure? Sandia National Laboratories – p. 20/5
  • SegmentationSubdividing a mesh into simpler, more manageablesub-partsTypes of segmentations [Attene et al. 2006]: [Katz et al. 2005] Geometric: clusters of similar geometry Semantic: how would a human segment the mesh?Salience [Hoffman 1997, Lee 2005, Golovinsky 2008]Geodesic distance [Tierny 2007, Berretti 2009]Medial/functional [de Goes 2008, Shapira 2008] [Berretti et al. 2009]Global part boundaries Sandia National Laboratories – p. 21/5
  • Skeleton extraction Simplifying a 3D surface to a 1D curve-skeleton [Dey and Sun 2006][Tierny 2006] Approaches [Cornea 2007]: Volumetric thinning [Svensson 2002, Siddiqi 2008] Geometric [Tierny 2006, Tagliasacchi 2008, Agathos 2010][Siddiqi 2008] Global connectivity of parts Sandia National Laboratories – p. 22/5
  • What do we want to do?We want to correspond part junctions (i.e. boundaries) andpart caps (i.e. part end-points). Then we can align thosecorrespondences in closed form.What do we need?Simple, robust algorithm for decomposing shape into part graphthat captures the part structures shared between two shapes ofthe same object classMust be: Invariant to isometry As minimal as possible Sandia National Laboratories – p. 23/5
  • InspirationConnectivity Shapes of Isenburg et al. [2001] Sandia National Laboratories – p. 24/5
  • ˆ Breath-first graph GLet the topology of the mesh do the talking, and this is what it says Sandia National Laboratories – p. 25/5
  • ˆBreath-first graph GGiven: Mesh M = {V, E, F } Seed vertices VS ⊆ VTraverse M in a breadth-first mannerEach frontier of the traversal becomes a node in a newgraph Sandia National Laboratories – p. 26/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆBreadth-first graph G Sandia National Laboratories – p. 27/5
  • ˆ Breadth-first graph Gmesh graph structure → simplified metagraph structure Sandia National Laboratories – p. 27/5
  • ˆSegmentation from G Cap segments: Contain a cap vertex Pipe segments: Connect 2 junctions Junction segments: Connect > 2 other seg- ments ˆSkeleton from G Use edges of BFG Vertex of skeleton is the centroid of mesh vertices it encodes Sandia National Laboratories – p. 28/5
  • Automatic BFG generationIssues: Seed vertices “Hairs” Sandia National Laboratories – p. 29/5
  • Seed verticesBreadth-first traversal tends to terminate at appendage tipsUse a “priming” run to find appendage tips.Problem: too many tips... Sandia National Laboratories – p. 30/5
  • “Hairs”Distilling the BFG: Trim cap segments with only one edge Get standard deviation (σ) of number of edges in all cap segments Iteratively trim segments of length < σUsually < 4 iterations Sandia National Laboratories – p. 31/5
  • Distilling the BFG Original BFG skeleton Trim segments of length 1 Iteratively trim Sandia National Laboratories – p. 32/5
  • Robust to...AutoBFG Start with mesh vertex closest to mesh ˆ centroid (vcent ), generate G0 ˆ Distill G0 ˆ ˆ Use caps of G0 , minus Vcent , to build G1 ˆ Distill G1 Sampling density ˆ Return G1TopoBFGPurely topological BFG. Same as above except that theinitial seed vertex is chosen randomly. Shape variation and pose Sandia National Laboratories – p. 33/5
  • Benchmark for Segmentation [Chen 2009] 400 different models, 20 classes, 20 meshes/class Based on human-generated segmentations Tested 7 other algorithms, plus two sanity checks Uses four different metrics (lower is always better) Cut discrepancy: Disagreement between segment boundaries and baseline Hamming distance: Disagreement between segment regions and baseline Consistency error: Disagreement between segment regions in a way that does not penalize differences in hierarchical granularity Rand index: Unlikelihood that a pair of faces will agree on segment identity Sandia National Laboratories – p. 34/5
  • Cut Discrepancy Sandia National Laboratories – p. 35/5
  • Hamming Distance Sandia National Laboratories – p. 36/5
  • Consistency Error Sandia National Laboratories – p. 37/5
  • Rand Index Sandia National Laboratories – p. 38/5
  • Shape Alignment Sandia National Laboratories – p. 39/5
  • → → P and Q Skeleton for P and Q →Correspondence Alignment Sandia National Laboratories – p. 40/5
  • Shape AlignmentTwo aspects of alignment:Correspondence Limit correspondence space to most semantically dissimilar vertices Junction (part boundaries) Cap (part tips) Greedy searchAlignmentGiven correspondence, the transform α which minimizes E(P, Q) can be solved for in closedform [Arun 1987]Real problem is correspondence Sandia National Laboratories – p. 41/5
  • CorrespondenceVery similar in spirit to greedy search of Mitra et al. [2005].Greedy Correspondence: ˆ ˆ ˆ ˆRequire: BFGs GP = {VP , EP } and GQ = {VQ , EQ }ˆ ˆ ˆ FP = junctions and caps of GP ˆ ˆ FQ = junctions and caps of GQ ˆ ˆ ˆ for all vP ∈ FP do ˆ ˆ Topologically sort FP w.r.t. graph distance from vP ˆ ˆ Find the vQ ∈ FQ (of the same type as vP ) whose topological sorted FQ most agrees ˆ ˆ ˆ with that of vP end forImagine two graphs as made of string. Grab both at a vertex, and if theknots line up, then it’s likely that they are corresponding points. Sandia National Laboratories – p. 42/5
  • ComplicationsWe’re extracting part structure from mesh topology. What if the topology doesn’t reflect the part structure? Sandia National Laboratories – p. 43/5
  • ComplicationsCorrespondence is based on similarity of internal graph distance, as well asEuclidean distance. Sandia National Laboratories – p. 44/5
  • Alignment Results Using Chen database, selected 10 object classes (200 meshes) Selected semantically relevant landmarks (tops of heads, tips of wings, front and back of body, etc.) For some mesh MP and MQ , whose landmark vertices are LP ⊂ VP and LQ ⊂ VQ , respectively (N = |LP | = |LQ |), and some aligning transform α, the error associated with (MP , MQ , α) is N 1 E(MP , MQ , α) = ||α(vi ) − wi ||, vi ∈ LP , wi ∈ LQ N i=1 Tested against Generalized-ICP of Segal et al. [2009] Results show mean error over each class Sandia National Laboratories – p. 45/5
  • Alignment Results 1.4 Baseline 1.2 AutoBFG TopoBFG 1 Gen ICP 0.8 0.6 0.4 0.2 0 0 50 100 150 200 Humans Sandia National Laboratories – p. 46/5
  • Alignment Results 1.4 Baseline 1.2 AutoBFG TopoBFG 1 Gen ICP 0.8 0.6 0.4 0.2 0 0 50 100 150 200 Hands Sandia National Laboratories – p. 47/5
  • Alignment Results 1.4 Baseline 1.2 AutoBFG TopoBFG 1 Gen ICP 0.8 0.6 0.4 0.2 0 0 50 100 150 200 Four legged animals Sandia National Laboratories – p. 48/5
  • Conclusions Shape alignment better with part information Startling amount of part information in topology Part information from breadth-first traversal ˆ Alignment using the breadth-first graph GFuture Work Shape reconstruction from BFG Applications to graph visualization Integration into Blender3D [www.blender.org] 1. Alignment 2. Auto-rigging Sandia National Laboratories – p. 49/5
  • Questions? Sandia National Laboratories – p. 50/5