Partial Shape Matching

1. Partial Shape Matching Hu Jianwei 2006-10-11

2. 3D Query Best Match 3D Database Shape Retrieval

3. Global Shape Matching • Skeleton Based Similarity • Reeb Graph Based Similarity • Shape Histograms • Light Field Descriptor • Extended Gaussian Image • ……

4. Partially Overlapping Scans Aligned Scans Merging

5. The same class with different overall shapes

6. Partial Shape Matching Partial matching is a much harder problem than global matching, since it needs to search for and define the subparts prior to measuring similarities. Exhaustive Search?

7. Partial Shape Matching Segmentation

8. Salient Geometric Features for Partial Shape Matching and Similarity ACM Transactions on Graphics, Vol. 25, No. 1, January 2006, Pages 130–150. Ran Gal & Daniel Cohen-Or Tel-Aviv University

10. Outline • Local Surface Descriptors • Salient Geometric Features • Indexing and Geometric Hashing

11. Quadric Fitting and Curvature Estimation • Quadric Fitting Douros and Buxton [2002] • Curvature Estimation Ohtake et al [2004]

12. Compact Representation • Defining the Local Patches • Error Measuring: Squared Algebraic Distances • Threshold: of the model bounding box diagonal length

13. Local Surface Descriptors • A Patch (Points) • A Point • A Curvature

14. Gaussian Curvature vs Max Curvature

15. Sample the Surface Randomly • Osada et al [2001] • Elad et al [2001]

16. Outline • Local Surface Descriptors • Salient Geometric Features • Indexing and Geometric Hashing

17. Salient Geometric Features • A set of descriptors that have: • A high curvature relative to their surroundings • A high variance of curvature values • A function of one free parameter: the scale

18. The curvature variance in the cluster The area of the patch associated with d relative to the sphere size The curvature associated with d The number of local minimums or maximums curvatures in the cluster Saliency Grade

19. The saliency of the region The degree of interestingness of the cluster Saliency Grade

20. Saliency Grade

21. Saliency Grade

22. Outline • Local Surface Descriptors • Salient Geometric Features • Indexing and Geometric Hashing

23. Indexing and Geometric Hashing • Each salient feature is associated with a vector index • The vector index is defined by the four terms: • Geometric hash table contains the vector indices

24. Best Transformation • Compute the transformation • Triplet of points • Voting system [Lamdan and Wolfson 1988]

25. Time and Storage

26. Self-Similarity

27. Shape Alignment

28. Partial Shape Retrieval

29. Retrieval Result

30. Spin Image vs SGFs

31. Registration of Synthetic Scans

32. Reconstruction

33. Local Feature Extraction and Matching Partial Objects Computer-Aided Design 38 (2006) 1020–1037 Dmitriy Bespalov, William C. Regli & Ali Shokoufandeh Drexel University

34. Watertight boundary-representation solid Implicit surfaces Analytic surfaces NURBS, etc Topologically and geometrically consistent Produced by kernel modelers and CAD systems Traditional CAD Representation

35. Usually a mesh or point cloud Usually an approximate representation Sometimes error prone Produced by CAD systems, animation tools, laser scanners, etc Traditional CAD Representation

36. Commonly used for Coarse-to-Fine representations of an object Very popular in Computer Vision Basic Idea: At each scale, topologically relevant components will decompose the object into so called salient parts Recursive application of this paradigm will create the object’s scale space hierarchy What is a Scale-Space Representation?

37. Why Scale-Space Representation? • A unified framework for matching • Different features can be parameterized as different scale-space decompositions • Robust & consistent across noisy and diverse data sets

38. Method • Start with CAD model • Perform geometry-based decomposition • Construct hierarchical “feature” graph • Use hierarchical matching to compare graphs

39. Algorithm Overview (I) • Given model P, compute mesh representation M • Define measurement function: • Our d is the maximum angle on • an angular shortest path distance • function between every two faces • on M will be captured in a pair-wise • distance matrix D. d(t1,t2)

40. Algorithm Overview (II) 3. DecomposeM into components relevant using a singular value decomposition of distance matrix D • Compute the SVD decomposition with • Compute the order-k compression matrix • Let denote the jth column of , • Form sub-feature as the union of faces with

41. Algorithm Overview (III) 4. Recursive feature decomposition using two principle components creates binary feature trees feature tree for simple_bracket feature tree for swivel

42. simple_bracket swivel Algorithm Overview (IV) 5. Compare feature trees (bottom up dynamic programming) using The Largest Common Subgraph Algorithm [Ullmann JR. 1976]

43. When to Stop? The feature is decomposed into sub-features and if the angular distance between components of and is large.

44. Example Decomposition

45. Feature Decomposition on Noisy Data

46. Partial Shape Matching A precision–recall graph for retrieval experiment using: 1. A Reeb Graph technique; 2. a Scale–Space technique with the max-angle distance function and simple sub-graph isomorphism for matching; 3. the original Scale–Space technique with a geodesic distance function; 4. a random retrieval technique.

47. Fidelity Experiments

48. Retrieval experiment

49. Partial Shape Matching of 3D Shapes with Priority-Driven Search Eurographics Symposium on Geometry Processing (2006) T.Funkhouser & P.Shilane Princeton University

50. System Execution • Preprocessing phase • Constructing Regions • Computing Shape Descriptors • Selecting Distinctive Features • Query phase • Creating Pairwise Feature Correspondences • Searching for the Optimal Multi-Feature Match