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9. 89. 13. 100. 80. 71. 64. 93. 59. 60. 50. 74. 65. 102. 57. 46. 48. 65. 91. 37. 59. 19. 76. 90. 70. 10. 0. 68. 99. 68. 56. 52. 69. 51. 51. G:. M:. G:. i. j. k. slack. i. a. a. b. c. j. k. b. c. d. slack. slack. d. curve. node. sheet. …….

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Match

9

89

13

100

80

71

64

93

59

60

50

74

65

102

57

46

48

65

91

37

59

19

76

90

70

10

0

68

99

68

56

52

69

51

51

G:

M:

G:

i

j

k

slack

i

a

a

b

c

j

k

b

c

d

slack

slack

d

curve

node

sheet

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Measuring 3D Shape Similarity by Matching the Medial Scaffolds

Ming-Ching Chang Benjamin B. Kimia

Laboratory for Engineering Man/Machine Systems (LEMS), Brown University, Providence RI, USA

Overview

Graph Matching Approach to Match the MS Hypergraphs

Define Compatibility Measures

  • Apply MS transforms to regularize the MS across low-cost transitions.

match

Extend the Graduated Assignment Scheme

  • Represent the match between graphs nodes in a matrix M.

  • Use slack row and column to handle missing/extra nodes.

  • Extend to hypergraph matching intuitively.

Main idea: Measure 3D shape similarity by matching a medial axis (MA) based representation ─ the Medial Scaffold (MS).

Node structural compatibility: 1−

Curve parametric compatibility:

Medial Representation

2D: shock graphs.

Correspondence via

dynamic programming

  • Energy minimization:

  • Shocks: medial axis points endowed with dynamics of flow.

  • Shape similarity is measured as the minimum extent of deformation necessary for one shape to match another.

  • Optimal deformation path as the transitions (discrete topological changes) between the two graphs.

Virtual Links to Improve Robustness

3D: Medial Scaffold hypergraphs.

  • Vertex: A14 or A1A3node

  • Link: A13 or A3curve

  • Hyperlink: A12sheet

Group two nodes into a virtual node.

Add two virtual links to

simulate such grouping.

Experimental Results

Matching carpal bones (hamates) across 6 patients.

Similarity in a 25 shape database.

Notation Akn:k+1 degree of contact at n distinct points.

Key Novelties

  • The Medial Scaffold representation is:

    • Hierarchical: capturing an intrinsic graph-like structure of shape, and

    • Complete: allowing a full reconstruction of shape.

  • We extend a popular graph matching scheme to match the MS hypergraphs.


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