I presented this overview of my work in Digital Geometry at ICVGIP ’04. Science City, Kolkata on 18-Dec-04

1. Digital Distance Geometry – Applications to Image Analysis Dr. P. P. Das [email_address] , [email_address] Interra Systems, Inc. www.interrasystems.com ICVGIP ’04. Science City. 18-Dec-04

4. What is Distance?

8. Distance Geometry

10. Taxicab Geometry Krause 1975

15. Digital Distance Geometry Basic Notions

17. Tessellations

21. Neighborhoods: 2D

22. Neighborhoods: 3D

26. 2D Paths

27. 3D Paths

30. Digital Distances in 2-D

32. Octagonal Distance Neighborhood sequence {1,2} d(a,b) = 10 a b 2 2 2 2 2 1 2 2 2 1 * 1 2 2 2 1 2 2 2 2 2

36. Simple Octagonal Distances

39. Fun Distances 

41. Knight’s Circle & Disk

43. Digital Distances in 3-D

45. Digital Distances in n-D

49. 2D 3D t-cost neighbors 1 1 1 1 o 1 1 1 1 2 1 2 1 o 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 2 2 3 2 3 2 2 2 3 2 3 1 1 1 1 o 1 1 1 1 2 1 2 1 o 1 2 1 2 2 1 2 1 o 1 2 1 2 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2 2 2 3 2 3 2 1 2 3 2 3

50. Path by t-Cost Distance

54. Generalized Octagonal Distances in 3-D

55. Digital Distance Computation & Approximation

56. Chamfering for computing Distance Transform o a b a Forward Scanning From Left to Right and Top to Bottom Backward Scanning From Right to Left and Bottom to Top b Distance at o = min (Distance value at Neighboring pixel + local distance between them) 1. Initialize all distance values to a Maximum Value. 2. At every point o compute the distance value from its visited neighbors as follows: Extend this concept with larger neighborhood and dimension. o a b a b

57. Chamfer/Weighted distances Borgefors and her colleagues 1984-2004

58. 2D 3D Weighted Distance Templates b a b a o a b a b c b c b a b c b c b a b a o a b a b c b c b a b c b c

62. Relative Error: t-cost distance … 3 2 … 2 2 1 1 t opt … 56 55 … 4 3 2 1 n

64. Benchmarking 2D Distances 0.030 0.073 <8,11> 0.035 0.083 <5,7> 0.025 0.081 <3,4> 0.068 0.134 <2,3> 0.043 0.118 {1,2} 0.056 0.187 {1,1,1,2} 0.026 0.087 {1,1,2} MSE MAE Distance Function

65. Benchmarking 3D Distances 0.034 0.107 <13,17,23> 0.043 0.107 <13,17,22> 0.043 0.107 <8,11,13> 0.043 0.118 <3,4,5> 0.070 0.269 {1,2} 0.034 0.146 {1,1,1,2,3} 0.027 0.105 {1,1,3} MSE MAE Distance Function

66. Good Weighted distances 0.184 0.167 0.167 0.155 0.143 - - - - - <3,4,5,6> <6,8,10,11> <6,9,10,11> <7,10,12,13> <8,11,13,15> 4 0.118 0.107 0.107 0.107 0.103 0.103 0.043 0.043 0.043 0.034 0.036 0.043 <3,4,5> <8,11,13> <13,17,22> <13,17,23> <16,21,27> <16,21,28> 3 0.134 0.081 0.083 0.073 0.068 0.025 0.035 0.03 <2,3> <3,4> <5,7> <8,11> 2 MAE MSE Weights Dimension

67. Neighborhood :: Digital Distance Position Independent Isotropic Unity Undefined Non-Proximal Others (Knight’s / Super-Knight’s) Position Dependent Isotropic Unity Graded Proximal Sequence-based (Octagonal) Position Independent Isotropic Unity Graded Proximal m-Neighbor Position Independent Position Independent Uniformity Isotropic Isotropic Isotropy & Symmetry F n of Separating Dimension Non-Unity Separating Cost Maximal Maximal Separating Dimension Proximal Proximal Proximity Chamfer / Weighted t-Cost

68. Glimpses of Applications

69. Distance Transform Minimum distance of a feature point from the back ground.

70. Medial Axis Transform A set of maximal blocks contained in the pattern.

72. Computation of minimal set of maximal disks 1. Compute Local Maximum Blocks from the distance transformed image. Form a relational table expressing the relationships between boundary pixels and individual disks. The problem is mapped to the covering of the list of boundary pixels with the optimal set of maximal blocks. 2 . 3. Nilson-Danielsson’96

73. Digital discs 2D 3D d 6 d 26 d 18 d 4 d 8 R R

74. Vertices of octagonal discs:

76. Thinning from Distance Transform Compute the set of Maximal Blocks. Use them as anchor points while iteratively deleting boundary points preserving the topology. Vincent ’91, Ragnemalm ’93, Svensson-Borgefors-Nystrom ’99

78. Normal Computations: Examples

80. Discrete Shading: Examples

81. Discrete Shading: Examples

82. Discrete Shading: Examples

83. Decomposition of 3D Objects Identification of seed of a component from inner layers of Distance Transformed Image. Seed-fusion by expansion and shrinking Region growing by reversed DT. Surface smoothing and merging. Svensson-Saniti di Baja’02

84. Cross-sectioning

85. Cross-sectioning with different distance functions.

86. A set of objects for experimentation

87. Cross-sectioning: Voxel data, MAT & Sphere Approx. Voxel Data MAT Euclidean Sphere Approximation

89. Thank you !

90. O(1) neighbors O(2) neighbors O(3) neighbors O(1) neighbors O(2) neighbors 2D 3D M-neighbors o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

93. Closed form expression:

94. Cross-sectioning using MAT

95. Cross-sectioning using Euclidean Sphere Approximation