1. Digital Camera and Computer Vision Laboratory Department of Computer Science and Information Engineering National Taiwan University, Taipei, Taiwan, R.O.C. Computer and Robot Vision I Chapter 6 Neighborhood Operators Presented by: 傅楸善 & 何育哲 0937 960 615 r94922131@ntu.edu.tw 指導教授 : 傅楸善 博士

2. DC & CV Lab. CSIE NTU 6.1 Introduction neighborhood operator: workhorse of low level vision neighborhood operator: performs conditioning, labeling, grouping

3. DC & CV Lab. CSIE NTU 6.1 Introduction The output of a neighborhood operator at a given pixel position is a function of the position, of the input pixel value at the position, of the values of the input pixels in some neighborhood around the given input position, and possibly of some values of previously generated output pixels

4. DC & CV Lab. CSIE NTU 6.1 Introduction numeric domain: arithmetic operations, +, -, min, max symbolic domain: Boolean operations, AND, OR, NOT, table-look-up nonrecursive neighborhood operators: output is function of input recursive neighborhood operators: output depends partly on previous output

5. DC & CV Lab. CSIE NTU 6.1 Introduction neighborhood might be small and asymmetric or large

6. DC & CV Lab. CSIE NTU 6.1 Introduction : set of neighboring pixel positions around position general nonrecursive neighborhood operator : input, output

7. DC & CV Lab. CSIE NTU 6.1 Introduction linear operator: one common nonrecursive neighborhood operator output possibly: position-dependent linear combination of inputs

8. DC & CV Lab. CSIE NTU 6.1 Introduction shift-invariant: position invariant, action same regardless of position composition of shift-invariant operators: shift- invariant

9. DC & CV Lab. CSIE NTU 6.1 Introduction cross-correlation of with : weight function: kernel or mask of weights : domain of

10. DC & CV Lab. CSIE NTU 6.1 Introduction common masks for noise cleaning, (a) box filter

11. DC & CV Lab. CSIE NTU 6.1 Introduction common masked for noise cleaning

12. DC & CV Lab. CSIE NTU 6.1 Introduction application of mask with weights to image 7.69

13. DC & CV Lab. CSIE NTU 6.1 Introduction convolution of with convolution: close relative to cross-correlation convolution: linear shift-invariant if mask symmetric, convolution, and correlation the same

14. DC & CV Lab. CSIE NTU 6.2 Symbolic Neighborhood Operators indexing of neighborhoods

15. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator : projection, outputs first or second argument : background background pixel labeled first nonbackground label

16. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator 4-connected output

17. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator 8-connected output

18. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator

19. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator

20. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator

21. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator

22. DC & CV Lab. CSIE NTU 6.2.1 Region-Growing Operator

23. DC & CV Lab. CSIE NTU 6.2.2 Nearest Neighbor Sets and Influence Zones influence zones: nearest neighbor sets influence zones: iteratively region-growing

24. DC & CV Lab. CSIE NTU 6.2.2 Nearest Neighbor Sets and Influence Zones 4-neighborhood for city-block distance e.g. ( i, j ), ( k, l ) | ( k – i ) | + | ( l – j ) | 8-neighborhood for max distance (of horizontal and vertical distances) e.g. ( i, j ), ( k, l ) max( | k – i |, | l – j | ) alternate 4, 8-neighborhood for Euclidean distance

25. DC & CV Lab. CSIE NTU Take a Break

26. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator region-shrinking: changes all border pixels to background region-shrinking: can change connectivity region-shrinking: can entirely delete region if repeatedly applied

27. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator : whether or not arguments identical : background border: has different neighbor and becomes background

28. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator 4-connected: output:

29. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator 8-connected: output:

30. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator region shrinking: related to binary erosion except on labeled region

31. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator

32. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator

33. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator

34. DC & CV Lab. CSIE NTU 6.2.3 Region-Shrinking Operator

35. DC & CV Lab. CSIE NTU 6.2.4 Mark-Interior/Border-Pixel Operator mark-interior/border-pixel operator marks all interior pixels with the label and all border pixels with the label

36. DC & CV Lab. CSIE NTU 6.2.4 Mark-Interior/Border-Pixel Operator : whether or not arguments identical : recognizes whether or not its argument is symbol

37. DC & CV Lab. CSIE NTU 6.2.4 Mark-Interior/Border-Pixel Operator 4-connected: output:

38. DC & CV Lab. CSIE NTU 6.2.4 Mark-Interior/Border-Pixel Operator 8-connected: output:

39. DC & CV Lab. CSIE NTU 6.2.5 Connectivity Number Operator connectivity number: nonrecursive and symbolic data domain connectivity number: classify the way pixel connected to neighbors six values of connectivity: five for border, one for interior border: isolated, edge, connected, branching, crossing

40. DC & CV Lab. CSIE NTU 6.2.5 Connectivity Number Operator 4-connectivity

41. DC & CV Lab. CSIE NTU 6.2.5 Connectivity Number Operator corner neighborhood

42. DC & CV Lab. CSIE NTU 6.2.5 Connectivity Number Operator

43. DC & CV Lab. CSIE NTU Yokoi Connectivity Number 4-connectivity : corner transition : corner all, no transition : center, neighbor, nothing will happen

44. DC & CV Lab. CSIE NTU Yokoi Connectivity Number 5: no transition all 8 neighbors 1, thus interior : 1 transition generates one connected component if center removed connectivity number

45. DC & CV Lab. CSIE NTU Yokoi Connectivity Number

46. DC & CV Lab. CSIE NTU Yokoi Connectivity Number

47. DC & CV Lab. CSIE NTU Yokoi Connectivity Number

48. DC & CV Lab. CSIE NTU Yokoi Connectivity Number lena.64*64

49. DC & CV Lab. CSIE NTU Yokoi Connectivity Number lena.yokoi

50. DC & CV Lab. CSIE NTU Yokoi Connectivity Number 8-connectivity, only slightly different : center and corner transition : corner no transition, all : itself and two 4-neighbors, corner

51. DC & CV Lab. CSIE NTU Yokoi Connectivity Number

52. DC & CV Lab. CSIE NTU Rutovitz Connectivity Number Rutovitz connectivity: number of transitions from one symbol to another Rutovitz connectivity number: sometimes called crossing number

53. DC & CV Lab. CSIE NTU Take a Break

54. DC & CV Lab. CSIE NTU 6.2.6 Connected Shrink Operator connected shrink: recursive operator, symbolic data domain connected shrink: deletes border pixels without disconnecting regions

55. DC & CV Lab. CSIE NTU 6.2.6 Connected Shrink Operator Top-down, left-right scan: deletes edge pixels not right-boundary

56. DC & CV Lab. CSIE NTU 6.2.6 Connected Shrink Operator : determines whether corner connected

57. DC & CV Lab. CSIE NTU 6.2.6 Connected Shrink Operator : background output symbol

58. DC & CV Lab. CSIE NTU 6.2.6 Connected Shrink Operator

59. DC & CV Lab. CSIE NTU 6.2.6 Connected Shrink Operator

60. DC & CV Lab. CSIE NTU 6.2.6 Connected Shrink Operator g g 11 1

61. DC & CV Lab. CSIE NTU 6.2.7 Pair Relationship Operator pair relationship: nonrecursive operator, symbolic data domain : determines whether first argument equals label

62. DC & CV Lab. CSIE NTU 6.2.7 Pair Relationship Operator 4-connected mode, output : not deletable if Yokoi number or no neighbor : possibly deletable if Yokoi number and some neighbor

63. DC & CV Lab. CSIE NTU 6.2.8 Thinning Operator thinning operator is composition of three operators: Yokoi connectivity, pair relationship, connected shrink

64. DC & CV Lab. CSIE NTU 6.2.8 Thinning Operator

65. DC & CV Lab. CSIE NTU 6.2.8 Thinning Operator lena.thin

66. DC & CV Lab. CSIE NTU 6.2.8 Thinning Operator lena.thin

67. DC & CV Lab. CSIE NTU Take a Break

68. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator distance transformation: produces distance to closest border pixel

69. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator nonrecursive, iterative, start with : background : border, because distance to border is : interior pixels

70. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator iteration: label with all with neighbor

71. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator equivalent algorithm: nonrecursive iterative : if no neighbor labeled, still interior, leave it alone : self interior but neighbor labeled : already labeled, won’t change value since later route farther 4-connected: output

72. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator distance transformation: produces distance to closest background recursive, two-pass

73. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator first pass: left-right, top-bottom : if background still background, since min: min of upper and left neighbors and add 4-connected output

74. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator second pass: right-left, bottom-up 4-connected output

75. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator

76. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator after pass 1

77. DC & CV Lab. CSIE NTU 6.2.9 Distance Transformation Operator

78. DC & CV Lab. CSIE NTU Take a Break

79. DC & CV Lab. CSIE NTU 6.2.10 Radius of Fusion The radius of fusion for any connected component of binary image is the radius of a disk satisfying the condition that if the binary image is morphologically closed with a disk of radius, then the given connected region will fuse with some other connected region

80. DC & CV Lab. CSIE NTU 6.2.10 Radius of Fusion

81. DC & CV Lab. CSIE NTU 6.2.11 Number of Shortest Paths number of shortest paths for each 0-pixel: to binary-1 pixel set given binary image : can be 4-neighborhood or 8-neighborhood : nonzero, stays unchanged since shortest paths counted : if zero then sum of neighboring shortest paths

82. DC & CV Lab. CSIE NTU 6.2.11 Number of Shortest Paths

83. DC & CV Lab. CSIE NTU 6.2.11 Number of Shortest Paths

84. DC & CV Lab. CSIE NTU 6.2.11 Number of Shortest Paths

85. DC & CV Lab. CSIE NTU Take a Break

86. DC & CV Lab. CSIE NTU 6.3 Extremum-Related Neighborhood Operators

87. DC & CV Lab. CSIE NTU 6.3.1 Non-Minima-Maxima Operator non-minima-maxima: nonrecursive operator, symbolic output

88. DC & CV Lab. CSIE NTU 6.3.1 Non-Minima-Maxima Operator a pixel can be neighborhood maximum not relative maximum

89. DC & CV Lab. CSIE NTU 6.3.1 Non-Minima-Maxima Operator

90. DC & CV Lab. CSIE NTU 6.3.1 Non-Minima-Maxima Operator 4-connected: output pixel

91. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator relative extrema operators: relative maximum and minimum operators relative extrema: recursive operator, numeric data domain relative extrema: input not changed output successively modified top-down, left-right scan, then bottom-up, right-left scan until no change

92. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator output value: of highest extrema reachable by monotonic path relative extrema pixels: output same as input pixels

93. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator pixel designations for the normal and reverse scans X0X1l0l1

94. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator

95. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator two primitive functions and : use new maximum value when ascending : keep original maximum when descending

96. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator 4-connected, output : top-down, left-right : bottom-up, right-left

97. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator 3 3,3,3,3 )

98. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator 5 5, 5, 5, 5)

99. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator

100. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator

101. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator

102. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator

103. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator

104. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator 6

105. DC & CV Lab. CSIE NTU 6.3.2 Relative Extrema Operator 6

106. DC & CV Lab. CSIE NTU 6.3.3 Reachability Operator successively propagating labels that can reach by monotonic paths descending reachability operator employs

107. DC & CV Lab. CSIE NTU 6.3.3 Reachability Operator g: pixels that are not relative extrema labeled background a: unchanged when neighbor is g or the same with neighbor b: become first propagated label if originally g c: common region more than one extremum can reach it by monotonic path c: already labeled and neighbor has different label then two extrema reach a: propagate from extremum

108. DC & CV Lab. CSIE NTU 6.3.3 Reachability Operator 4-connected, output

109. DC & CV Lab. CSIE NTU 6.3.3 Reachability Operator

110. DC & CV Lab. CSIE NTU 6.3.3 Reachability Operator

111. DC & CV Lab. CSIE NTU 6.3.3 Reachability Operator

112. DC & CV Lab. CSIE NTU Take a Break

113. DC & CV Lab. CSIE NTU 6.4 Linear Shift-Invariant Neighborhood Operators convolution: commutative, associative, distributor over sums, homogeneous

114. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation convolution of an image f with kernel w

115. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

116. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

117. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

118. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

119. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

120. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

121. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

122. DC & CV Lab. CSIE NTU 6.4.1 Convolution and Correlation

123. DC & CV Lab. CSIE NTU 6.4.2 Separability straightforward computation: multiplications, additions decomposition: multiplications, additions

124. DC & CV Lab. CSIE NTU 6.4.2 Separability

125. DC & CV Lab. CSIE NTU Project due Nov. 22 Write a program to generate Yokoi connectivity number

126. DC & CV Lab. CSIE NTU Project due Nov. 29 Write a program to generate thinned image

127. DC & CV Lab. CSIE NTU Midterm Nov. 23 extent: whatever covered up to Nov. 16 (inclusive)