1. Chapter 6 Neighborhood Operators
Computer Vision
Chapter 6
Neighborhood Operators
Presented by: 黃士展
傅楸善 教授
Digital Camera and Computer Vision Laboratory
Department of Computer Science and Information Engineering
National Taiwan University, Taipei, Taiwan, R.O.C.

2. 6.1 Introduction Visit each point p in the image data and do {
N = a neighborhood or region of the image data
around the point p
result(p) = f(N) }
After applying Neighborhood Operation
Same dimension
Might be interpreted as image
Element type based on the functions
RAW image
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3. 6.1 Introduction Domain type: Numeric, Symbolic.
Neighbor type: 4-neighbor, 8-neighbor, etc.
Recursive type: Rec, Non-Rec.
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4. 6.1 Introduction: Domain Type
Numeric domain: arithmetic operations, +, -, min, max
Symbolic domain: Boolean operations, AND, OR, NOT, table-look-up
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5. 6.1 Introduction: Neighbor Type
Neighborhood might be asymmetric
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6. 6.1 Introduction: Neighbor Type
(X ± 1, Y) and (X, Y ± 1)
(X ± 1 , Y ± 1)
(X or X ± 1, Y or Y ± 1) – (X, Y)
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7. 6.1 Introduction: Recursive Type
Non-recursive neighborhood operators: output is function of input (Parallel)
Recursive neighborhood operators: output depends partly on previous output (Sequential)
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8. 6.1 Introduction common masks for noise cleaning
 Box filter = unweighted filter
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9. 6.1 Introduction common masked for noise cleaning DC & CV Lab.
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10. 6.1 Introduction Different N x N box filter effects comparison
Original Image
3 x 3
5 x 5
9 x 9
15 x 15
35 x 35
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11. 6.1 Introduction
7.69
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12. 6.2 Symbolic Neighborhood Operators
indexing of neighborhoods
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13. 6.2.0 Dilation and Erosion (Revisit)
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14. 6.2.0 Dilation
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17. DC & CV Lab.
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18. 6.2.0 Erosion
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19. 6.2.0 Erosion
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20. 6.2.0 Erosion
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21. DC & CV Lab.
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22. 6.2.0 Opening & Closing
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23. 6.2.0 Opening
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24. 6.2.0 Opening
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25. 6.2.0 Closing
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26. 6.2.0 Closing
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27. 6.2.0 Application
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28. Opening
Closing

29. 6.2.1 Region Segmentation Region Segmentation:
Image into many regions
Connected, Similarity
Grayscale, Color, Texture, etc.
Spatial Continuity
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30. 6.2.1 Region Segmentation 整張影像的每一像素都必須被分區 每一區塊必須是一個完整的連結區塊
沒有任一像素可以被分類到兩不同區塊
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31. 6.2.1 Region Segmentation 同一區塊內的像素都具備相似的特性 不同特性的像素一定分到不同的區塊
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32. 6.2.1 Region-Growing Operator
: projection, outputs first or second argument
: background
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33. 6.2.1 Region-Growing Operator
4-connected:
output:
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34. 6.2.1 Region-Growing Operator
8-connected:
output:
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35. 6.2.1 Region-Growing Operatorx7 x2 X6 X3 x0 X1 X8 X4 x5
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36. 6.2.1 Region-Growing Operator
Step1. Initial Seed Sets -> New Region
Step2. Neighboring pixels similarity.
Step3. Pixels remaining -> Step1, or stop
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37. 6.2.1 Region-Growing Operator
Seed sets choice is based on user criterion, thus image info is important
Grayscale -> Histogram
Texture -> Orientation
And so on
Keep examining another pixels with seed points, classify them.
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38. 6.2.1 Region-Growing Operator
Some important issues :
The suitable selection of seed points is important
More information of the image is better
The value, “minimum area threshold”
The value, “similarity threshold value“
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39. 6.2.1 Region-Growing Operator
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40. 6.2.1 Region-Growing Operator
Grayscale lightning image. The grayscale value of this image is from 0 to 255
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41. 6.2.1 Region-Growing Operator
All the points with grayscale 255 are considered as seed points
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42. Threshold: (a) >225 (b) >190 (c) >155
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43. 6.2.1 Region-Growing Operator
Advantages：
Can provide the original images which have clear edges with good segmentation results.
Only need a small number of seed points to grow the region we want.
We can determine the seed points and the criteria we want to make.
We can choose the multiple criteria at the same time.
It performs well with respect to noise.
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44. 6.2.1 Region-Growing Operator
The differences between Dilation and Region-Growing are ?
Dilation: Initial Area, Structure elements ,Binary Image
Region-Growing: Seeds, criterion ,Labeled Regions
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45. Take a Break
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46. 6.2.2 Nearest Neighbor Sets and Influence Zones
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47. 6.2.3 Region-Shrinking Operator
Can change all border pixels to background
Can change connectivity
Can entirely delete region if repeatedly applied
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48. 6.2.3 Region-Shrinking Operator
: whether or not arguments identical
: background
border: has different neighbor and becomes background
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49. 6.2.3 Region-Shrinking Operator
4-connected:
output:
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50. 6.2.3 Region-Shrinking Operator
8-connected:
output:
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51. 6.2.3 Region-Shrinking Operator
region shrinking: related to binary erosion except on labeled region
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52. 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
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53. 6.2.4 Mark-Interior/Border-Pixel Operator
: whether or not arguments identical
: recognizes whether or not its argument is symbol
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54. 6.2.4 Mark-Interior/Border-Pixel Operator
4-connected:
output:
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55. 6.2.4 Mark-Interior/Border-Pixel Operator
8-connected:
output:
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56. 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
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57. 6.2.5 Connectivity Number Operator
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58. 6.2.5 Connectivity Number Operator
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59. 6.2.5 Connectivity Number Operator
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60. Yokoi Connectivity Number
4- Connectivity number (= y)
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61. Yokoi Connectivity Number
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62. Yokoi Connectivity Number
5: all corners are consistent, interior
: same as neighbor, but corner
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63. Yokoi Connectivity Number
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64. Yokoi Connectivity Number
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65. Yokoi Connectivity Number
lena.64*64
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66. Yokoi Connectivity Number
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67. Yokoi Connectivity Number
8-connectivity, only slightly different
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68. Yokoi Connectivity Number
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69. Rutovitz Connectivity Number
Rutovitz connectivity number: sometimes called crossing number
h(a,b,c) = { 1, if (a=b and a != c) or (a != b and a =c)
0, otherwise }
a1 = h( X0, X1, X6)
a2 = h( X0, X6, X2)
a3 = h( X0, X2, X7)
a4 = h( X0, X7, X3)
a5 = h( X0, X3, X8)
a6 = h( X0, X8, X4)
a7 = h( X0, X4, X5)
a8 = h( X0, X5, X1)
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70. Take a Break
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71. 6.2.6 Connected Shrink Operator
connected shrink: recursive operator, symbolic data domain
connected shrink: deletes border pixels without disconnecting regions
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72. 6.2.6 Connected Shrink Operator
Top-down, left-right scan: deletes edge pixels not right-boundary
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73. DC & CV Lab.
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74. 6.2.6 Connected Shrink Operator
: determines whether corner connected
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75. 6.2.6 Connected Shrink Operator
: background
output symbol
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76. 6.2.6 Connected Shrink Operator
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77. 6.2.6 Connected Shrink Operator
h (b, c, d, e) = { 1, if b = c and (d != b or e != b)
0, otherwise }
Output symbol
y = f (1 , 0 , 0 , 0 , 1) = g
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78. 6.2.6 Connected Shrink Operatorg 1
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79. 6.2.7 Skeleton / Medial Axis (Revisit)
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80. 6.2.7 Skeleton / Medial Axis
Foreground regions are made of some uniform slow-burning material
Light fires simultaneously at all points along the boundary of this region
Watch the fire move into the interior
Fire meets -> extinguished
The so called `quench line‘ which is the skeleton arises. 
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81. 6.2.7 Skeleton / Medial Axis
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82. 6.2.7 Skeleton / Medial Axis
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83. 6.2.7 Skeleton / Medial Axis
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84. 6.2.7 Skeleton / Medial Axis Take this image for example DC & CV Lab.
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85. 6.2.8 Thinning Operator
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86. 6.2.8 Thinning Operator Goal :
Thin the regions in binary image into a one-pixel-width line figure.
Thin != Skeleton
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87. 6.2.8 Thinning Operator The 4 constraints on thinning operator:
The subtle edge-end must be reserved
Connected part couldn’t be broken
Connected areas must lead to continuous line
Thinned graph must be similar with the medial axis
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88. 6.2.8 Thinning Operator Algorithm:
It can be separated into 2 parts A and B. Also, we use 8-neighbor here.
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89. 6.2.8 Thinning Operator
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90. 6.2.8 Thinning Operator Goal of part A:
Eliminate the up-left corner, right, bottom pixels.
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91. Fig. After applying algo part A to original input image
6.2.8 Thinning Operator
Fig. After applying algo part A to original input image
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92. 6.2.8 Thinning Operator
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93. 6.2.8 Thinning Operator Goal of part B:
Eliminate the bottom-right corner, left, top pixels.
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94. Fig. After applying algo part B to previously output image
6.2.8 Thinning Operator
Fig. After applying algo part B to previously output image
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95. Fig. Apply thinning algorithm once again
6.2.8 Thinning Operator
Fig. Apply thinning algorithm once again
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96. 6.2.8 Thinning Operator
During the thinning algo, some rules should be satisfied:
Connected regions are thinned to 8-connected paths.
Thinning results be as close to skeleton as possible.
Extra spurs (short branches) caused by thinning must be minimized.
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97. 6.2.8 Thinning Operator
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98. 6.2.8 Thinning Operator
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99. Take a Break
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100. 6.2.9 Distance Transformation Operator
distance transformation: produces distance to closest border pixel
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101. 6.2.9 Distance Transformation Operator
nonrecursive, iterative, start with
: background
: border, because distance to border is
: interior pixels
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102. 6.2.9 Distance Transformation Operator
iteration: label with all with neighbor
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103. 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
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104. 6.2.9 Distance Transformation Operator
distance transformation: produces distance to closest background
recursive, two-pass
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105. 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
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106. 6.2.9 Distance Transformation Operator
second pass: right-left, bottom-up
4-connected output
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107. 6.2.9 Distance Transformation Operator
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108. 6.2.9 Distance Transformation Operator
after pass 1
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109. 6.2.9 Distance Transformation Operator
pass 1
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110. Take a Break
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111. Radius of Fusion
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112. Radius of Fusion
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113. 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
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114. 6.2.11 Number of Shortest Paths
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115. 6.2.11 Number of Shortest Paths
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116. 6.2.11 Number of Shortest Paths
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117. Take a Break
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118. 6.3 Extremum-Related Neighborhood Operators
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119. 6.3.1 Non-Minima-Maxima Operator
non-minima-maxima: nonrecursive operator, symbolic output
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120. 6.3.1 Non-Minima-Maxima Operator
a pixel can be neighborhood maximum not relative maximum
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121. 6.3.1 Non-Minima-Maxima Operator
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122. 6.3.1 Non-Minima-Maxima Operator
4-connected:
output pixel
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123. 6.3.2 Relative Extrema Operator
relative extrema operators
relative maximum and minimum operators
relative extrema
recursive operator, numeric data domain
input not changed；output modified
top-down, left-right scan
then bottom-up, right-left scan
until no change
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124. 6.3.2 Relative Extrema Operator
output value: of highest extrema reachable by monotonic path
relative extrema pixels: output same as input pixels
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125. 6.3.2 Relative Extrema Operator
pixel designations for the normal and reverse scans
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126. 6.3.2 Relative Extrema Operator
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127. 6.3.2 Relative Extrema Operator
two primitive functions and
: use new maximum value when ascending
: keep original maximum when descending
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128. 6.3.2 Relative Extrema Operator
4-connected, output
: top-down , left-right
: bottom-up , right-left
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129. 6.3.2 Relative Extrema Operator
a0 = l0
a1 = h (x0, x1, a0, l1)
a2 = h (x0, x2, a1, l2) 3
3,3,3,3 )
x2 , l2
x1 , l1
a0 l0, x0
x2 , l2
x1 , l1
a0 l0, x0
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130. 6.3.2 Relative Extrema Operator
a0 = l0
a1 = h (x0, x1, a0, l1)
a2 = h (x0, x2, a1, l2) 5
5, 5, 5, 5)
x2 , l2
x1 , l1
a0 l0, x0
x2 , l2
x1 , l1
a0 l0, x0
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131. 6.3.2 Relative Extrema Operator
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132. 6.3.2 Relative Extrema Operator
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133. 6.3.2 Relative Extrema Operator
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134. 6.3.2 Relative Extrema Operator
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135. 6.3.2 Relative Extrema Operator4
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136. 6.3.2 Relative Extrema Operator
a0 = l0
a1 = h (x0, x1, a0, l1)
a2 = h (x0, x2, a1, l2) 6 x2 1 9 1 9 x1 l1
a0 l0 x0 l2 x0 2 x2 x1 l2
a0 l0
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137. 6.3.2 Relative Extrema Operator
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138. 6.3.3 Reachability Operator
successively propagating labels that can reach by monotonic paths descending reachability operator employs
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139. 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
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140. 6.3.3 Reachability Operator
4-connected, output
a0 = l0
a1 = h (a0 , l1 , x0 , x1)
a2 = h (a1 , l2 , x0 , x2)
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141. 6.3.3 Reachability Operator
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142. 6.3.3 Reachability Operator
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143. 6.3.3 Reachability Operator
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144. Take a Break
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145. 6.4 Linear Shift-Invariant Neighborhood Operators
convolution: commutative, associative, distributor over sums, homogeneous
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146. 6.4.1 Convolution and Correlation
convolution of an image f with kernel w
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147. 6.4.1 Convolution and Correlation
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148. 6.4.1 Convolution and Correlation
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149. 6.4.1 Convolution and Correlation
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150. 6.4.1 Convolution and Correlation
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151. 6.4.1 Convolution and Correlation
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152. 6.4.1 Convolution and Correlation
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153. 6.4.1 Convolution and Correlation
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154. 6.4.1 Convolution and Correlation
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155. 6.4.2 Separability straightforward computation:
multiplications, additions
decomposition:
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156. 6.4.2 Separability
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157. Project due Nov. 6
Write a program to generate Yokoi connectivity number
You can down sample lena.bmp from 512*512 to 64*64 first.
Sample pixels positions at (r, c) = (0,0), (0,8) ……, so everyone will get the same answer .
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158. Project due Nov. 20 Write a program to generate thinned image.
You can down sample lena.bmp from 512*512 to 64*64 first.
Sample pixels positions at (r, c) = (0,0), (0,8) ……, so everyone will get the same answer .
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159. Midterm Nov. 13 extent: whatever covered up to Nov. 6 (inclusive)
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