1. Computer and Robot Vision I
Chapter 6
Neighborhood Operators
Presented by: 林德垣
指導教授: 傅楸善 博士
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 neighborhood operator: workhorse of low- level vision
neighborhood operator: performs conditioning, labeling, grouping
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3. 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
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4. 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
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5. 6.1 Introduction neighborhood might be small and asymmetric or large
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6. 6.1 Introduction : set of neighboring pixel positions around position
general nonrecursive neighborhood operator : input , output
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7. 6.1 Introduction
linear operator: one common nonrecursive neighborhood operator
output: possibly position-dependent linear combination of inputs
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8. 6.1 Introduction
shift-invariant: position invariant: action same regardless of position
composition of shift-invariant operators: shift-invariant
[e.g.] N satisfies a Shift invariance:
(r’, c’) ∈ N(r,c)
(r’- u, c’-v) ∈ N(r-u, c-v) , for all (u,v)
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9. 6.1 Introduction cross-correlation of with
: weight function: kernel or mask of weights
: domain of
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10. 6.1 Introduction common masks for noise cleaning, (a) box filter
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11. 6.1 Introduction common masked for noise cleaning DC & CV Lab.
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12. 6.1 Introduction Different N x N filter effects comparison
Original Image
3 x 3
5 x 5
9 x 9
15 x 15
35 x 35
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13. 6.1 Introduction application of mask with weights to image 7.69
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14. 6.1 Introduction convolution of with
convolution: close relative to cross-correlation
convolution: linear shift-invariant
if mask symmetric, convolution, and correlation the same
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15. 6.2 Symbolic Neighborhood Operators
indexing of neighborhoods
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16. 6.2.1 Region-Growing Operator
: projection, outputs first or second argument
: background
background pixel labeled first nonbackground label
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17. 6.2.1 Region-Growing Operator
4-connected:
output:
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18. 6.2.1 Region-Growing Operator
8-connected:
output:
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19. 6.2.1 Region-Growing Operator
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20. 6.2.1 Region-Growing Operator
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21. 6.2.1 Region-Growing Operator
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22. 6.2.1 Region-Growing Operator
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23. 6.2.1 Region-Growing Operator
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24. 6.2.2 Nearest Neighbor Sets and Influence Zones
influence zones: nearest neighbor sets
influence zones: iteratively region-growing
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25. 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
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26. Take a Break
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27. 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
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28. 6.2.3 Region-Shrinking Operator
: whether or not arguments identical
: background
border: has different neighbor and becomes background
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29. 6.2.3 Region-Shrinking Operator
4-connected:
output:
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30. 6.2.3 Region-Shrinking Operator
8-connected:
output:
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31. 6.2.3 Region-Shrinking Operator
region shrinking: related to binary erosion except on labeled region
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32. 6.2.3 Region-Shrinking Operator
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33. 6.2.3 Region-Shrinking Operator
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34. 6.2.3 Region-Shrinking Operator
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35. 6.2.3 Region-Shrinking Operator
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36. 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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37. 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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38. 6.2.4 Mark-Interior/Border-Pixel Operator
4-connected:
output:
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39. 6.2.4 Mark-Interior/Border-Pixel Operator
8-connected:
output:
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40. 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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41. 6.2.5 Connectivity Number Operator
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42. 6.2.5 Connectivity Number Operator
corner neighborhood
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43. 6.2.5 Connectivity Number Operator
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44. Yokoi Connectivity Number
: corner transition
: corner all , no transition
: center , neighbor , nothing will happen
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45. Yokoi Connectivity Number
5: no transition all 8 neighbors 1, thus interior
: 1 transition generates one connected component if center removed
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46. Yokoi Connectivity Number
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47. Yokoi Connectivity Numberd b c e d b c
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48. Yokoi Connectivity Numberd b c e d b c
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49. Yokoi Connectivity Number
lena.64*64
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50. Yokoi Connectivity Number
lena.yokoi
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51. Yokoi Connectivity Number
8-connectivity, only slightly different
: center and corner transition
: corner no transition, all
: itself and two 4-neighbors , corner
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52. Yokoi Connectivity Number
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53. Rutovitz Connectivity Number
Rutovitz connectivity: number of transitions from one symbol to another
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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54. Take a Break
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55. 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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56. 6.2.6 Connected Shrink Operator
Top-down, left-right scan: deletes edge pixels not right-boundary
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57. 這邊是示意圖，後面有計算範例
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58. 6.2.6 Connected Shrink Operator
: determines whether corner connected
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59. 6.2.6 Connected Shrink Operator
: background
output symbol
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60. 6.2.6 Connected Shrink Operator
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61. 6.2.6 Connected Shrink Operator1
h (b, c, d, e) = { 1, if c= b and (d <> b or e<>b)
0, otherwise }
Output symbol
y = f (1 , 0 , 0 , 0 , 1) = g
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62. 6.2.6 Connected Shrink Operatorg 1
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63. 6.2.7 Pair Relationship Operator
pair relationship: nonrecursive operator, symbolic data domain
: determines whether first argument equals label
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64. 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
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65. 6.2.8 Thinning Operator
thinning operator is composition of three operators:
Mark-Interior/Border-Pixel
The pair relationship
Connected shrink
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66. 6.2.8 Thinning Operator
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67. 6.2.8 Thinning Operator
lena.thin
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68. 6.2.8 Thinning Operator
lena.thin
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69. Take a Break
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70. 6.2.9 Distance Transformation Operator
distance transformation: produces distance to closest border pixel
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71. 6.2.9 Distance Transformation Operator
nonrecursive, iterative, start with
: background
: border, because distance to border is
: interior pixels
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72. 6.2.9 Distance Transformation Operator
iteration: label with all with neighbor
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73. 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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74. 6.2.9 Distance Transformation Operator
distance transformation: produces distance to closest background
recursive, two-pass
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75. 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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76. 6.2.9 Distance Transformation Operator
second pass: right-left, bottom-up
4-connected output
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77. 6.2.9 Distance Transformation Operator
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78. 6.2.9 Distance Transformation Operator
after pass 1
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79. 6.2.9 Distance Transformation Operator
pass 1
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80. Take a Break
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81. Radius of Fusion
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82. Radius of Fusion
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83. 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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84. 6.2.11 Number of Shortest Paths
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85. 6.2.11 Number of Shortest Paths
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86. 6.2.11 Number of Shortest Paths
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87. Take a Break
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88. 6.3 Extremum-Related Neighborhood Operators
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89. 6.3.1 Non-Minima-Maxima Operator
non-minima-maxima: nonrecursive operator, symbolic output
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90. 6.3.1 Non-Minima-Maxima Operator
a pixel can be neighborhood maximum not relative maximum
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91. 6.3.1 Non-Minima-Maxima Operator
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92. 6.3.1 Non-Minima-Maxima Operator
4-connected:
output pixel
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93. 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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94. 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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95. 6.3.2 Relative Extrema Operator
pixel designations for the normal and reverse scans
P285 課本的ab圖有錯
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96. 6.3.2 Relative Extrema Operator
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97. 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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98. 6.3.2 Relative Extrema Operator
4-connected, output
: top-down , left-right
: bottom-up , right-left
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99. 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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100. 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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101. 6.3.2 Relative Extrema Operator
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102. 6.3.2 Relative Extrema Operator
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103. 6.3.2 Relative Extrema Operator
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104. 6.3.2 Relative Extrema Operator
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105. 6.3.2 Relative Extrema Operator4
重複以上動作，一直到最後，把每一個Pixel都走過一遍，至此，完成了Left-Right, Top-Down Scan
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106. 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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107. 6.3.2 Relative Extrema Operator
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108. 6.3.3 Reachability Operator
successively propagating labels that can reach by monotonic paths descending reachability operator employs
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109. 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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110. 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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111. 6.3.3 Reachability Operator
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112. 6.3.3 Reachability Operator
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113. 6.3.3 Reachability Operator
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114. Take a Break
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115. 6.4 Linear Shift-Invariant Neighborhood Operators
convolution: commutative, associative, distributor over sums, homogeneous
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116. 6.4.1 Convolution and Correlation
convolution of an image f with kernel w
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117. 6.4.1 Convolution and Correlation
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118. 6.4.1 Convolution and Correlation
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119. 6.4.1 Convolution and Correlation
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120. 6.4.1 Convolution and Correlation
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121. 6.4.1 Convolution and Correlation
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122. 6.4.1 Convolution and Correlation
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123. 6.4.1 Convolution and Correlation
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124. 6.4.1 Convolution and Correlation
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125. 6.4.2 Separability straightforward computation:
multiplications, additions
decomposition:
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126. 6.4.2 Separability
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127. 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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128. 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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129. Midterm Nov. 13 extent: whatever covered up to Nov. 6 (inclusive)
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