1. Computer and Robot Vision I
Chapter 5 Mathematical Morphology
Presented by: 傅楸善 & 王彥鈞
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. 5.1 Introduction mathematical morphology works on shape
shape: prime carrier of information in machine vision
morphological operations: simplify image data,
preserve essential shape characteristics, eliminate
irrelevancies
shape: correlates directly with decomposition of
object, object features, object surface defects, assembly defects
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3. 5.2 Binary Morphology
set theory: language of binary mathematical morphology
sets in mathematical morphology: represent shapes
Euclidean N-space: EN
discrete Euclidean N-space: ZN
N=2: hexagonal grid, square grid
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4. 5.2 Binary Morphology (cont’)
dilation, erosion: primary morphological operations
opening, closing: composed from dilation, erosion
opening, closing: related to shape representation, decomposition, primitive extraction
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6. 5.2.1 Binary Dilation
dilation: combines two sets by vector addition of set elements
dilation of A by B:
addition commutative  dilation commutative:
binary dilation: Minkowski addition
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7. 5.2.1 Binary Dilation (cont’)
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8. 5.2.1 Binary Dilation (cont’)
A: referred as set, image
B: structuring element: kernel
dilation by disk: isotropic swelling or expansion
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9. 5.2.1 Binary Dilation (cont’)
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10. 5.2.1 Binary Dilation (cont’)
dilation by kernel without origin: might not have common pixels with A
translation of dilation: always can contain A
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11. 5.2.1 Binary Dilation (cont’)
=lena.bin.128=
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12. 5.2.1 Binary Dilation (cont’)
=lena.bin.dil=
By structuring
element :
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13. 5.2.1 Binary Dilation (cont’)
N4: set of four 4-neighbors of (0,0) but not (0,0)
4-isolated pixels removed
only points in I with at least one of its 4-neighbors remain
At: translation of A by the point t
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14. 5.2.1 Binary Dilation (cont’)
dilation: union of translates of kernel
addition associative dilation associative
associativity of dilation: chain rule: iterative rule
dilation of translated kernel: translation of dilation
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15. 5.2.1 Binary Dilation (cont’)
dilation distributes over union
dilating by union of two sets: the union of the dilation
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16. 5.2.1 Binary Dilation (cont’)
dilating A by kernel with origin guaranteed to contain A
extensive: operators whose output contains input dilation extensive when kernel contains origin
dilation preserves order
increasing: preserves order
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17. 5.2.1 Binary Dilation (cont’)
Reader’s Digest Oct.1994, p .73
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18. 5.2.2 Binary Erosion erosion: morphological dual of dilation
erosion of A by B: set of all x s.t.
erosion: shrink: reduce:
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19. 5.2.2 Binary Erosion (cont’)
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20. 5.2.2 Binary Erosion (cont’)
=lena.bin.128=
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21. 5.2.2 Binary Erosion (cont’)
=Lena.bin.ero=
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22. 5.2.2 Binary Erosion (cont’)
erosion of A by B: set of all x for which B translated to x contained in A
if B translated to x contained in A then x in A B
erosion: difference of elements a and b
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23. 5.2.2 Binary Erosion (cont’)
dilation: union of translates
erosion: intersection of negative translates
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24. 5.2.2 Binary Erosion (cont’)
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25. 5.2.2 Binary Erosion (cont’)
Minkowski subtraction: close relative to erosion
Minkowski subtraction:
erosion: shrinking of the original image
antiextensive: operated set contained in the original set
erosion antiextensive: if origin contained in kernel
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26. 5.2.2 Binary Erosion (cont’)
if then because
eroding A by kernel without origin can have nothing in common with A
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27. 5.2.2 Binary Erosion (cont’)
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28. 5.2.2 Binary Erosion (cont’)
dilating translated set results in a translated dilation
eroding by translated kernel results in negatively translated erosion
dilation, erosion: increasing
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29. 5.2.2 Binary Erosion (cont’)
eroding by larger kernel produces smaller result
Dilation, erosion similar that one does to foreground, the other to background
similarity: duality
dual: negation of one equals to the other on negated variables
DeMorgan’s law: duality between set union and intersection
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30. 5.2.2 Binary Erosion (cont’)
Joke
999555
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31. 5.2.2 Binary Erosion (cont’)
negation of a set: complement
negation of a set in two possible ways in morphology
logical sense: set complement
geometric sense: reflection: reversing of set orientation
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32. 5.2.2 Binary Erosion (cont’)
complement of erosion: dilation of the complement by reflection
Theorem 5.1: Erosion Dilation Duality
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33. 5.2.2 Binary Erosion (cont’)
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34. 5.2.2 Binary Erosion (cont’)
Corollary 5.1:
erosion of intersection of two sets: intersection of erosions
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35. 5.2.2 Binary Erosion (cont’)
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36. 5.2.2 Binary Erosion (cont’)
erosion of a kernel of union of two sets: intersection of erosions
erosion of kernel of intersection of two sets: contains union of erosions
no stronger
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37. DC & CV Lab.
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38. 5.2.2 Binary Erosion (cont’)
chain rule for erosion holds when kernel decomposable through dilation
duality does not imply cancellation on morphological equalities
containment relationship holds
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39. 5.2.2 Binary Erosion (cont’)
genus g(I): number of connected components minus number of holes of I, 4-connected for object, 8-connected for background
8-connected for object, 4-connected for background
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40. 5.2.2 Binary Erosion (cont’)
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41. 5.2.2 Binary Erosion (cont’)
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42. Joke
Frisvr
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43. 5.2.3 Hit-and-Miss Transform
hit-and-miss: selects corner points, isolated points, border points
hit-and-miss: performs template matching, thinning, thickening, centering
hit-and-miss: intersection of erosions
J,K kernels satisfy
hit-and-miss of set A by (J,K)
hit-and-miss: to find upper right-hand corner
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44. 5.2.3 Hit-and-Miss Transform (cont’)
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45. DC & CV Lab.
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46. 5.2.3 Hit-and-Miss Transform (cont’)
J locates all pixels with south, west neighbors part of A
K locates all pixels of Ac that have south, west neighbors in Ac
J and K displaced from one another
Hit-and-miss: locate particular spatial patterns
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47. 5.2.3 Hit-and-Miss Transform (cont’)
hit-and-miss: to compute genus of a binary image
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48. 5.2.3 Hit-and-Miss Transform (cont’)
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49. 5.2.3 Hit-and-Miss Transform (cont’)
hit-and-miss: thickening and thinning
hit-and-miss: counting
hit-and-miss: template matching
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50. The_lion_sleeps_tonight
Joke
The_lion_sleeps_tonight
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51. 5.2.4 Dilation and Erosion Summary
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52. 5.2.4 Dilation and Erosion Summary (cont’)
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53. 5.2.5 Opening and Closing
dilation and erosions: usually employed in pairs
B K: opening of image B by kernel K
B K closing of image B by kernel K
B open under K: B open w.r.t. K: B= B K
B closed under K: B closed w.r.t. K: B= B K
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54. 5.2.5 Opening and Closing (cont’)
=lena.bin.128=
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55. 5.2.5 Opening and Closing (cont’)
=lena.bin.open=
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56. 5.2.5 Opening and Closing (cont’)
morphological opening, closing: no relation to topologically open, closed sets
opening characterization theorem
A K: selects points covered by some translation of K, entirely contained in A
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57. 5.2.5 Opening and Closing (cont’)
opening with disk kernel: smoothes contours, breaks narrow isthmuses
opening with disk kernel: eliminates small islands, sharp peaks, capes
closing by disk kernel; smoothes contours, fuses narrow breaks, long, thin gulfs
closing with disk kernel: eliminates small holes, fill gaps on the contours
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58. 5.2.5 Opening and Closing (cont’)
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59. 5.2.5 Opening and Closing (cont’)
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60. 5.2.5 Opening and Closing (cont’)
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61. 5.2.5 Opening and Closing (cont’)
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62. 5.2.5 Opening and Closing (cont’)
=lena.bin.128=
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63. 5.2.5 Opening and Closing (cont’)
=lena.bin.close=
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64. 5.2.5 Opening and Closing (cont’)
unlike erosion and dilation: opening invariant to kernel translation
opening antiextensive
like erosion and dilation: opening increasing
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65. 5.2.5 Opening and Closing (cont’)
A K: those pixels covered by sweeping kernel all over inside of A
F: shape with body and handle
L: small disk structuring element with radius just larger than handle width extraction of the body and handle by opening and taking the residue
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66. 5.2.5 Opening and Closing (cont’)
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67. 5.2.5 Opening and Closing (cont’)
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68. joke
無間道之我想補考
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69. 5.2.5 Opening and Closing (cont’)
extraction of trunk and arms with vertical and horizontal kernels
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70. DC & CV Lab.
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71. 5.2.5 Opening and Closing (cont’)
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72. 5.2.5 Opening and Closing (cont’)
extraction of base trunk horizontal and vertical areas
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73. 5.2.5 Opening and Closing (cont’)
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74. 5.2.5 Opening and Closing (cont’)
noisy background line segment removal
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75. DC & CV Lab.
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76. 5.2.5 Opening and Closing (cont’)
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77. DC & CV Lab.
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78. 5.2.5 Opening and Closing (cont’)
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79. 5.2.5 Opening and Closing (cont’)
decomposition into parts
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80. 5.2.5 Opening and Closing (cont’)
closing: dual of opening
like opening: closing invariant to kernel translation
closing extensive
like dilation, erosion, opening: closing increasing
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81. 5.2.5 Opening and Closing (cont’)
opening idempotent
closing idempotent
if L K not necessarily follows that
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82. 5.2.5 Opening and Closing (cont’)
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83. 5.2.5 Opening and Closing (cont’)
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84. 5.2.5 Opening and Closing (cont’)
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85. 5.2.5 Opening and Closing (cont’)
closing may be used to detect spatial clusters of points
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86. Joke
超Q雙胞胎
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87. 5.2.6 Morphological Shape Feature Extraction
morphological pattern spectrum: shape-size histogram [Maragos 1987]
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88. 5.27 Fast Dilations and Erosions
decompose kernels to make dilations and erosions fast
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89. Joke
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90. 5.3 Connectivity morphology and connectivity: close relation
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91. 5.3.1 Separation Relation
S separation if and only if S symmetric, exclusive, hereditary, extensive
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92. 5.3.2 Morphological Noise Cleaning and Connectivity
images perturbed by noise can be morphologically filtered to remove some noise
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93. 5.3.3 Openings Holes and Connectivity
opening can create holes in a connected set that is being opened
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94. 5.3.4 Conditional Dilation
select connected components of image that have nonempty erosion conditional dilation J ,
defined iteratively J0 = J
J are points in the regions we want to select
conditional dilation J =Jm
where m is the smallest index Jm=Jm-1
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95. DC & CV Lab.
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96. 5.4 Generalized Openings and Closings
generalized opening: any increasing, antiextensive, idempotent operation
generalized closing: any increasing. extensive, idempotent operation
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97. Joke
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98. 5.5 Gray Scale Morphology
binary dilation, erosion, opening, closing naturally extended to gray scale
extension: uses min or max operation
gray scale dilation: surface of dilation of umbra
gray scale dilation: maximum and a set of addition operations
gray scale erosion: minimum and a set of subtraction operations
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99. 5.5.1Gray Scale Dilation and Erosion
top: top surface of A: denoted by
umbra of f: denoted by
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100. 5.5.1Gray Scale Dilation and Erosion (cont’)
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101. 5.5.1Gray Scale Dilation and Erosion (cont’)
gray scale dilation: surface of dilation of umbras
dilation of f by k: denoted by
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102. 5.5.1Gray Scale Dilation and Erosion (cont’)
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103. 5.5.1Gray Scale Dilation and Erosion (cont’)
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104. 5.5.1Gray Scale Dilation and Erosion (cont’)
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105. 5.5.1Gray Scale Dilation and Erosion (cont’)
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106. 5.5.1Gray Scale Dilation and Erosion (cont’)
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107. DC & CV Lab.
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108. 5.5.1Gray Scale Dilation and Erosion (cont’)
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109. 5.5.1Gray Scale Dilation and Erosion (cont’)
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110. Joke
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111. 5.5.1Gray Scale Dilation and Erosion (cont’)
gray scale erosion: surface of binary erosions of one umbra by the other umbra
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112. 5.5.1Gray Scale Dilation and Erosion (cont’)
=lena.im=
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113. 5.5.1Gray Scale Dilation and Erosion (cont’)
=lena.im.dil=
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114. 5.5.1Gray Scale Dilation and Erosion (cont’)
Structuring Elements:
Value=0 *
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115. 5.5.1Gray Scale Dilation and Erosion (cont’)
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116. 5.5.1Gray Scale Dilation and Erosion (cont’)
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117. 5.5.1Gray Scale Dilation and Erosion (cont’)
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118. 5.5.1Gray Scale Dilation and Erosion (cont’)
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119. 5.5.1Gray Scale Dilation and Erosion (cont’)
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120. 5.5.1Gray Scale Dilation and Erosion (cont’)
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121. 5.5.1Gray Scale Dilation and Erosion (cont’)
=lena.im.ero=
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122. 5.5.1Gray Scale Dilation and Erosion (cont’)
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123. 5.5.1Gray Scale Dilation and Erosion (cont’)
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124. 5.5.2 Umbra Homomorphism Theorems
surface and umbra operations: inverses of each other, in a certain sense
surface operation: left inverse of umbra operation
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125. 5.5.2 Umbra Homomorphism Theorems
Proposition 5.1
Proposition 5.2
Proposition 5.3
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126. 5.5.3 Gray Scale Opening and Closing
gray scale opening of f by kernel k denoted by f k
gray scale closing of f by kernel k denoted by f k
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127. 5.5.3 Gray Scale Opening and Closing (cont’)
=lena.im.open=
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128. 5.5.3 Gray Scale Opening and Closing (cont’)
=lena.im.close=
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129. 5.5.3 Gray Scale Opening and Closing (cont’)
duality of gray scale, dilation erosion duality of opening, closing
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130. 5.5.3 Gray Scale Opening and Closing (cont’)
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131. joke
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132. 5.6 Openings Closings and Medians
median filter: most common nonlinear noise-smoothing filter
median filter: for each pixel, the new value is the median of a window
median filter: robust to outlier pixel values leaves, edges sharp
median root images: images remain unchanged after median filter
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133. 5.7 Bounding Second Derivatives
opening or closing a gray scale image simplifies the image complexity
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134. 5.8 Distance Transform and Recursive Morphology
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135. 5.8 Distance Transform and Recursive Morphology (cont’)
Fig 5.39 (b) fire burns from outside but burns only downward and right-ward
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136. 5.9 Generalized Distance Transform
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137. 5.10 Medial Axis
medial axis transform medial axis with distance function
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138. 5.10.1 Medial Axis and Morphological Skeleton
morphological skeleton of a set A by kernel K ,where
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139. 5.10.1 Medial Axis and Morphological Skeleton (cont’)
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140. 5.10.1 Medial Axis and Morphological Skeleton (cont’)
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141. 5.10.1 Medial Axis and Morphological Skeleton (cont’)
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142. 5.11 Morphological Sampling Theorem
Before sets are sampled for morphological processing, they must be morphologically simplified by an opening or a closing .
Such sampled sets can be reconstructed in two ways: by either a closing or a dilation.
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143. =========== Joke===========
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144. 5.12 Summary
morphological operations: shape extraction, noise cleaning, thickening
morphological operations: thinning, skeletonizing
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145. Homework
Write programs which do binary morphological dilation, erosion, opening, closing, and hit-and-miss transform on a binary image (kernel r=2) (Due Nov. 7)
Write programs which do gray scale morphological dilation, erosion, opening, and closing on a gray scale image (Due Nov. 21)
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