CMT107 Visual Computing
Image Morphology
Copyright By PowCoder代写 加微信 powcoder
School of Computer Science and Informatics
Cardiff University
• Morphology
• Dilation
• Duality of Dilation and Erosion
• Hit-or-Miss transformation
Acknowledgement
The majority of the slides in this section are from Saha at University of Iowa
Morphology
• Morphological operators often take a binary image and a
structuring element as input and combine them using a
set operator (intersection, union, inclusion, complement).
• The structuring element is shifted over the image. At each
pixel of the image, its elements are compared with the
set of the underlying pixels.
• If the two sets match the condition defined by the set
operator (e.g., if the set of pixels in the structuring
element is a subset of the underlying image pixels), the
pixel underneath the origin of the structuring element is
set to a predefined value (0 or 1 for binary images).
• A morphological operator is defined by its structuring
element and the applied set operator.
Morphology Applications
• Image pre-processing
• Noise filtering
• shape simplification
• Enhancing object structures
• Skeletonisation
• Thinning
• Convex hull
• Object marking
• Segmentation of the object from background
• Quantitative descriptors of objects
• Perimeter
Example: Morphological Operation
• Let ⊕ denote a morphological operator
𝑋⊕𝐵 = 𝑝 ∈ 𝑍2 𝑝 = 𝑥 + 𝑏, 𝑥 ∈ 𝑋, 𝑏 ∈ 𝐵}
𝐵 = 0,0 , 1,0
𝑋 = 0,0 , 1,0 , 1,1 , 1,2 , 2,2 , 0,3 , (0,4)
𝑋⊕𝐵 = { 0,0 , 1,0 , 2,0 , 1,1 , 2,1 , 1,2 , 2,2 ,
3,2 , 0,3 , 1,3 , 0,4 , (1,4)}
0,0 , (1,0) 1,1 , (2,1) 2,2 , (3,2) 0,4 , (1,4)
1,0 , (2,0) 1,2 , (2,2) 0,3 , (1,3)
• Morphological dilation ‘⊕’ combines two sets using vector addition of set
𝑋⊕𝐵 = 𝑝 ∈ 𝑍2 𝑝 = 𝑥 + 𝑏, 𝑥 ∈ 𝑋, 𝑏 ∈ 𝐵}
• Commutative:
• Associative:
𝑋⊕𝐵⊕𝐷 = 𝑋⊕(𝐵⊕𝐷)
• Invariant of translation:
𝑋ℎ⊕𝐵 = (𝑋⊕𝐵)ℎ
𝑋ℎ = 𝑝 ∈ 𝑍
2 𝑝 = 𝑥 + ℎ, 𝑥 ∈ 𝑋}
• If 𝑋 ⊆ 𝑌, then 𝑋⊕𝐵⊆ 𝑌⊕𝐵
• Morphological erosion ‘⊖’ combines two sets using vector subtraction of set
elements, and is a dual operator of dilation
𝑋⊖𝐵 = 𝑝 ∈ 𝑍2 ∀𝑏 ∈ 𝐵, 𝑝 + 𝑏 ∈ 𝑋}
• Not Commutative:
• Not Associative:
𝑋⊖𝐵⊖𝐷 ≠ 𝑋⊖(𝐵⊖𝐷)
• Invariant of translation:
𝑋ℎ⊖𝐵 = (𝑋⊖𝐵)ℎ, and
𝑋⊖𝐵ℎ = (𝑋⊖𝐵)−ℎ
• If 𝑋 ⊆ 𝑌, then 𝑋⊖𝐵⊆ 𝑌⊖𝐵
Duality: Dilation and Erosion
• The transpose ෘ𝐵 of a structuring element 𝐵 is the structuring element
mirrored in the origin: ෘ𝐵 = −𝑏 𝑏 ∈ 𝐵}
• Duality between morphological dilation and erosion operators:
(𝑋⊖ 𝐵)𝑐= 𝑋𝑐 ⊕ ෘ𝐵 (𝑐 means complement)
• Erosion and dilation are not inverse transforms. An erosion followed by a
dilation leads to an interesting morphological operation, called opening
𝑋 ∘ 𝐵 = 𝑋⊖𝐵 ⊕𝐵
• Erosion and dilation are not inverse transforms. An erosion followed by a
dilation leads to an interesting morphological operation, called opening
𝑋 ∘ 𝐵 = 𝑋⊖𝐵 ⊕𝐵
• Erosion and dilation are not inverse transforms. An erosion followed by a
dilation leads to an interesting morphological operation, called opening
𝑋 ∘ 𝐵 = 𝑋⊖𝐵 ⊕𝐵
• Erosion and dilation are not inverse transforms. An erosion followed by a
dilation leads to an interesting morphological operation, called opening
𝑋 ∘ 𝐵 = 𝑋⊖𝐵 ⊕𝐵
• A dilation followed by an erosion leads to the interesting morphological
operation, called closing
𝑋 • 𝐵 = 𝑋⊕𝐵 ⊖𝐵
• A dilation followed by an erosion leads to the interesting morphological
operation, called closing
𝑋 • 𝐵 = 𝑋⊕𝐵 ⊖𝐵
• A dilation followed by an erosion leads to the interesting morphological
operation, called closing
𝑋 • 𝐵 = 𝑋⊕𝐵 ⊖𝐵
• Hit-or-miss is a morphological operator for finding local patterns of
foreground and background pixels. Unlike dilation and erosion, this operation
is defined using a composite structuring element 𝐵 = 𝐵1, 𝐵2 . The hit-or-
miss operator is defined as follows
𝑋⊗𝐵 = 𝑥 𝐵1 𝑥 ⊂ 𝑋 𝑎𝑛𝑑 (𝐵2)𝑥 ⊂ 𝑋
• Relation with erosion
𝑋⊗𝐵 = (𝑋⊖ 𝐵1) ∩ (𝑋
Hit-or-Miss transformation
Hit-or-miss transformation: examples
Hit-or-miss transformation: examples
• What is morphology? What are the applications of morphology?
• What are the dilation, erosion, opening and closing operators?
• What is hit-or-miss transformation?
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