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Image Processing and Analysis (ImagePandA)
7 – Morphological Image Processing Christoph Lampert / Chris Wojtan Based on content from “Digital Image Processing” by Gonzalez and Woods TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.:
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Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation Some Basic Morphological Algorithms Gray-Scale Morphology
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Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation Some Basic Morphological Algorithms Gray-Scale Morphology
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Preliminaries Set theory 𝐵 = 𝑤 𝑤=−𝑏, for 𝑏∈𝐵 Reflection
𝐵 𝑧 = 𝑐 𝑐=𝑏+𝑧, for 𝑏∈𝐵 Translation
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Preliminaries Set theory Structuring elements (SEs)
Small sets or subimages used to probe an image SEs are padded to rectangular images
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Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation Some Basic Morphological Algorithms Gray-Scale Morphology
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Erosion and Dilation Erosion 𝐴⊖𝐵={𝑧| 𝐵 𝑧 ⊆𝐴} 𝐴⊖𝐵={𝑧| 𝐵 𝑧 ∩ 𝐴 𝑐 =∅}
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Erosion and Dilation Erosion 𝐴⊖𝐵={𝑧| 𝐵 𝑧 ⊆𝐴} 𝐴⊖𝐵={𝑧| 𝐵 𝑧 ∩ 𝐴 𝑐 =∅}
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Erosion and Dilation Erosion 𝐴⊖𝐵={𝑧| 𝐵 𝑧 ⊆𝐴} 𝐴⊖𝐵={𝑧| 𝐵 𝑧 ∩ 𝐴 𝑐 =∅}
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Erosion and Dilation 𝐴⊕𝐵={𝑧| 𝐵 𝑧 ∩𝐴≠∅} Dilation
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Erosion and Dilation 𝐴⊕𝐵={𝑧| 𝐵 𝑧 ∩𝐴≠∅} Dilation
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Erosion and Dilation Duality
Erosion is just like dilating the compliment Dilation is just like eroding the compliment (𝐴⊖𝐵) 𝑐 = 𝐴 𝑐 ⊕ 𝐵 (𝐴⊕𝐵) 𝑐 = 𝐴 𝑐 ⊖ 𝐵
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Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation Some Basic Morphological Algorithms Gray-Scale Morphology
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Opening and Closing Opening A∘𝐵= 𝐴⊖𝐵 ⊕𝐵 Closing A∘𝐵= 𝐴⊕𝐵 ⊖𝐵
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Opening and Closing Opening A∘𝐵= 𝐴⊖𝐵 ⊕𝐵 Closing A∘𝐵= 𝐴⊕𝐵 ⊖𝐵
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Opening and Closing
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Opening and Closing
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Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation Some Basic Morphological Algorithms Gray-Scale Morphology
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The Hit-or-Miss Transformation
Given disjoint structuring elements 𝐶 and 𝐷 𝐴⊛𝐵= 𝐴⊖𝐶 ∩( 𝐴 𝑐 ⊖𝐷) Point 𝑧 belongs to the hit-or-miss transform output if: 𝐶 translated to 𝑧 fits in 𝐴, and 𝐷 translated to 𝑧 misses 𝐴 (fits the background of 𝐴) Result is the set of positions where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely. It is used to detect patterns or shapes
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The Hit-or-Miss Transformation
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Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation Some Basic Morphological Algorithms Gray-Scale Morphology
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Boundary Extraction 𝛽 𝐴 =𝐴−(𝐴⊖𝐵)
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Boundary Extraction 𝛽 𝐴 =𝐴−(𝐴⊖𝐵)
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Hole Filling Pick a pixel inside of a hole, call it 𝑋 0
Iterate 𝑋 𝑘 =( 𝑋 𝑘−1 ⊕𝐵)∩ 𝐴 𝑐
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Hole Filling Pick a pixel inside of a hole, call it 𝑋 0
Iterate 𝑋 𝑘 =( 𝑋 𝑘−1 ⊕𝐵)∩ 𝐴 𝑐
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Extracting Connected Components
Pick a pixel inside of a shape, call it 𝑋 0 Iterate 𝑋 𝑘 =( 𝑋 𝑘−1 ⊕𝐵)∩𝐴
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Thinning 𝐴⊗𝐵=𝐴− 𝐴⊛𝐵 =𝐴∩ (𝐴⊛𝐵) 𝑐 Repeat until no change
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Thickening Morphological dual of thinning 𝐴⊙𝐵=𝐴∪ 𝐴⊛𝐵
Repeat until no change More common approach: Thin the background 𝐴 𝑐 and then compliment the result
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Thickening
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Pruning
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Geodesic Dilation
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Geodesic Erosion
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Morphological Reconstruction
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Morphological Reconstruction
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Border clearing Find all boundary pixels in A
Morphologically reconstruct all of them Subtract those from the original
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Border clearing
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Outline Preliminaries Erosion and Dilation Opening and Closing
The Hit-or-Miss Transformation Some Basic Morphological Algorithms Gray-Scale Morphology
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Gray-Scale Morphology
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Gray-Scale Morphology
Erosion 𝑓⊖𝑏 𝑥,𝑦 = min 𝑠,𝑡 ∈𝑏 𝑓(𝑥+𝑠,𝑦+𝑡) Dilation 𝑓⨁𝑏 𝑥,𝑦 = max 𝑠,𝑡 ∈𝑏 𝑓(𝑥−𝑠,𝑦−𝑡)
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Gray-Scale Morphology
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Gray-Scale Morphology
Erosion 𝑓⊖𝑏 𝑥,𝑦 = min 𝑠,𝑡 ∈𝑏 𝑓(𝑥+𝑠,𝑦+𝑡) Dilation 𝑓⨁𝑏 𝑥,𝑦 = max 𝑠,𝑡 ∈𝑏 𝑓(𝑥−𝑠,𝑦−𝑡) Opening f∘𝑏= 𝑓⊖𝑏 ⊕𝑏 Closing f∘𝑏= 𝑓⊕𝑏 ⊖𝑏
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Gray-Scale Morphology
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Gray-Scale Morphology
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Gray-Scale Morphology
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Gray-Scale Morphology
Mostly analogous to binary morphology Lots of potential, but lots of heuristics/hacks Not easy to find general tricks that work on all problems
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Gray-Scale Morphology
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