morphological image processing
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Mathematical morphology is a framework for image analysis using set theory operations. It is used for tasks like noise filtering, shape analysis, and segmentation. Basic operations include erosion, dilation, opening, and closing using a structuring element. Erosion shrinks objects while dilation expands them. Opening eliminates small objects and closing fills small holes. Together these operations can filter images while preserving overall shapes. Morphological operations also enable extracting object boundaries, thinning images to skeletons, and finding connected components.
( XWAXABA c
−−∩−=∗](https://image.slidesharecdn.com/chapter9morphologicalimageprocessing-130613013223-phpapp02/85/morphological-image-processing-40-320.jpg)
morphological image processing
- 1. Chapter 9: Morphological Image Processing Digital Image Processing
- 2. 2 Mathematic Morphology used to extract image components that are useful in the representation and description of region shape, such as boundaries extraction skeletons convex hull morphological filtering thinning pruning
- 3. 3 Mathematic Morphology mathematical framework used for: pre-processing noise filtering, shape simplification, ... enhancing object structure skeletonization, convex hull... Segmentation watershed,… quantitative description area, perimeter, ...
- 4. 4 Z2 and Z3 set in mathematic morphology represent objects in an image binary image (0 = white, 1 = black) : the element of the set is the coordinates (x,y) of pixel belong to the object Z2 gray-scaled image : the element of the set is the coordinates (x,y) of pixel belong to the object and the gray levels Z3
- 6. 6 Reflection and Translation },|{ˆ Bfor bbwwB ∈−∈= },|{)( Afor azaccA z ∈+∈=
- 8. 8 Example
- 9. Structuring element (SE) 9 small set to probe the image under study for each SE, define origo shape and size must be adapted to geometric properties for the objects
- 10. Basic idea in parallel for each pixel in binary image: check if SE is ”satisfied” output pixel is set to 0 or 1 depending on used operation 10
- 11. How to describe SE many different ways! information needed: position of origo for SE positions of elements belonging to SE 11
- 12. Basic morphological operations Erosion Dilation combine to Opening object Closening background 12 keep general shape but smooth with respect to
- 13. Erosion Does the structuring element fit the set? erosion of a set A by structuring element B: all z in A such that B is in A when origin of B=z shrink the object 13 }{ Az|(B)BA z ⊆=−
- 14. Erosion 14
- 15. Erosion 15
- 16. 16 Erosion }{ Az|(B)BA z ⊆=−
- 17. Dilation Does the structuring element hit the set? dilation of a set A by structuring element B: all z in A such that B hits A when origin of B=z grow the object 17 }ˆ{ ΦA)Bz|(BA z ≠∩=⊕
- 18. Dilation 18
- 19. Dilation 19
- 20. 20 Dilation }ˆ{ ΦA)Bz|(BA z ≠∩=⊕ B = structuring element
- 21. 21 Dilation : Bridging gaps
- 22. useful erosion removal of structures of certain shape and size, given by SE Dilation filling of holes of certain shape and size, given by SE 22
- 23. Combining erosion and dilation WANTED: remove structures / fill holes without affecting remaining parts SOLUTION: combine erosion and dilation (using same SE) 23
- 24. 24 Erosion : eliminating irrelevant detail structuring element B = 13x13 pixels of gray level 1
- 25. Opening erosion followed by dilation, denoted ∘ eliminates protrusions breaks necks smoothes contour 25 BBABA ⊕−= )(
- 26. Opening 26
- 27. Opening 27
- 28. 28 Opening BBABA ⊕−= )( })(|){( ABBBA zz ⊆∪=
- 29. Closing dilation followed by erosion, denoted • smooth contour fuse narrow breaks and long thin gulfs eliminate small holes fill gaps in the contour 29 BBABA −⊕=• )(
- 30. Closing 30
- 31. Closing 31
- 33. 33 Properties Opening (i) A°B is a subset (subimage) of A (ii) If C is a subset of D, then C °B is a subset of D °B (iii) (A °B) °B = A °B Closing (i) A is a subset (subimage) of A•B (ii) If C is a subset of D, then C •B is a subset of D •B (iii) (A •B) •B = A •B Note: repeated openings/closings has no effect!
- 34. Duality Opening and closing are dual with respect to complementation and reflection 34 )ˆ()( BABA cc =•
- 35. 35
- 36. 36
- 37. Useful: open & close 37
- 39. Hit-or-Miss Transformation (HMT)⊛ find location of one shape among a set of shapes ”template matching composite SE: object part (B1) and background part (B2) does B1 fits the object while, simultaneously, B2 misses the object, i.e., fits the background? 39
- 40. 40 Hit-or-Miss Transformation )]([)( XWAXABA c −−∩−=∗
- 41. 41 Boundary Extraction )()( BAAA −−=β
- 42. 42 Example
- 43. 43 Region Filling ,...3,2,1)( 1 =∩⊕= − kABXX c kk
- 44. 44 Example
- 46. 46 Example
- 47. Convex hull A set A is is said to be convex if the straight line segment joining any two points in A lies entirely within A. i i DAC 4 1 )( = ∪= ,...3,2,1and4,3,2,1)( ==∪∗= kiABXX ii k i k 47
- 48. 48
- 51. 51 Skeletons K k k ASAS 0 )()( = ∪= BkBAkBAASk )()()( −−−= })(|max{ Φ≠−= kBAkK ))(( 0 kBASA k K k ⊕∪= =
- 52. 52
- 53. 53 Pruning }{1 BAX ⊗= AHXX ∩⊕= )( 23 314 XXX ∪= H = 3x3 structuring element of 1’s )( 1 8 1 2 k k BXX ∗∪= =
- 54. 54
- 55. 55
- 56. 56
- 57. 57
- 58. 58 5 basic structuring elements
Editor's Notes
- #2: Department of Computer Engineering, CMU