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morphological image processing

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morphological image processing

  1. 1. Chapter 9: MorphologicalImage ProcessingDigital Image Processing
  2. 2. 2Mathematic Morphology used to extract image components that areuseful in the representation and description ofregion shape, such as boundaries extraction skeletons convex hull morphological filtering thinning pruning
  3. 3. 3Mathematic Morphologymathematical framework used for: pre-processing noise filtering, shape simplification, ... enhancing object structure skeletonization, convex hull... Segmentation watershed,… quantitative description area, perimeter, ...
  4. 4. 4Z2and Z3 set in mathematic morphology representobjects in an image binary image (0 = white, 1 = black) : theelement of the set is the coordinates (x,y)of pixel belong to the object  Z2 gray-scaled image : the element of the setis the coordinates (x,y) of pixel belong to theobject and the gray levels  Z3
  5. 5. 5Basic Set Theory
  6. 6. 6Reflection and Translation},|{ˆ Bfor bbwwB ∈−∈=},|{)( Afor azaccA z ∈+∈=
  7. 7. 7Logic Operations
  8. 8. 8Example
  9. 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 geometricproperties for the objects
  10. 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 onused operation10
  11. 11. How to describe SE many different ways! information needed: position of origo for SE positions of elements belonging to SE11
  12. 12. Basic morphological operations Erosion Dilation combine to Opening object Closening background12keep general shape butsmooth with respect to
  13. 13. Erosion Does the structuring element fit theset?erosion of a set A by structuring elementB: all z in A such that B is in A whenorigin of B=zshrink the object13}{ Az|(B)BA z ⊆=−
  14. 14. Erosion14
  15. 15. Erosion15
  16. 16. 16Erosion}{ Az|(B)BA z ⊆=−
  17. 17. Dilation Does the structuring element hit theset? dilation of a set A by structuringelement B: all z in A such that B hits Awhen origin of B=z grow the object17}ˆ{ ΦA)Bz|(BA z ≠∩=⊕
  18. 18. Dilation18
  19. 19. Dilation19
  20. 20. 20Dilation}ˆ{ ΦA)Bz|(BA z ≠∩=⊕B = structuring element
  21. 21. 21Dilation : Bridging gaps
  22. 22. useful erosion removal of structures of certain shape andsize, given by SE Dilation filling of holes of certain shape and size,given by SE22
  23. 23. Combining erosion anddilation WANTED: remove structures / fill holes without affecting remaining parts SOLUTION: combine erosion and dilation (using same SE)23
  24. 24. 24Erosion : eliminating irrelevantdetailstructuring element B = 13x13 pixels of gray level 1
  25. 25. Openingerosion followed by dilation, denoted ∘ eliminates protrusions breaks necks smoothes contour25BBABA ⊕−= )(
  26. 26. Opening26
  27. 27. Opening27
  28. 28. 28OpeningBBABA ⊕−= )(})(|){( ABBBA zz ⊆∪=
  29. 29. Closingdilation followed by erosion, denoted • smooth contour fuse narrow breaks and long thin gulfs eliminate small holes fill gaps in the contour29BBABA −⊕=• )(
  30. 30. Closing30
  31. 31. Closing31
  32. 32. 32ClosingBBABA −⊕=• )(
  33. 33. 33PropertiesOpening(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 °BClosing(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 •BNote: repeated openings/closings has no effect!
  34. 34. Duality Opening and closing are dual with respectto complementation and reflection34)ˆ()( BABA cc=•
  35. 35. 35
  36. 36. 36
  37. 37. Useful: open & close37
  38. 38. Application: filtering38
  39. 39. Hit-or-Miss Transformation(HMT)⊛ find location of one shape among a set of shapes”template matching composite SE: object part (B1) and backgroundpart (B2) does B1 fits the object while, simultaneously,B2 misses the object, i.e., fits the background?39
  40. 40. 40Hit-or-Miss Transformation)]([)( XWAXABA c−−∩−=∗
  41. 41. 41Boundary Extraction)()( BAAA −−=β
  42. 42. 42Example
  43. 43. 43Region Filling,...3,2,1)( 1 =∩⊕= − kABXX ckk
  44. 44. 44Example
  45. 45. 45Extraction of connectedcomponents
  46. 46. 46Example
  47. 47. Convex hull A set A is issaid to beconvex ifthe straightline segmentjoining anytwo pointsin A liesentirelywithin A.iiDAC41)(=∪=,...3,2,1and4,3,2,1)( ==∪∗= kiABXX iikik47
  48. 48. 48
  49. 49. 49ThinningcBAABAABA)()(∗∩=∗−=⊗
  50. 50. 50Thickening)( BAABA ∗∪=•
  51. 51. 51SkeletonsKkk ASAS0)()(=∪=BkBAkBAASk )()()( −−−=})(|max{ Φ≠−= kBAkK))((0kBASA kKk⊕∪==
  52. 52. 52
  53. 53. 53Pruning}{1 BAX ⊗=AHXX ∩⊕= )( 23314 XXX ∪=H = 3x3 structuring element of 1’s)( 1812kkBXX ∗∪==
  54. 54. 54
  55. 55. 55
  56. 56. 56
  57. 57. 57
  58. 58. 585 basic structuring elements

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