Used to extract image components that are useful in the representation and description of region shape, such as

Transcription

1 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

2 Sets 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 Z 2 gray-scaled image : the element of the set is the coordinates (x,y) of pixel belong to the object and the gray levels Z 3

3 Preliminary: A: A set in Z 2 with elements of a=(a 1,a 2 ) a A a A A B C = A B D= A B A B= { } c A w w A = {, } A B= w w A w B = A B c

4 Operators by examples:

5 Preliminary: A: A set in Z 2 with elements of a=(a 1,a 2 ) ˆ =, for, Reflection B { w w b b B} ( A) { c c a z a A} = +, for, Translation z

6 Reflection and Translation by examples: Need for a reference point. Reference point

7 Basic Logic Operations:

8 Examples

9 Two Fundamental Morphological Operators: Dilation: Set B: A structural elements. Relation to Convolution mask: Flipping Overlapping { ( ˆ) } ( ˆ) Other names: Grow, Expand z { } A B= z B A = z B A A z

10 Dilation by example:

11 Dilation by example: B: a 3 3 mask.

12 Application: Gap filling

13 Two Fundamental Morphological Operators: Erosion: ( ) { } A B= z B A Set B: A structural elements. Other banes: Shrink, Reduce z

14 { ( ) } A B= z B A z

15 Erosion by example: B: a 3 3 mask.

16 Erosion by example: 11 pixels (Diameter)

17 Dilation-Erosion Duality: c ( ) ( ) z ( ) Remember: { } c { c ( ) } A B = z B A = z B A = c = z B A = A Bˆ { c } z { ( ˆ ) } A B= z B A z z c

18 Erosion Application: Remove details 1,3,5,7,9, and 15 Erode with 13 Dilate with 13

19 Opening and Closing: Dilation expands and Erosion shrinks. Opening: Smooth contour (تنگه (Means: Break narrow isthmuses Remove thin protrusion Closing: Smooth contour Fuse narrow breaks, and long thin gulfs. Remove small holes, and fill gaps.

20 Opening and Closing: Dilation expands and Erosion shrinks. Opening: A erosion followed by a dilation using the same structuring element for both operations. Closing: ( ) ( ) ( ) { } z z A B = AΘB B= B B A A Dilation followed by a erosion using the same structuring element for both operations. ( ) A B= A B ΘB

21 Opening Illustration:

22 Opening Illustration:

23 Closing Illustration:

24 Closing Illustration:

25 Opening and Closing

26 Medical Application: Closing 5 5 Opening 5 5 Original Segmentation

27 Opening and Closing Duality: c ( ) ( c A B = A Bˆ ) Opening Properties: A B is a subset (subimage) of A If C is a subset of D, then C B is a subset of D B (A B) B = A B Multiple apply has no effect. Closing Properties: A is a subset (subimage) of A B If C is a subset of D, then C B is a subset of D B (A B) B = A B Multiple apply has no effect.

28 Noise Reduction Opening Dilation of Opening Closing of Opening

29 Hit-or-Miss Shape Detection X-Y-X shape X enclosed by W c ( ) ( ) c ( ) A* B= AΘX A Θ W X A* B= AΘB A ΘB 1 2 ( ) ( ˆ ) 1 2 A* B= AΘB A B B 1 : Object related B 2 : Background related

30 Hit-or-Miss: Another application: Corners

31 Morphological Operator Applications: Boundary Extraction: β ( A) = A ( AΘB) Eroded Difference

32 Boundary Extraction:

33 Region Filling: Start form p inside boundary. X 0 k = Until: p ( ) X = X B A X k k+ 1 = X k c

34 Region Filling Example: Semi-automated to cancel reflection effect

35 Connected components Extraction: Start from p belong to desired region. X 0 k = Until: p ( ) X = X B A X k k+ 1 = X k

36 Original Thr Connected Components with size Erosion (5 5)

37 Convex Hull of S: Smallest Convex set H, containing S Define Four basic structural elements, B i, i=1,2,3,4 X = i 0 ( ) ( ) k i i i X = X B A i = 1, 2,3, 4 and k = 1, 2,3, k A 4 C A = D, D = X i= 1 i i i converged

38 C C Converged: C C C C

39 Shortcoming of previous algorithm: Grow more than minimum required convex size. Limit to vertical-horizontal expansion.

40 Thinning: Another approach: { } { 1 2 n B = B, B,, B } ( * ) ( * ) A B= A A B = A A B (( ( ) ) ) 1 2 n { B} ( ) A = A B B B Repeat until convergence c

41 Convert to m- connection

42 Thickening: A B= A ( A* B) (( ( 1 2 ) ) ) n { B} = ( ) A A B B B Structural elements are as before.

43 Skeletonization A with notation S(A): For z belong to S(A) and (D) z, the largest disk centered at z and contained in A, one can not find a larger disk containing (D) z and included in A. Disk(D) z touches the boundary of A at two or more different points.

44 Skeleton by example:

45 Formulation: K S( A) = S ( A) k = 0 k ( ) = ( Θ ) ( Θ ) ( ) ( ) S A A kb A kb B, :Opening K (( ( ) ) ) AΘ kb = AΘB ΘB ΘB : k times K=max k ( AΘkB) { } Reconstruction: A= ( ) ( S ) k A kb k=0 k (( ( ) ) ) ( ) ( ) A kb = A B B B : k times

46

47

48

49

50

51

52

53 Extension to Gray-Level images: f(x,y): the input image b(x,y): a structuring element (a subimage function) (x,y) are integers. f and b are functions that assign a gray-level value (real number or real integer) to each distinct pair of coordinate (x,y)

54 Extension to Gray-Level images: Dilation: D f and D b are the domains of f and b, respectively. ( f b)( st, ) = max{ f( s xy, t) + bxy (, ) ( s x),( t y) D ;( x, y) D } condition (s-x) and (t-y) have to be in the domain of f and (x,y) have to be in the domain of b is similar to the condition in binary morphological dilation where the two sets have to overlap by at least one element f b

55 Dilation Similarity with Convolution: f(s-x) : f(-x) is simply f(x) mirrored with respect to the original of the x axis. the function f(s-x) moves to the right for positive s, and to the left for negative s. Maxoperation replaces the sums of convolution Addition operation replaces with the products of convolution. General effect Ifall the values of the structuring element are positive, the output image tends to be brighter than the input Dark details either are reduced or eliminated, depending on how their values and shapes relate to the structuring element used for dilation.

56 1D example: ( f b)( s) = max{ f( s x) + b( x) ( s x) D ; x D } f b

57 Erosion: ( fθ b)( st, ) = min{ f( s+ xy, + t) bxy (, ) ( s+ x),( t+ y) D ;( x, y) D } Condition (s+x) and (t+y) have to be in the domain of f and (x,y) have to be in the domain of b is similar to the condition in binary morphological erosion where the structuring element has to be completely contained by the set being eroded. f b

58 Similarity to 2D correlation f(s+x) moves to the left for positive s and to the right for negative s. General effect If all the elements of the structuring element are positive, the output image tends to be darker than the input The effect of bright details in the input image that are smaller in area than the structuring element is reduced, with the degree of reduction being determined by the gray-level values surrounding the bright detail and by the shape and amplitude values of the structuring element itself.

59 1D case: ( fθ b)( s) = min{ f( s+ x) b( x) ( s+ x) D ; x D } f b

60 Erosion-Dilation Duality: c c ( f Θ b) ( s, t) = ( f bˆ )( s, t) where c f = f( x, y) and bˆ = b( x, y)

61 Dilation-Erosion Dilation Original Erosion

62 Opening-Closing: Same (binary) relation with Dilation-Erosion: f b= ( f b) b f b= ( f b) b ( f b) c = f c bˆ

63 1D Example: One scan line Rolling ball for opening Opening results Rolling ball for Closing Closing Results

64 Opening-Closing Properties: Opening: () i ( f b) f ( ii) if f f, then ( f b) ( f b), Closing: ( iii) ( f b) b = ( f b) () i f ( f b) ( ii) if f f, then ( f b) ( f b) ( iii) ( f b) b = ( f b) e r indicates that the domain of e is a subset of the domain of r, and also that e(x,y) r(x,y) for any (x,y) in the domain of e

65 Opening The structuring element is rolled underside the surface of f All the peaks that are narrow with respect to the diameter of the structuring element will be reduced in amplitude and sharpness So, opening is used to remove small light details, while leaving the overall gray levels and larger bright features relatively undisturbed. The initial erosion removes the details, but it also darkens the image. The subsequent dilation again increases the overall intensity of the image without reintroducing the details totally removed by erosion.

66 Closing The structuring element is rolled on top of the surface of f Peaks essentially are left in their original form (assume that their separation at the narrowest points exceeds the diameter of the structuring element) So, closing is used to remove small dark details, while leaving bright features relatively undisturbed. The initial dilation removes the dark details and brightens the image The subsequent erosion darkens the image without reintroducing the details totally removed by dilation

67 Opening Closing Example Opening: Decreased size of small bright details. No changes to dark region Closing: Decreased size of small dark details. No changes to bright region

68 Smoothing: Gradient: Laplacian: Gray Level Morphological Examples: g = (( f b) b) g = ( f b) ( f Θb) g = ( f b) + ( f Θb) 2 f Dilation Erosion Smoothing Gradient Laplacian

69 Smoothing

70 Gradient:

71 Top-hat: h= f ( f b) Enhancing details in presence of shades

72

73