Morphological Operations. GV12/3072 Image Processing.

Transcription

1 Morphological Operations 1

2 Outline Basic concepts: Erode and dilate Open and close. Granulometry Hit and miss transform Thinning and thickening Skeletonization and the medial axis transform Introduction to gray level morphology. 2

3 What Are Morphological Operators? Local pixel transformations for processing region shapes Most often used on binary images Logical transformations based on comparison of pixel neighbourhoods with a pattern. 3

4 Simple Operations - Examples Eight-neighbour erode a.k.a. Minkowsky subtraction Erase any foreground pixel that has one eight-connected neighbour that is background. 4

5 8-neighbour erode Threshold Erode 1 Erode 2 Erode 5 5

6 8-neighbour dilate Eight-neighbour dilate a.k.a. Minkowsky addition Paint any background pixel that has one eight-connected neighbour that is foreground. 6

7 8-neighbour dilate Dilate 1 Dilate 2 Dilate 5 7

8 Why? Smooth region boundaries for shape analysis. Remove noise and artefacts from an imperfect segmentation. Match particular pixel configurations in an image for simple object recognition. 8

9 Structuring Elements Morphological operations take two arguments: A binary image A structuring element. Compare the structuring element to the neighbourhood of each pixel. This determines the output of the morphological operation. 9

10 Structuring elements The structuring element is also a binary array. A structuring element has an origin

11 Binary images as sets We can think of the binary image and the structuring element as sets containing the pixels with value I = {(1,1), (2,1), (3,1), (2,2), (3,2), (4,4)} 11

12 Some sets notation Union and intersection: I I 1 1 I I 2 2 x : x : Complement I C x : x x I x I I 1 1 or x I 2 and x I 2 Difference I 1 2 \ I x : xi and xi We use for the empty set

13 More sets notation The symmetrical set of S with respect to point o is S o x : x S

14 Fitting, Hitting and Missing S fits I at x if { y : y x s, ss} I S hits I at x if { y : y x s, ss} I S misses I at x if { y : y x s, ss} I 14

15 Fitting, Hitting and Missing Image Structuring element

16 Erosion The image E = I S is the erosion of image I by structuring element S. E( x) 1 0 if S fits I at otherwise x E x : x si for every ss 16

17 Implementation (naïve) % I is the input image, S is a structuring element % with origin (ox, oy). E is the output image. function E = erode(i, S, ox, oy) [X,Y] = size(i); [SX, SY] = size(s); E = ones(x, Y); for x=1:x; for y=1:y for i=1:sx; for j=1:sy if(s(i,j)) E(x,y) = E(x,y) & ImageEntry(I, x+i-ox, y+j-oy); end end; end end; end 17

18 Example Structuring element

19 Example Structuring element

20 Example Structuring element

21 Dilation The image D = I S is the dilation of image I by structuring element S. D( x) 1 0 if S hits I at otherwise x D x: x s, yi and s S 21

22 Example Structuring element

23 Example Structuring element

24 Example Structuring element

25 26 Erosion and dilation Erosion and dilation are dual operations: Commutativity and associativity S I S I C C ) ( I S S I I S S I ) ( ) ( ) ( ) ( T S I T S I T S I T S I

26 Opening and Closing The opening of I by S is I S ( I S) S The closing of I by S is I S ( I S) S 27

27 Opening and Closing Opening and closing are dual transformations: ( I S) C I C S Opening and closing are idempotent operations: I S I S ( I S) S ( I S) S 28

28 Example close open Structuring element

29 Example close open Structuring element

30 Example close open Structuring element

31 Morphological filtering To remove holes in the foreground and islands in the background, do both opening and closing. The size and shape of the structuring element determine which features survive. In the absence of knowledge about the shape of features to remove, use a circular structuring element. 32

32 Example Close then open Structuring element Open then close

33 Example Close then open Structuring element Open then close

34 Example Close then open Structuring element Open then close

35 Count the Red Blood Cells 36

36 Granulometry Provides a size distribution of distinct regions or granules in the image. We open the image with increasing structuring element size and count the number of regions after each operation. Creates a morphological sieve. 37

37 Granulometry function gspec = granulo(i, T, maxrad) % Segment the image I B = (I>T); % Open the image at each structuring element size up to a % maximum and count the remaining regions. for x=1:maxrad O = imopen(b,strel( disk,x)); numregions(x) = max(max(connectedcomponents(o))); end gspec = diff(numregions); 38

38 Count the Red Blood Cells 39

39 Threshold and Label 40

40 Disc(11) 41

41 Disc(19) 42

42 Disc(59) 43

43 Number of regions Number of Regions Struct. Elt. radius 44

44 Number of regions Granulometric Pattern Spectrum Struct. Elt. radius 45

45 Hit-and-miss transform Searches for an exact match of the structuring element. H = I S is the hit-and-miss transform of image I by structuring element S. Simple form of template matching. 46

46 Hit-and-miss transform = = *

47 Upper-Right Corner Detector 48

48 Thinning and Thickening Defined in terms of the hit-and-miss transform: The thinning of I by S is I S I \ ( I S) The thickening of I by S is I S I ( I S) Dual operations: C ( I S) I C S 49

49 Sequential Thinning/Thickening These operations are often performed in sequence with a selection of structuring elements S 1, S 2,, S n. Sequential thinning: I S i 1,..., n ((( I S ) S )... S ) i : 1 2 n Sequential thickening: I S i 1,..., n ((( I S ) S )... S ) i : 1 2 n 50

50 Sequential Thinning/Thickening Several sequences of structuring elements are useful in practice These are usually the set of rotations of a single structuring element. Sometimes called the Golay alphabet. 51

51 Golay element L * * 0 * 1 * L 1 * 1 * L L L * 1 * 1 * 0 * * 1 * 0 * * L 5 * 1 * L L L * 0 * 1 * 1 * 52

52 Sequential Thinning See bwmorph in matlab. 0 iterations 1 iteration 2 iterations 5 iterations Inf iterations 53

53 Sequential Thickening 1 iteration 5 iterations 2 iterations Inf iterations 54

54 Skeletonization and the Medial Axis Transform The skeleton and medial axis transform (MAT) are stick-figure representations of a region X 2. Start a grassfire at the boundary of the region. The skeleton is the set of points at which two fire fronts meet. 55

55 Skeletons 56

56 Medial axis transform Alternative skeleton definition: The skeleton is the union of centres of maximal discs within X. A maximal disc is a circular subset of X that touches the boundary in at least two places. The MAT is the skeleton with the maximal disc radius retained at each point. 57

57 Medial axis transform 58

58 Skeletonization using morphology Use structuring element B = The n-th skeleton subset is S n ( X ) ( X n B) \ The skeleton is the union of all the skeleton subsets: ( X n B) B n denotes n successive erosions. S( X ) n1 S n ( X ) 59

59 Reconstruction We can reconstruct region X from its skeleton subsets: X n0 S n ( X ) n B We can reconstruct X from the MAT. We cannot reconstruct X from S(X). 61

60 DiFi: Fast 3D Distance Field Computation Using Graphics Hardware Sud, Otaduy, Manocha, Eurographics

61 MAT in 3D from Transcendata Europe Medial Object Price, Stops, Butlin Transcendata Europe Ltd 63

62 Applications and Problems The skeleton/mat provides a stick figure representing the region shape Used in object recognition, in particular, character recognition. Problems: Definition of a maximal disc is poorly defined on a digital grid. Sensitive to noise on the boundary. Sequential thinning output sometimes preferred to skeleton/mat. 64

63 Example Skeletons: Thinned: 65

64 Gray-level Morphology Erosion, dilation Opening and closing The image and the structuring element are gray level arrays. Used to remove speckle noise. Also for smoothing, edge detection and segmentation. 66

65 Umbra U( f ) ( x, y): y f ( x) f U ( f ) 67

66 Top surface T( R) ( x, y) : y z for all( x, z) R R T(R) 68

67 Gray level erode and dilate Erosion: f k T( U( f ) U( k)) Dilation: f k T( U( f ) U( k)) Open and close defined as before. 69

68 f U ( f ) k U(k) U( f ) U( k) T( U( f ) U( k)) 70

69 Erosion S=ones(3,3) S=ones(5,5) S=ones(7,7) 72

70 Dilation S=ones(3,3) S=ones(5,5) S=ones(7,7) 73

71 Speckle removal Erode Open Salt and pepper noise Close Dilate 74

72 Favorites E = medfilt2() [D,L] = bwdist() 75

73 Summary Simple morphological operations Erode and dilate Open and close Applications: Granulometry Thinning and thickening Skeletons and the medial axis transform Gray level morphology 76

74 77

75 Find the Letter e 78