Some Basic Morphological Algorithms (4)

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2 Some Basic Morphological Algorithms (4) Convex Hull A set A is said to be convex if the straight line segment joining any two points in A lies entirely within A. The convex hull H or of an arbitrary set S is the smallest convex set containing S. 2/27/

3 Some Basic Morphological Algorithms (4) Convex Hull i Let B, i = 1, 2, 3, 4, represent the four structuring elements. The procedure consists of implementing the equation: with X = A. i 0 i i X = ( X * B ) A k k 1 i = 1, 2,3, 4 and k = 1, 2,3,... When the procedure converges, or X = X,let D = X, the convex hull of A is 4 C( A) = D i= 1 i i i i i k k 1 k 2/27/

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6 Some Basic Morphological Algorithms (5) Thinning The thinning of a set A by a structuring element B, defined A B= A ( A* B) = A ( A* B) c 2/27/

7 Some Basic Morphological Algorithms (5) A more useful expression for thinning A symmetrically is based on a sequence of structuring elements: { } { n B = B, B, B,..., B } where B i is a rotated version of B i-1 The thinning of A by a sequence of structuring element { B} A B = A B B B 1 2 n { } ((...(( ) )...) ) 2/27/

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9 Some Basic Morphological Algorithms (6) Thickening: The thickening is defined by the expression ( * ) A B= A A B The thickening of A by a sequence of structuring element { B} A B A B B B 1 2 n { } = ((...(( ) )...) ) In practice, the usual procedure is to thin the background of the set and then complement the result. 2/27/

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11 Some Basic Morphological Algorithms (7) Skeletons A skeleton, S( A) of a set A has the following properties a. if z is a point of S( A) and ( D) is the largest disk centered at z and contained in A, one cannot find a larger disk containing ( D ) and included in A. The disk ( D) is called a maximum disk. b. The disk ( D) touches the boundary of A at two or more different places. z z z z 2/27/

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13 Some Basic Morphological Algorithms (7) The skeleton of A can be expressed in terms of erosion and openings. S( A) = S ( A) k = 0 with K = max{ k A kb φ}; K k S ( A) = ( A kb) ( A kb) B k where B is a structuring element, and ( A kb) = ((..(( A B) B)...) B) k successive erosions of A. 2/27/

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16 Some Basic Morphological Algorithms (7) A can be reconstructed from the subsets by using K A = ( S ( A) kb) k = 0 k where S ( A) kb denotes k successive dilations of A. k ( S ( A) kb) = ((...(( S ( A) B) B)... B) k k 2/27/

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18 Some Basic Morphological Algorithms (8) Pruning a. Thinning and skeletonizing tend to leave parasitic components b. Pruning methods are essential complement to thinning and skeletonizing procedures 2/27/

19 Pruning: Example X = A { B} 1 2/27/

20 Pruning: Example k = 1 ( * k ) X = X B 2/27/

21 Pruning: Example ( ) X = X H A 3 2 H : 3 3 structuring element 2/27/

22 Pruning: Example X4 = X1 X3 2/27/

23 Pruning: Example 2/27/

24 Some Basic Morphological Algorithms (9) Morphological Reconstruction It involves two images and a structuring element a. One image contains the starting points for the transformation (The image is called marker) b. Another image (mask) constrains the transformation c. The structuring element is used to define connectivity 2/27/

25 Morphological Reconstruction: Geodesic Dilation Let F denote the marker image and G the mask image, F G. The geodesic dilation of size 1 of the marker image (1) with respect to the mask, denoted by G ( ), is defined as (1) G ( ) D ( F) = F B G The geodesic dilation of size n of the marker image F ( n) with respect to G, denoted by DG ( F), is defined as (0) with DG ( F) F. 2/27/ D ( n) (1) ( n 1) DG ( F) = DG ( F) DG ( F) = F

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27 Morphological Reconstruction: Geodesic Erosion Let F denote the marker image and G the mask image. The geodesic erosion of size 1 of the marker image with (1) respect to the mask, denoted by EG ( F), is defined as (1) G ( ) E ( F) = F B G The geodesic erosion of size n of the marker image F ( n) with respect to G, denoted by EG ( F), is defined as (0) with EG ( F) F. ( n) (1) ( n 1) EG ( F) = EG ( F) EG ( F) = 2/27/

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29 Morphological Reconstruction by Dilation Morphological reconstruction by dialtion of a mask image D G from a marker image F, denoted R ( F), is defined as the geodesic dilation of F with respect to G, iterated until stability is achieved; that is, G D ( k) G( ) = G ( ) R F D F ( k) ( k 1) with k such that DG ( F) = DG ( F). 2/27/

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31 Morphological Reconstruction by Erosion Morphological reconstruction by erosion of a mask image E G from a marker image F, denoted R ( F), is defined as the geodesic erosion of F with respect to G, iterated until stability is achieved; that is, G E ( k) G( ) = G ( ) R F E F ( k) ( k 1) with k such that EG ( F) = EG ( F). 2/27/

32 Opening by Reconstruction The opening by reconstruction of size n of an image F is defined as the reconstruction by dilation of F from the erosion of size n of F; that is ( ) ( n) D OR ( F) = RF F nb where ( F nb) indicates n erosions of F by B. 2/27/

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34 Filling Holes Let Ixy (, ) denote a binary image and suppose that we form a marker image F that is 0 everywhere, except at the image border, where it is set to 1- I; that is Fxy (, ) then = 1 Ixy (, ) if ( xy, ) is on the border of 0 otherwise I D H = R c ( F) I c 2/27/

35 SE : 3 3 1s. 2/27/

36 D H = R c ( F) I c 2/27/

37 Border Clearing It can be used to screen images so that only complete objects remain for further processing; it can be used as a singal that partial objects are present in the field of view. The original image is used as the mask and the following marker image: Ixy (, ) if ( xy, ) is on the border of I Fxy (, ) = 0 otherwise D X = I R ( F) I 2/27/

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39 Summary (1) 2/27/

40 Summary (2) 2/27/

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43 Gray-Scale Morphology f( xy, ) : gray-scale image bxy (, ): structuring element 2/27/

44 Gray-Scale Morphology: Erosion and Dilation by Flat Structuring [ f b] ( xy, ) = min { f( x+ sy, + t) } ( st,) b [ f b] ( xy, ) = max { f( x sy, t) } ( st,) b 2/27/

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46 Gray-Scale Morphology: Erosion and Dilation by Nonflat Structuring [ f b ]( xy, ) = min { f( x+ sy, + t) b ( st, )} N ( st,) b [ f b ]( xy, ) = max { f( x sy, t) + b ( st, )} N ( st,) b N N 2/27/

47 Duality: Erosion and Dilation c c f b ( xy, ) = f b ( xy, ) [ ] c where f = f( xy, ) and b= b( x, y) c c f b = f b [ ] c f b f b ( ) c = ( ) 2/27/

48 Opening and Closing f b= f b b ( ) f b= f b b ( ) c c f b = f b= f b ( ) c c f b = f b= f b ( ) 2/27/

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50 Properties of Gray-scale Opening ( a) f b f ( b) if f f, then f b f b ( ) ( ) ( ) ( c) f b b= f b where e r denotes e is a subset of r and also exy (, ) rxy (, ). 2/27/

51 Properties of Gray-scale Closing ( a) f f b ( b) if f f, then f b f b ( ) ( ) ( ) ( c) f b b= f b 2/27/

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53 Morphological Smoothing Opening suppresses bright details smaller than the specified SE, and closing suppresses dark details. Opening and closing are used often in combination as morphological filters for image smoothing and noise removal. 2/27/

54 Morphological Smoothing 2/27/

55 Morphological Gradient Dilation and erosion can be used in combination with image subtraction to obtain the morphological gradient of an image, denoted by g, g = ( f b) ( f b) The edges are enhanced and the contribution of the homogeneous areas are suppressed, thus producing a derivative-like (gradient) effect. 2/27/

56 Morphological Gradient 2/27/

57 Top-hat and Bottom-hat Transformations The top-hat transformation of a grayscale image f is defined as f minus its opening: T ( f) = f ( f b) hat The bottom-hat transformation of a grayscale image f is defined as its closing minus f: B ( f) = ( f b) f hat 2/27/

58 Top-hat and Bottom-hat Transformations One of the principal applications of these transformations is in removing objects from an image by using structuring element in the opening or closing operation 2/27/

59 Example of Using Top-hat Transformation in Segmentation 2/27/

60 Granulometry Granulometry deals with determining the size of distribution of particles in an image Opening operations of a particular size should have the most effect on regions of the input image that contain particles of similar size For each opening, the sum (surface area) of the pixel values in the opening is computed 2/27/

61 Example 2/27/

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63 Textual Segmentation Segmentation: the process of subdividing an image into regions. 2/27/

64 Textual Segmentation 2/27/

65 Gray-Scale Morphological Reconstruction (1) Let f and g denote the marker and mask image with the same size, respectively and f g. The geodesic dilation of size 1 of f with respect to g is defined as ( ) D (1) ( f) = f g g g where denotes the point-wise minimum operator. The geodesic dilation of size n of f with respect to g is defined as ( n) (1) ( n 1) (0) Dg ( f) = D g Dg ( f) with Dg ( f) = f 2/27/

66 Gray-Scale Morphological Reconstruction (2) The geodesic erosion of size 1 of f with respect to g is defined as ( ) E (1) ( f) = f g g g where denotes the point-wise maximum operator. The geodesic erosion of size n of f with respect to g is defined as ( n) (1) ( n 1) (0) Eg ( f) = Eg Eg ( f) with Eg ( f) = f 2/27/

67 Gray-Scale Morphological Reconstruction (3) The morphological reconstruction by dilation of a grayscale mask image g by a gray-scale marker image f, is defined as the geodesic dilation of f with respect to g, iterated until stability is reached, that is, D R ( f) = D f with k such that g ( k) g ( ) ( ) = k+ ( ) D f D f ( k) ( 1) g g The morphological reconstruction by erosion of g by f is defined as with k such that E R ( f) = E f g ( k) g ( ) k+ ( ) = ( ) E f E f ( k) ( 1) g g 2/27/

68 Gray-Scale Morphological Reconstruction (4) The opening by reconstruction of size n of an image f is defined as the reconstruction by dilation of f from the erosion of size n of f; that is, [ ] ( n ) D O ( f ) = R f nb R f The closing by reconstruction of size n of an image f is defined as the reconstruction by erosion of f from the dilation of size n of f; that is, [ ] ( n ) E C ( f ) = R f nb R f 2/27/

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70 Steps in the Example 1. Opening by reconstruction of the original image using a horizontal line of size 1x71 pixels in the erosion operation [ ] ( n ) D O ( f ) = R f nb R 2. Subtract the opening by reconstruction from original image f f O f ( n) ' = R ( ) 3. Opening by reconstruction of the f using a vertical line of size 11x1 pixels 4. Dilate f1 with a line SE of size 1x21, get f2. f [ ] f O f R f nb ( n) D 1 = R ( ') = f ' ' 2/27/

71 Steps in the Example 5. Calculate the minimum between the dilated image f2 and and f, get f3. 6. By using f3 as a marker and the dilated image f2 as the mask, R ( f3) = D f3 D ( k) f 2 f 2 ( ) with k such that D f3 = D f3 ( ) k+ ( ) ( k) ( 1) f 2 f 2 2/27/