11. Gray-Scale Morphology. Computer Engineering, i Sejong University. Dongil Han

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1 Computer Vision 11. Gray-Scale Morphology Computer Engineering, i Sejong University i Dongil Han Introduction Methematical morphology represents image objects as sets in a Euclidean space by Serra [1982], Sternberg [1986] The extension of the morphological transformatons from binary to grayscale processing by Haralick, Sternberg, and Zuang [1987] introduced a natural morphological generalization of the dilation and erosion operations by Heijams [1991] how to use binary morphological loperators and threholding h techniques to build a large class of grayscale morphological operators 2/47

2 Extensions to Gray-Scale Images Gray-scale morphology morphological gradient operation - boundary extraction region partitioning smoothing sharpening texture segmentation, etc 3/47 Grayscale Dilation and Erosion Grayscale images can be represented as binary images in three-dimensional space can be viewed as three-dimensional surfaces - height: brightness value Dialation, erosion, and the morphological gradient Dilation and erosion are often used in combination to implement image processing operations 4/47

3 Dilation gray-scale dilation f(x,y) by b(x,y) is defined as (f b)(s,t) = max{f(s-x, t-y) + b(x,y) (s-x),(t-y) D f ; (x,y) D b } where D f and D b are domains of f and b, respectively Dilation vs. convolution Similar to 2-D convolution convolution : sum of product dilation : maximum of sum 5/47 Dilation In case of simple 1-D function, gray-scale dilation f(x) by b(x) is (f b)(s) = max{f(s-x) + b(x) (s-x) D f ; x D b } where D f and D b are domains of f and b, and can be rewritten as (f b)(s) ) = max{f(x) + b(s-x) ) x D f ; (s-x) D b } Concept of gray-scale dilation b(-x) : simply b(x) mirrored with respect to the origin b(s-x) : moves to the right for positive s, and vice versa At each position s, find the maximum of the sum of f and b f sliding by b = b sliding by f 6/47

4 Dilation 7/47 Dilation Effects of dilation If all 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 8/47

5 Erosion gray-scale erosion f(x,y) by b(x,y) is defined as (f b)(s,t) = min{f(s + x, t+ y) - b(x,y) (s+x),(t+y) (t+ ) D f ; (x,y) D b } where D f and D b are domains of f and b,, respectively Erosion vs. convolution Also, similar to 2-D convolution correlation : sum of product erosion : minimum of subtraction 9/47 Erosion 10/47

6 Effects of erosion Erosion If all the values of the structuring element are positive, the output image tends to be darker than the input Bright details that are smaller than the structuring element is reduced Duality c c ( f b) ( s, t) ( f bˆ)( s, t) where f c = -f(x,y) and bˆ =b(-x,-y) (, 11/47 Dilation and Erosion 12/47

7 Opening and Closing The opening of image f by subimage(structuring g element) b is defined as f b ( f b) b The opening f by b is the erosion of f by b, followed by a dilation of the result by b The closing of image f by subimage(structuring element) b is defined as f b ( f b) The closing f by b is the dilation of f by b, followed by a erosion of the result by b Duality b c ( f b) c f bˆ 13/47 Opening Grayscale Opening and Closing can be interpreted as rolling a ball as the structuring element under the signal's s surface the highest values of the ball that can reach the top at each location constitute the opening result tends to remove bright objects that are small size and break narrow connections between two bright objects FIGURE 3.2 The grayscale opening using a ball-structuring element. 14/47

8 Closing Grayscale Opening and Closing can be interpreted as rolling the ball above the signal's surface the lowest values of the ball that can reach the bottom at each location constitute the closing result tends to preserve small objects that are brighter that the background and connect bright objects with small gaps in between FIGURE 3.3 The grayscale closing using a ball-structuring element. 15/47 Opening and Closing original opening closing 16//47

9 Opening and Closing 17/47 Opening Opening and Closing Remove small light details(with respect to the size of structuring element) Leaving the overall gray levels and larger bright features Initial erosion removes the small details and darkens the image The subsequent dilation again increases the overall intensity of the image without reintroducing the details Closing Remove small dark details Leaving the overall gray levels and bright features Initial dilation removes the dark details and brightens the image The subsequent erosion darkens the image without reintroducing the details removed by dilation 18/47

10 Gray-Scale Morphology Some Applications of Gray-scale morphology morphological smoothing morphological gradient Top-hat transformation Textural segmentation Granulometry 19/47 Morphological lsmoothing Perform morphological lopening fll followed dby a closing Remove or attenuate both bright and dark artifacts or noise s ( f b) b 20/47

11 Morphological l Gradient The difference of dilation and erosion Highlights sharp gray-level transitions in the input image Tends to depend less on edge directionality g ( f b ) ( f b ) 21/47 Morphological l Gradient The concept of morphological gradient 22/47

12 Top-hat httransformation ti The difference between the original image and its opening Use cylindrical or parallelepiped structuring element function with flat top Enhancing detail in the presence of shading Example : enhancement of detail in the background region below the lower part of the horse s head h f ( f b) 23/47 Textural segmentation ti Objective : yields the boundary between the two textural regions Closing Tends to remove dark detail When the size of the structuring t element corresponds to that t of the small blubs, b they are removed Opening Removes the light patches between blobs Leaving dark region on the right A simple threshold yields the boundary 24/47

13 Granulometry Objective : determining the size distribution of particles in an image Idea : opening operations of a particular size have the most effect on regions of the input image that contain particles of similar size Method Perform opening operations with increasing the size of structuring element Compute the difference between the original image and its opening Difference are normalized and construct a histogram of particle-size distribution 25/47 Morphological limage Processing Basic Morphological l Algorithms Boundary Extraction Region Filling Extraction of Connected Components Convex Hull Thinning Thickening Skeletons pruning, etc. 26/47

14 Boundary Extraction (A) : the boundary of a set A ( A ) A ( A B ) where B is a suitable structuring element 27/47 Boundary Extraction Depends on the size of structuring t element B, the width of boundary could be changed ( A) A ( A B) 28/47

15 Region filling Use set dilation, complementation, and intersections X = 0 p (point p : inside the boundary) X = c ( X B) A k 1,2,3, = k k terminates at iteration step k if X k X k 1 Final result : set union of X k and A 29/47 Region filling 30/47

16 Region filling 31/47 Extraction of Connected Components Let Y represent a connected component contained in a set A X = k ( X k - 1 B) A k = 1,2,3,,,... X = p 0 (point p : on the boundary) terminates at iteration step k if X k X then k 1 Y X k 32/47

17 Extraction of Connected Components 33/47 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 of an arbitrary set S is the smallest convex set containing S i i i X = ( X B ) A k k-1 * i = 1,2,3,4 k = 1,2,3,... X i A 0 i i D X ; conv indicates convergence conv Convex hull of A : i i X k X k 1 4 i= 1 i C ( A ) = D 34/47

18 Convex Hull x: don t care 35/47 Thinning The thinning of a set A by a structuring element B A B = A ( A B ) = A ( A B ) c * * More useful expression for thinning n { B } = { B, B, B,, B } B = A {B} = (( ((A B 1 ) B 2 ) ) B n ) for n structuring element, repeat until no further changes occur 36/47

19 Thinning 37/47 Thickening Morphological dual of thinning thickening : A B = A ( A * B) n { B } = { B, B, B,, B } A {B} = (( ((A B 1 ) B 2 ) ) B n ) May ygenerates disconnected points Usual procedure is to thin the background of the set 38/47

20 Thickening 39/47 Skeletons Skeleton generation steps 40/47

21 Skeletons Skeleton of A S: S k K ( A) S k ( A) k 0 ( A) ( A kb) [( A kb) B)] ( A kb ) ((...( ( A B ) B )...) B K max{ k ( A kb ) } Reconstruct : ( S k K k 0 A ( S ( A) kb) ( A) kb) ((...( S ( A) B) B)...) B k k 41/47 Skeletons 42/47

22 Pruning Essential complement to thinning and skeletonizing algorithms Clean up pparasitic components Procedure Thinning ( detect only end points) : X 1 A {B } Forming a set X 2 containing all end points in X 1 : Dilation of the end points : 8 k X 2 ( X1* B ) k 1 3 ( X 2 H A X ) Finally, the union of X 3 and X 1 : X 4 X1 X 3 43/47 Pruning 44/47

23 Summary of morphological operators 45/47 Summary of morphological operators 46/47

24 Summary of morphological operators 47/47 Summary of morphological operators 48/47

25 Summary Morphology : powerful set of tools for extracting features of interest in an image Most appealing aspects : extensive set-theoretical foundation Very simple, implementation is easy, widely used Play a major role in procedures for image segmentation and image description 49/47

26 Hit-or-Miss Transformation Morphological lhit-or-miss Transformation Basic tool for shape detection Objective : find the location of certain shape, say, X Definition Let X be enclosed by a small window W A c * B ( A X ) [ A ( W X )] If B 1 = X, B 2 = (W-X) A * c B ( A B1 ) [ A B2 ] By using the definition of set differences and the dual relationship A * B ( A B ) [ A Bˆ ( 1 B2 ] 51/47 Hit-or-Miss Transformation 52/47

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