9.4 The hit-or-miss transformation Illustration...
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1 Morphological image processing(9.4 to 9.5.8) SLIDE 1/ The hit-or-miss transformation Illustration...
2 Morphological image processing(9.4 to 9.5.8) SLIDE 2/18 Objectiveistofindadisjointregion(set)inanimage If B denotes the set composed of D and its background, the match/hit (orsetofmatches/hits)ofb ina,is Generalizednotation: B=(B 1,B 2 ) B 1 : A B=(A D) [A c (W D)] Set formed from elements of B associated with an object B 2 : Set formed from elements of B associated with the corresponding background [Preceedingdiscussion: B 1 =D andb 2 =(W D)] More general definition: A B=(A B 1 ) [A c B 2 ] A Bcontainsalltheoriginpointsatwhich,simultaneously,B 1 found ahitinaandb 2 foundahitina c
3 Morphological image processing(9.4 to 9.5.8) SLIDE 3/18 Alternative definition: A B=(A B 1 ) (A ˆB 2 ) A background is necessary to detect disjoint sets Whenwe onlyaimtodetectcertainpatterns withinaset, a background is not required, and simple erosion is sufficient 9.5 Some basic morphological algorithms When dealing with binary images, the principle application of morphology is extracting image components that are useful in the representation and description of shape Boundary extraction Theboundaryβ(A)ofasetAis β(a)=a (A B), where B is a suitable structuring element
4 Morphological image processing(9.4 to 9.5.8) SLIDE 4/18 Illustration... Example 9.5: Morphological boundary extraction
5 Morphological image processing(9.4 to 9.5.8) SLIDE 5/ Hole filling A setwhoseelementsare8-connectedboundariesthatencloseabackground region(hole) Givenapointpineachhole,theobjectiveistofillalltheholeswith1 s All non-boundary(background) points are labeled 0 Begin by forming an array X 0 of 0 s, except at the locations in X 0 that correspondtothepointspineachhole,whichissetto1... Thefollowingprocedurefillsalltheholeswith1 s, X k =(X k 1 B) A c, k=1,2,3,..., where B is the symmetric structuring element in figure 9.15(c) ThealgorithmterminatesatiterationstepkifX k =X k 1 ThesetunionofX k andacontainsthefilledsetanditsboundary NotethattheintersectionateachstepwithA c limitsthedilationresultto inside the region of interest
6 Morphological image processing(9.4 to 9.5.8) SLIDE 6/18 Example 9.6: Morphological hole filling
7 Morphological image processing(9.4 to 9.5.8) SLIDE 7/ Extraction of connected components Let A be a set containing one or more connected components, and form anarrayx 0 (withthesamesizeasa)whoseelementsare0(background), exceptateachlocationknowntocorrespondtoapointineachconnected componentina,whichissetto1(foreground) ThefollowingiterativeprocedurestartswithX 0 andfindalltheconnected components X k =(X k 1 B) A k=1,2,3,..., where B is a suitable structuring element. When X k = X k 1, with X k containing all the connected components, the procedure terminates This algorithm is applicable to any finite number of sets of connected componentscontainedina,assumingthatapointisknownineachconnected component
8 Morphological image processing(9.4 to 9.5.8) SLIDE 8/18
9 Morphological image processing(9.4 to 9.5.8) SLIDE 9/18 Example 9.7
10 Morphological image processing(9.4 to 9.5.8) SLIDE 10/ Convex hull Morphological algorithm for obtaining the convex hull, C(A), of a set A... LetB 1,B 2,B 3 andb 4 representthefourstructuringelementsinfig9.19(a), and then implement the equation... X i k=(x k 1 B i ) A, i=1,2,3,4, k=1,2,..., X i 0=A Now let D i =Xconv, i where conv indicates convergence in the sense that Xk i=xi k 1. ThentheconvexhullofAis 4 C(A)= D i Procedure illustrated in Fig 9.19: entries indicate don t care conditions Shortcoming of above algorithm: convex hull can grow beyond the minimum dimensions required to guarantee convexity Possible solution: Limit growth so that it does not extend past the vertical and horizontal dimensions of the original set of points Boundaries of greater complexity can be used to limit growth even further in images with more detail i=1
11 Morphological image processing(9.4 to 9.5.8) SLIDE 11/18
12 Morphological image processing(9.4 to 9.5.8) SLIDE 12/ Thinning: ThethinningofasetAbyastructuringelementB: A B=A (A B)=A (A B) c Symmetric thinning: Sequence of SEs, {B} = { B 1,B 2,B 3,...,B n}, where B i isarotatedversionofb i 1 A {B}=((...((A B 1 ) B 2 )...) B n )
13 Morphological image processing(9.4 to 9.5.8) SLIDE 13/ Thickening: Thickening is the morphological dual of thinning and is defined by: A B=A (A B), where B is a structuring element Similartothinning: A {B}=((...((A B 1 ) B 2 )...) B n ) Structuring elements for thickening are similar to those of Fig 9.21(a), but with all 1 s and 0 s interchanged Aseparatealgorithmforthickeningisseldomusedinpractice wethinthe background instead, and then complement the result
14 Morphological image processing(9.4 to 9.5.8) SLIDE 14/ Skeletons The algorithm proposed in this section is similar to the medial axis transformation (MAT). The MAT transformation is discussed in section and is far inferior to the skeletonization algorithm introduced in section The skeletonization algorithm proposed in this section also does not guarantee connectivity. We therefore do not discuss this algorithm. Illustration of the above comments...
15 Morphological image processing(9.4 to 9.5.8) SLIDE 15/18 A further illustration...
16 Morphological image processing(9.4 to 9.5.8) SLIDE 16/ Pruning Cleans up parasitic components left by thinning and skeletonization Use combination of morphological techniques Illustrative problem: Hand-printed character recognition Analyze shape of skeleton of character Skeletons characterized by spurs( parasitic components) Spurs caused during erosion of non-uniformities in strokes We assume that the length of a parasitic component does not exceed a specified number of pixels
17 Morphological image processing(9.4 to 9.5.8) SLIDE 17/18
18 Morphological image processing(9.4 to 9.5.8) SLIDE 18/18 Anybranchwiththreeorlesspixelsistobeeliminated (1) Three iterations of: X 1 =A {B} (2)FindalltheendpointsinX 1 : X 2 = 8 k=1 (X 1 B k ) (3)Dilateendpointsthreetimes,usingAasadelimiter: X 3 =(X 2 H) A, H= (4) Finally: X 4 =X 1 X 3
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