Chapter 9: Morphological image processingIntroduction: the language and the tool of mathematical morphologicalis set theory simplify image data hold basic shape property eliminate irrelevant structureChapter 9: Morphological image processingu preliminariesu dilation and erosionu opening and closingu the hit-or-miss transformationu some basic morphological algorithms9.1 preliminaries1. Some basic concepts(1) subset:(2) union: (3) Intersection: (4) Disjoint or exclusive:(7) reflection:(5) complement:(6) difference:(8) translation:2. Logic operation9.1 preliminariesNOT 、AND、OR9.2 dilation and erosiondilationerosionduals of dilation and erosion9.2 dilation and erosion 1. dilationThe dilation of A and B is denotedInterpretation： u obtaining the reflection of B about its origin u shifting the reflection of B by z u the set of all displacements that and A overlap by at least one element i.e.(1) definitionE.g.(2) Implement steps:1) Design a structuring element and itsreflection BOriginal image regionA(2) Implement steps:2) Shifting origin of the reflection to background element adjacent to theobjective border3) if and A overlap by at least one element,the origin of is set objective element4) Repeat and repeat 2) and 3), until it is overOriginal image regionA(2) Implement steps:Dilation of A and BE.g.B将断裂开的目标物进行合并，便于对其整体的提取。(a) 原图 (b) 膨胀一次结构元素?2. erosion(1) definitionThe erosion of A and B is denoted A BA BInterpretation：ushifting the origin of B by z u the set of all displacements that B and A overlap completely(2) Implement steps:1) Design a structuring elementBE.g.Original image regionA2) Shifting origin of B to objective element3) if B and A overlap completely, the origin ofB is still set objective element(2) Implement steps:E.g.BOriginal image regionA Erosion of A and B(a) 原图 (b)腐蚀两次腐蚀处理可以将粘连在一起的不同目标物分 离，并可以将小的颗粒噪声去除。3. duals of dilation and erosionDilation and erosion are duals of each other with respect to set complementation and reflect.(A B) cAcand(a )A(b )B(c )(d )A B(e )Ac(f)(g )Ac(h )9.3 opening and closingopeningclose轮廓变得光滑、断 开狭窄的间断和消 除细的突出物轮廓变得光滑、消 弥狭窄的间断和长 细的鸿沟、消除小 的孔洞，并填补轮 廓线中的断裂1. openingthe opening A by B is the erosion of A by B, followed by a dilation of the result by B.the closing A by B is the dilation of A by B, followed erosion by a of the result by B.2. closing腐蚀膨胀(b)开运算结果 (a) 原图 (c) 腐蚀运算结果腐蚀膨胀(a) 原图(c) 膨胀运算结果(b)闭运算结果9.4 the hit-or-miss transformation前面介绍了数学形态学的4个基本运算。有人将击中-击不中变换也看作二值数学形态学的基本运算，若将击中-击不中变换与前4种基本运算结合，还可以组成另一些形态分析的组合运算和基本运算。9.4 the hit-or-miss transformation数学形态学里的击中-击不中变换是形状检测 的一种基本工具。它实际上对应两个操作，所以 用到两个结构元素。设A为原始图像，E和F是一对 不重合的集合，则击中-击不中变换用 表示，定 义为： (a)(b)(c)(d)(e)(f)(a)(b)(c)(d)9.5 some basic morphological algorithmsu Boundary Extraction u Region Filling u Extraction of Connected Component u Convex Hull u Thinning u Thickening u Skeletons1. Boundary ExtractionThe boundary of a set A , denoted by , can be obtained by first eroding A by B and then performing the set difference between A and its erosion. That is,1. Boundary Extraction2. Region FillingA simple algorithm for region filling is based on set dilations, complementation, and intersection.steps:(1) Choice a point p inside the boundary, let X0=p.(2)Be eroded X0 by structuring element B, and(1) then performing the set intersection between(2) its dilation and Ac.(3)(3)Repeat and repeat (2),until Xk=Xk-1.(4)(4)Obtains the set union of Xk and A.2. Region Filling2. Region Filling3. Extraction of Connected ComponentLet Y represent a connected component contained in a set A and assume that a point p of Y is known. The following iterative expression yields all the elements of Y:steps:(1) Choice a point p inside Y, let X0=p.(2)Be eroded X0 by structuring element B, and(1) then performing the set intersection between(2) its dilation and A.(3)(3)Repeat and repeat (2),until Xk=Xk-1.3. Extraction of Connected Component3. Extraction of Connected Component4. Convex HullA 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. The set difference H-S is called the convex deficiency of S. Let , then the convex hull of A is:4. Convex Hull5. ThinningThe thinning of a set A by a structuring element B, denoted ,can be defined in terms of the hit-or-miss transform: 结构元 素序列5. Thinning6. ThickeningThickening is the morphological dual of thinning and is defined by expression结构元 素序列6. Thickening7. SkeletonsThe skeleton of A can be expressed in terms of erosion and openings, that is:7. SkeletonsAlso, A can be reconstructed from these subsets by using the equation7. Skeletons7. Skeletons