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1、,Mathematical Morphology,A Geometric Approach to Image Processing and Analysis,2,Image Analysis and Processing,Geometry Space Abstract SpaceLinearNon Linear,结构元素,数学形态学研究几何结构的基本思想是利用“结构元素”（structuring element）探测图像，看能否将这个结构元素很好地填放在图像的内部，同时验证填放结构元素的方法是否有效。结构元素的设计在处理实际问题中是非常重要的，它决定了抽取信息的结果，构造不同的结构元素，就可以
2、完成不同的分析任务。,B,二值图像的表示,一个矩阵图像中位于原点处的像素值用带“”号下标的字体表示，并约定用“1”表示活动（前景）像素，用“0”表示不活动（背景）像素。处理图像时，假定所有不在矩阵边框内的像素均为“0”值。如有界矩阵S，其中含有一个23的矩形带下标的元素0表示 原点的位置,图像形态学初步,腐蚀膨胀膨胀与腐蚀的代数意义膨胀与腐蚀的滤波特点小结作业,基础平移概念,将一个集合A平移距离x，表示为A+x,1、腐蚀（erode）定义,集合A被集合B“腐蚀”，表示为,其中A为输入图像，B为结构元素,腐蚀的结果由将B平移x，但仍然包含在A内的所有x点组成。如果将B看作模板，则由在平移过程中，
3、所有可以填入A 内部的模板的原点组成。腐蚀还有几种常用表示：E(A,B)，ERODE(A,B),腐蚀的性质,1、如果原点在结构元素的内部，则腐蚀后的图像为原图像的一个子集，即腐蚀具有收缩图像的作用，也就是可以去除比模板小的噪声；2、如果原点不在结构元素的内部，则腐蚀后的图像可能不在原图像的内部，反而可能具有填充图像内孔洞的作用。,数值举例,原点不在结构元素内,2、膨胀（dilate）,A被B膨胀表示为,：,Ac表示A 的补集。膨胀还可以用D(A,B)，DILATE(A,B)表示,性质,1、对前景的外部作了平滑滤波,2、满足交换律,3、膨胀的等效表达式,膨胀,小结,1、膨胀可以实现图像缝隙的连接
4、；2、腐蚀可以去除小颗粒噪声或毛刺；3、多种组合，实现开、闭、击中、击不中；4、典型的非线性滤波，滤波效果可交互控制；5、模板设计与算法设计,膨胀、腐蚀的组合滤波效果,应用,边界提取 骨架抽取 极限腐蚀 Top-hat变换 流域变换 灰度形态变换,Basic Morphology OperatorsDilation,Erosion,Opening,Closing Basic Morphology AlgorithmsBoundary extractionRegion fillingHit-or-Miss transformationThinningThickeningPruning,Appli
5、cations,FilteringSegmentationCoding&Compression Object detectionComputer vision,Question,What is Mathematical Morphology?,A Commercial Answer,Mathematical Morphology is FAST!Mathematical Morphology is CHEAP!,Physical,Signal analysis techniques based on set theory aiming at the study of relations bet
6、ween physical and structural properties,Signal Processing,Non linear signal processing techniques based on minimum and maximum operations,Engineering,Algorithm and software/hardware tools for developing signal processing applications,An(imprecise)Mathematical Answer,A mathematical tool for investiga
7、ting geometric structure in binary and grayscale images.,Shape Processing and Analysis,Visual perception requires transformation of images so as to make explicit particular shape information.Goal:Distinguish meaningful shape information from irrelevant one.The vast majority of shape processing and a
8、nalysis techniques are based on designing a shape operator which satisfies desirable properties.,Example,Image analysis consists of obtaining measurements characteristic to images under consideration.Geometric measurements(e.g.,object location,orientation,area,length of perimeter),Grayscale Images,B
9、inary Images,Morphological Shape Operators,Objects are opaque and shape information is not additive!Shapes are usually combined by means ofSet Union(overlapping objects)：Set Intersection(occluded objects):,Morphological Shape Operators,Shape operators should distribute over set-unions and set-inters
10、ections(a type of“linearity”)!,MorphologicalDilation,MorphologicalErosion,Morphological Operators,Erosions and dilations are the most elementary operators of mathematical morphology.More complicated morphological operators can be designed by means of combining erosions and dilations.,Question,What i
11、s Mathematical Morphology?,A(precise)Mathematical Answer,Algebra Complete Lattices,Operators Erosions-Dilations,Mathematical Morphology,Topology Hit-or-Miss,Geometry Convexity-Connectivity Distance,A mathematical tool that studies operators on complete lattices,Mathematical,Lattice theory for object
12、s or operators in continuous or discrete spaces,Topology and stochastic models,Translation Invariant Operators,Morphological Erosion,“LINEARITY”,TRANSLATION INVARIANCE,Morphological Erosion,Morphological Erosion,Pablo Picasso,Pass with the Cape,1960,StructuringElement,Morphological Dilation,“LINEARI
13、TY”,TRANSLATION INVARIANCE,Morphological Dilation,Morphological Dilation,Pablo Picasso,Pass with the Cape,1960,Morphological Dilation,Morphological Opening,Morphological Opening,Pablo Picasso,Pass with the Cape,1960,Morphological Opening,Is a smoothing filter!Amount and type of smoothing is determin
14、ed by the shape and size of the structuring element.Approximates a shape from below,since,Morphological Opening&Closing,Dilation,Erosion,Opening,Closing,Morphological Opening&Closing,OpeningSmoothes the contourBreaks narrow isthmusesEliminates thin protrusionsX B is a subset of XClosing Smoothes the
15、 contourFuses narrow breaksEliminates small hollFill gaps in the contourX B is a subset of X,Filtering Example,Boundary Extraction,Question,Henri Matisse,Woman with Amphoraand Pomegranates,1952,Can we automatically extract the largest connected component(the womans body)in this image?,Answer,ORIGINA
16、L,MARKER,MARKER,MARKER,MARKER,This is a morphological operator that filters out connected image components of a certain size and shape CONNECTED OPERATORS!,Connected Component,Reconstruction,Geodesic Reconstruction,Region Filling,8-connected boundaryBeginning with a point P inside X and letDo Until,
17、Important Results,+,Main Idea,Examine the geometrical structure of an image by matching it with small patterns at various locations.By varying the size and shape of the matching patterns,called structuring elements,one can extract useful information about the shape of the different parts of the imag
18、e and their interrelations.Results in image operators which are well suited for the analysis of the geometrical and topological structure of an image.,Question,What about gray-scale images?,Greyscale Erosion,Grayscale Dilation,Grayscale Dilation&Erosion,Grayscale Opening&Closing,Greyscale Opening,Grayscale Morphology,ORIGINAL,EROSION,DILATION,OPENING,Remark,An Application-Target Detection,DATA,An Application:Target Detection,DATA,An Application:Target Detection,DATA,FINAL RESULT,
