1、Mathematical MorphologyA Geometric Approach to Image Processing and Analysis2Image Analysis and Processing Geometry Space Abstract SpaceLinearNon LinearLinearlConvolutionlFourier, WaveletlTomographlSplinesStatisticallMultivariate analysislNeural SetslStereologyMorphologicallMorphological FilteringlG

2、ranulometrylRandom setslWatershedsSyntacticallSemantic approachlGrammarslIndexation結構元素4數學形態學研究幾何結構的基本思想是利用“結構元素”（structuring element）探測圖像，看能否將這個結構元素很好地填放在圖像的內部，同時驗證填放結構元素的方法是否有效。4結構元素的設計在處理實際問題中是非常重要的，它決定了抽取信息的結果，構造不同的結構元素，就可以完成不同的分析任務。AB二值圖像的表示4一個矩陣4圖像中位于原點處的像素值用帶“”號下標的字體表示，并約定用“1”表示活動（前景）像素，用“0”表

3、示不活動（背景）像素。處理圖像時，假定所有不在矩陣邊框內的像素均為“0”值。4如有界矩陣S，其中含有一個23的矩形4帶下標的元素0表示 原點的位置 0000011101110000S圖像形態學初步4腐蝕4膨脹4膨脹與腐蝕的代數意義4膨脹與腐蝕的濾波特點4小結4作業基礎平移概念將一個集合A平移距離x ，表示為A+x :AaxaxAaxa+xA+xA二值圖像的平移1、腐蝕（erode）定義集合A被集合B“腐蝕”，表示為BA:AxBxBA其中A為輸入圖像，B為結構元素 腐蝕的結果由將B平移x，但仍然包含在A內的所有x點組成。如果將B看作模板，則由在平移過程中，所有可以填入A 內部的模板的原點組成。腐

4、蝕還有幾種常用表示：E(A,B)，ERODE(A,B) 腐蝕的性質1、如果原點在結構元素的內部，則腐蝕后的圖像為原圖像的一個子集，即腐蝕具有收縮圖像的作用，也就是可以去除比模板小的噪聲；2、如果原點不在結構元素的內部，則腐蝕后的圖像可能不在原圖像的內部，反而可能具有填充圖像內孔洞的作用。AB原點在結構元素內部時的腐蝕AB原點不在結構元素內部時的腐蝕數值舉例010101001101 ,1 1011 10AB011000010000000BA原點不在結構元素內11111111101111,101111101110101111111101AB01011100101010001111001101000

5、111110BA11111111111111111111111111111111111)(BAA2、膨脹（、膨脹（dilate）A被B膨脹表示為BA： ccBABA)(Ac表示A 的補集。膨脹還可以用D(A,B)， DILATE(A,B)表示 ABAB利用圓盤對矩形膨脹，尖角被磨圓性質1、對前景的外部作了平滑濾波 2、滿足交換律 ABBA:BbbABA3、膨脹的等效表達式:AaaBBA膨脹ABAB離散情況下的明克夫斯基和（膨脹）小結小結1、膨脹可以實現圖像縫隙的連接；2、腐蝕可以去除小顆粒噪聲或毛刺；3、多種組合，實現開、閉、擊中、擊不中；4、典型的非線性濾波，濾波效果可交互控制；5、模板設計

6、與算法設計膨脹、腐蝕的組合濾波效果應用4邊界提取 4骨架抽取 4極限腐蝕 4Top-hat變換 4流域變換 4灰度形態變換 Basic Morphology OperatorspDilation, Erosion, Opening, Closing Basic Morphology AlgorithmspBoundary extractionpRegion fillingpHit-or-Miss transformationpThinningpThickeningpPruningApplicationsFilteringSegmentationCoding & Compression Obje

7、ct detectionComputer visionQuestionWhat is Mathematical Morphology ?A Commercial Answer Mathematical Morphology is FAST ! Mathematical Morphology is CHEAP !PhysicalSignal analysis techniques based on set theory aiming at the study of relations between physical and structural propertiesSignal Process

8、ingNon linear signal processing techniques based on minimum and maximum operationsEngineeringAlgorithm and software/hardware tools for developing signal processing applicationsAn (imprecise) Mathematical AnswerA mathematical tool for investigating geometric structure in binary and grayscale images.S

9、hape 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 analysis techniques are based on designing a shape opera

10、tor which satisfies desirable properties. ExampleZImage analysis consists of obtaining measurements characteristic to images under consideration.ZGeometric measurements (e.g., object location, orientation, area, length of perimeter)Grayscale ImagesBinary ImagesMorphological Shape Operators Objects a

11、re opaque and shape information is not additive ! Shapes are usually combined by means of Set Union (overlapping objects)： Set Intersection (occluded objects):XX12X1X2XXXXc2112X2X1Morphological Shape Operators Shape operators should distribute over set-unions and set-intersections (a type of “linear

12、ity”) !()=()()XXXX1212MorphologicalDilation()=()()XXXX1212MorphologicalErosionMorphological 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.QuestionWha

13、t is Mathematical Morphology ?A (precise) Mathematical AnswerAlgebra Complete LatticesOperators Erosions-DilationsMathematical MorphologyTopology Hit-or-MissGeometry Convexity - Connectivity DistanceApplications Image Processing and AnalysisA mathematical tool that studies operators on complete latt

14、icesMathematicalLattice theory for objects or operators in continuous or discrete spacesTopology and stochastic modelsTranslation Invariant Operators()=( )XXhhXXhhMorphological Erosion()=()()XXXX1212“LINEARITY”()=( )XXhhTRANSLATION INVARIANCE|)(XBhBXXhMorphological ErosionBhXBX|)(XBhBXXhMorphologica

15、l ErosionPablo Picasso, Pass with the Cape, 1960StructuringElementMorphological Dilation()=()()XXXX1212“LINEARITY”()=( )XXhhTRANSLATION INVARIANCE|)(XBhBXXhMorphological Dilation|)(XBhBXXhXBXhBMorphological DilationPablo Picasso, Pass with the Cape, 1960StructuringElementMorphological DilationMorpho

16、logical Opening|)(XBBBBXBXhhBhBXXBBXBX)(Morphological OpeningPablo Picasso, Pass with the Cape, 1960StructuringElementMorphological Opening Is a smoothing filter ! Amount and type of smoothing is determined by the shape and size of the structuring element. Approximates a shape from below, since XBXM

17、orphological Opening & ClosingDilation, Erosion, Opening, Closing Morphological Opening & Closing Opening Smoothes the contour Breaks narrow isthmuses Eliminates thin protrusions X B is a subset of X Closing Smoothes the contour Fuses narrow breaks Eliminates small holl Fill gaps in the contour X B

18、is a subset of XFiltering ExampleBoundary Extraction)()(XXXBQuestionHenri Matisse, Woman with Amphoraand Pomegranates, 1952Can we automatically extract the largest connected component (the womans body) in this image ?AnswerORIGINALEROSION(MARKER)ORIGINAL B MARKERMARKERMARKERMARKERThis is a morpholog

19、ical operator that filters out connected image components of a certain size and shape CONNECTED OPERATORS !Connected Component,|)(XCCxCXxC)()(XXXxxReconstruction)()()(XXxXx Geodesic Reconstruction)()(XXxMxM )()(0XXMM ),()(xBXMXxM Region Filling 8-connected boundary Beginning with a point P inside X

20、and let Do UntilPX 0ckkABXX)(11kkXXImportant ResultsXXXX1212()()IncreasingOperator+! !)(BXBXXBBTranslationInvariantOperator()=( )XXhhMain 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

21、, called structuring elements, one can extract useful information about the shape of the different parts of the image and their interrelations. Results in image operators which are well suited for the analysis of the geometrical and topological structure of an image. QuestionWhat about gray-scale images ?Greyscale Erosion()=()()FFFF1212“LINEARITY”MINIMUM)()()(