1、Mathematical MorphologyA Geometric Approach to Image Processing and Analysis2Image Analysis and Processing Geometry Space Abstract SpaceLinearNon LinearLinearConvolutionFourier, WaveletTomographSplinesStatisticalMultivariate analysisNeural SetsStereologyMorphologicalMorphological FilteringGranulomet

2、ryRandom setsWatershedsSyntacticalSemantic approachGrammarsIndexation結構元素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。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二值圖像的表示一個矩陣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”表示不活動（背景）像素。處理圖像時，假

3、定所有不在矩陣邊框內的像素均為“0”值。如有界矩陣S，其中含有一個23的矩形帶下標的元素0表示 原點的位置 圖像形態學初步腐蝕膨脹膨脹與腐蝕的代數意義膨脹與腐蝕的濾波特點小結作業基礎平移概念將一個集合A平移距離x ，表示為A+x axa+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

4、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原點不在結構元素內2、膨脹（dilate）A被B膨脹表示為： 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 3、膨脹的等效表達式膨脹ABAB離散情況下的明克夫斯基和（膨脹）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；

5、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、模板設計與算法設計膨脹、腐蝕的組合濾波效果應用邊界提取 骨架抽取 極限腐蝕 Top-hat變換 流域變換 灰度形態變換 Basic Morphology OperatorsDilation, Erosion, Opening, Closing Basic Morphology AlgorithmsBoundary extractionRegion fillingHit-or-Miss transformationThinningThickeningPruningApplicationsFilteringSeg

6、mentationCoding & Compression Object detectionComputer visionQuestionWhat is Mathematical Morphology ?A Commercial AnswerMathematical Morphology is FAST !Mathematical Morphology is CHEAP !PhysicalSignal analysis techniques based on set theory aiming at the study of relations between physical and str

7、uctural propertiesSignal ProcessingNon 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

8、in binary and grayscale images.Shape Processing and AnalysisVisual 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 bas

9、ed on designing a shape operator which satisfies desirable properties. ExampleImage analysis consists of obtaining measurements characteristic to images under consideration.Geometric measurements (e.g., object location, orientation, area, length of perimeter)Grayscale ImagesBinary ImagesMorphologica

10、l Shape OperatorsObjects are opaque and shape information is not additive !Shapes are usually combined by means ofSet Union (overlapping objects)：Set Intersection (occluded objects):Morphological Shape OperatorsShape operators should distribute over set-unions and set-intersections (a type of “linea

11、rity”) !MorphologicalDilationMorphologicalErosionMorphological OperatorsErosions and dilations are the most elementary operators of mathematical morphology.More complicated morphological operators can be designed by means of combining erosions and dilations.QuestionWhat is Mathematical Morphology ?A

12、 (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 latticesMathematicalLattice theory

13、for objects or operators in continuous or discrete spacesTopology and stochastic modelsTranslation Invariant OperatorsMorphological Erosion“LINEARITY”TRANSLATION INVARIANCEMorphological ErosionMorphological ErosionPablo Picasso, Pass with the Cape, 1960StructuringElementMorphological Dilation“LINEAR

14、ITY”TRANSLATION INVARIANCEMorphological DilationMorphological DilationPablo Picasso, Pass with the Cape, 1960StructuringElementMorphological DilationMorphological OpeningMorphological OpeningPablo Picasso, Pass with the Cape, 1960StructuringElementMorphological OpeningIs a smoothing filter !Amount a

15、nd type of smoothing is determined by the shape and size of the structuring element.Approximates a shape from below, since Morphological Opening & ClosingDilation, Erosion, Opening, Closing Morphological Opening & ClosingOpeningSmoothes the contourBreaks narrow isthmusesEliminates thin protrusionsX

16、B is a subset of XClosing Smoothes the contourFuses narrow breaksEliminates small hollFill gaps in the contourX B is a subset of XFiltering ExampleBoundary ExtractionQuestionHenri Matisse, Woman with Amphoraand Pomegranates, 1952Can we automatically extract the largest connected component (the woman

17、s body) in this image ?AnswerORIGINALEROSIONMARKERMARKERMARKERMARKERThis is a morphological operator that filters out connected image components of a certain size and shape CONNECTED OPERATORS !Connected ComponentReconstructionGeodesic ReconstructionRegion Filling8-connected boundaryBeginning with a

18、 point P inside X and letDo UntilImportant ResultsIncreasingOperator+TranslationInvariantOperatorMain IdeaExamine 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 image and their interrelations.Results in image operators which are well suited for the analysis of the geometrical and topological