Image Analysis Morphological Image Analysis
14/2/2011Image AnalysisMorphological Image AnalysisChristophoros Nikoucnikou@cs.uoi.grImages taken from: R. Gonzalez and R. Woods. Digital Image Processing, Prentice Hall, 2008University of Ioannina - Department of Computer Science2Morphological Image Processingand AnalysisIn form and feature, face and limb,I grew so like my brother,That folks got taking me for himAnd each for one another.Henry Sambrooke Leigh,Carols of Cockayne, The TwinsC. Nikou – Image Analysis (T-14)1
14/2/2011Contents3Mathematical morphology provides tools forthe representationpand descriptionpof imagegregions (e.g. boundary extraction, skeleton,convex hull).It provides techniques for pre- and postprocessing of an image (morphologicalthi ithinning,pruning,ifiltfiltering).i )Its principles are based on set theory.Applications to both binary and graylevelimages.C. Nikou – Image Analysis (T-14)Preliminaries4The four horizontal and vertical neighbours ofpixel p are called 4-neighboursgof p and areapdenoted by N4(p).The four diagonal neighbours of a pixel p aredenoted by ND(p).Together N4(p) and ND(p) are called the 8neighbours of pixel p and are denoted byN8(p).C. Nikou – Image Analysis (T-14)2
14/2/20115Preliminaries (cont.)Adjacency of pixelsLet V be the set of intensity values used todefine the adjacency (e.g. V {1} for binaryimages).4-adjacency. Two pixels p and q with values inV are 4-adjacent if q is in N4(p).8-adjacency. Two pixels p and q with values inV are 8-adjacent if q is in N8(p).C. Nikou – Image Analysis (T-14)6Preliminaries (cont.)Adjacency of pixelsm-adjacencymadjacency (mixed adjacency)adjacency). Two pixels pand q with values in V are m-adjacent if q is in N4(p), or q is in ND(p) and the set N4(p) N4(q) hasno ppixels whose values are in V .Mixed adjacency is a modification of the 8adjacency. It is introduced to eliminateambiguities of 8-adjacency.C. Nikou – Image Analysis (T-14)3
14/2/2011Preliminaries (cont.)7Adjacency of pixelsPixels in abinary image8-adjacencyNot m-connected. Theyhave a common 4connected neighbor.m-adjacencym-connected. They do not have anycommon 4-connected neighbor.The role of m-adjacency is to define a singlepath between pixels. It is used in many imageanalysis and processing algorithms.C. Nikou – Image Analysis (T-14)Preliminaries (cont.)8Basic set operations.A B {w w A OR w B}A B {w w A AND w B}A B {w w A, w B} A B cAc {w w A}C. Nikou – Image Analysis (T-14)4
14/2/2011Preliminaries (cont.)9The above operations assume that theimagesg containingg the sets are binaryy andinvolve only the pixel location.Union and intersection are different when wedefine set operations involving intensityvalues:{}A B {min(a, b) a A, b B}A B max(a, b) a A, b BzThe elements of the setsare gray values on thesame location z.zC. Nikou – Image Analysis (T-14)Preliminaries (cont.)10Set reflection:S t translationSettl ti byb z:Bˆ {w w b, for b B}( B) z {c c b z , ffor b B}C. Nikou – Image Analysis (T-14)5
14/2/201111Preliminaries (cont.)Set reflection and translationare employed to structuringelements (SE).SE Are small sets or subimagesused to examine the imageunder study for properties ofNote: ggrayy colourinterestinterest.represents a value ofThe origin must be specified. one and white colour aZeros are appended to SE to zero value.give them a rectangular form.C. Nikou – Image Analysis (T-14)12Preliminaries (cont.)The origin of the SE B visits every pixel in an image A.It performs an operation (generally non linear) betweenits elements and the pixels under it.It is then decided if the pixel will belong to the resultingset or not based on the results of the operation.Zero padding is necessary (like in convolution) toensure that all of the elements of A are processedprocessed.C. Nikou – Image Analysis (T-14)6
14/2/2011Preliminaries (cont.)13For example, it marks the pixel under its centeras belonging to the result if B is completelycontained in A (Aœ 2, Bœ 2).C. Nikou – Image Analysis (T-14)Morphological Operations14 Some basic operations– Erosion.– Dilation.– Opening.– Closing. Applications– Morphological filtering.– The hit-or-miss transformation.C. Nikou – Image Analysis (T-14)7
14/2/2011Erosion15The erosion of a set A by a SE B is defined asA B {z ( B) z A}The result is the set of all points z such that Btranslated by z is contained in A.Equivalently:A B {z ( B) z Ac }C. Nikou – Image Analysis (T-14)16Erosion (cont.)Erosion is a shrinking operationC. Nikou – Image Analysis (T-14)8
14/2/2011Erosion (cont.)17C. Nikou – Image Analysis (T-14)Erosion (cont.)18Erosion by a square SE of varying size11x11Original image45x4515x15C. Nikou – Image Analysis (T-14)9
14/2/201119Erosion (cont.)Erosion can split apart joined objectsErosion can strip away extrusionsWatch out: Erosion shrinks objectsC. Nikou – Image Analysis (T-14)Dilation20The dilation of a set A by a SE B is defined asA B {z ( Bˆ ) z A }The result is the set of all points z such that thereflected B translated overlap with A at at leastone elementelement.Equivalently: A B {z [( Bˆ ) z A] A}C. Nikou – Image Analysis (T-14)10
14/2/201121Dilation (cont.)Dilation is a thickening operationC. Nikou – Image Analysis (T-14)22Dilation (cont.)Dilation bridges gaps.Contrary to low pass filtering it produces a binary image.C. Nikou – Image Analysis (T-14)11
14/2/2011Dilation (cont.)23Dilation can repair breaksDilation can repair intrusionsWatch out: Dilation enlarges objectsC. Nikou – Image Analysis (T-14)Duality24Erosion and dilation are dual operations withrespect to set complementation and reflection:Also,( A B)c Ac Bˆ( A B )c Ac BˆThe duality is useful when the SE is symmetric:The erosion of an image is the dilation of itsbackground.C. Nikou – Image Analysis (T-14)12
14/2/2011Compound Operations25More interesting morphological operationscan be pperformed byy combiningg erosionsand dilations in order to reduce shrinking orthickening.The most widely used of these compoundoperations are:– Opening– ClosingC. Nikou – Image Analysis (T-14)Opening26The opening of set A by structuring element Bis defined asA B ( A B) Bwhich is an erosion of A by B followed by adilation of the result byy B.C. Nikou – Image Analysis (T-14)13
14/2/2011Opening (cont.)27Geometric interpretation: The boundary ofthe openingpg is defined byyppoints of the SEthat reach the farthest into the boundary of Aas B is “rolled” inside of this boundary.A B {( B) z ( B ) z A}C. Nikou – Image Analysis (T-14)Opening (cont.)28Notice the difference with the simple erosion:A B {z ( B) z A}A B {( B) z ( B) z A}If B translated by z lies inside A, then the resultcontains the whole set of points covered by the SEand not only its center as it is done in the erosion.C. Nikou – Image Analysis (T-14)14
14/2/2011Closing29The closing of set A by structuring element Bis defined asA B ( A B) Bwhich is an dilation of A by B followed by anerosion of the result byy B.C. Nikou – Image Analysis (T-14)30Closing (cont.)It has a similar geometric interpretationp that B is rolled on the outside of theexceptboundary:A B {w ( B) z A , for all translates of ( B ) z containing w}C. Nikou – Image Analysis (T-14)15
14/2/2011Opening and Closing31Erosion: elements where thedisk can not fit are eliminated.Opening: outward corners arerounded.Dilation: inward intrusions arereduced in depth.Closing: inward corners arerounded.C. Nikou – Image Analysis (T-14)Duality32Opening and closing are dual operations.Erosion-Dilation dualityOpening-Closing duality( A B)c Ac Bˆ( A B)c Ac Bˆ( A B )c Ac Bˆ( A B )c Ac BˆC. Nikou – Image Analysis (T-14)16
14/2/201133Properties of Opening and ClosingOpening:A B AC D C B D B( A B) B A BClosing:A A BC D C B D B( A B) B A BThe last properties, in each case, indicate thatmultiple openings or closings have no effect afterthe first application of the operatorC. Nikou – Image Analysis (T-14)34Morphological Filtering ExampleThe image contains noise: Light elements on darkbackground. Dark elements on the lightcomponents of thefingerprint.Objective: Eliminate noise while distorting theimage as little as possible.We will apply an opening followed by closing.C. Nikou – Image Analysis (T-14)17
14/2/201135Morphological Filtering Example(cont.)AA BBackground noise completely removed (noisecomponents smaller than the SE).The size of the dark noise elements in the fingerprintstructure increased (inner dark structures).C. Nikou – Image Analysis (T-14)36Morphological Filtering Example(cont.)A B( A B) B A BThe dilation reduced the size of the inner noise oreliminated it completely.However, new gaps were created by the openingbetween the fingerprint ridges.C. Nikou – Image Analysis (T-14)18
14/2/201137Morphological Filtering Example(cont.)A BA B BThe dilation reduces the new gaps between theridges but it also thickens the ridges.C. Nikou – Image Analysis (T-14)38Morphological Filtering Example(cont.)A B B[ A B B] B ( A B) BThe final erosion (resulting to a closing of theopened image) makes the ridges thinner.C. Nikou – Image Analysis (T-14)19
14/2/201139Morphological Filtering Example(cont.)A( A B) BThe final result is clean of noise but some ridges werenot fully repaired.We should impose conditions for maintaining theconnectivity (we will see a more advanced algorithm).C. Nikou – Image Analysis (T-14)40The Hit-or-Miss TransformationBasic tool for shape detection.y B: the set of all locations of theErosion of A byorigin of B that B is completely contained in A.Alternatively, it is the set of all locations that Bfound a match (hit) in A.C. Nikou – Image Analysis (T-14)20
14/2/201141The Hit-or-Miss Transformation(cont.)There are many possible locations for the shapewe search (the SE!). If we are looking for disjoint(disconnected) shapes it is natural to assume abackground for it.Therefore, we seek to match B in A andsimultaneously we seek to match the backgroundof B in Ac.Mathematically, the hit-or-miss transformation is:A * B ( A B) ( Ac Bb )C. Nikou – Image Analysis (T-14)42The Hit-or-Miss Transformation(cont.)We seek to locate the shapeD in the image A.We define a thin backgroundW for the shape.We takeWt k theth intersectioni tti offthe two resultsA * D ( A D) Ac (W D) C. Nikou – Image Analysis (T-14)21
14/2/201143Morphological AlgorithmsUsing these morphological operations we mayextract image components for shape representation: Shape boundaries. Region filling. Connected components Convex hull. Shape thinning and thickening.thickening Skeletons.We may also accomplish a morphological imagereconstruction.C. Nikou – Image Analysis (T-14)44Boundary ExtractionThe boundary of a set A, denoted by β(A),mayy be obtained by:yβ ( A) A ( A B)C. Nikou – Image Analysis (T-14)22
14/2/201145Boundary Extraction (cont.)The boundary is one pixel thick due to the3x3 SE. Other SE would result in thickerboundaries.Original ImageExtracted BoundaryC. Nikou – Image Analysis (T-14)Region Filling46Given a pixel inside a boundary, regiong attemptsp to fill the area surrounded byyfillingthat boundary with 1s.Given a point insidehere, can we fillf thewhole circle?C. Nikou – Image Analysis (T-14)23
14/2/201147Region Filling (cont.)Form a set X0 with zeros everywhere exceptat the seed ppoint of the region.gThen,X k ( X k 1 B) Ac , k 1, 2,3,.Where B is a 3x3 crosscross-shapedshaped SE.The algorithm terminates when Xk Xk-1.The set union of Xk and A contains all thefilled holes and their boundaries.C. Nikou – Image Analysis (T-14)48Region Filling (cont.)This is a first example where the morphologicalp((dilation)) is conditioned.operationThe intersection of the result with Ac limits theresult inside the region of interest.X k ( X k 1 B) Ac , k 1, 2,3,.C. Nikou – Image Analysis (T-14)24
14/2/2011Region Filling (cont.)49C. Nikou – Image Analysis (T-14)Region Filling (cont.)50Original ImageOne RegionFilledAll RegionsFilledC. Nikou – Image Analysis (T-14)25
14/2/201151Extraction of connected components Given a pixel on a connected component, find the rest ofthe pixels of that component. The algorithm may be applied to many connectedcomponents provided we know a pixel on each one ofthem. Disadvantage: we have to provide a pixel on the connectedcomponent. There are more sophisticated algorithms that detect thenumber of components without manual interaction. Thepurpose here is to demonstrate the flexibility ofmathematical morphology.C. Nikou – Image Analysis (T-14)52Extraction of connected components(cont.)Form a set X0 with zeros everywhere except atthe seed ppoint of the connected components.pThen,X k ( X k 1 B ) A, k 1, 2,3,.Where B is a 3x3 square-shapedsquare shaped SE.The algorithm terminates when Xk Xk-1.Xk contains all the connected components.C. Nikou – Image Analysis (T-14)26
14/2/201153Extraction of connected components(cont.)Note the similarity with region filling. The onlydifference is the use of A instead of Ac.This is not surprising as we search for foregroundobjects.C. Nikou – Image Analysis (T-14)54Extraction of connected components(cont.)Image of chicken filetcontaining bone fragmentsResult of simple thresholdingImage erosion to retain onlyobjects of significant size.15 connected components detected with four of them beingsignificant in size. This is an indication to remove the chickenfilet from packaging.C. Nikou – Image Analysis (T-14)27
14/2/2011Convex Hull55A set A is convex if the straight line segment joiningany two points in A lies entirely within A.The convex hull H of an arbitrary set S the smallestconvex set containing S.The difference H-S is called convex deficiency.The convex hull and the convex deficiency are usefulquantities to characterize shapes.qpWe present here a morphological algorithm to obtainthe convex hull C(A) of a shape A.C. Nikou – Image Analysis (T-14)Convex Hull (cont.)56The procedure requires four SE Bi, i 1, 2, 3, 4, andimplements the following equation:X ki ( X k 1 * B i ) A, i 1, 2,3, 4 and k 1, 2,3,.with X 0i Awith i referring to the SE and k to iteration.Then lettingThen,D i X kiThe convex hull of A is4C ( A) D ii 1C. Nikou – Image Analysis (T-14)28
14/2/2011Convex Hull (cont.)57X ki ( X k 1 * B i ) Ai 1, 2,3, 4 and k 1, 2,3,. with X 0i A4D i X ki , C ( A) D ii 1The method consists of iteratively applying the hit-or misstransform to A with B1.When no changes occur we perform the union with A andsave the result to D1.The procedure is then continued with B2 (applied to A) and soon.The union of the results is the convex hull of A.Note that a simple implementation of the hit or miss is applied(no background match is required).C. Nikou – Image Analysis (T-14)58Convex Hull (cont.)The hit-or-miss transform tries to find (“hit”) these structuresin the image.The SE has points with “don’t care” condition. For all the SE,a match is found in the image when these conditions hold: the central pixel in the 3x3 region in the image is 0. the three shaded pixels under the mask are 1s.The remaining pixels do not matter.C. Nikou – Image Analysis (T-14)29
14/2/2011Convex Hull (cont.)59- Problem.P blThe result is convexbut greater than thetrue convex hull.C. Nikou – Image Analysis (T-14)Convex Hull (cont.)60Solution: limit the growth so that it does not extend past thehorizontal and vertical limits of the original set of points.Original imageInitial convex hullRefined convex hullMore complex boundaries have been imposed to images withfiner details in their structure (e.g. the maximum of thehorizontal vertical and diagonal dimensions could be used).C. Nikou – Image Analysis (T-14)30
14/2/2011Thinning61The thinning of a set A, by a SE B may bedefined in terms of the hit-or-miss transform:A B A ( A * B) A ( A * B)cNo background match is required and the hitor-miss part is reduced to simple erosion.A more advanced expression is based on asequence of SE {B} {B1 , B 2 , B 3 ,., B n },where each Bi is a rotated version of Bi-1.C. Nikou – Image Analysis (T-14)62Thinning (cont.)The thinning by a sequence of SE is defined by:A {B} ((.(((( (( A B1 ) B 2 )).)) B n )The process is to thin A by one pass by B1,then thin the result with one pass of B2, andso on, until we employ Bn.Th entireTheti process iis repeatedt d untiltil no ffurtherthchanges occur. Each individual thinning isperformed by:A B A ( A * B)C. Nikou – Image Analysis (T-14)31
14/2/2011Thinning (cont.)63 No change between theresult of B7 and B8 at thefirst pass. No change between theresults of B1, B2, B3, B4 atthe second pass. No change occurs afterthe second pass by B6 . The final result isconverted tom-connectivity to have aone pixel thick structure.C. Nikou – Image Analysis (T-14)Thickening64Thickening is a morphological dual of thinning:AB A ( A * B)The SE have the same form as the ones usedfor thinning with the 1s and 0s interchanged.It may also be defined by a sequence ofoperations:A {B} ((.(( AB1 )B 2 ).)Bn )C. Nikou – Image Analysis (T-14)32
14/2/2011Thickening (cont.)65In practice, a separate algorithm is seldom used forthickening.gThe usual process is to thin the background of theset in question and then take the complement of theresult.The advantage is that the thinned background formsa boundaryy for the thickeninggpprocess. Directimplementation of thickening has no stoppingcriterion.A disadvantage is that there may be isolated pointsneeding post-processing.C. Nikou – Image Analysis (T-14)Thickening (cont.)66Original set AThinning ofAcThickened setobtained bycomplementing theresult of thinning.AcElimination of disconnected points.C. Nikou – Image Analysis (T-14)33
14/2/2011Skeletons67The notion of a skeleton S(A) of aset A, intuitively, has the followingproperties: If z is a point belonging to S(A)and (D)z is the largest diskcentered at z and contained in A:one cannot find a larger disk (notnecessarily centered at z)containingt i i (D)( )z andd iincludedl d d iin A. (D)z is then called maximum disk. The maximum disk touches theboundary of A at two or moredifferent points.C. Nikou – Image Analysis (T-14)Skeletons (cont.)68It may be shown that a definition of the skeletonmayy be givengin terms of erosions and openings:pgKS ( A) Sk ( A), with Sk ( A) ( A kB) ( A kB) Bk 0with ( A kB) ((.( A B ) B) .) B)k successive erosionsK is the last iterative step before A erodes to anempty set:K max{k A kb }C. Nikou – Image Analysis (T-14)34
14/2/2011Skeletons (cont.)69The previous formulation allows the iterativereconstruction of A from the sets forming itsskeleton by:KA ( S k ( A) kB ) ,k 0with S k ( A) B ((.( Sk ( A) B ) B ) .) B)k successive dilations of the set Sk ( A )C. Nikou – Image Analysis (T-14)70Skeletons (cont.)The skeleton is thicker than essential. disconnected.The morphologicalformulation does notguarantee connectivity.More assumptions areneeded to obtain amaximally thin andconnected skeleton.C. Nikou – Image Analysis (T-14)35
14/2/201171Morphological Reconstruction The morphological algorithms discussed sofar involve an imageg and a SE. Morphological reconstruction involves twoimages and a SE.– The marker image containing the startingpoint of the transformation.– TheTh maskk image,iwhichhi h constraintst i t thethtransformation.– The SE is used to define connectivity.C. Nikou – Image Analysis (T-14)72Morphological Reconstruction (cont.)The geodesic dilation of size 1 of a marker imageF by a SE B, with respect to a mask image G isdefined by:(1)DG ( F ) ( F B) GSimilarly, the geodesic dilation of size n is defined by:DG( n ) ( F ) DG(1) DG( n 1) ( F ) with DG(0) ( F ) FThe intersection operator at each step guaranteesthat the growth (dilation) of marker F is limited bythe mask G.C. Nikou – Image Analysis (T-14)36
14/2/201173Morphological Reconstruction (cont.)Geodesic dilation of size 1.The result will not contain elements not belonging tothe mask G.C. Nikou – Image Analysis (T-14)74Morphological Reconstruction (cont.)The geodesic erosion of size 1 of a marker imageF by a SE B, with respect to a mask image G isdefined by:(1)EG ( F ) ( F B) GSimilarly, the geodesic erosion of size n is defined by:EG( n ) ( F ) EG(1) EG( n 1) ( F ) with EG(0) ( F ) FThe union operator guarantees that the geodesicerosion of marker F remains greater than or equalto the mask G.C. Nikou – Image Analysis (T-14)37
14/2/201175Morphological Reconstruction (cont.)Geodesic erosion of size 1.The result will at least contain the mask G.C. Nikou – Image Analysis (T-14)76Morphological Reconstruction (cont.) The geodesic dilation and erosion are dualspto set complementation.pwith respect They always converge after a finite numberof steps:– Geodesic dilation: propagation of the marker isconstrained by the mask image.– GeodesicG d i erosion:ishrinkingh i ki off ththe markerk iisconstrained by the mask.C. Nikou – Image Analysis (T-14)38
14/2/201177Morphological Reconstruction (cont.)The morphological reconstruction by dilation ofmask image G from a marker image F is defined asthe geodesic dilation of F with respect to G, iterateduntil stability s achieved:RGD ( F ) DG( k ) ( F )with k such that:DG( k ) ( F ) DG( k 1) ( F )C. Nikou – Image Analysis (T-14)78Morphological Reconstruction (cont.)Example of morphologicalreconstruction by dilation.The mask,, marker,, SE and thefirst step of the algorithm arefrom the example of geodesicdilation.C. Nikou – Image Analysis (T-14)39
14/2/201179Morphological Reconstruction (cont.)The morphological reconstruction by erosion ofmask image G from a marker image F is defined asthe geodesic erosion of F with respect to G, iterateduntil stability is achieved:RGE ( F ) EG( k ) ( F )with k such that:EG( k ) ( F ) EG( k 1) ( F )The example is left as an exercise!C. Nikou – Image Analysis (T-14)80ApplicationsOpening by ReconstructionIn morphological opening, erosion removes smallobjects and dilation attempts to restore the shape ofthe objects that remain without the small objects.This is not accurate as it depends on the similaritybetween the shapes to be removed and the SE.Opening by reconstruction restores exactly theshapes of the objects that remain after erosion.C. Nikou – Image Analysis (T-14)40
14/2/201181ApplicationsOpening by Reconstruction (cont.)The opening by reconstruction of size n of an imageF is defined as the reconstruction by dilation of Ffrom the erosion of size n of F:OR( n ) ( F ) RFD [ ( F nB) ]The imageThiF isi usedd as ththe maskk andd ththe nerosions of F by B are used as the initial markerimage.C. Nikou – Image Analysis (T-14)82ApplicationsOpening by Reconstruction (cont.)We are interested in extracting characters with long verticalstrokes ( 50 pixels high).Original imageOpeningOne erosion by a 51x1 SEOpening by reconstructionC. Nikou – Image Analysis (T-14)41
14/2/2011ApplicationsRegion filling83No starting point is needed to be provided.gimageg I(x,y)( y) is used as a mask.The originalThe marker image is 1 I ( x, y ) if ( x, y ) is on the border of IF ( x, y ) 0otherwise Only dark pixels of I(x,y) touching the borderhave a value of 1 in F(x,y).F(x y)The binary image with all regions (holes) filled iscgiven by:H RIDc ( F ) C. Nikou – Image Analysis (T-14)84ApplicationsRegion filling (cont.)We wish to fill the hole of the image I.The complement builds a wall around the hole.The marker image F is one at the border exceptfrom border pixels of the original image.C. Nikou – Image Analysis (T-14)42
14/2/2011ApplicationsRegion filling (cont.)85The dilation of the marker F starts from theborder and grows inward.The complement is used as an AND mask: itprotects all foreground pixels (including the wall)from changing during the iterations.The last operation provides only the hole points.C. Nikou – Image Analysis (T-14)ApplicationsRegion filling (cont.)86Original imageComplement of original imageResult of hole fillingMarker image (1s almost everywhereon the border apart of some pointson the right border)C. Nikou – Image Analysis (T-14)43
14/2/201187ApplicationsBorder ClearingThe extraction of objects from an image is afundamental task in automated imageg analysis.yAn algorithm for removing objects that touch(are connected) to the image border is usefulbecause only complete objects remain for furtherprocessing. it is a signal that partial objects remain in thefield of view.C. Nikou – Image Analysis (T-14)88ApplicationsBorder Clearing (cont.)The original image is used as a mask.The marker image is I ( x, y ) if ( x, y ) is on the border of IF ( x, y ) otherwise 0The border clearing algorithm first computes theDmorphological reconstruction RI ( F ),which simply extracts the objects touching theborder and then obtains the new image with noobjects touching the borders I RID ( F ).C. Nikou – Image Analysis (T-14)44
14/2/2011ApplicationsBorder Clearing (cont.)89Original image IReconstruction by dilation ofthe 1s touching the borderReconstructed image I RID ( F )C. Nikou – Image Analysis (T-14)90Gray-Scale Morphology The image f (x,y) and the SEb(x,y) take real or integervalues.l SE may be flat or nonflat. Due to a number of difficulties(result interpretation, erosion isnot bounded by the image,etc ) symmetrical flat SE withetc.)origin at the center areemployed. Set reflection: bˆ( x, y ) b( x, y )C. Nikou – Image Analysis (T-14)45
14/2/201191Gray-Scale ErosionThe erosion of image f by a SE b at anylocation ((x,y),y) is defined as the minimum valueof the image in the region coincident with bwhen the origin of b is at (x,y):f ( x s, y t )}[ f b] ( x, y) (min{s ,t ) bIn practice, we place the center of the SE atevery pixel and select the minimum value ofthe image under the window of the SE.C. Nikou – Image Analysis (T-14)92Gray-Scale DilationThe dilation of image f by a SE b at any,y) is defined as the maximum valuelocation ((x,y)of the image in the window outlined by b:f ( x s, y t )}[ f b] ( x, y) max{( s ,t ) bThe SE is reflected as in the binary case.C. Nikou – Image Analysis (T-14)46
14/2/201193Gray-Scale Erosion and DilationOriginal imageErosion by a flat disk SE ofradius 2:Darker background,small bright dots reduced,dark features grew.Dilation by a flat disk SE ofradius 2:Lighter background,small dark dots reduced,light features grew.C. Nikou – Image Analysis (T-14)94Gray-Scale Morphology (nonflat SE)The erosion of image f by a nonflat SE bN isdefined as:{ f ( x s, y t ) bN ( s, t )}[ f bN ] ( x, y) ( smin,t ) bNThe dilation of image f by a nonflat SE bN isdefined as:[ f bN ] ( x, y) max { f ( x s, y t ) bN (s, t )}( s ,t ) bNWhen the SE is flat the equations reduce tothe previous formulas up to a constant.C. Nikou – Image Analysis (T-14)47
14/2/2011Duality95As in the binary case, erosion and dilation aredual operations with respect to functioncomplementation and reflection:( f b)c ( x, y ) ( f c bˆ)( x, y )Similarly,( f b)c ( x, y ) ( f c bˆ)( x, y )In what follows, we omit the coordinates forsimplicity.C. Nikou – Image Analysis (T-14)96Gray-Scale Opening and ClosingThe opening of image f by SE b is:f b ( f b) bThe closing of image f by SE b is:f b ( f b) bThey are also duals with respect to functioncomplementation and reflection:( f b)c f c bˆ( f b)c f c bˆC. Nikou – Image Analysis (T-14)48
14/2/201197Gray-Scale Opening and Closing(cont.)Geometric interpretation of opening:It is the highest value reached by any part of the SE as itpushes up against the under-surface of the image (up to thepoint it fits completely).It removes small bright details.Notice the similarity with binary opening (smooths outwardcorners from the inside).C. Nikou – Image Analysis (T-14)98Gray-Scale Opening and Closing(cont.)Geometric interpretation of closing:It is the lowest value reached by any part of the SE as itpushes down against the upper side of the image intensitycurve.It highlights small dark regions of the image.Notice the similarity with binary closing (smooths inwardcorners from the outside).C. Nikou – Image Analysis (T-14)49
14/2/201199Gray-Scale Opening and Closing(cont.)Properties of opening:(1)f b f(2) If f1 f 2 , then f1 b f 2 b(3) ( f b) b f bThe first property indicates that: the domain of the opening is a subset of thedomain of f and [ f b ]( x, y ) f ( x, y )C. Nikou – Image Analysis (T-14)100Gray-Scale Opening and Closing(cont.)Properties of closing:(1) f f b(2) If f1 f 2 , then f1 b f 2 b(3) ( f b) b f bThe first property indicates that: the domain of f is a subset of the domain of theclosing and f ( x, y ) [ f b ]( x, y )C. Nikou – Image Analysis (T-14)50
14/2/2011101Gray-Scale Opening and Closing(cont.)O i i l iimageOriginalOpeningOi byb a flflatt diskdi k SEof radius 3:Intensities of brightfeatures decreased,Effects on background arenegligible (as opposed toerosion).ClosingCli byb a flatfl t didiskk SE offradius 5:Intensities of dark featuresincreased,Effects on background arenegligible (as opposed todilation).C. Nikou – Image Analysis (T-14)102Gray-ScaleMorphological Algorithms Morphological smoothing Morphological gradient Top-hat transformation Bottom-hat transformation Granulometry Textural segmentationC. Nikou – Image Analysis (T-14)51
14/2/2011Morphological Smoothing103Opening suppresses light details smaller than the SEand closing suppresses (makes lighter) dark detailssmaller than the SE.They are used in combination as morphological filtersto eliminate undesired structures.Cygnus Loop sup
14/2/2011 3 5 Preliminaries (cont.) Adjacency of pixels Let V be the set of intensity values used tobe the set of intensity values used to define the adjacency (e.g. V {1} for binary images). 4-adjacency.Two pixels p and q with values in V are 4-adjacent if q is in N4(p). C. Nikou – Image Analysis (T-14) 8-adjacency.Two pixels p and q with values in V are 8-adjacent if q is in N8(p).
Using Morphological Analysis in Business Markets Kaj Storbacka & Suvi Nenonen To cite this article: Kaj Storbacka & Suvi Nenonen (2012) Competitive Arena Mapping: Market Innovation Using Morphological Analysis in Business Markets, Journal of Business-to-Business Marketing
Phylogeny between Molecular and Morphological Taxonomy Our orders sampling included representatives of the 4 data of Acantharia, including the first sequence criteria, for Holacanthida. Based on morphological genera we identified 24 (of the 49 described) and and 14 (of 18) families. Clades
between morphological irregularity and frequency was considered. In English, irregular verbs are more frequent than regular ones (Ullman, 1999 and Michel et al., 2011). We tested whether this frequency-irregularity relation observed in English would also hold in a more complex morphological system like Spanish.
L2: x 0, image of L3: y 2, image of L4: y 3, image of L5: y x, image of L6: y x 1 b. image of L1: x 0, image of L2: x 0, image of L3: (0, 2), image of L4: (0, 3), image of L5: x 0, image of L6: x 0 c. image of L1– 6: y x 4. a. Q1 3, 1R b. ( 10, 0) c. (8, 6) 5. a x y b] a 21 50 ba x b a 2 1 b 4 2 O 46 2 4 2 2 4 y x A 1X2 A 1X1 A 1X 3 X1 X2 X3
Morphological Image Processing and Analysis In form and feature, face and limb, . Its principles are based on set theory. Applications to both binary and graylevel images. 4 C.
The term isderived from "image metamorphosis" and should not be confused with morphological image processing operators which detect image features. Morphing is an image processing technique typically used as an animation tool for the metamorphosis from one image to another.
using image analysis Morphological Analysis of Yarrowia lipolytica under Stress Conditions through Image Processing Automatic counting of viable/non-viable yeasts by epifluorescence microscopy with acridine orange as dying agent Characterization of bubbles in a bubble column by image analysis Simultaneous monitoring of lactic acid bacteria andAuthor: Eugénio C. FerreiraPublish Year: 2003
Actual Image Actual Image Actual Image Actual Image Actual Image Actual Image Actual Image Actual Image Actual Image 1. The Imperial – Mumbai 2. World Trade Center – Mumbai 3. Palace of the Sultan of Oman – Oman 4. Fairmont Bab Al Bahr – Abu Dhabi 5. Barakhamba Underground Metro Station – New Delhi 6. Cybercity – Gurugram 7.
