Morphological Image Processing

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Morphological Image Processig Biary dilatio ad erosio Set-theoretic iterpretatio Opeig, closig, morphological edge detectors Hit-miss filter Morphological filters for gray-level

1 Morphological Image Processig Biary dilatio ad erosio Set-theoretic iterpretatio Opeig, closig, morphological edge detectors Hit-miss filter Morphological filters for gray-level images Cascadig dilatios ad erosios Rak filters, media filters, majority filters Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 1

2 Biary images are commo Biary image processig l Itermediate abstractio i a gray-scale/color image aalysis system Thresholdig/segmetatio Presece/absece of some image property l Text ad lie graphics, documet image processig Represetatio of idividual pixels as 0 or 1, covetio: l foregroud, object = 1 (white) l backgroud = 0 (black) Processig by logical fuctios is fast ad simple Shift-ivariat logical operatios o biary images: morphological image processig Morphological image processig has bee geeralized to gray-level images via level sets Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 2

3 Shift-ivariace Assume that digital images f [x,y] ad g[x,y] have ifiite support [ x, y] {, 2, 1,0,1,2, } {, 2, 1,0,1,2, }... the, for all itegers a ad b [ ] g[ x, y] f x, y Shiftivariat system [ ] g[ x a, y b] f x a, y b Shiftivariat system Shift-ivariace does ot imply liearity (or vice versa). Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 3

4 Structurig elemet Neighborhood widow operator { } = f x W f x, y 180 degree rotatio { } x, y y : x, y Π xy structurig elemet ˆ W f x, y { } = { f x + x ʹ, y + yʹ x, yʹ } The hat otatio idicates Π xy Example structurig elemets : : ʹ Π xy upright structurig elemet (i.e., ot rotated!) 5x5 square x x cross y y Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 4

5 Biary dilatio (expadig foregroud) { } g x, y = OR W f x, y := dilate( f,w ) x f [x, y] Π y xy g[x, y] Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 5

6 Biary dilatio with square structurig elemet { } g x, y = OR W f x, y := dilate( f,w ) Expads the size of 1-valued objects Smoothes object boudaries Closes holes ad gaps Origial (701x781) dilatio with 3x3 structurig elemet dilatio with 7x7 structurig elemet Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 6

7 Biary erosio (shrikig foregroud) g x, y = AND W ˆ f x, y { } := erode( f,w ) x f [x, y] Π y xy g[x, y] Caveat: There is aother defiitio of erosio i the literature, which flips the structurig elemet, as for dilatio. The Laguita olie videos use that alterative defiitio. Matlab fuctio imerode uses the defiitio o this slide. To the best of our kowledge, there is o such discrepacy defiig dilatio. Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 7

8 Biary erosio with square structurig elemet g x, y = AND W ˆ f x, y { } := erode( f,w ) Origial (701x781) erosio with 3x3 structurig elemet erosio with 7x7 structurig elemet Shriks the size of 1-valued objects Smoothes object boudaries Removes peisulas, figers, ad small objects Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 8

9 Relatioship betwee dilatio ad erosio Duality: erosio is dilatio of the backgroud dilate f,w erode f,w ( ) ( ) = NOT erode NOT f ( ) = NOT dilate NOT f, Wˆ ( ), Wˆ But: erosio is ot the iverse of dilatio f ( ) ( ) x, y erode dilate( f,w ),W dilate erode( f,w ),W Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 9

10 Example: blob separatio/detectio by erosio Origial biary image Circles (792x892) Erosio by 30x30 structurig elemet Erosio by 70x70 structurig elemet Erosio by 96x96 structurig elemet Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 10

Example: blob separatio/detectio by erosio Origial biary image Circles (792x892) Erosio by disk-shaped structurig elemet Diameter=15 Erosio by disk-shaped structurig

11 Example: blob separatio/detectio by erosio Origial biary image Circles (792x892) Erosio by disk-shaped structurig elemet Diameter=15 Erosio by disk-shaped structurig elemet Diameter=35 Erosio by disk-shaped structurig elemet Diameter=48 Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 11

12 Example: chai lik fece hole detectio Origial grayscale image Fece (1023 x 1173) Fece thresholded usig Otsu s method Erosio with 151x151 cross structurig elemet Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 12

13 Set-theoretic iterpretatio Set of object pixels F {( x, y): f ( x, y) = 1} Backgroud: complemet of foregroud set F c {( x, y): f ( x, y) = 0} Dilatio is Mikowski set additio G = F Π xy Cotiuous (x,y). Works for discrete [x,y] i the same way. Commutative ad associative! { ( ) F, ( p x, p ) y Π } xy = ( x + p x, y + p y ): x, y ( p x,p y ) Π xy = F + px,p y ( ) traslatio of F ( by vector p x, p ) y Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 13

14 Set-theoretic iterpretatio: dilatio G F y x Π xy G = F Π xy { ( ) F, ( p x, p ) y Π } xy = ( x + p x, y + p y ): x, y = F + px,p y ( p x,p y ) Π xy ( ) Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 14

15 Set-theoretic iterpretatio: erosio G F Mikowski set subtractio Not commutative! Not associative! y x Π xy G = ( p x, p y ) Π xy F + px,p y ( ) = FΘΠ xy Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 15

16 Opeig ad closig Goal: smoothig without size chage Ope filter Close filter ( ) = dilate erode f,w ( ) = erode dilate f,w ope f,w close f,w ( ( ),W ) ( ),W ( ) Ope filter ad close filter are biased l Ope filter removes small 1-regios l Close filter removes small 0-regios l Bias is ofte desired for ehacemet or detectio! Ubiased size-preservig smoothers close ope f,w ope close f,w ( ( ),W ) ( ( ),W ) ( ) = close ope f,w ( ) = ope close f,w close-ope ad ope-close are duals, but ot iverses of each other. Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 16

17 Small hole removal by closig Origial biary mask Dilatio 10x10 Closig 10x10 Differece to origial mask Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 17

Digital Image Processig: Berd Girod, 2013-2018

18 Morphological edge detectors f x, y dilate( f,w) f erode( f,w) f dilate( f,w) erode( f,w) Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 18

Recogitio by erosio Biary image f ( ) dilate erode( NOT f,w ),W 1400 2000 Structurig elemet W 44 34

19 Recogitio by erosio Biary image f ( ) dilate erode( NOT f,w ),W Structurig elemet W Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 19

20 Recogitio by erosio Structurig elemet W Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 20

Recogitio by erosio Biary image f ( ) dilate erode( NOT f,w ),W 1400 2000 Structurig elemet W 62 18

21 Recogitio by erosio Biary image f ( ) dilate erode( NOT f,w ),W Structurig elemet W Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 21

22 Recogitio by erosio Structurig elemet W Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 22

Hit-miss filter Biary image f ( ( ) & erode( f,w ),W ) dilate erode NOT f,v 1400 2000 Structurig elemet V 62 62

23 Hit-miss filter Biary image f ( ( ) & erode( f,w ),W ) dilate erode NOT f,v Structurig elemet V Structurig elemet W Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 23

Hit-miss filter Structurig elemet V 62 18 Structurig elemet W 62 18 Digital Image

24 Hit-miss filter Structurig elemet V Structurig elemet W Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 24

25 Morphological filters for gray-level images Threshold sets of a gray-level image f [x,y] ( ) = x, y Τ θ f x, y { : f x, y θ}, < θ < + Recostructio of origial image from threshold sets f x, y { ( )} = sup θ : x, y Τ θ f x, y Idea of morphological operators for multi-level (or cotiuous-amplitude) sigals l Decompose ito threshold sets l Apply biary morphological operator to each threshold set l Recostruct via supremum operatio l Gray-level operators thus obtaied: flat operators! Flat morphological operators ad thresholdig are commutative Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 25

26 Dilatio/erosio for gray-level images Explicit decompositio ito threshold sets ot required i practice Flat dilatio operator: local maximum over widow W g x, y = max{ W { f x, y }}:= dilate f,w Flat erosio operator: local miimum over widow W ( ) g x, y = mi{ Wˆ { f x, y } }:= erode f,w ( ) Biary dilatio/erosio operators cotaied as special case Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 26

27 1-d illustratio of erosio ad dilatio Structurig elemet legth = Dilatio Erosio Origial Structurig elemet: horizotal lie Amplitude Sample o. Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 27

28 Image example Origial 394 x 305 Dilatio 10x10 square Erosio 10x10 square Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 28

29 Flat dilatio with differet structurig elemets Origial Diamod Disk 20 degree lie 9 poits 2 horizotal lies Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 29

30 Example: coutig cois thresholded 1 coected compoet Origial 20 coected compoets dilatio thresholded after dilatio Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 30

31 Example: chai lik fece hole detectio Origial grayscale image Fece (1023 x 1173) Flat erosio with 151x151 cross structurig elemet Biarized by Thresholdig Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 31

32 Morphological edge detector origial f dilatio g g f g-f g f thresholded Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 32

33 Beyod flat morphological operators Geeral dilatio operator g x, y = sup α,β Like liear covolutio, with sup replacig summatio, additio replacig multiplicatio Dilatio with uit impulse d α,β does ot chage iput sigal: { f x α, y β + w α,β } = sup = 0 α = β = 0 else α,β { } w x α, y β + f α,β f x, y = sup α,β { f x α, y β + d α,β } Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 33

34 Fid such that Aswer: w α,β f x, y Flat dilatio as a special case = sup α,β { f x α, y β + w α,β } = dilate f,w w α,β = 0 α,β Π xy else ( ) Hece, write i geeral g!" x, y# $ = sup{ f!" x α, y β # $ + w!" α,β # $ } α,β = dilate f,w ( ) = dilate( w, f ) Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 34

35 Geeral erosio for gray-level images Geeral erosio operator g x, y = if α,β Dual of dilatio g x, y = if α,β = sup α,β f x +α, y + β w α,β { } = erode( f,w) f x +α, y + β w α,β { } f x +α, y + β + w α,β { } = dilate( f,ŵ) Flat erosio cotaied as a special case Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 35

36 Cascaded dilatios f dilate( f,w 1 ) dilate ( f,w,w 1 2 ) dilate ( f,w,w,w 1 2 3) dilate dilate( f,w 1 ),w 2 = dilate( f,w) where w = dilate( w 1,w ) 2 Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 36

37 Cascaded erosios Cascaded erosios ca be lumped ito sigle erosio erode erode( f,w 1 ),w 2 = erode dilate ( f,ŵ 1 ),w 2 = dilate dilate( f,ŵ 1 ),ŵ 2 = dilate( f,ŵ) = erode( f,w) where w = dilate( w 1,w ) 2 New structurig elemet (SE) is ot the erosio of oe SE by the other, but dilatio. Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 37

38 Fast dilatio ad erosio Idea: build larger dilatio ad erosio operators by cascadig simple, small operators Example: biary erosio by 11x11 widow f x, y f x, y Erosio with 1x3 widow Erosio with 1x3 widow Erosio with 11x11 widow 5 stages g x, y g x, y Erosio with 3x1 widow Erosio with 3x1 widow 5 stages 120 AND per pixel 2x10 = 20 AND per pixel Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 38

39 Rak filters Geeralisatio of flat dilatio/erosio: i lieu of mi or max value i widow, use the p-th raked value Icreases robustess agaist oise Best-kow example: media filter for oise reductio Cocept useful for both gray-level ad biary images All rak filters are commutative with thresholdig Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 39

40 Media filter Gray-level media filter Biary images: majority filter Self-duality g x, y = media W f x, y { } g x, y = MAJ W f x, y media f,w majority f,w ( ) = media( f,w ) { } ( ) = NOT majority NOT f := media( f,w ) := majority( f,w ) (,W ) Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 40

41 Majority filter: example Biary image with 5% Salt&Pepper oise 3x3 majority filter 20% Salt&Pepper oise 3x3 majority filter Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 41

42 Media filter: example Origial image 5% Salt&Pepper oise 3x3 media filterig 7x7 media filterig Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 42

43 Example: o-uiform lightig compesatio Origial image 1632x1216 pixels Dilatio (local max) Rak filter 61x61 structurig elemet 10st brightest pixel 61x61 structurig elemet Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 43

Image Processig: Berd Girod, 2013-2018 Staford

44 Example: o-uiform lightig compesatio Backgroud origial image After global thresholdig Digital Image Processig: Berd Girod, Staford Uiversity -- Morphological Image Processig 44

Structuring Element Representation of an Image and Its Applications

Structuring Element Representation of an Image and Its Applications Iteratioal Joural of Cotrol Structurig Automatio Elemet ad Represetatio Systems vol. of a o. Image 4 pp. ad 50955 Its Applicatios December 004 509 Structurig Elemet Represetatio of a Image ad Its Applicatios