6.3. DIFFERENCEOPERATOR
59
Area 1:
1 2 2 3
4 5 5 6
7 8 8 9
Output of f homogeneity y edge detector is:
max of {
| 5 - - 1 1 |
| 5 5 - - 2 2 |
| 5 5 - - 3 3 |
| 5 - - 4 4 |
| 5 5 - - 6 6 |
| 5 5 - - 7 7 |
| 5 - - 8 8 |
| 5 5 - - 9 9 |
} = 4
Area 2:
10 10 0 10
10 10 0 10
10 10 0 1
Output of f homogeneity y edge detector is:
max of {
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
| 10 0 - - 1 1 |
} = 9
Area 3:
10 5 5 3
10 5 5 3
10 5 5 3
Output of f homogeneity y edge detector is:
max of {
| 5 - - 10|
| 5 5 - - 5 5 |
| 5 5 - - 3 3 |
| 5 - - 10|
| 5 5 - - 3 3 |
| 5 5 - - 10|
| 5 - - 5 5 |
| 5 5 - - 3 3 |
} = 5
Figure6.2:AnExampleoftheHomogeneityOperator
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60
CHAPTER6. ADVANCEDEDGEDETECTION
Figure6.3:ResultofHomogeneityEdgeDetector
over the e input image array andcalculates the absolute values ofthe four
dierences. As s in n the e homogeneity case, the dierence operator requires
thresholding.
Figure6.5showstheresult ofthe dierence edgedetector. . Thisresult
issimilartothatshowninFigure6.3. Thedierenceedgedetectordidde-
tectmoreofthebrickandmortarlinesthanthehomogeneityoperator. The
choicebetweenthetwoedgedetectorsdependingonthedesireddetail. The
dierenceoperatorisfasterthanthehomogeneityoperator. Thedierence
operatorusesonlyfourintegersubtractionsperpixel,whilethehomogeneity
operatoruseseightsubtractionsperpixel. Thesecomparetotheninemul-
tiplicationsandadditionsfortheconvolution-basededgedetectorsdiscussed
inChapter5.
6.4 DierenceofGaussians
ThenextoperatortoexamineisthedierenceofGaussiansedgedetector,
which allows s varying g the width h of a a convolution mask k and adjusting the
detailintheoutput[6.2,6.3]. TheresultsinFigures6.3and6.5aregood.
Suppose,however,wewantedtodetectonlytheedges ofthelargeobjects
inthehouse image(door,windows,etc.) ) andnotdetectthesmallobjects
(bricks,leaves,etc.).
Varying the width of convolution masks eliminates details. . If f a mask
iswide,convolvinganimagewillsmoothoutdetails,muchlikeaveraging.
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6.4. DIFFERENCEOFGAUSSIANS
61
Area 1:
1 2 2 3
4 5 5 6
7 8 8 9
Output of f difference e edge e detector is:
max of {
| 1 - - 9 9 |
| 7 7 - - 3 3 |
| 4 - - 6 6 |
| 2 2 - - 8 8 |
} = 8
Area 2:
10 10 0 10
10 10 0 10
10 10 0 1
Output of f difference e edge e detector is:
max of {
| 10 0 - - 1 1 |
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
| 10 0 - - 10 0 |
} = 9
Area 3:
10 5 5 3
10 5 5 3
10 5 5 3
Output of f difference e edge e detector is:
max of {
| 10 0 - - 3|
| 10 - 3|
| 10 0 - - 3|
| 5 5 - - 5 5 |
} = 7
Figure6.4:AnExampleoftheDierenceOperator
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62
CHAPTER6. ADVANCEDEDGEDETECTION
Figure6.5:ResultofDierenceEdgeDetector
Stockmarketpricesvarygreatlybytheminute.Thevariationslessonwhen
the prices s are e examined hourly. . Examining g the prices s weekly causes the
variationstodisappearandgeneraltrendstoappear. Convolvinganimage
withawide,constantmask,smoothestheimage. Narrower,varyingmasks,
permitthedetailstoremain.
Figure 6.6 shows two o example e masks. . These e masks s are \dierence of
Gaussians"or\Mexicanhat"functions.Thecenterofthemasksisapositive
peak(16inthe7x7masks|19inthe9x9mask).Themasksslopedownward
toasmallnegativepeak(-3inbothmasks)andbackuptozero.Thecurve
inthe9x9maskiswiderthanthatinthe3x3mask.Noticehowthe9x9mask
hitsitsnegativepeakthreepixelsawayfromthecenterwhilethe7x7masks
hitsitspeaktwopixelsawayfromthecenter. Also,noticethesemasksuse
integervalues. Mostedgedetectorsofthistypeuse oatingpointnumbers
thatpeakat+1. Usingintegersgreatlyincreasesthespeed.
Figure6.7illustrateshowthenarrowermaskwilldetectsmalledgesthe
widemaskmisses.EachareainFigure6.7hasasmallpatternsimilartothe
brickandmortarpatterninthehouseimage.Thispatternhassmallobjects
(bricks)withmanyedges. Convolvingthe7x7maskinFigure6.6withthe
7x7areainFigure6.7,resultsina+40;the7x7maskdetectedanedgeat
thecenterofthe7x7area.Doingthesamewiththe9x9maskinFigure6.6
withthe9x9areainFigure6.7,producesa-20;the9x9maskdidnotdetect
anyedges. The\hat"inthe9x9maskwaswideenoughtosmoothoutthe
edgesandnotdetectthem.
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6.4. DIFFERENCEOFGAUSSIANS
63
7x7 mask
0
0
-1
-1
-1
0
0
0
-2
-3
-3
-3
-2
0
-1
-3
5
5
5
-3
-1
-1
-3
5
16
5
-3
-1
-1
-3
5
5
5
-3
-1
0
-2
-3
-3
-3
-2
0
0
0
-1
-1
-1
0
0
9x9 mask
0
0
0
-1
-1
-1
0
0
0
0
-2
-3
-3
-3
-3
-3
-2
0
0
-3
-2
-1
-1
-1
-2
-3
0
-1
-3
-1
9
9
9
-1
-3
-1
-1
-3
-1
9
19
9
-1
-3
-1
-1
-3
-1
9
9
9
-1
-3
-1
0
-3
-2
-1
-1
-1
-2
-3
0
0
-2
-3
-3
-3
-3
-3
-2
0
0
0
0
-1
-1
-1
0
0
0
Figure6.6: Gaussian\MexicanHat"Masks
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64
CHAPTER6. ADVANCEDEDGEDETECTION
7x7 area with lines
0 10 0 0 0 10 0 0 0 10 0 0
0 0 0 0 0 0 0 0 0 0 0
10 0 0 10 0 0 0 10 0 0 0 10
0 0 0 0 0 0 0 0 0 0 0
0 10 0 0 0 10 0 0 0 10 0 0
0 0 0 0 0 0 0 0 0 0 0
10 0 0 10 0 0 0 10 0 0 0 10
result of f convolution n with 7x7 mask = = 40
9x9 area with lines
0 0 0 0 0 0 0 0 0 0 0 0 0 0
10 0 0 10 0 0 0 10 0 0 0 10 0 0 0 10
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 10 0 0 0 10 0 0 0 10 0 0 0 10 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
10 0 0 10 0 0 0 10 0 0 0 10 0 0 0 10
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 10 0 0 0 10 0 0 0 10 0 0 0 10 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
result of f convolution n with 9x9 mask = = -20
Figure6.7:DetectingSmallEdges
6.5. MOREDIFFERENCES
65
Therst sectionof Listing6.1 shows thetwoGaussianmasksandthe
functiongaussian
edge. gaussian
edgehasthesameformastheotheredge
detectors.Anadditionalsizeparameter(either7or9)speciesmaskwidth.
The inner r loop p over a and b varies s with h this s parameter. . The e processing
portionisthesameastheotherconvolutionmaskedgedetectorspresented
inChapter 5. . Withgaussian
edge, thresholding canproduce e aclear edge
imageorleaveitotoshowthestrengthoftheedges.
Figure6.8showstheresultofedgedetectionusingthenarrower7x7mask
andFigure6.9showstheresultofthewider9x9mask. Thenarrowermask
(Figure6.8) detectedalltheedgesofthe bricks,roofshingles,andleaves.
Thewidermask(Figure6.9)didnotdetecttheedgesofsmallobjects,only
edges ofthelarger objects. . Figure e 6.8mightbe too cluttered,so use the
widermask. Ifnedetailisdesired,thenarrowermaskistheonetouse.
Figure6.8:ResultofGaussianEdgeDetectorwith7x7Mask
6.5 MoreDierences
Theotheredgedetectorspresentedsofarcandetectedgesondierentsize
objects.Thehomogeneityoperatorcantakethedierenceofthecenterpixel
andapixelthatistwoorthreepixels away. . Thedierenceedgedetector
cantakethedierenceofoppositepixelsina5x5or7x7areainsteadofa
3x3area.ThequickmaskinChapter5canchangefrom3x3to5x5withthe
centervalueequalto4andthefourcornersequalto-1. Trythesechanges
asanexercise.
66
CHAPTER6. ADVANCEDEDGEDETECTION
Figure6.9: ResultofGaussianEdgeDetectorwith9x9Mask
6.6 Contrast-basedEdgeDetector
Oneproblemwithdetectingedges involvesunevenlightinginimages. . The
contrast-basededgedetector[6.4]helpstakecareofthisproblem.Inwelllit
areasofanimagetheedgeshavelargedierencesingraylevels.Ifthesame
edgeoccursinapoorlylitareaoftheimage,thedierenceingraylevelsis
muchsmaller.Mostedgedetectorsresultinastrongedgeinthewelllitarea
andaweakedgeinthepoorlylitarea.
Thecontrast-basededgedetectortakestheresultofanyedgedetectorand
dividesitbytheaveragevalueofthearea. Thisdivisionremovestheeect
ofunevenlightingintheimage. Theaveragevalueofanareaisavailableby
convolvingtheareawithamaskcontainingallonesanddividingbythesize
ofthearea.
Figure6.10illustratesthecontrast-basededgedetector.Almostanyedge
detectorcanbethebasisforthisoperation.Figure6.10usesthequickedge
detector from Chapter 5. . The e edgein thewelllit areais anobvious and
strongedge.Convolvingthequickmaskwiththisareayields100.Theedge
inthe poorly lit area is easy tosee, , but t convolving withthe quick mask
resultsin10,aweakedgethatthresholdingwouldprobablyeliminate.This
distinction should be avoided. . The e well l lit and poorly lit edges s are very
similar;bothchangefromonegrayleveltoanothergraylevelthatistwice
asbright.
Dividingbytheaveragegraylevelintheareacorrectsthisdiscrepancy.
Figure6.10showsthesmoothingmaskthatcalculatestheaveragegraylevel.
6.6. CONTRAST-BASEDEDGEDETECTOR
67
Edge Detector r Mask
-1 0 0 -1
0 4 4 0
-1 0 0 -1
Edge in n well lit area
50 100 100
50 100 100
convolution with edge mask yields:
50 100 100
400 - - 50 0 - - 50 0 - - 100 0 - - 100 = = 100
Edge in n poorly y lit t area
5 10 0 10
5 10 0 10
convolution with edge mask yields:
5 10 0 10
40 - 5 5 - - 5 5 - - 10 0 - - 10 0 = = 10
Smoothing mask
1 1 1
1/9 * * 1 1 1 1
1 1 1
convolution of f smoothing mask with edge in n well lit area yields:
50+50+50+100+100+100+100+100+100 / / 9 9 = 750/9 9 = = 83
convolution of f smoothing mask with edge in n poorly y lit area yields:
5+5+5+10+10+10+10+10+10 / / 9 9 = = 75/9 = = 8
dividing original convolution n by y the smoothing mask result:
edge in n well lit area: 100 / 83 = 1
edge in n poorly y lit t area: : 10 0 / / 8 8 = 1
Figure6.10:Contrast-BasedEdgeDetector
68
CHAPTER6. ADVANCEDEDGEDETECTION
Convolvingthewelllit areayieldsanaveragevalueof83. . Convolvingthe
poorlylitareayieldsanaveragevalueofeight. Dividing(integerdivision)
thestrongedgeinthewelllitareaby83yieldsone.Dividingtheweakedge
byeightalsogives aresult ofone. . Thetwoedges s from unevenly lit areas
yieldthesameanswerandyouhaveconsistentedgedetection.
The next section of Listing 6.1 1 shows the e contrast
edge function that
followsthesamestepsastheotheredgedetectorfunctions.Thedierenceis
intheprocessingloopoveraandb,whichcalculatestwoconvolutions:sum
n
(the numerator orquick edge output)andsum
d(the smoothingoutput).
Aftertheloopsoveraandb,dividesum
dbynineanddividesum
nbythis
result. Thee
maskatthebeginningofListing6.1replacesthequickmask
fromChapter5,withevery elementinthe quick maskmultipliedbynine.
The largervalues are necessary becausedividingbytheaveragegray level
couldreducethestrengthofalledgestozero.
Figure 6.11 shows the result of the regular quick k edge e detector. . Fig-
ure 6.12showsthe result of dividingthe quickedgeresultby the average
valuetoproducecontrast-basededgedetection. Noticetheinsideoftheup-
stairsanddownstairswindows. Figure6.11(quickedge)showsedgesinside
thedownstairswindows,but notinsidetheupstairswindows. . Figure e 6.12
(contrast-based)showsdetailsinsidethedownstairsandupstairswindows.
Refertotheoriginalimage(Figure6.1).Thedownstairswindowsareshaded
andtheupstairswindowsarenot.Contrast-basededgedetectiongivesbetter
resultsinunevenlighting.
Figure6.11: ResultofQuickEdgeDetector
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