CHAPTER 3 EDE DETECTION USIN CLASICAL EDE DETECTORS Edge detection is one o te most importnt opertions in imge nlsis. An edge is set o connected piels tt lie on te boundr between two regions. Te clssiiction o edge detectors discussed in tis cpter is bsed on te beviorl stud o tese edges wit respect to te ollowing opertors: rdient edge detectors Lplcin o ussin ussin edge detectors 3. RADIENT EDE DETECTORS Te irst derivtive ssumes locl mimum t n edge. For grdient imge t loction were nd re te row nd column coordintes respectivel we tpicll consider te two directionl derivtives. Te two unctions tt cn be epressed in terms o te directionl derivtives re te grdient mgnitude nd te grdient orienttion.
Te grdient mgnitude is deined b [ ] 3. Tis quntit give te mimum rte o increse o per unit distnce in te grdient orienttion o. Te grdient orienttion is lso n importnt quntit. Te grdient orienttion is given b tn 3. were te ngle is mesured wit respect to te - is. Te direction o te edge t is perpendiculr to te direction o te grdient vector t tt point. Te oter metod o clculting te grdient is given b estimting te inite dierence. lim 3.3 lim tereore we cn pproimte tis inite dierence s 3. 5
3.5 Using te piel coordinte nottion nd considering tt j corresponds to te direction o nd i corresponds to te direction i i j 3.6 i i j or i j i j 3.7 Te most populr clssicl grdient-bsed edge detectors re Roberts cross grdient opertor Sobel opertor nd te Prewitt opertor. 3.. ROBERTS EDE DETECTOR Te clcultion o te grdient mgnitude nd grdient mgnitude o n imge is obtined b te prtil derivtives nd t ever piel loction. Te simplest w to implement te irst order prtil derivtive is b using te Roberts cross grdient opertor. Tereore i j i j 3.8 i j i j 3.9 6
Te bove prtil derivtives cn be implemented b pproimting tem to two msks. Te Roberts opertor msks re Tese ilters ve te sortest support tus te position o te edges is more ccurte but te problem wit te sort support o te ilters is its vulnerbilit to noise. It lso produces ver wek responses to genuine edges unless te re ver srp. 3.. PREWITT EDE DETECTOR Te Prewitt edge detector is muc better opertor tn te Roberts opertor. Tis opertor ving 33 msks dels better wit te eect o noise. An pproc using te msks o size 33 is given b considering te below rrngement o piels bout te piel [i j] o 7 6 [ i j] 5 3 Te prtil derivtives o te Prewitt opertor re clculted s c c 3. 3 7 6 c c 3. 6 5 7
Te constnt c implies te empsis given to piels closer to te center o te msk. nd re te pproimtions t [i j]. Setting c te Prewitt opertor is obtined. Tereore te Prewitt msks re s ollows Tese msks ve longer support. Te dierentite in one direction nd verge in te oter direction so te edge detector is less vulnerble to noise. 3..3 SOBEL EDE DETECTOR Te Sobel edge detector is ver muc similr to te Prewitt edge detector. Te dierence between te bot is tt te weigt o te center coeicient is in te Sobel opertor. Te prtil derivtives o te Sobel opertor re clculted s 3. 3 7 6 3.3 6 5 8
Tereore te Sobel msks re Altoug te Prewitt msks re esier to implement tn te Sobel msks te lter s better noise suppression crcteristics. 3.. FREI-CHEN EDE DETECTOR Te Frei-Cen edge detector is lso irst order opertion like te previousl discussed opertors. Edge detection using te Frei-Cen msks is implemented b mpping te intensit vector using liner trnsormtion nd ten detecting edges bsed on te edges bsed on te ngle between te intensit vector nd its projection onto te edge subspce. Frei-Cen edge detection is relized wit te normlized weigts. Frei-Cen msks re unique msks wic contin ll o te bsis vectors. Tis implies tt 33 imge re is represented wit te weigted sum o nine Frei-Cen msks. Primril te imge is convolved wit ec o te nine msks. Ten n inner product o te convolution results o ec msk is perormed. Te Frei-Cen re 3 9
5 6 6 7 6 8 3 9 Te irst our Frei-Cen msks bove re used or edges nd te net our re used or lines nd te lst msk is used to compute verges. For edge detection pproprite msks re cosen nd te imge is projected onto it. Te projection equtions re given b S M e ` cos 3. were nd } { e k s k M 9 k s k S 3. LAPLACIAN OF AUSSIAN LO Te principle used in te Lplcin o ussin metod is te second derivtive o signl is zero wen te mgnitude o te derivtive is mimum. Te Lplcin o -D unction is deined s 3.5 3
3.. THE LO OPERATOR Te two prtil derivtive pproimtions or te Lplcin or 33 region re given s 7 5 3 8 3.6 8 7 6 5 3 8 3.7 Te msks or implementing tese two equtions re s ollows 8 Te bove prtil derivtive equtions re isotropic or rottion increments o 9 nd 5 respectivel. Edge detection is done b convolving n imge wit te Lplcin t given scle nd ten mrk te points were te result ve zero vlue wic is clled te zero-crossings. Tese points sould be cecked to ensure tt te grdient mgnitude is lrge. Mrr nd Hildret develop tis metod. Mrr nd Hildret metod Te edge piels in n imge re determined b single convolution opertion. Te bsic principle o tis metod is to ind te position in n imge were te second derivtives become zero. Tese positions correspond to edge positions. Te ussin unction irstl smootens or blurs n edge s sown in te igure 3.. Blurring is 3
dvntgeous ere becuse Lplcin would be ininit t unsmooted edge nd tereore edge position is still preserved. LO opertor is still susceptible to noise but b ignoring zero-crossings produced b smll cnges in imge intensit cn reduce te eects o noise. b ussin smooting unction o c Lplcin o ussin o Figure 3.: Te smooting o signl wit ussin unction 3
LO opertor gives edge direction inormtion s well s edge points determined rom te direction o te zero-crossing. Hence te purpose o te ussin unction in te LO ormultion is to smoot te imge nd te purpose o te Lplcin opertor is to provide n imge wit zero crossings used to estblis te loction o edges. Some o te disdvntges o LO re te LO being second derivtive opertor te inluence o noise is considerble. It lws genertes closed contours wic is not relistic. Te Mrr-Hildret opertor will mrk edges t some loctions tt re not edges. 33