508 ntegated Detonal Devatve Gadent Opeato OSCAR A. ZUNGA, MEMBER, EEE, AND ROBERT M. FELLOW, EEE HARALCK, Abstat -Auate edge deton nfomaton s equed n many mage poessng applatons. A vaety of opeatos foomputng loal edge deton have been poposed, many of them estmatng a knd of gadent. These opeatos fae two majo poblems. One poblem s the nheent bas n the estmate of edge deton. The bas tself s a funton of edge deton. Anothe poblem s the senstvty to the pesene of nose n the mage data. The seond poblem an be allevated by an nease n the poessng neghbohood sze but usually at the expense of an nease n estmate bas and also nefos n the poessng ofsmall o thn objets. An opeato based on the ub faet model s dsussed, whh edues shaply both estmate bas and nose senstvty wth no nease n omputatonal omplexty. The measue of gadent stength s the maxmum value of the ntegal of the fst detonal devatve taken ove a etangula o squae neghbohood, the maxmum beng taken ove all possble detons fo the detonal devatve.the lne deton whh maxmzes the ntegal defnes the new estmate of gadent deton. Expemental esults show the supeoty of ths opeato to othes suh as the Robets opeato, the Pewtt opeato, the Sobel opeato, and the standad ub faet gadent opeato fo step edges and amp edges. Unde zeo-nose ondtons the 7x 7 ntegated detonal devatve gadent opeato has a wost bas of less than 0.090, and the 5 x 5 ntegated detonal devatve gadent opeato has a wost bas of less than 0.26 on amp edges. Foompason puposes the 7 X 7 standad ub faet gadent opeato has a wost bas of about.20, and the 5x 5 standad ub faet gadent opeato has a wost bas of 0.50.The 7x 7 Pewtt opeato has a wost bas of 50, and the 5x 5 Pewtt opeato has a wost bas of4. Ths mpovement n wost bas stays wth the ontamnaton of edges by addtve ndependent zeo-mean Gaussan nose.. NTRODUCTON The omputaton of edge deton s a equed step n many mage poessng tasks. Hough tansfomaton tehnques [3] have been used extensvely togethe wth edge deton nfomaton to detetlnes [3], les [0], and abtay shapes []. Matell [2] and Rame [5] eah use edge deton nfomaton to pefom edge lnkng. Kthen and Rosenfeld [] and Zunga and Haalk [7] use edge deton nfomaton n shemes to detet ones. A vaety of opeatos foomputng loal edge deton have been poposed, many of them estmatng a knd of gadent of the ptue funton. Loal edge deton s then estmated as that deton whh s othogonal to the gadent deton. Knowledge of the detonal devatves D and D2 n any two othogonal detons s suffent to ompute the detonal devatve n any abtay deton. The gadent magntude, whh s defned as the maxmum suh detonal devatve, s omputed as VD2+ D2 and ts deton as tan D2/D,. A numbe of loal opeatos have been utlzed to estmate these detonal devatves. Examples ae the Robets opeato [6], the Pewtt opeato [4], the Sobel opeato [3], and the Huekel opeato [8]. Of speal nteest ae those opeatos whh esult fom a loal sufae ft to the gaytone mage data and subsequent omputaton of detonal devatves n two othogonal detons fom the analyt despton of the estmated sufae. Thus the ognal Robets opeato has been shown to esult fom a lnea sufae ft wthn a 2x 2 pxel neghbohood [5]. The Manuspt eeved June 20, 985; evsed Otobe 22, 986. 0. A. Zunga s wth the Depatment of Eletal Engneeng, Vgna Polytehn nsttute and State Unvesty, Blaksbug, VA 2406. R. M. Haalk s wth the Depatment of Eletal Engneeng, Unvesty of Washngton, Seattle, WA 9895. EEE Log Numbe 874385. EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC -7, NO. 3, MAY/JUNE 987 ognal Pewtt opeato was deved fom a quadat sufae ft wthn a 3 X 3 pxel neghbohood. Haalk [6] and Zunga and Haalk [7] ompute edge deton nfomaton fom a ub polynomal sufae ft. Ths fttng dea an also be extended to abtay szed neghbohoods [2, [5]. These gadent opeatos fae two majo poblems. One poblem s that the estmate of edge deton s nheently based as a funton of tue edge deton and dsplaement of the tue edge fom the pxel's ente. The seond majo poblem s the senstvty of these opeatos to data nose. An obvous appoah to deease nose senstvty s to nease the neghbohood sze. Howeve, ths eates poblems wth small o thn objets. n addton, fo all the gadent opeatos aleady mentoned an nease n neghbohood sze auses an nease n edge deton estmate bas. n ths ofespondene we use an opeato based on a ub polynomal sufae ft. Ths ub faet model has been suessfully used n the past to detet edges [6], topogaph featues [7], and ones [7]. nstead of omputng detonal devatves detly fom the ftted sufae as n the ase of the standad ub faet gadent opeato mentoned eahe, we desbe an opeato that measues gadent stength as the mnmum value of the ntegal of fst detonal devatve taken ove lnes gong n all possble detons. The lne deton whh maxmzes the ntegal defnes the new estmate of gadent deton. We pesent expemental evdene that ths opeato possesses two man haatests. ) Edge deton estmate bas s shaply edued as ompaed wth the bas of the standad ub faet, Sobel, and Pewtt gadent opeatos. Nose senstvty s ompaable to that of the Sobel and Pewtt opeatos and muh bette than the standad ub faet opeato. 2) Unlke the standad ub faet, Sobel, and Pewtt opeatos, neasng the neghbohood sze deeases both estmate bas and nose senstvty. Fo amp edges the ntegated opeato s vey nealy unbased. The wost bas fo the 7 x 7 opeato s less than 0.092, and the wost bas fo the 5 x 5 opeato s less than 0.260. Seton desbes the standad ub faet gadent opeato. Seton desbes the mathematal analyss neessay to deve the new gadent estmate. Seton V pesents expemental esults and povdes a ompason of the ntegated detonal devatve gadent opeato aganst the standad ub faet gadent opeato, Pewtt opeato, and Sobel opeato fo step edges and amp edgees ontamnated by zeo-mean Gaussan nose. The appendx desbes the mathematal analyss of the ub faet.. THE STANDARD CuBC FACET GRADENT OPERATOR Gven a gaytone ntensty funton f defned n the ow and olumn oodnate system of a gven pxel neghbohood, the gadent veto funton vf s gven by Vt = (df df () Unde the ub faet model eah sufae faet enteed about a gven pxel s appoxmated by the bvaate ub fom f(, ) K2 + K3+K4 2 008-9472/87/0500-0508$0.00 987 EEE K5 + K62 ± K73 + K82C + K92 + K03 (2) whee the K oeffents hange fom neghbohood to neghbohood and ae estmated usng a least-squae-eo sufae ft. A detaled analyss of ths estmaton poedue s povded n the Appendx. Evaluatng the fst ow and olumn patal devatves of (2) at the neghbohood ente (0,0) and eplang the values n (), the gadent veto at the neghbohood ente
EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 509 beomes vf =(K2, K3) The magntude and deton of the standad ub faet gadent opeato ae, theefoe, gven by MK + K2 and tan- K27K3, espetvely, wth the gadent deton beng measued as a lokwse angle fom the olumn axs.. THE NTEGRATED DRECTONAL DERVATVE GRADENT OPERATOR Fo a gven deton veto (sn 6, os 6), t s well-known that the fst detonal devatve f6'(, ) of f n the deton 6 an be evaluated as the omponent of the gadent vf along the deton veto, that s, f )=-dfsn6±-dfos60. df d (3) Let F6 be defned as Fo= 4LWf f_fo(pos 6+ t sn6, -psn6+wos#)dpdw (4) fo a gven N x N neghbohood. F6 epesents the ntegated fst detonal devatve along lnes othogonal to the deton 6 fomng a etangle of length 2L and wdth 2W enteed at the ogn of the oodnate system. The poposed gadent estmate s G = F6,. ux " M.x(5) whee Fa. = max6 F6 and U6M s a unt veto n the deton that maxmzes Fo. Usng (2) and (3), fo'(p os 6+ w sn 0, - p sn + X os6) edues to fq'( p os 6 + X sng, - p sng + w os 6) = [3( K9 - K7) sn3±+ 3(K8 - Klos) os3 + (3K7-2K9) sn6 + (3Ko - 2K8) os 0] p2 + 2[-K8 sn36 + (3K7-2K9) sn26 os6 + (2K8-3Ko) snsos26 + K9 os3 ] Poo + [3(K7 - Kg)sn36 +3(Klo - K8)os3 6 + 3K9 sng + 3K8 os6 ] X2 + [- K5 sn26+ 2(K4- K6) sn6os 6+ K5 os2 J p + 2[ K4 sn2 + Ks sn 0 os6 + K6 COS2 6 l + K2sn6 + K3os 6. (6) Substtutng (6) n (4) esults n Fe = 4 W J (Ap2+Bpw+Cw2+Dp+Ew+F)dpdw whee A, B, C, D, E, and F ae the oeffents of the quadat equaton (6). Evaluatng ths ntegal esults n F6 2+ CW3 =-AL ±CW+ F. 3 3 Fnally, F6 = (Kg - K7)(L2 - W2) sn3' + [L2K7 - -(2L2-3W2)K ± K2} sns + (K8 - Klo)(L2 - W2)os36 + [L2KO -(2L2-3W2)K8 + K3] os6. (7) Thus Fe edues to a tgonomet expesson n sn 0 and os 6. Note that f L = W, the ub tems sn3 6 and os3 6 dop, and F,, s maxmzed when Then 2 L K7+ -L2Kg+K2 0 = OMAX = tan 3 L2K0 + -L2K8 + K3 3 FeMA = VD, + D2 (9) whee D, and D2 ae the numeato and denomnato of the agument of the tangent funton n (8). The omputaton of the gadent stength gven by (9) and gadent detonal gven by (8) an be smplfed by peomputng "ow devatve" and "olumn devatve" masks fom the numeato and denomnato of (8), espetvely, usng the masks fo the K paametes deved n the Appendx. As a speal ase, when L = W=0, then (8) and (9) edue to the standad ub faet opeato defned n Seton. f L * W, then the maxmzaton of FO to obtan the estmated gadent stength FOMAX and estmated gadent deton max an be aed out by the followng poedue. Fom (7), F6 an be expessed as Fe= u sn3 6 + v os3 6 + w sn A + z os 6 (0) whee u, v, w, and z ae funtons of the K paametes. Then Fo = 3usn26 os6-3vos26sn±+ wos6 - zsn6. Equatng F6' to zeo to obtan an extemum esults n [3u(l-os26)+ w] os6 = (3vos2' + z)sno. () Let x-os26; then (9) beomes 3u( - x) + w = (3vx + z) tang. Squang ths expesson, we obtan (3u(-x) + w)2 = (3vx + z)2tan26. Replang tan2 6 (8) by (- x)/x and afte some algeba manpulaton, we fnally obtan the ub expesson n x: 9( u2 + v2)x3 -(8u2 +6uw +9v -6vz)x2 +(9u2+6uw+w2-6vz+z2)x -z2=0. (2) Then F6, and 6MAx an be obtaned by the followng steps: ) fnd all the oots x of (2) between 0 and ; 2) onvet these x's to all possble 6's fom x = os2 6; 3) evaluate F6 at all these 6's; 4) hoose maxmum value FQ. and mamx V. EXPERMENTAL RESULTS The expements wee pefomed wth step and amp edges ontamnated by zeo-mean Gaussan nose. Step edges wee geneated n a etangula gd wth oentatons 6 fom zeo to 90 and wth andom dsplaement fom the gd's ente un-
50 EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 fomly dstbuted wthn the ange (- D, D) wth the maxmum dsplaement D gven by D{=(05os, 0o 9 < 45 45s< <90, assumng a unt dstane between two fou-neghbo pxels n the gd. A step edge passng though a pxel dvdes t nto two pats havng aeas A and A2, wth A + A2 =. Let the oespondng gaytone ntenstes to eah sde of the edge be, and 2. The pxel s then assgned a gaytone ntensty aodng to the ule = haj + 2A2. The expements wee pefomed wth values fo, and 2 equal to 00 and 200, espetvely. That s, the edge ontast was set to 00 Ṙamp edges wee geneated by defousng step edges wth a 3 x 3 aveagng flte. Fnally, both step and amp edges wee ontamnated by addng zeo-mean Gaussan nose wth a gven standad devaton. Tunng Up the ntegated Detonal Devatve Gadent Opeato The pefomane of the ntegated detonal devatve gadent opeato depends on the hoe of ntegaton lmts L and W. As seen n Seton, ths opeato s equvalent to the standad ub faet gadent opeato when L = W = 0. We expet ts pefomane to mpove as L and W move away fom zeo and to eah a maxmum fo some value of L * 0 and some value of W * 0. We ae pmaly nteested n mpovng the edge deton estmate. We use two pefomane measuements, edge deton estmate bas and edge deton estmate standad devaton. The latte measues nose senstvty. The estmate bas s defned as the dffeene between the estmate mean and the tue edge deton. A sngle pefomane measuement to ompae two sets of values of L and W s obtaned by ombnng the pevous two measuements nto the oot-mean-squae eo of the estmate whh an be shown to be equal to the squae oot of the sum of the squae bas and the estmate vaane. t was obseved that the opeato aheved best pefomane n the oot-mean-squae eo sense when L = W=.8 fo a 5 x 5 neghbohood sze and L = W = 2.5 fo a 7 x 7 neghbohood sze fo both step and amp edges and fo a vaety of nose levels. Compang Dffeent Gadent Opeatos The followng gadent opeatos wee tested: 5 x 5 extended Sobel [9], 5 x 5 and 7 x 7 Pewtt, 5 x 5 and 7 x 7 standad ub faet, and 5 x 5 and 7 x 7 ntegated detonal devatve. Fg. shows the 5 x 5 ow devatve masks fo eah of the opeatos tested, and Fg. 2 shows the 7 x 7 ow devatve mask fo the ntegated detonal devatve gadent opeato. The olumn devatve masks an be obtaned fom the ow masks by tansposton. Fo a step o amp edge of a gven oentaton and nose standad devaton, eah opeato was appled to the gd's ente 0 000 tmes, eah tme wth a dffeent nose sample and a dffeent edge dsplaement fom the gd's ente. Unde zeonose ondtons the opeatos wee appled only 00 tmes. Edge oentatons vaed fom zeo to 900 and nose standad devaton fom zeo to 00. Edge ontast was set to 00. The edge deton estmates podued by eah opeato wee plotted as follows: fo a fxed nose standad devaton, estmate bas aganst tue edge deton, and estmate standad devaton aganst tue edge deton; fo a fxed edge deton, estmate bas aganst nose standad devaton, and estmate standad devaton aganst nose standad devaton. We ompae fst the standad ub faet gadent opeato and the ntegated detonal devatve gadent opeato. Unde zeo-nose ondtons, Fgs. 3 and 4 show estmate bas aganst tue edge deton fo step and amp edges. Thee thngs an be obseved. Fst, the ntegated opeato s lealy supeo 8 0 6[ 0 8 3-5 -7-5 3-6 -2s28-25 -6-44 -62-68 -62-44 /300 o o o o /420 0 0 0 0 0 6 25 28 25 6 44 62 68 62 44-8 lo 6 lo -8.3 s 7 s 3 (a) -2-2 2.2.2 - -t /50 0 0 0 0 0-2 22 _ -2 2 () (d) Fg.. Row devatve masks fo gadent opeatos n 5 x 5 neghbohood sze (a) ntegated detonal devatve. (b) Standad ub faet. () Pewtt. (d) Extended Sobel. /42 (b) s5 8.*0.8 s 4 -lo -20 -lo 4 /240 o o o o oj 4 0 ( 0 4 s 8 8 s 32 36-35 -92-35 36 32-206 -396-50 -548-50 -396-206 -229-324 -38-400 -38-324 -229 0 0 0 0 0 0 0 229 324 38 400 38 324 229 206 396 50 548 50 396 206-32 -36 35 92 35-36 -32 Fg. 2. Row devatve mask fo ntegated detonal devatve gadent opeato fo 7 x 7 neghbohood sze. 3.388 A n td :=4-300... 8.8n8o TRUE ANGLE (DEGREES) K8888 A 5x5 ntegated Detonal Devatve El 5x5 Standad Cub Faet X 7x7 ntegated Detonal Devatve + 7x7 Standad Cub Faet Fg. 3. Estmate bas as funton of tue edge deton fo step edges unde zeo-nose ondtons. to the standad ub faet gadent opeato. Unde zeo-nose ondtons the 7 x 7 ntegated detonal devatve gadent opeato has a wost bas of less than 0.09, and the 5 x 5 ntegated detonal devatve gadent opeato has a wost bas of less than 0.260 on amp edges. Foompason puposes, the 7 x 7 standad ub faet gadent opeato has a wost bas
EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 5 508 - U, -4 L 8.88ass TRUE ANGLE (DEGREES) 98.888 A 5x5 ntegated Detonal Devatve O 5x5 Standad Cub Faet X 7x7 ntegated Detonal Devatve + 7x7 Standad Cub Faet Fg. 4. Estmate bas as funton of tue edge deton fo amp edges unde Fg. 7. Estmate standad devaton as funton of nose standad devaton zeo-nose ondtons. fo step edge. Edge oentaton s 22.5 Edge ontast s 00. W -t-, -. v X. 80888 NOSE ST BEY 80.800 A 5x5 ntegated Detonal Devatve 0 5x5 Standad Cub Faet X 7x7 ntegated Detonal Devatve + 7x7 Standad Cub Faet 7.808 P f 4 va C: -: L3NOSE ST 0 A 5x5 ntegated Detonal Devatve 3 5x5 Standad Cub Faet X?x7 ntegated Detonal Devatve 4-7x7 Standad Cub Faet 8O.888 NOSE ST 0EV 08 08 A 5x5 ntegated Detonal Devatve o 5x5 Standad Cub Faet x 7x7 ntegated Detonal Devatve + 7x7 Standad Cub Faet Fg. 5. Estmate bas as funton of nose standad devaton fo step edge. Edge oentaton s 22.5. Edge ontast s 00. Fg. 8. Estmate standad devaton as funton of nose standad devaton fo amp edge. Edge oentaton s 22.5e. Edge ontast s 00. C,) -7. 00 F-~F ^3 / \~~ L B000 j 88 NOSE ST DEY 88.080 A 5x5 ntegated Detonal Devatve O3 5x5 Standad Cub Faet X 7x7 ntegated Detonal Devatve + 7x7 Standad Cub Faet 0.00000 TRUE ANGLE (DEGREES) 98.8088 A 5x5 ntegated Detonal Devatve o 5x5 Sobel 0 5x5 Pewtt x 7x7 ntegated Detonal Devatve * 7x7 Pewtt Fg. 6. Estmate bas as funton of nose standad devaton fo amp edge. Fg. 9. Estmate bas as funton of tue edge deton fo step edges unde Edge oentaton s 22.5". Edge ontast s 00. zeo-nose ondtons.
52 EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 6.000- l BBB n -6. Ad _,:O / B.00000 TRUE ANGLE (DEGREES) 90.0000 A 5x5 ntegated Detonal a 5x5 Sobel Devatve 0 5x5 Pewtt x 7x7 ntegated Detonal Devatve + 7x7 Pewtt Fg. 0. Estmate bas as funton of tue edge fo amp edges unde zeonose ondtons. -6.... - B. 00000 TRUE ANGLE (DEGREES) 90.0000 A 5x5 ntegated Detonal Devatve O 5x5 Sobel 0 5x5 Pewtt x 7x7 ntegated Detonal Devatve + 7x7 Pewtt L Fg. 2. Estmate bas as funton of tue edge deton fo amp edge. Nose standad devaton s 25. Edge ontast s 00. -, t. BOO m -. XL - B. 00000 TRUE ANGLE (DEGREES) 90.B00B A 5x5 ntegated Detonal Devatve O 5x5 Sobel 0 5x5 Pewtt X 7x7 ntegated Detonal Devatve + 7x7 Pewtt B. 80000 TRUE ANGLE (DEGREES) a90.bb 5x5 ntegated Detonal Devatve a 5x5 Sobel 0 5x5 Pewtt x 7x7 ntegated Detonal Devatve + 7x7 Pewtt Fg.. Estmate bas as funton of tue edge deton fo step edge. Nose standad devaton s 25. Edge ontast s 00. of about.20, and the 5 x 5 standad ub faet gadent opeato has a wost bas of 0.50. Ths mpovement n wost bas stays wth the ontamnaton of edges by addtve ndependent zeo-mean Gaussan nose. Seond, fo the ntegated opeato, estmate bas deeases as the neghbohood sze neases whle the opposte happens wth the standad ub faet gadent opeato. Thd, both opeatos pefom bette wth amp edges than wth step edges. Fgs. 5-8 show estmate bas and estmate standad devaton aganst nose standad devaton fo a fxed-edge oentaton of 22.50 and addtve ndependent Gaussan nose. Agan, the ntegated opeato s unfomly supeo to the standad ub faet gadent opeato fo both step and amp edges. Next we ompae the ntegated detonal devatve opeato wth the Pewtt and extended Sobel opeato. Unde zeo-nose ondtons, Fgs. 9 and 0 show estmate bas as a funton of tue edge deton fo step and amp edges. The 7 x 7 ntegated opeato has the smallest bas followed by the 5 x 5 ntegated opeato, the 5 x 5 extended Sobel, and the 5 x 5 and 7 x 7 Pewtt opeatos. Note that fo amp edges the esponse of the n- Fg. 3. Estmate standad devaton as funton of tue edge deton fo step edge. Nose standad devaton s 25. Edge ontast s 00. tegated opeato s nealy flat about zeo, that s, the opeato s nealy unbased. Fo the 7 x 7 ntegated opeato the wost bas s less than 0.090, and fo the 5 x 5 ntegated opeato the wost bas s less than 0.260. Foompason puposes the wost bas n the 7 x 7 Pewtt opeato s about 50, and the wost bas n the 5 x 5 Pewtt opeato s about 4. Agan, the ntegated opeato s the only one fo whh bas deeases as neghbohood sze neases. Only the 5 x 5 Sobel opeato s shown, but as pevously demonstated by lannno and Shapo [9], the 3 x 3 Sobel opeato has a smalle bas than the 5 x 5 extended Sobel, but t s stll sgnfantly lage than the bas of the ntegated opeato and wth a muh wose nose senstvty. lannno and Shapo [9] also show esults wth 3 x 3 and 5 x 5 teated Sobel opeatos. The bas fo these teated opeatos s stll lage than the bas of the ntegated opeato, and they ae moe expensve omputatonally. Fgs. 0-4 show estmate bas and estmate standad devaton as a funton of tue edge deton fo step and amp edges when the nose standad devaton s equal to 25. The bas fo all the opeatos shown s nealy dental to the bas unde zeo-nose
EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 53 34.00 '< 000, ) >- C-, en tl) 4 tl P -',,xa ~* " 0. 00000 TRUE ANGLE (DEGREES) 90. 0000 5x5 ntegated Detonal Devatve o 5x5 Sobel 0 5x5 Pewtt x 7x7 ntegated Detonal Devatve + 7x7 Pewtt Fg. 4. Estmate standad devaton as funton of tue edge deton fo amp edge. Nose standad devaton s 25. Edge ontast s 00. 0.00000 NOSE ST DEV. -_ --T, - - o x 5x5 Devat ntegated Detonal 5x5 Sobe 5x5 Pewtt ve 7x7 ntegated Detonal Devat 7x7 Pewtt ve!~~~~~~~~~~~~~~ _!~~~~~~~~~~~~~~~ Fg. 7. Estmate standad devaton as funton of nose standad devaton fo step edge. Edge oentaton s 22.5. Edge ontast s 00. M.-.6 0.3- C-D L) - 0.- LO) t~~~~~~~~~~~~~~~~ ~ ~~~~t 4 - E. E00= 0. 0000 NOSE S DEV 00.000 A 5x5 ntegated Detonal Devatve D 5x5 Sobel 0 5x5 Few tt x 7x7 ntegated D etonal Devatve ~~~+7x7 Pew ttl Fg. 5. Estmate bas as funton of nose standad devaton fo step edge. Edge oentaton s 22.5. Edge ontast s 00. 3.000 L 00000 NOSE ST 0EV 00.000 A 5x5 ntegated Detonal Devatve C 5x5 Sobel 0 5x5 Pewtt v 7x7 ntegated Detonal Devatve + 7x7 Pewtt Fg. 6. Estmate bas as funton of nose standad devaton fo amp edge. Edge oentaton s 22.5. Edge ontast s 00. L 0. 00000 NOSE ST DEV 00. 00 A 5x5 ntegated Detonal Devatve 0 5x5 Sobel 9 5x5 Pewtt x 7x7 ntegated Detonal Devatve + 7x7 Pewtt Fg. 8. Estmate standad devaton as funton of nose standad devaton fo amp edge. Edge oentaton s 22.5. Edge ontast s 00. ondtons. t an be seen fom the plots of estmate standad devaton that, as expeted, the 7 X 7 opeatos ae less senstve to nose than the 5 x 5 opeatos. The estmate standad devatons fo the ntegated opeato and the Pewtt opeato ae about the same. The Sobel opeato has a slghtly lage estmate standad devaton. Fgs. 5-8 show estmate bas and estmate standad devaton as a funton of nose standad devaton fo a fxed edge oentaton of 22.50. Seveal thngs an be obseved fom these plots. Fst, estmate bas fo all the opeatos emans nealy flat as the nose level neases up to about a standad devaton of 90. Some of the opeatos show an nease n estmate bas at ths pont. The smallest bas oesponds to the ntegated opeato followed by the Sobel. The Pewtt opeato shows the lagest bas. Seond, the estmate standad devaton fo all opeatos neases lnealy wth an nease n nose standad devaton. The Pewtt opeato has the smallest estmate standad devaton, followed losely by the ntegated opeato and the Sobel opeato. The 7 x 7 opeatos have a muh smalle standad devaton than the 5 x 5 opeatos. Fo all the opeatos, amp edges podue a smalle estmate bas than step edges, whle step edges podue a smalle estmate standad devaton. Fnally, we show the gadent stength esponse of eah of the 5 x 5 opeatos on the two mages shown n Fg. 9. The fst
54 EEE TRANSACTONS ON SYSTEMS, MAN, AND (a) Fg. 9. Fg. 20. CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 (b) mages used to ompae gadent opeatos. (a) Synthet mage. (b) Aeal sene. (a) (b) () (d) Gadent stength esponse fo 5 x 5 opeatos on synthet mage. (a) ntegated detonal devatve. (b) Standad ub faet. () Pewtt. (d) Extended Sobel.
EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 55 (a) (b) () (d) Fg. 2. Gadent stength esponse fo 5 x 5 opeatos on aeal sene. (a) ntegated detonal devatve. (b) Standad ub faet. () Pewtt. (d) Extended Sobel. mage s a 64 x 64 synthet mage onsstng of a bght tangle on a dak bakgound wth zeo-mean Gaussan nose added to t. The bakgound gay level s 50, the objet gay level s 50, and the standad devaton of the nose s 30. The seond mage s a 64 x 64 aeal sene. Fgs. 20 and 2 show the gadent stength esponse fo eah of the 5 x 5 opeatos. The ntegated detonal devatve gadent opeato and the Sobel opeatos yeld edges wth smla amounts of blu, and both opeatos podue a good pefomane n the pesene of nose. The Pewtt opeato has also a good pefomane n the pesene of nose but podues a lage amount of edge blu. The standad ub faet gadent opeato yeld edges wth the least amount of blu but has a poo pefomane n the pesene of nose. V. CONCLUSON A gadent opeato based on an ntegated detonal devatve on a ub faet has been nvestgated. Expemental esults wth step and amp edges ontamnated by zeo-mean Gaussan nose show that ths opeato possesses the followng haatests. ) Edge deton estmate bas s shaply edued as ompaed wth the bas of the standad ub faet, Sobel, and Pewtt gadent opeatos. Nose senstvty s ompaable to that of the Sobel and Pewtt opeatos and muh bette than the standad ub faet gadent opeato. 2) Unlke the standad ub faet, Sobel, and Pewtt gadent opeatos, neasng the neghbohood sze deeases both estmate bas and nose senstvty. Fo amp edges the ntegated opeato s vey nealy unbased. The wost bas fo the 7 x 7 opeato s less than 0.09, and the wost bas fo the 5 x 5 opeato s less than 0.260. n ompason, the wost bas fo the 7 x 7 Pewtt opeato s about 50, and the wost bas fo the 5 x 5 Pewtt opeato s about 40. 3) Edge stength esponse n the pesene of nose s as good as that of the Sobel opeato and bette than the esponse of the Pewtt and standad ub faet gadent opeatos. 4) Computatonal omplexty s the same as the omplexty of the Sobel, Pewtt, and standad ub faet gadent opeatos sne t only nvolves the applaton of peomputed ow and olumn devatve masks.
56 EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 APPENDX mage neghbohood. We wll assume a etangula-shaped THE CuB FACET neghbohood whose ow ndex set s R, whose olumn ndex set The ub faet s desbed by the two-dmensonal poy- s C, and whose ente s taken to be at (0,0). Note that fo an nomal sufae funton even-szed neghbohood the ente falls at the pont whee the fouente pxels meet. The squaed fttng eo ove ths Jf(, ) = K + K2 + K3 + K42 + K5 neghbohood s gven by + K2 + K73 + Kg2 + Kg2+ KoC3. e2= Z E (K+K2 + K3 + K42 + K + K6C2 Xe.- *. Ths polynomal s * ft - to the -s gaytones f(, ) nsde.. a gven R (ec + K73 + K2 + Kg2 + K3 -f(,c)) Takng the patal devatves of e2 wth espet to the paametes K* * K esults n de2/k de2/dk2 de2/dk3 de2/3dk4 de2/dk5 2(-K + K2 + K3 + K42 + K5 + K62 + K73±+K52 + K9 2 + Ko3 - f(, )) ; de2/ak ERE de2/ak7 3 de2/dk8 2 ae2/akg 2 de2/dko C3 Beause the sum s between symmet lmts, a onsdeable amount of anellaton ous when ayng out the summaton: de2/ak, K42+K62+K-f(,) de2/dk2 K74 + K922 + K22 - f(, ) de2/dk3 K822 + K04 + K32 - f(, ) de2/dk4 K44 + K622 + K,2 2f(, ) de2/dk =2 K5 22- f(, ) de2/dk6 erec K422+ K6C4 + Kl2 2f(, ) ae2/ak7 K76 + Kg4C2 + K24-3f(, ) de2/dk8 K842 + Klo24 + K32-2f(, ) de/8dk9 K74 + K94 + K222 d2f(, ) de2/8ko K824 + Klo6 + K34 _3f(, ) Settng the patals to zeo and solvng, we mmedately obtan the least-squaes estmate K' of K5: Y E ef(,c) K _ R E C S E E 22 er ec The least-squaes estmates fo the emanng paametes ae obtaned by solvng the followng thee systems of equatons: E [ E2 [ Z 2,E4 Y. Y, 42 EE2 Y. Y, 4 EE 22 C Y, E 42 Y.EE22 Z42 Y. El 42 Y. 24 ZZ2 -FK, F Z4 flk j 2K7 2 E Y.2 K6 Y. ES22 K2 4 E2 4 _ Y.2EC6 J K3 K o E EfJ(, ) F f(, ) f(,) E f(, ) EZ 3f(, ) E E d2f(, ) EZ f(, ) Y22Cf(, ) Z EC3f(, )
EEE TRANSACTONS ON SYSTEMS, MAN, AND CYBERNETCS, VOL. SMC-7, NO. 3, MAY/JUNE 987 57 Fo n = 0,, 2, o 3, let R,, and C,, be defned as R,= E 2n C= E C2n. er ec Futhemoe, let G= RORCC2 - RC A = R R3COC2 - B=ROR B RRCC3 2 R2ClC - l22 C Q =CO(R0R2 - R ) T= R (C0C2 - C2) U=o(RR3 - R ) V=C(RoR2- R ) W = R(CoC2 - C2) Z= RO(ClC3 - C2) The soluton of the foegong systems of equatons s then gven by K =- (G-TR2-QC2)f(, ) QT K = E (A-WR22- UC2) f (, ) KS, = > o(b -ZR2_VC2C2)f(,) Kg =-WE, (Ro2 - Rl)f(,) ( K' =- Y(.C2 VZ - C)f(, ) K7 Pat ER t l2) f n, ) U Q KP, K ~~~~~ V 8, p27(ro-r)f(,) -~ (C2- )f) C2)f(, ). K{0=-EF(C Z REFERENCES 93 [] D. H. Ballad, "Genealzng the Hough tansfom to detet abtay shapes," Patten Reognton, vol. 3, no. 2, pp. -22, 98. [2] M. Books, "Ratonalzng edge detetos," Comput. Gaphs mage Poessng, vol. 8, pp. 277-285, 978. [3] R. Duda and P. Hat, "Use of Hough tansfomaton to detet lnes and uves n ptues," Commun. Ass. Comput. Mah., vol. 5, p., 972. [4 93. Patten Classfaton and Sene Analyss. New Yok: Wley, [5] R. M. Haalk, "Edge and egon analyss fo dgtal mage data," Comput. Gaphs mage Poessng, vol. 2, pp. 60-73, 980. [6], "Dgtal step edges fom zeo-ossng of seond detonal devatves." EEE Tans. Patten Anal. Mah. ntell., vol. PAM-6, pp. 58-68, Jan. 984. [7 R. M. Haalk, L. T. Watson, and T. Laffey, "The topogaphal pmal sketh," to be publshed, 983. [8] M. Huekel, "An opeato whh loates edges n dgtal ptues," J. Ass. Comput. Mah., vol. 8, pp. 3-25,97. [9] A. annno and S. D. Shapo, "An teatve genealzaton of the Sobel edge deteton opeato," n Po. EEE Patten Reognton and mage Poessng Conf., 979, pp. 30-37. [0] C. Kmme, D. Ballad, and J. Sklansky, "Fndng les by an aay of aumulatos," Commun. Ass. Comput. Mah., vol. 8, no. 2, pp. 20-22, 975. [] L. Kthen and A. Rosenfeld, "Gay level ome deteton," Compute Sene Cente, Unv. Mayland, College Pak, MD 20742, Teh. Rep. 887, Ap. 980. [2] A. Matell, "Edge deteton usng heust seah methods," Comput. Gaphs mage Poessng, vol., pp. 69-82, 972. [3] F. O'Goman and M. B. Clowes, "Fndng ptue edges though ollneaty of featue ponts," EEE Tans. Comput., vol. C-25, pp. 449-456, 976. [4] J. M. S. Pewtt, "Objet enhanement and extaton," n Ptue Poessng and Psyhoptos, B. S. Lpkn and A. Rosenfeld, Eds. New Yok: Aadem, 970, pp. 75-49. [5] E. Rame, "Tansfomaton of photogaph mages nto stoke aays," EEE Tans. Cuts Syst., vol. CAS-22, pp. 363-373, 975. [6 L. G. Robets, "Mahne peepton of thee dmensonal solds," n Optal and Eleto-optal nfomaton Poessng, J. T. Tppet et al., Eds. Cambdge, MA: MT Pess, 965, pp. 59-97. [7] 0. A. Zunga and R. M. Haalk, "Cone deteton usng the faet model," n Po. EEE Compute Vson and Patten Reognton Conf., 983, pp. 30-37. Entopy and Coelaton: Some Comments TARALD 0. KVALSETH, MEMBER, EEE Abstat-Fo measung the degee of assoaton ooelaton between two nomnal vaables, a measue based on nfomatonal entopy s pesented as beng pefeable to that poposed eently by Hobe [l. Asymptot developments ae also pesented that may be used fo makng appoxmate statstal nfeenes about the populaton measue when the sample sze s easonably lage. The use of ths methodology s lustated usng a numeal example. NTRODUCTON n a eent atle n ths TRANSACTONS, Hobe [] poposed a measue of assoaton (oelaton) between two nomnal vaables based on nfomaton-theoet mets. The pupose of ths oespondene s essentally theefold. Fst, ths autho wants to take ssue wth Hobe's measue, spefally beause of mpope nomng used to esale the measue to the [0,] nteval. Seond, a moe easonable nfomatonal measue wll be dentfed whh s also symmet and s smply the weghted aveage of asymmet measues fo the two nomnal vaables. Thd, asymptot (lage-sample) methods of statstal nfeene wll be outlned sne, n addton to obtanng pont estmates fo the assoaton measue, t s geneally of nteest to be able to onstut onfdene ntevals and test hypotheses about the populaton measue. The esults wll be exemplfed usng a numeal example. MEASURE FoRMULATON To defne the notaton to be used, onsde that paj s the populaton pobablty of that ell n the x J ontngeny (oss-lassfaton) table whh oesponds to ategoy of the ow vaable X and ategoy j of the olumn vaable Y. Futhemoe, let p-+ and p,+j. denote the magnal totals fo ow and olumn j, espetvely. The nfomaton onveyed about X by Y, (X; Y), and ve vesa, (Y; X), also alled tansnfomaton, s gven by the well-known fomula = (X; Y) = H(X) - H(XY) = ( Y; X) = H( Y) -H( YX) () Manuspt eeved May 25, 986; evsed Januay 25, 987. The autho s wth the Depatment of Mehanal Engneeng, Unvesty of Mnnesota, Mnneapols, MN 55455. EEE Log Numbe 874205. 008-9472/87/0500-057$0.00 987 EEE