Planar Curve Representation and Matching

Transcription

1 Plana Cuve Repesentaton and Matchng Mahe Al-Khayat and Fahad Kamanga Compute Scence and Engneeng Depatment Unvesty of Texas at Alngton Alngton, Texas 7609, USA [al-khay Abstact In ths pape, we dscuss a method fo epesentng and matchng plana cuves. The technque s based on usng calculatons fom concentc ccles to epesent each cuve by two sets of angles. The angles ae defned by vectos constucted fom the cente pont of the ccles and the ponts on the cuve tace that ntesect each ccle. The ccles have ncementally nceasng ad epesented by the mnmum adus and the adus ncement value. The numbe of ccles used specfes the level of abstacton at whch the cuves ae epesented. Ths epesentaton s nvaant to tanslaton and otaton tansfomatons. Expements wth dffeent classes of cuves have shown that ou technque s obust to dgtzaton eos and nose effects, and can pefom well when the numbe of concentc ccles ae elatvely small. In patcula, we descbe the potental applcablty of ths technque to fngepnt dentfcaton poblem. Intoducton Cuve matchng s the appoach used to fnd the best ft between two cuves. By best ft, we mean that two cuves can be algned togethe such that a secton o subcuve fom one cuve s geometcally smla to that of the othe cuve. The mpotance of cuve matchng s due to ts wde applcablty n compute vson and object ecognton systems. Cuves epesent boundaes of objects, lnes n fngepnts, and contous n maps. Relatonshps between dffeent objects can be mapped nto elatonshps between the cuves extacted fom these objects. Ths pape pesents an appoach to epesentaton and matchng of two-dmensonal plana cuves usng concentc ccles nfomaton. The appoach s nvaant to tanslaton and otaton tansfomatons and does not eque the cuves to possess ctcal ponts o beakponts. In addton, the computatonal complexty of matchng phase s O M whee M s the numbe of ccles used fo the epesentaton and s much less than n (numbe of ponts on the cuve). It has a heachcal natue n whch cuves can be epesented usng heachy of abstacts. Ths popety pomotes the paallelzaton of matchng to mpove pefomance and elmnate unnecessay testng. Pevous eseach on cuve epesentaton and matchng ([]-[]) can be categozed nto seveal classes. Wolfson [] pesented two algothms fo cuve matchng that fall unde the cuvatue-elated [] class of algothms. Hs algothms used the chaactestc stngs to epesent the cuves. The chaactestc stngs of two cuves ae matched togethe to fnd a set of possble canddate matches. The canddate BMVC 998 do:0.5/c..8

2 Btsh Machne Vson Confeence 75 matches ae mapped back to the ognal cuves whee eo s calculated fo each pa. The best esult s etuned. The methods ae tanslaton and otaton nvaant. The two algothms dffe n matchng of chaactestc stngs. Whle the fst one uses a smple stng-matchng algothm that eques one moe step to quantze the chaactestc stngs, the second method mantans shft accumulatos and consdes a toleance measue. The latte s moe obust but has an O n wost-case complexty. Ou matchng algothm uses the moe compact concentc ccles epesentaton fo matchng and ts complexty depends lnealy on the length of the epesentaton vecto whch s less than that of the chaactestc stngs. Anothe appoach suggested by Feeman [5] apples to closed o open cuves. We only consde the open cuve poblem n ou appoach, but suggest scene-matchng appoach that can apply to closed cuves. Feeman's appoach belongs to the featuecalculaton class of algothms. Fo open cuves, hs appoach uses dscontnutes n cuvatue (DICs) [] to beak cuves nto segments and establsh coespondence between dffeent segments fom dffeent cuves. Fo each segment, the featues calculated ae:. Length of segment dvded by the dstance between end ponts of segment.. Total "bay" aea dvded by squae the dstance between end ponts of segment.. Total "pennsula" aea dvded by squae the dstance between end ponts of segment.. Maxmum "bay" depth dvded by the dstance between end ponts of segment. 5. Maxmum "pennsula" depth dvded by the dstance between end ponts of segment. The above featues ae tanslaton, otaton and scale nvaant. The nvaance to scalng esults fom nomalzng by the dstance between end ponts fo the length featues and squae that dstance fo the aea featues. Ou concentc ccles epesentaton fnds beakponts as a devatve of the algothm and does not eque the exstence befoehand. The thd class, template-based appoach, was ntoduced by Tuney et al. [] fo ecognzng objects usng the bounday cuves. Objects exst n a scene and may be patally occluded. Evey object s epesented by a set of templates. A template s a two-dmensonal bounday cuve of the object and s epesented by a set of "salent featues". Each salent featue (also known as subtemplate) s dstnct on the template and dstngushable fom othe templates. It has a weght facto (called "salency measue") that measues ts dstnctveness. If s a subtemplate of template T, and w s the weght assocated wth t, then w, w whee s the numbe of subtemplates of T. Fndng the salency s accomplshed by coelatng evey template aganst all othe possble templates. A database s mantaned to keep all templates and subtemplates n the system. These featues ae not local to the object tself. Thus, the weghts need to be updated wheneve a new object s consdeed and only those n the system can patcpate n the matchng and ecognton phases. The method s useful n poduct assembly lnes whee objects avalable to the system ae known. In the followng sectons, we pesent the concentc ccles epesentaton and matchng algothms. We wll also pesent fngepnt ecognton wok as a potental applcaton of the algothm and conclude wth ongong eseach and futue wok. 0, ()

3 76 Btsh Machne Vson Confeence Concentc Ccles. Pepocessng In the pepocessng phase, captued gay-scale mages wee smoothed usng an "edgepesevng" smoothng technque. Fou spatal aveagng fltes n the neghbohood of w w wee used. These fltes calculate the aveages along the hozontal lne, vetcal lne, and the dagonals passng though the pxel whose value s to be updated. The algothm updates the pxel value wth the aveage that esults n mnmum vaance. Ths technque s sometmes called "aveagng by mnmum vaance". Bnazaton s acheved by a theshold value. We adopted a local thesholdng appoach. Thnnng o skeletonzaton of cuves s used to obtan -pxel wde cuves. The thnnng algothm peseves the 8-neghbohood connectedness whle elmnatng edundant pxels.. Repesentaton Concentc ccles epesentaton uses 8-neghbohooh chan codes that encode tace of cuves as nput. Fom that, a cuve s epesented by two sets of angles : M and : M coespondng to angles between vectos centeed at a pont, p 0, on the cuve tace. The pocedue s to daw ccles centeed at p 0 wth ncementally nceasng ad. The ponts at whch these ccles ntesect the cuve ae maked and used as the tp ponts of vectos ntated fom p 0. Each vecto wll have a magntude equal to the adus of the ccle at whch the ntesecton occued. It s mpotant that the cuve ntesects any ccle at two ponts only. When a stuaton occus that a cuve ntesects a ccle at moe than one locaton on each sde of the cente pont, the cuve wll be segmented. In fact, ths s the cteon we wll use to segment the cuves, and each segment wll be epesented by the two sets of angles between the vectos at evey ccle. The ccles wll have a mnmum adus, a maxmum adus, and an ncement by whch two consecutve ccles dffe n adus. The angles fo a gven ccle c whose adus s ae calculated as follows: fnd two ponts on the cuve segment ntesectng ths ccle. Mak these ponts p and dependng on the locaton wth espect to p 0. The angle between vectos p p 0 p and p 0 p s the th element (ndexed by ) n the fst set. Fo the th element of the second set, the angle between p p and 0 p 0 p s chosen. p p s the vecto 0 fom the ccle whose adus s ( and s consdeed constant, but t s not equed to be so). The angles ae gaphcally epesented n Fgue. Gven a segment of some cuve, the algothm conssts of two steps. The fst step s to choose the cente pont of all the ccles. The second step s to calculate the epesentaton of the cuve segment. The followng ae the two steps, and the pocedues nvolved n both of them.

4 Btsh Machne Vson Confeence 77 p - p 0 p - + p + Fgue : Gaphcal demonstaton of the concentc ccles epesentaton angles. Detemne p 0 on the cuve segment.. Connect the two end ponts of the cuve segment.. At the md pont of the connectng lne, constuct a pependcula lne. p 0 s the pont on the cuve segment ntesectng that pependcula. Detemne the cuve epesentaton.. Intalze mn.. Repeat the followng untl max o no moe ponts ae found p to the ght of p 0 such that the length of a. fnd the futhest pont s less than o equals to. If t s less than, then stop. b. fnd the futhest pont s less than o equal to. If t s less than, then stop. c. calculate as the angle between d. calculate as the angle between e. ncement by. p to the left of p 0 such that the length of p 0 p and p p and 0 p 0 p. p 0 p. p 0 p segment p 0 p segment Fgue demonstates thee ccles that wee used fo the epesentaton of a cuve. In ths example, we ovelad the ccles on the ognal cuve at pont 0 p. Usng the same fgue, one can dentfy the vectos and calculate the adus-angle nfomaton based on the above algothm. We calculated the angles and tabulated the epesentaton n Fgue (d). Accodng to ths nfomaton, we wee able to eceate the ponts on the cuve that ntesected the ccles. To econstuct the cuve, we used staght-lne segments to connect evey two consecutve ponts (Fgue (c)). No nfomaton about the cuve segment between these ponts can be nfeed fom the epesentaton. A lage ncement value of adus means that moe nfomaton about the cuve segment s abstacted out o lost. Ths adus ncemental value s mpotant to peseve featues of the cuve and mnmze eo. When t s small, featues wll be detected and we wll be able to econstuct them, but edundant nfomaton such as nose wll be ncluded. When t s lage, some featues ae lost o abstacted out but nose s suppessed. Fgue shows a heachcal

5 78 Btsh Machne Vson Confeence epesentaton of a cuve to demonstate the effect of adus ncement value. The cuve s fst epesented usng one ccle. The adus s chosen so that the cuve s enclosed by p 0 (a) (b) (c) Fgue : Concentc ccles epesentaton. (a) Ognal cuve. (b) Repesentaton ccles. (c) Reconstucted cuve. (d) Repesentaton sets. the ccle. Then, the adus s dvded by two and the cuve s epesented usng two ccles. Afte that, each adus s agan dvded by two and the cuve s epesented usng fou ccles, and so on. Each case s equvalent to epesentng the cuve at one level of abstacton. The fst case, epesentaton usng one ccle, s the most abstact epesentaton. We used fou levels of abstacton to epesent the cuve, and econstucted t at each level. Note that the level numbe s nvesely elated to the adus ncement value. Thus, we can contol the abstacton of the epesentaton by ths value, and adapt t to dffeent applcatons and envonments. (d) (a) (b) (e) (f) (g) (h) (c) (d) Fgue : Heachcal epesentaton of a cuve at dffeent levels. (a) and (b) Ognal and econstucted cuves usng one-ccle epesentaton, (c) and (d) usng two-ccle epesentaton, (e) and (f) usng fou-ccle epesentaton, and (g) and (h) usng eghtccle epesentaton.

6 Btsh Machne Vson Confeence 79. Matchng We say that some cuve s a coveed cuve f and only f the followng two condtons ae satsfed. Fstly, t does not coss any epesentaton ccle at moe than two ponts wth the cente pont between them. Secondly, the cuve's end ponts ae at a dstance less than o equal to twce the maxmum adus of the epesentaton ccles. A coveed cuve has a sngle epesentaton by the sets and and t has one cente pont. When a cuve s not a coveed cuve, then t wll be dvded nto two o moe coveed subcuves o segments. We developed pocedues to match two coveed cuves, and one coveed cuve wth a geneal cuve (possbly coveed). We wll stat by dscussng the fst case and move towads the geneal one. The pocedues ae based on matchng the epesentatons of cuves povded they wee calculated usng the same mn and paametes. Gven two cuves C and D wth epesentatons C and D, espectvely, both cuves can be epesented by nfomaton fom an equal numbe of concentc ccles, wth C R C, C, C () and D R D, D, D. () Fo completeness, we nclude the ad of the ccles at whch each angle element was calculated. Fo example, C and C ae the epesentaton angles calculated at adus C mn C ( ). Ths epesentaton suppots futue extensons to cases whee coespondng ad of dffeent cuves ae not equal. The numbe of elements n each of these sets s M. When the two cuves match, the coespondng angles n the sets must be equal. Fst C s tested aganst D by compang the coespondng elements,.e. C s compaed wth D fo,..., M. If any pa s not equal, the match fals. Ths test s suffcent to detect match falue, but n ode to conclude that the two cuves match, one moe test s equed. Fst, we calculate anothe set of angles,. Each element of ths set epesents the angle between the vecto whose tp s at p and the one whose tp s at p, wth ad and, espectvely, and calculated as gven by Equaton () fo,..., M and 0. () Then, we compae the coespondng sum n both cuves. Ths sum measues the total devaton of angles at adus fom those at adus. Geometcally, ths test s mpotant to dffeentate between cuves that ae symmetc about p o. It s pefeed to stat testng angles coespondng to ccles wth lage ad, and then the ones coespondng to smalle ad. The eason s that angles at vey small ad ae moe lkely to be equal even f the cuves don t match. Thus, we wll be able to decease the tme equed to epot a case whee the matchng fals. A compason that uses equalty s appopate n pefect stuatons. Pactce suggests that many types of eos pesent n mages of cuves causes a test that s based on

7 80 Btsh Machne Vson Confeence equalty to fal, even f the two cuves ae dentcal n eal wold. A modfcaton to ths matchng algothm that s toleant to eos s moe obust. Instead of testng coespondng angles fo equalty, a elaxed condton would be to test f the absolute dffeence between them s less than some toleance. In ths case, a measue of epesentatons test s used. Equatons (5) and (6) epesent the mean squae eos between the coespondng epesentaton sets. MSE CD M M max (5) M MSECD C D C D. (6) M max In ou mplementaton, we chose a vaable toleance that s a functon of the adus. Expements showed that t s ecommended to have toleance values that get smalle as the adus nceases. Fo both tests, we used functons of the fom gven by Equaton (7) whee K s a constant that can be dffeent n both tests. T( ) K (7) mn ( ) Note that the toleance fo the smallest ccle ( ) s gven by Equaton (8) whch s the lagest value. K (8). Pefomance mn C D O n computatons. If the numbe of pxels on a cuve s n, then pepocessng eques Anothe O n computatons ae equed to epesent the cuve usng concentc ccles nfomaton because evey pont on the cuve tace s vsted once to detemne ts dstance fom the cente pont. When matchng s conducted on coveed cuves (cuves epesented by two sets of angles), the matchng s OM -complex, whee M s the numbe of ccles used n the epesentaton. In some cases, the ntenton s to fnd a subcuve wthn a geneal cuve. The subcuve can be epesented by the two sets of angles, but the geneal cuve cannot be epesented. To enable such testng, the concentc ccles ae calculated fo a subset of ponts fom the geneal cuve and tested aganst those calculated fom the subcuve. Ths eques nm O computatons. The numbe of ccles, M, epesentng a cuve plays a sgnfcant ole n computatons. A wost-case stuaton occus when M s equal to n /,.e. the adus ncement s vey small that evey pont on the cuve ntesects a ccle. In ths case, the complexty s O n. On the aveage, M s much less than n. Expemental Results The algothms dscussed n the pevous secton wee mplemented usng Khoos.0 unde UNIX opeatng system on a SUN/SPARC machne.

8 Btsh Machne Vson Confeence 8 Expements wee conducted on dffeent classes of cuves. We shall pesent cuves fom fngepnt mages to demonstate the effect of concentc ccles cuve matchng. The notaton fpx.y s used to epesent fngepnt mage y obtaned fom peson x. Each fngepnt wll geneate few hundeds of cuves whose epesentatons become the database of known cuves. Fgue shows fou cuves extacted fom fngepnt fp7.. Assume that ths set of cuves can unquely dentfy the fngepnt. Fo llustaton puposes, we maked the cente of the concentc ccles epesentaton of each cuve wth a dffeent numbe. The cuves wee matched aganst fngepnts fp., fp., and fp. epesentng dffeent ndvduals. They wee also tested aganst anothe fngepnt (fp7.) of the same ndvdual. The matchng esults ae shown n Fgue 5. The matchng cuves ae maked to dentfy the cuve fom Fgue (b) aganst whch they match. When a cuve matches moe than one cuve fom fp7., t s maked by (*). (a) Fgue : Fngepnt fp7.. (a) Selected cuves. (b) Cente ponts maked. (b) A hgh-level pocedue can analyze the aangement of the maked centes of cuves fom fp7. to encode the patten made by the cuves. Ths patten can be encoded by the followng tanslaton and otaton featues:. The elatve postons of each cuve's cente pont to that of the othe cuves.. The elatve oentaton of each cuve to the othe cuves. Ths so-called patten analyss pocedue wll test the esults of cuve matchng to decde whethe the same patten exsts. The nspecton of the cuves n Fgue 5 shows that the matchng cuves of fp7. have a patten smla to that of fngepnt fp7.. The pevous dscusson demonstates one of many methods that can be used to study the pattens of the matchng cuves. A bette appoach to the same poblem may assgn weghts to dffeent cuves based on the dstnctveness. Then, usng the weghts of cuves and the matchng pattens, a decson can be made to whethe the two fngepnts belong to the same ndvdual. Othe cuve matchng technques can be used. The concentc ccles technque has many featues that can be utlzed to speed up the fngepnt dentfcaton. Most cuves extacted fom fngepnts do not possess dstnctve featues. So featue-based appoaches wll not pefom well hee. Cuve matchng usng concentc ccles can be pefomed at hgh level of abstacton to select a set of canddate cuves. Then, subsequent tests can be done on the set at lowe levels of abstacton to fne-tune the match.

9 8 Btsh Machne Vson Confeence * * (a) fp. (b) fp. * (c) fp. (d) fp7. Fgue 5: Cuves matchng the ones selected fom fp7. n dffeent fngepnts. (a) fp.. (b) fp.. (c) fp.. (d) fp7.. Conclusons and Futue Wok In ths eseach, concentc ccles nfomaton was used to epesent and match open cuves. Ths epesentaton s nvaant to tanslaton and otaton. The cente of the ccles s the pont whose dstances fom the end ponts of the cuve ae equal. Fom ths cente, vectos wth tps on the cuve ponts that ntesect the ccles ae constucted. The epesentaton conssts of two sets of angles calculated between vectos. The angles ae ndexed by the ccles ad. The paametes of the epesentaton nclude mnmum adus, adus ncement, and maxmum adus. The adus ncement value s the adus dffeence between two consecutve ccles and s constant fo a epesentaton. Matchng examnes the epesentatons of dffeent cuves wthn some toleance. The toleance s a deceasng functon of adus. Ths s justfed by the fact that at hghe ad, small changes n angles esult n damatc changes on the cuves. In pactce, toleance s mpotant to ensue the obustness of the matchng test aganst nose and dgtzaton eos. The concentc ccles appoach has many advantages ove the methods dscussed n the ntoducton. In the template-based appoach, a po knowledge of the objects

10 Btsh Machne Vson Confeence 8 appeang n the applcaton s essental fo the calculaton of salent featues. Ou appoach does not eque any po knowledge of the cuves. It does not eque cuves to possess any ctcal ponts. Featue-calculaton appoach uses ctcal ponts to segment cuves. When cuves ae extacted fom patally occluded objects, these ctcal ponts may o may not appea on the cuves. The advantage of the featuecalculaton appoach s that the epesentaton s nvaant to scalng. Ou algothms ae effectve when used n a system wth lage numbe of cuves to eject msmatches at ealy stages. The numbe of ccles epesentng a cuve detemnes the level of abstacton at whch t s epesented. The test can be conducted at hgh levels of abstacton and epeated agan at lowe levels fo cuves that passed the pevous ones. Thus, msmatches ae flteed as we poceed to lowe levels of abstacton. We conducted tests on dffeent classes of cuves. The potental applcaton on fngepnt ecognton was dscussed and showed pomsng esults. Concentc ccles s a new appoach. It can be mpoved to automate the selecton of the adus ncement value to enable cuves to be epesented at the hghest possble abstacton. We ae nvestgatng buldng the epesentaton as a heachcal stuctue, and studyng paallel algothms fo matchng. In addton, we ae lookng nto developng scale nvaant sets fom the concentc ccles. Refeences [] H. J. Wolfson, On cuve matchng, IEEE Tansactons on Patten Analyss and Machne Intellgence, vol. PAMI-, no. 5, pp. 8-89, May 990. [] J. T. Schwatz and M. Sha, Identfcaton of patally obscued objects n two and thee dmensons by matchng nosy chaactestc cuves, The Intenatonal Jounal of Robotcs Reseach, vol. 6, no., pp. 9-, 987. [] J. Hong and H. J. Wolfson, An mpoved model-based matchng method usng footpnts, n Poceedngs of the Intenatonal Confeence on Patten Recognton, Rome, Italy, pp. 7-78, Novembe 988. [] A. Kalvn, E. Schonbeg, J. T. Schwatz, and M. Sha, Two-dmensonal, model-based, bounday matchng usng footpnts, The Intenatonal Jounal of Robotcs Reseach, vol. 5, no., pp. 8-55, 986. [5] H. Feeman, Shape descpton va the use of ctcal ponts, Patten Recognton vol. 0, pp , 978. [6] P. Wene, Lnea patten matchng algothms, n Poceedngs of the th Annual Symposum on Swtchng and Automata Theoy, IEEE Compute Socety, pp. -, 97. [7] J. W. Mckee and J. K. Aggawal, Compute ecognton of patal vews of cuved objects, IEEE Tansactons on Computes, vol. C-6, no. 8, pp , August 977. [8] P. L. Rosn, Repesentng cuves at the natual scales, Patten Recognton, vol. 5, no., pp. 5-5, 99. [9] I. Wess, Nose-esstant nvaants of cuves, IEEE Tansactons on Patten Analyss and Machne Intellgence, vol. 5, no. 9, pp. 9-98, Septembe 99. [0] H. Teh and R. T. Chn, On the detecton of domnant ponts on dgtal cuves, IEEE Tansactons on Patten Analyss and Machne Intellgence, vol., no. 8, pp , August 989.

11 8 Btsh Machne Vson Confeence [] A. Pkaz and I. Dnsten, Usng smple decomposton fo smoothng and featue pont detecton of nosy dgtal cuves, IEEE Tansactons on Patten Analyss and Machne Intellgence, vol. 6, no. 8, pp , August 99. [] M. Wong and A. W. M. Smeuldes, Dgtal cuvatue estmaton, CVIP: Image Undestandng, vol. 58, no., pp. 66-8, Novembe 99. [] H. Feeman and L. S. Davs, A cone-fndng algothm fo chan-coded cuves, IEEE Tansactons on Computes, pp. 97-0, Mach 977. [] J. L. Tuney, T. N. Mudge, and R. A. Volz, Recognzng patally occluded pats, IEEE Tansactons on Patten Analyss and Machne Intellgence, vol. PAMI-7, no., pp. 0-, July 985.