Region Matching by Optimal Fuzzy Dissimilarity

Transcription

1 Rego Matchg by Optmal Fuzzy Dssmlarty Zhaggu Zeg, Ala Fu ad Hog Ya School of Electrcal ad formato Egeerg The Uversty of Sydey Phoe: Fax: Emal: Abstract: Ths paper presets a approach to rego feature extracto ad rego matchg for cartoo mage processg. By troducg the erta coordate system, the varat features of regos have bee extracted. Fuzzy dssmlarty s proposed as a matchg varable. To smplfy the optmal rego matchg, a optmal matchg par theorem s proposed ad proved. Expermets are coducted for verfyg the feature extracto ad the optmal matchg par theorem. Keywords: Rego Feature; Rego Matchg; Fuzzy Dssmlarty; Computer Amato; Cartoo mage Processg.. troducto Rego matchg s mportat mage processg, useful for the producto of the betwee frames of computer amato [0], obect recogto [8] ad vdeo compresso. There are maly two categores of rego or pot matchg methods:. Patter classfcato such as earest eghborhood [3], adacecy costrat [], relaxato labelg [6] ad logcal tree classfcato [].. Matchg path by dyamc programmg [7], [], [5]. Methods of the frst category are quck ad smple but sestve to parameters ad thresholds. addto, msmatches are expected frequetly due to the uavalablty of dstgushg features []. They caot guaratee the global optmal matchg results. Methods of the secod category, however, ca always reach global optmal matchg results, wth fewer msmatches but t s slow ad very tme cosumg. Moreover, exstg matchg methods always use a measuremet of smlarty [], [0], [3]. The smlarty s the reverse of the sum of a preset costat ad the feature dfferece of two regos. Ths causes olearty ad creases computato tme the matchg process. Ths paper presets a method to extract momet varat features, troduces a otato of fuzzy dssmlarty to smplfy matchg computato ad proposes a optmal matchg par theorem whch ca greatly decrease the computato complexty of searchg for a optmal matchg path. The paper s orgazed as follows. Feature extracto s descrbed secto. Secto 3 proposes a optmal matchg par theorem based o dyamc programmg. Secto 4 provdes several expermets to evaluate the proposed feature extracto ad optmal matchg par theorem. Coclusos are preseted secto 5.. Feature Extracto To match the regos of two obects effectvely, the features of those regos should frst be extracted. Although formato loss s allowed feature extracto, key formato of those regos should be retaed so that t s possble to recostruct the dstctve regos to

2 some extet. Although there are o uform extracto rules yet to rego matchg, a good feature extracto method should be depedet of traslato, rotato ad the scale of obects. ths paper, momet varat features are used. We select the relatve posto, oretato ad sze of the mmal costrat rectagle of the regos to characterze the regos.. Obect Features Suppose that obects are comprsed of twodmesoal regos. Regos are made of polyles ad the pot coordates of the polyles are show as Fgure. The feature extracto of other kds of regos would be smlar to the followg. To extract rego features depedet of traslato, rotato ad scalg of a obect, a ew coordate system s establshed. The org of the ew coordate system ca be set at the mass ceter of the obect. et X, Y be the coordates of the mass ceter of th rego, the rego ceter ca be gve by X l x ad Y l y where l x x ) + ( y y s ( ( + ) ( + ) ) the legth of the le betwee the th ad the x + x( +) ( + )th pots; x ad y + y( +) y are the mdpot of the le from the th to the ( + ) th pots; x, y are the orgal coordates of the th pot of the th rego; s the le umber of the th rego. et X ad Y be the coordates of mass ceter of the obect or the org of the ew coordate system of the obect. The mass ceter ca be detfed by the followg equatos: X X ad Y Y,,,...,,,,..., where l th rego; s the legth of les of the s the legth of all les of the obect; s the rego umber of the obect. The oretato of the obect ca be detfed by the agle φ betwee the ma erta axs x (8)(0) ad the orgal axs x : xx ( X X ), yy ( Y X ), xy ( X X )( Y φ arct g yy xy xx Y ), where xx, yy ad xy are the erta momet of the obect to the x ad y axes. The ew axes x ad y are based o the ma erta axes. Thus the ew coordates of all pots ca be trasformed by the followg equatos: x x cosφ + y s φ X ad y x s φ + y cosφ Y. Fgure depcts the obect after the coordate trasformato. The sze of the obect ca be detfed by ts mmal costrat rectagle [] called the feature rectagle whch has a mmal area ad cotas the obect based o the ew coordate system. The feature rectagle R of the obect ca be defed by ts coordates of the top-left ad bottom-rght corers: R ( x, y ; x, ), y { M x where x,..., ;,..., }, y Max{ y,..., ;,..., }, x Max{ x,..., ;,..., }, y M{ y,..., ;,..., }.

3 et x x y y w be the wdth ad h be the heght of the feature rectagle respectvely ad w ad h represet the sze of the obect. They ca be used to ormalze the features of regos order to separate the scale factor of the rego features. ow the mass ceter, the oretato ad the sze of the obect are ready for the extracto of rego features.. Rego Features Based o the ew coordate system of the obect, the features of the regos ca be obtaed usg a smlar method to the above (as llustrated Fgure 3). The org of the th rego s X X cosφ + Y s φ, Y X s φ + Y cosφ. The oretato of the th rego ca be detfed by the agle φ betwee the ma erta axs x of th rego ad the ma erta axs x of the obect as the followg equatos: xx l ( x X ), yy l ( y Y ) xy l ( x X xy arct g yy xx φ where, ad xx yy xy, )( y the th rego to the x ad Y ), are the erta torque of y axes. The ew axes x ad y are based o the ma erta axes of the rego. Thus the ew coordates of all pots of the th rego ca be obtaed from the trasformato equatos: x x cosφ + y s φ X, y x s φ + y cosφ Y. Fgure 4 depcts the regos after the coordate trasformato. The sze of the th rego ca be detfed by the mmal costrat rectagle or the feature rectagle based o the ew coordate system of the th rego. The feature rectagle R of the th rego ca be measured by the coordates of ts top-left ad bottom-rght corers: R ( x, y ; x, y ), { M x where x,..., ;,..., }, y Max{ y,..., ;,..., }, x Max{ x,..., ;,..., }, y M{ y,..., ;,..., }. et w x x be the wdth ad h y y be the heght of the feature rectagle of the th rego respectvely. w ad h represet the sze of the th rego. They may be used further to ormalze the features of polyles of regos order to separate the scale factor of the polyles features..3 Vectorzato of Rego Features As s show above, a rego of a obect ca be characterzed by ts mass ceter, oretato ad sze relatve to a obect. ow a ormalzed feature vector of the th rego ca be defed as v ( v, v,..., v6 ) X Y w where v ad v ; v 3 ad w h w h w cosφ w s φ v 4 ; v5 ad v6. h w h v ad v represet the ormalzed mass ceter of the th rego, v 3 ad v 4 represet the ormalze sze of the th rego, v 5 ad v 6 represet the ormalzed oretato of the th rego. Obvously, these features are varat to traslato, rotato ad scalg of a obect. To cofrm the results of rego feature extracto, the regos of a obect may be 3

4 recostructed by usg ther extracted features. Fgure 5 shows a cartoo of a eagle. Fgure 6 llustrates the rego recostructo of the eagle based o the orgal cartoo. 3. Rego Matchg Rego matchg s carred out to fd a set of matchg pars of the regos of two obects accordg to some matchg crtera. Some researchers [7], [], [] troduced fuzzy matchg varables such as smlarty ad membershp of two regos. Ther goal was to fd a optmal path or a set of matchg pars whch make the sum of the smlarty or membershp maxmal. However, usg the rego feature vector preseted, such fuzzy varables are coveet for the matchg process. cotrast, we troduce a varable of fuzzy dssmlarty of two regos or vectors ad set the optmal goal as the matchg path wth the mmal sum of dssmlarty. 3. Fuzzy Dssmlarty of Regos Assume that vector set V { v, v,..., v,..., v} represets the regos of the source obect ad vector set V { v, v,..., v,..., v } represets m m regos of the target obect. The Eucldea dstace betwee the th vector of the source obect ad the th vector of target obect s 6 d v v ( vk vk),,...,,,..., m. k Suppose the maxmal dstace D Max{ d,...,,,..., m}, the the fuzzy dssmlarty of the th vector of the source obect relatve to the th vector of the target obect ca be defed as d σ, 0 σ,,...,,,..., m. D Fgure 7 shows the fuzzy dssmlarty wth respect to the Eucldea dstace. Whe σ 0, the mplcato s that the th vector of the source obect exactly matches the th vector of the target obect. Whe σ, the mplcato s that the th vector of the source obect s totally dssmlar to the th vector of the target obect. 3. Optmal Matchg Path A matchg map shows all possble matchg pars of the regos of two obects. Fgure 8 s a example of a matchg map. As show the map, the rows represet the regos of a source obect. The colums represet the regos of the target obect. The tersectos of rows ad colums represet possble matchg pars. Each matchg par s assocated wth a matchg error measured by ts fuzzy dssmlarty. A matchg path stads for a set of matchg pars {(, ),...,(, ),...,(, ),..., ; m}, where the matchg par, ) dcates that ( the th rego of the source obect matches the rego of target obect. For smplcty, a matchg path ca be deoted by {,...,,..., }. t s a le lkg tersectos the matchg map. The legth of a path ca be measured by the sum of the fuzzy dssmlartes o ts tersectos. The matchg path whch satsfes... s defed as a vald matchg path or a urepeatable matchg path. The optmal matchg path s the oe whch reaches the mmal sum of fuzzy dssmlarty all vald matchg paths. Some researchers [7], [], [] have already appled dyamc programmg for pot matchg or rego matchg. Assumg that m, the frst ad the last matchg par are kow as (, p) ad (, q), a searchg algorthm for the optmal matchg path s suggested below: Set the tal sum of fuzzy dssmlarty k 0 ad the tal matchg path path {}, k,..., m. ( + ) k 4

5 Set σ k to k p, σ k to k q order to force the path start at (, p) ad ed at (, q). oop ( ; > 0; ) f p, q, cotue. oop ( ; < m; + + ) f, p, q, cotue. oop ( k ; k < m; k + + ) f k,, p, q, cotue. } k M { ( + ) k Ed loop σ + ( + ) k path {, path( + ) k } Ed loop Ed loop The output of the searchg would be the sum of fuzzy dssmlarty ad the optmal path p would be path p. f the frst matchg par ad the last matchg par are ot kow, the aother searchg for the best frst ad last matchg par s ecessary. f > m, the searchg process s the same as above except for terchagg, ad settg k,...,. As we ca see from the searchg algorthm gve above, t s very tme cosumg. The searchg tmes are proporto to O( m ) whe the frst ad last matchg pars are kow or eve O ( m 3 ) whe the frst ad last matchg pars are ot kow. For those regos of two obects, whch are statc relatve to the obect mass ceter ad dstct from ther features, we fd that t s ot ecessary for them to partcpate such tme cosumg searchg. They ca be matched drectly accordg to the followg theorem. 3.3 Optmal Matchg Par Theorem Theorem: f the matchg error of a matchg par (, ) s the uque mmum ts row ad colum of a matchg map, the the matchg par (, ) s the global optmal matchg path. The theorem ca be descrbed as follows. et ) vector set V { v,..., v,..., v } matches vector set V { v,.., v,..., v } ; m ) the matchg error of the matchg par (, ) s fuzzy dssmlarty σ ; 3),...,,..., ) dcates a vald matchg ( path where... ; 4) the mmal matchg errors of theth row are σ M{ σ,..., m} o the matchg map; 5) the mmal matchg error of the th # colum s σ M{ σ,..., } o the matchg map; 6) the optmal matchg path s (,,...,,..., ). The theorem ca be stated as: # f σ σ σ ad # σ σ to ay, σ σ to ay, the whe the optmal path (,,...,,..., ). The theorem ca be proved as follows. Proof: Assume that ) The optmal matchg paths start at S ad eds at E as llustrated Fgure 9. ) Back trackg method s used to follow the optmal path wth a mmal sum of matchg errors. 3) The partal optmal path betwee row + ad row s kow as a-e. The theorem ca be proved two steps: frstly, the optmal path may ot go to colum at row +,.e. +,...,. Secodly, the optmal path must go to colum at row based o the partal optmal path betwee row + ad row,.e. at. 5

6 Part Suppose that the optmal path goes by a at track step + as show Fgure 9. ext, at the track step +, the optmal path has at least two possbltes. Oe path s from a to b.e. the optmal path reaches colum at step +. t wll be prove that ths case s mpossble. Aother path s from a to b where b s assumed as the pot wth the mmal matchg error at step +. Accordg to the codtos set the theorem, b s mpossble o because the pot wth the mmal match error s at colum oly whe row s row. ow observe the optmal path at step. Sce t s a urepeatable matchg, t s ot allowed to go o from b where o dcates the matchg par (, ). Select a arbtrary pot o but o o row. Sce the matchg error at b s less tha at pot b, the sum of the matchg errors of path a-b-o s less tha of path a-b-o. Therefore, the local optmal path betwee row ad row + caot go from a-b-o,.e. t s mpossble for the global optmal path to go by b. Cosderg the same stuato the rage from row + to row + 3, the result s cosstet utl o the row. Ths s to say, the global optmal path caot reach colum the rage from row + to row. Part ow assume that the optmal path reaches b at step +. Please ote that b may ot be the pot wth the mmal matchg error as assumed above. The optmal path o row may go to o whch the theorem suggests or to a arbtrary pot o. ext, measure the match path o the row. Select a attrbute vald pot c. The path co-b has less matchg error tha the path co-b because o s the mmal matchg error pot o row. Therefor, for ay vald arbtrary pot c o row, path c-o-b s selected. Thus the global optmal path must go by o,.e. rego of the source obect matches rego of the target obect. By summarzg the two parts above, we prove the theorem proposed. The theorem s geeralzed to all matchg applcatos whch use matchg dssmlarty of two regos. f the matchg varable s the smlarty of two regos or pots some applcatos, the mmum of the theorem should be chaged to maxmum. Applyg ths theorem, oe ca dramatcally save the matchg tme wthout losg matchg accuracy because some regos of the two obects are relatvely statc or ther features extracted are dstct from those of other regos. Our expermets (see below) support ths vewpot. Therefore, we suggest that the optmal matchg par theorem should be appled before the dyamc programmg s used. For those matchg pars whch caot be satsfed by the codtos of the theorem, dyamc programmg has to be employed. 4. Expermets Based o the rego extracto ad optmal matchg algorthm preseted, four expermets have bee coducted to evaluate ts performace. Fgure 0 shows sx frames of a flyg balloo. The balloo s comprsed of two regos, a ellpse ad a strp. kg les across Frame ad Frame 6 detfy the matched regos. Frame to Frame 5 are the -betwee frames based o the rego matchg results. Judgg from lkg les of the matched regos ad the -betwee frames, the rego matchg of the balloo s successful. Fgures ad llustrate a source ad a target frame of a smlg grl. Each cludes 6 regos. Fgure 3 shows the matchg results by terpolato frames betwee Fgure ad Fgure. Fgure 4 shows a target frame of a eagle. The source frame of the eagle s show Fgure 5. Each frame s comprsed of regos. Fgure 6

7 5 depcts the rego matchg results. The matchg results are cofrmed by the two betwee frames. Fgures 6 ad 7 show a source frame ad a target frame. Each frame s comprsed of 8 regos. Fgure 8 llustrates the matchg results of the two key frames. t ca be see from the -betwee frames that the rego matchg results are rght. Table shows the effects of applyg the optmal matchg par theorem the sx expermets. After employg the optmal matchg par theorem, the umber of regos eedg optmal search s dramatcally reduced. Hece the matchg tme s greatly shorteed. Table : The Results of Applyg the Optmal Matchg Par Theorem. Pcture Total of Optmal matchg Rego pars eedg optmal regos pars of search regos Balloo 0 Smlg 6 4 grl Eagle 0 Woma Coclusos ths paper, a erta coordate systems are troduced to extract the rego features varat to traslato, rotato ad scalg. cotrast to other researchers [], [], [], we use fuzzy dssmlarty of regos our optmal matchg of regos. addto, a optmal matchg par theorem s proposed ad prove. The expermets cofrm the feature extracto ad the optmal matchg par theorem. The method preseted has the followg propertes: Computg complexty. To fd all optmal matchg pars, the computg complexty s O (m) where ad m are rego umbers of the source ad the target obects respectvely. cotrast, the computg complexty of covetoal optmal searchg s O( m ) or eve O { m 3 ). Robustess. The matchg results are uque ad optmal to the rego feature vectors gve. Geeralty. The feature extracto ad matchg algorthm are geeralzed pot or rego matchg. However, some applcatos where two regos are smlar ad terchage ther posto two key frames, the rego features preseted ths paper may ot be eough to dstgush regos from each other ad msmatches may be caused. More rego features such as the posto of adacet regos should be extracted ad used the matchg process. Refereces. M. E. Asar,. Masmoud ad. Radouae, A ew rego matchg method for stereoscopc mages, pp.83-94, Patter Recogto etter, Z. X. Cha ad G. Y. Xu, tellgece ad Applcato, Qghua Uversty Publshg, J. Flusser, Obect matchg by meas of matchg lkehood coeffcets, pp , Patter Recogto etter 5, M. K. Hu, Vsual patter recogto by momet varats, RE Tras. form. Theory 8, 79-87, A. Goshtasby ad.g. Stockma, Pot patter matchg usg covex hull edges, , EEE Tras. Syst. Ma. Cyberet S. Raade ad A. Rosefeld, Pot patchg by relaxato, 69-75, Patter Recogto, T. W. Sederberg ad E. Greewood, A physcally based approach to -D shape bledg pp.5-34, Computer Graphcs, July D. S. Shu ad Y. Huag, Cotrol Techques of Robotcs, Mechacal dustry Publshg, J. A. Vetura,. Y. a ad W. Wa, Optmal matchg of geeral polygos base 7

8 o the mmum zoe error, Patter Recogto etters 6, 5-36, J. H. Yu ad Y. B., Computer Amato Theory ad Applcatos, Qghua Uversty Publshg, Y. Zhag, A fuzzy approach to dgtal mage warpg EEE Computer Graphcs ad Applcatos, pp.35-4, July Y. Zhag, Fuzzy theory techques ad ther applcato dgtal mage trasformato pp , Vol. 3, Fuzzy Theory System: Techques ad Applcatos, Z. Ch, H. Ya ad T. D. Pham, Fuzzy Algorthms, wth Applcatos to mage Processg ad Patter Recogto, World Scetfc Publshg, Z. Ch ad H. Ya, "D3-derved fuzzy rules ad optmzed defuzzfcato for hadwrtte umeral recogto," EEE Tras. o Fuzzy Systems, 4(): 4-3, B. Rchard, Dyamc Programmg, Prceto Uversty Press, Fgure 3. ew coordate systems of regos. Fgure 4. Regos after coordate trasformato. Fgure. ew coordate system of obect. Fgure 5. A eagle. Fgure. Obect after coordate trasformato. Fgure 6. The recostructo of the eagle. 8

9 Fuzzy Dssmlarty.0 D Dstace Fgure 7. The Dssmlarty va Eucldea dstace. Fgure. The source frame of a grl. 3 V 3 m p V q Fgure 8. The matchg map. Fgure. The target frame of a grl S 3 m c o o b b a E Fgure 9. The partal optmal path. Fgure 3. The matched grl. Fgure 0. The matched balloo. Fgure 4. The target frame of the eagle. 9

10 Fgure 5. The matched eagle. Fgure 7. The woma secod frame. Fgure 6. The woma frst frame. Fgure 8. The matched woma. 0